University of Southern California Dissertations and Theses

Utility functions induced by certain and uncertain incentive schemes

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Utility functions induced by certain and uncertain incentive schemes

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Content UTILITY FUNCTIONS INDUCED BY CERTAIN AND UNCERTAIN INCENTIVE SCHEMES by Maximilian Zellner A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY INDUSTRIAL AND SYSTEMS ENGINEERING August 2022 Copyright 2022 Maximilian Zellner Contents List of Tables v List of Figures ix Abstract xvii Preface xix 1 Scope and Approach 1 1.1 Research Goals and Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Research Approach and Methodology. . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Structure of Dissertation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2 Literature Review on Decision-Making Under Incentives 5 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 Literature Review Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.3 Results of Algorithmic Literature Search . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.4 Research Observations on Decision-Making under Incentives . . . . . . . . . . . . . . 9 2.5 Review of Elicitation and Simulation Methods. . . . . . . . . . . . . . . . . . . . . . 13 2.5.1 Review of Utility Elicitation Methods . . . . . . . . . . . . . . . . . . . . . . 14 i 2.5.2 Review of Simulation Methods to Model Human Behavior . . . . . . . . . . . 16 2.6 Research Placement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3 Comparison of Utility Elicitation Methods 19 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.2 Overview of Assessment Methods and Background . . . . . . . . . . . . . . . . . . . 21 3.2.1 Indierence Utility Elicitation Methods . . . . . . . . . . . . . . . . . . . . . 21 3.2.2 Utility Functions and Methods to Determine Parameters. . . . . . . . . . . . 23 3.3 Results of Behavioral Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.3.1 Results of Curve-Fitting Procedures and Monotonicity of Assessments . . . . 26 3.3.2 Results of Fitting One - Parametric Utility Functions . . . . . . . . . . . . . 29 3.3.3 Results of Fitting Two - Parametric Utility Functions . . . . . . . . . . . . . 37 3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 4 LotteryDesignAlgorithmstoElicitUtilityFunctionsviaPreferenceStatements 44 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 4.2 Motivational Example for Decision-Making under Independent Target Schemes . . . 48 4.2.1 Assumptions of the Motivational Example . . . . . . . . . . . . . . . . . . . . 49 4.2.2 Setup and Structure of Agent-Based Simulation Model (ABM) . . . . . . . . 50 4.2.3 Agent-Based Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . 52 4.2.4 Summary of Results and Implications . . . . . . . . . . . . . . . . . . . . . . 56 4.3 Lottery Design Algorithms to Elicit Utility Functions via Preference Statements . . 58 4.3.1 Rationale of Utility-Elicitation Algorithms . . . . . . . . . . . . . . . . . . . . 58 4.3.2 Assumptions of Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 4.3.3 Algorithm for Step Utility Functions and Example of Application . . . . . . . 61 ii 4.3.4 Algorithm for Exponential Utility Functions and Example of Application . . 64 4.3.5 Algorithm for Logistic Utility Functions and Example of Application . . . . . 66 4.3.6 Testing for Uniqueness of the Obtained Solution . . . . . . . . . . . . . . . . 71 4.3.7 Application Areas, Conclusion, and Limitations . . . . . . . . . . . . . . . . . 73 5 Human Decision-Making Under Certain and Uncertain Incentives 75 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 5.2 Determination of Personal Utility Function, Terminology, and Assumptions . . . . . 78 5.2.1 Elicitation of Personal Utility Functions . . . . . . . . . . . . . . . . . . . . . 78 5.2.2 Terminology and Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . 78 5.2.3 Background on Calculated Measures . . . . . . . . . . . . . . . . . . . . . . . 80 5.3 Experiment Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 5.3.1 Results of Choice Under Fixed Targets . . . . . . . . . . . . . . . . . . . . . . 83 5.3.2 Results of Choice Under Binary Targets . . . . . . . . . . . . . . . . . . . . . 86 5.3.3 Results of Choice Under Averse Targets . . . . . . . . . . . . . . . . . . . . . 91 5.3.4 Results of Choice Under Seeking Targets. . . . . . . . . . . . . . . . . . . . . 94 5.3.5 Results of Choice Under Logistic Targets . . . . . . . . . . . . . . . . . . . . 98 5.3.6 Summary Analysis of Third Factors and Subject Feedback . . . . . . . . . . . 102 5.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 6 Conclusion 115 6.1 Consistency of Human Decision-Makers’ Utility Assessments Across Elicitation Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 6.2 Normative Behavior of Human Decision-Makers Under Incentives . . . . . . . . . . . 118 6.3 Future Research Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 iii Bibliography 122 Appendix 132 Appendix I - Chapter 2: Utility Elicitation Methods . . . . . . . . . . . . . . . . . . . . . 132 Appendix I.I - Curve fitting procedure to determine utility functions . . . . . . . . . 132 Appendix I.II - Determination of wealth parameter using approximation methods M 1 and M 2 ..................................... 133 Appendix I.III - Determination of parameters using approximation methodsM 1 and M 2 for linear risk tolerance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 Appendix II - Chapter 3 ABM and algorithms . . . . . . . . . . . . . . . . . . . . . . . . 135 Appendix II.I - Using utility functions as target generating functions . . . . . . . . . 135 Appendix II.II - Background on incentive schemes with certain target . . . . . . . . 137 Appendix II.III - Background on normalized exponential utility functions . . . . . . 138 Appendix II.IV - Background on normalized logistic utility function . . . . . . . . . 138 Appendix II.V - Proof of algorithm to elicit exponential utility functions . . . . . . . 142 Appendix II.VI - Proof of algorithm to elicit logistic utility functions . . . . . . . . . . . . 143 Appendix III - Chapter 4: Behavioral experiment to assess decision-making behavior under incentives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 Appendix III.I - Experiment setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 Appendix III.II - Ten-item inventory for Big 5 character trait elicitation . . . . . . . 146 Appendix III.III - Lotteries presented to experiment participants . . . . . . . . . . . 149 Appendix III.IV - Fitting procedure for utility functions under incentives . . . . . . 149 Appendix III.V - Individual responses under dierent imposed target schemes . . . . 154 Appendix III.VI - Visualizations used during the behavioral experiment . . . . . . . 170 iv List of Tables 1.1 Overview of methods and their purpose . . . . . . . . . . . . . . . . . . . . . . . . . 2 3.1 Assessed utility functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.2 Individual responses to certain equivalence questions . . . . . . . . . . . . . . . . . . 28 3.3 Individual responses to indierence probability questions . . . . . . . . . . . . . . . . 29 3.4 Risk tolerance fit (Method: Certain Equivalence) . . . . . . . . . . . . . . . . . . . . 34 3.5 Risk tolerance fit (Method: Indierence probability) . . . . . . . . . . . . . . . . . . 35 3.6 Wealth fit (Method:Certain Equivalence). . . . . . . . . . . . . . . . . . . . . . . . . 38 3.7 Wealth fit (Method:Indierence Probability) . . . . . . . . . . . . . . . . . . . . . . . 39 3.8 Comparison parameters derived by approximation and curve fitting - Linear Risk Tolerance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.9 Comparisonparametersderivedbyapproximationandcurvefitting-LogisticUtility Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 4.1 Overview of notation and its meaning . . . . . . . . . . . . . . . . . . . . . . . . . . 47 4.2 Components of the ABM and their attributes and methods . . . . . . . . . . . . . . 51 4.3 Parameter values used in ABM simulation . . . . . . . . . . . . . . . . . . . . . . . . 52 4.4 Summary of algorithm assuming normalized exponential utility function . . . . . . . 67 v 4.5 Parameter settings logistic utility elicitation example . . . . . . . . . . . . . . . . . . 71 4.6 Summary of algorithm assuming normalized logistic utility function . . . . . . . . . 71 4.7 Classification errors between fitted induced utility and target generating function using lottery generating algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 5.1 Summary of tasks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 5.2 Aggregation of responses under fixed target . . . . . . . . . . . . . . . . . . . . . . . 85 5.3 Aggregation of responses under binary target . . . . . . . . . . . . . . . . . . . . . . 90 5.4 Aggregation of responses under exponential (averse) target . . . . . . . . . . . . . . 93 5.5 Aggregation of responses under exponential (seeking) target . . . . . . . . . . . . . . 97 5.6 Aggregation of responses under logistic target . . . . . . . . . . . . . . . . . . . . . . 101 5.7 Summaryofconsistencywithnormativebehaviorunderdierenttargetschemesand demographic data per subject . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 5.8 Coecient-level estimates for model estimating the variation in adherence to prob- ability maximizing behavior (hypothesis H 1 ). . . . . . . . . . . . . . . . . . . . . . . 106 5.9 Coecient-level estimates for model estimating the variation in adherence to prob- ability maximizing behavior (hypothesis H 2 ). . . . . . . . . . . . . . . . . . . . . . . 107 5.10 Coecient-level estimates for model estimating the variation in adherence to prob- ability maximizing behavior (hypothesis H 3 ). . . . . . . . . . . . . . . . . . . . . . . 107 5.11 Membersofgroupspertargetschemeaccordingtok-meansclustering(k=2, outlier subject 8 removed) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 5.12 Consistency with probability and expected utility maximization per group under dierent target schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 5.13 Group 1: Coecient-level estimates for certainty and task type . . . . . . . . . . . . 112 5.14 Group 2: Coecient-level estimates for certainty and task type . . . . . . . . . . . . 112 vi 5.15 Aggregate and groupwise feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 6.1 Overview of utility functions fitted to elicited statements . . . . . . . . . . . . . . . . 133 6.2 Lotteries to elicit step utility function . . . . . . . . . . . . . . . . . . . . . . . . . . 149 6.3 Lotteries to elicit exponential utility function (risk averse) . . . . . . . . . . . . . . . 150 6.4 Lotteries to elicit exponential utility function (risk seeking) . . . . . . . . . . . . . . 150 6.5 Lotteries to elicit logistic utility function . . . . . . . . . . . . . . . . . . . . . . . . . 151 6.6 Binary target infromation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 6.7 Averse target information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 6.8 Seeking target information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 6.9 Logistic target infromation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 6.10 Responses per subject to assessments 1-4 under fixed target . . . . . . . . . . . . . . 155 6.11 Responses per subject to assessments 5-8 under fixed target . . . . . . . . . . . . . . 156 6.12 Responses per subject to validation questions under step targets . . . . . . . . . . . 157 6.13 Responses per subject to assessments 1-4 under binary target . . . . . . . . . . . . . 158 6.14 Responses per subject to assessments 5-8 under binary target . . . . . . . . . . . . . 159 6.15 Responses per subject to assessments 1-4 under averse target . . . . . . . . . . . . . 160 6.16 Responses per subject to assessments 5-8 under averse target . . . . . . . . . . . . . 161 6.17 Responses per subject to assessments 9-10 under averse target . . . . . . . . . . . . . 162 6.18 Responses per subject to assessments 1-4 under seeking target . . . . . . . . . . . . . 163 6.19 Responses per subject to assessments 5-8 under averse target . . . . . . . . . . . . . 164 6.20 Responses per subject to assessments 9-10 under averse target . . . . . . . . . . . . . 165 6.21 Responses per subject to assessments 1-4 under logistic target . . . . . . . . . . . . . 166 6.22 Responses per subject to assessments 5-8 under logistic target . . . . . . . . . . . . . 167 6.23 Responses per subject to assessments 9-12 under logistic target . . . . . . . . . . . . 168 vii 6.24 Responses per subject to assessments 13-16 under logistic target . . . . . . . . . . . 169 viii List of Figures 2.1 Literature review process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2 Title wordcloud for search "Decision-making under incentives" . . . . . . . . . . . . . 8 2.3 Abstract wordcloud for search "Decision-making under incentives" . . . . . . . . . . 8 2.4 Title wordcloud for search "Incentive eects on decision-making" . . . . . . . . . . . 9 2.5 Abstract wordcloud for search "Incentive eects on decision-making" . . . . . . . . . 10 3.1 Example: Visualization of certain equivalence method . . . . . . . . . . . . . . . . . 22 3.2 Example: Visualization of indierence probability method . . . . . . . . . . . . . . . 23 3.3 Visualization of approximation method M1 . . . . . . . . . . . . . . . . . . . . . . . 24 3.4 Visualization of approximation method M2 . . . . . . . . . . . . . . . . . . . . . . . 24 3.5 Subjects’ 1-6 responses and fit curves . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.6 Subjects’ 7-12 responses and fit curves . . . . . . . . . . . . . . . . . . . . . . . . . . 31 4.1 Principal’s utility gap introduced by dierent target schemes . . . . . . . . . . . . . 53 4.2 Principal’s utility gap introduced by dierent target schemes . . . . . . . . . . . . . 54 4.3 Lower and Upper Bound Approximation of Step-Utility Function . . . . . . . . . . . 55 4.4 Lower and Upper Bound Approximation of Exponential Utility Function . . . . . . . 55 4.5 Lower and Upper Bounds on Utility Function . . . . . . . . . . . . . . . . . . . . . . 56 ix 4.6 Heatmap of classification error for dierent parameter setting of X0 and k . . . . . . 57 4.7 Permissible logistic utility functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 4.8 Preference ordering of two lotteries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 4.9 Preference ordering of three lotteries . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 4.10 Convergence of step-utility algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 4.11 Convergence of step-utility function algorithm . . . . . . . . . . . . . . . . . . . . . . 64 4.12 Convergence of bounds towards real risk aversion . . . . . . . . . . . . . . . . . . . . 66 4.13 Bounds on Risk Aversion Coecient at dierent iterations of algorithm . . . . . . . 67 4.14 Convergence of algorithm to parameters . . . . . . . . . . . . . . . . . . . . . . . . . 70 4.15 Proposed and real logistic utility function including outcomes of lotteries. . . . . . . 72 5.1 Personal, induced (fitted), induced (normative) utility functions under fixed target (Subjects 1-6) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 5.2 Personal, induced (fitted), induced (normative) utility functions under fixed target (Subjects 7-12) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 5.3 Personal, induced (fitted), induced (normative) utility functions under averse target (Subjects 1-6) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 5.4 Personal, induced (fitted), induced (normative) utility functions under averse target (Subjects 7-12) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 5.5 Personal,induced(fitted),induced(normative)utilityfunctionsunderseekingtarget (Subjects 1-6) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 5.6 Personal,induced(fitted),induced(normative)utilityfunctionsunderseekingtarget (Subjects 7-12) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 5.7 Personal,induced(fitted),induced(normative)utilityfunctionsunderlogistictarget (Subjects 1-6) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 x 5.8 Personal,induced(fitted),induced(normative)utilityfunctionsunderlogistictarget (Subjects 7-12) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 5.9 Consistency vs. Time clusters of subjects under fixed, binary, and averse targets . . 108 5.10 Consistency vs. Time of subjects under seeking and logistic target . . . . . . . . . . 109 5.11 Consistency with prob.max. along groups . . . . . . . . . . . . . . . . . . . . . . . . 111 6.1 Step utility function with dierent targets . . . . . . . . . . . . . . . . . . . . . . . . 137 6.2 Logistic utility function for dierent parameters x0 and k . . . . . . . . . . . . . . . 140 6.3 Risk aversion function for dierent values of parameter k. . . . . . . . . . . . . . . . 141 6.4 Process flow of the experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 6.5 Screenshot: Introductory video . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 6.6 Screenshot: Age Demographics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 6.7 Screenshot: Gender Demographics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 6.8 Screenshot: Work Experience Demographics . . . . . . . . . . . . . . . . . . . . . . . 172 6.9 Screenshot: Education Demographics . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 6.10 Screenshot: Decision Theory Demographics . . . . . . . . . . . . . . . . . . . . . . . 173 6.11 Screenshot: Probability Theory Demographics . . . . . . . . . . . . . . . . . . . . . . 173 6.12 Screenshot: Big Five Character Traits Question 1 . . . . . . . . . . . . . . . . . . . . 174 6.13 Screenshot: Big Five Character Traits Question 2 . . . . . . . . . . . . . . . . . . . . 174 6.14 Screenshot: Big Five Character Traits Question 3 . . . . . . . . . . . . . . . . . . . . 175 6.15 Screenshot: Big Five Character Traits Question 4 . . . . . . . . . . . . . . . . . . . . 175 6.16 Screenshot: Big Five Character Traits Question 5 . . . . . . . . . . . . . . . . . . . . 176 6.17 Screenshot: Big Five Character Traits Question 6 . . . . . . . . . . . . . . . . . . . . 176 6.18 Screenshot: Big Five Character Traits Question 7 . . . . . . . . . . . . . . . . . . . . 177 6.19 Screenshot: Big Five Character Traits Question 8 . . . . . . . . . . . . . . . . . . . . 177 xi 6.20 Screenshot: Big Five Character Traits Question 9 . . . . . . . . . . . . . . . . . . . . 178 6.21 Screenshot: Big Five Character Traits Question 10 . . . . . . . . . . . . . . . . . . . 178 6.22 Screenshot: Indierence Probability Method Explanatory Video Screen 1 . . . . . . 179 6.23 Screenshot: Indierence Probability Method Explanatory Video Screen 2 . . . . . . 180 6.24 Screenshot: Indierence Probability Method Explanatory Video Screen 3 . . . . . . 181 6.25 Screenshot: Indierence Probability Method Assessment 1 . . . . . . . . . . . . . . . 182 6.26 Screenshot: Indierence Probability Method Assessment 1 Validation . . . . . . . . . 182 6.27 Screenshot: Indierence Probability Method Assessment 2 . . . . . . . . . . . . . . . 183 6.28 Screenshot: Indierence Probability Method Assessment 2 Validation . . . . . . . . . 183 6.29 Screenshot: Indierence Probability Method Assessment 3 . . . . . . . . . . . . . . . 184 6.30 Screenshot: Indierence Probability Method Assessment 3 Validation . . . . . . . . . 184 6.31 Screenshot: Indierence Probability Method Assessment 4 . . . . . . . . . . . . . . . 185 6.32 Screenshot: Indierence Probability Method Assessment 4 Validation . . . . . . . . . 185 6.33 Screenshot: Indierence Probability Method Assessment 5 . . . . . . . . . . . . . . . 186 6.34 Screenshot: Indierence Probability Method Assessment 5 Validation . . . . . . . . . 186 6.35 Screenshot: Certain Equivalence Method Explanatory Video Screen 1 . . . . . . . . 187 6.36 Screenshot: Certain Equivalence Method Explanatory Video Screen 2 . . . . . . . . 187 6.37 Screenshot: Certain Equivalence Method Explanatory Video Screen 3 . . . . . . . . 188 6.38 Screenshot: Certain Equivalence Method Explanatory Video Screen 4 . . . . . . . . 188 6.39 Screenshot: Certain Equivalence Method Assessment 1 . . . . . . . . . . . . . . . . . 189 6.40 Screenshot: Certain Equivalence Method Assessment 1 Validation . . . . . . . . . . 189 6.41 Screenshot: Certain Equivalence Method Assessment 2 . . . . . . . . . . . . . . . . . 190 6.42 Screenshot: Certain Equivalence Method Assessment 2 Validation . . . . . . . . . . 190 6.43 Screenshot: Certain Equivalence Method Assessment 3 . . . . . . . . . . . . . . . . . 191 xii 6.44 Screenshot: Certain Equivalence Method Assessment 3 Validation . . . . . . . . . . 191 6.45 Screenshot: Certain Equivalence Method Assessment 4 . . . . . . . . . . . . . . . . . 192 6.46 Screenshot: Certain Equivalence Method Assessment 4 Validation . . . . . . . . . . 192 6.47 Screenshot: Certain Equivalence Method Assessment 5 . . . . . . . . . . . . . . . . . 193 6.48 Screenshot: Certain Equivalence Method Assessment 5 Validation . . . . . . . . . . 193 6.49 Screenshot: Risk Tolerance Assessment Introductory Video . . . . . . . . . . . . . . 194 6.50 Screenshot: Risk Tolerance Assessment Introductory Video Validation . . . . . . . . 194 6.51 Screenshot: Risk Tolerance Assessment 1 . . . . . . . . . . . . . . . . . . . . . . . . 195 6.52 Screenshot: Risk Tolerance Assessment 1 Validation . . . . . . . . . . . . . . . . . . 195 6.53 Screenshot: Risk Tolerance Assessment 2 . . . . . . . . . . . . . . . . . . . . . . . . 196 6.54 Screenshot: Risk Tolerance Assessment 2 Validation . . . . . . . . . . . . . . . . . . 196 6.55 Screenshot: Introduction to Decision Making under Targets Section. . . . . . . . . . 197 6.56 Screenshot: Introduction to Decision Making under Targets Section Validation . . . 198 6.57 Screenshot: Introduction to decision making under fixed targets . . . . . . . . . . . . 199 6.58 Screenshot: Introduction to decision making under fixed targets validation . . . . . . 200 6.59 Screenshot: Decision making under fixed targets assessment 1 . . . . . . . . . . . . . 201 6.60 Screenshot: Decision making under fixed targets assessment 2 . . . . . . . . . . . . . 202 6.61 Screenshot: Decision making under fixed targets assessment 3 . . . . . . . . . . . . . 203 6.62 Screenshot: Decision making under fixed targets assessment 4 . . . . . . . . . . . . . 204 6.63 Screenshot: Decision making under fixed targets assessment 5 . . . . . . . . . . . . . 205 6.64 Screenshot: Decision making under fixed targets assessment 6 . . . . . . . . . . . . . 206 6.65 Screenshot: Decision making under fixed targets assessment 7 . . . . . . . . . . . . . 207 6.66 Screenshot: Decision making under fixed targets assessment 8 . . . . . . . . . . . . . 208 6.67 Screenshot: Decision making under fixed targets validation choice 1. . . . . . . . . . 209 xiii 6.68 Screenshot: Decision making under fixed targets validation choice 2. . . . . . . . . . 210 6.69 Screenshot: Introduction to decision making under targets with two outcomes . . . . 211 6.70 Screenshot: Introduction to decision making under targets with two outcomes vali- dation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 6.71 Screenshot: Decision making under targets with two outcomes assessment 1 . . . . . 212 6.72 Screenshot: Decision making under targets with two outcomes assessment 2 . . . . . 213 6.73 Screenshot: Decision making under targets with two outcomes assessment 3 . . . . . 214 6.74 Screenshot: Decision making under targets with two outcomes assessment 4 . . . . . 215 6.75 Screenshot: Decision making under targets with two outcomes assessment 5 . . . . . 216 6.76 Screenshot: Decision making under targets with two outcomes assessment 6 . . . . . 217 6.77 Screenshot: Decision making under targets with two outcomes assessment 7 . . . . . 218 6.78 Screenshot: Decision making under targets with two outcomes assessment 8 . . . . . 219 6.79 Screenshot: Introduction to decision making under targets with three outcomes . . . 220 6.80 Screenshot: Introduction to decision making under targets with three outcomes val- idation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 6.81 Screenshot: Decision making under targets with three outcomes assessment 1 - averse222 6.82 Screenshot: Decision making under targets with three outcomes assessment 2 - averse223 6.83 Screenshot: Decision making under targets with three outcomes assessment 3 - averse224 6.84 Screenshot: Decision making under targets with three outcomes assessment 4 - averse225 6.85 Screenshot: Decision making under targets with three outcomes assessment 5 - averse226 6.86 Screenshot: Decision making under targets with three outcomes assessment 6 - averse227 6.87 Screenshot: Decision making under targets with three outcomes assessment 7 - averse228 6.88 Screenshot: Decision making under targets with three outcomes assessment 8 - averse229 6.89 Screenshot: Decision making under targets with three outcomes assessment 9 - averse230 xiv 6.90 Screenshot: Decision making under targets with three outcomes assessment 10 - averse231 6.91 Screenshot: Decision making under targets with three outcomes assessment 1 - seeking232 6.92 Screenshot: Decision making under targets with three outcomes assessment 2 - seeking233 6.93 Screenshot: Decision making under targets with three outcomes assessment 3 - seeking234 6.94 Screenshot: Decision making under targets with three outcomes assessment 4 - seeking235 6.95 Screenshot: Decision making under targets with three outcomes assessment 5 - seeking236 6.96 Screenshot: Decision making under targets with three outcomes assessment 6 - seeking237 6.97 Screenshot: Decision making under targets with three outcomes assessment 7 - seeking238 6.98 Screenshot: Decision making under targets with three outcomes assessment 8 - seeking239 6.99 Screenshot: Decision making under targets with three outcomes assessment 9 - seeking240 6.100Screenshot: Decision making under targets with three outcomes assessment 10 - seeking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 6.101Screenshot: Introduction to decision making under targets with four outcomes . . . 242 6.102Screenshot: Introduction to decision making under targets with four outcomes vali- dation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 6.103Screenshot: Decision making under targets with four outcomes assessment 1 . . . . . 244 6.104Screenshot: Decision making under targets with four outcomes assessment 2 . . . . . 245 6.105Screenshot: Decision making under targets with four outcomes assessment 3 . . . . . 246 6.106Screenshot: Decision making under targets with four outcomes assessment 4 . . . . . 247 6.107Screenshot: Decision making under targets with four outcomes assessment 5 . . . . . 248 6.108Screenshot: Decision making under targets with four outcomes assessment 6 . . . . . 249 6.109Screenshot: Decision making under targets with four outcomes assessment 7 . . . . . 250 6.110Screenshot: Decision making under targets with four outcomes assessment 8 . . . . . 251 6.111Screenshot: Decision making under targets with four outcomes assessment 9 . . . . . 252 xv 6.112Screenshot: Decision making under targets with four outcomes assessment 10 . . . . 253 6.113Screenshot: Decision making under targets with four outcomes assessment 11 . . . . 254 6.114Screenshot: Decision making under targets with four outcomes assessment 12 . . . . 255 6.115Screenshot: Decision making under targets with four outcomes assessment 13 . . . . 256 6.116Screenshot: Decision making under targets with four outcomes assessment 14 . . . . 257 6.117Screenshot: Decision making under targets with four outcomes assessment 15 . . . . 258 6.118Screenshot: Decision making under targets with four outcomes assessment 16 . . . . 259 6.119Screenshot: Elicitation of email address . . . . . . . . . . . . . . . . . . . . . . . . . 260 6.120Screenshot: Elicitation of survey feedback . . . . . . . . . . . . . . . . . . . . . . . . 261 xvi Abstract Utility functions encode a decision-maker’s preference under uncertainty. This research inves- tigates human decision-maker’s utility functions under certain and uncertain incentive schemes. Previous decision analysis research has focused primarily on deriving the form of a fully rational decision-maker’s utility function under a particular incentive or target. This work focuses on three distinct areas of utility research with human subjects: the consistency of human decision-makers’ assessed utilities under common elicitation methods without incentives, the applicability of prefer- ence methods, and humans’ adherence to normative probability maximizing behavior and induced utility theory under incentives. Whencomparingthemostcommonutilityelicitationmethods, certainequivalenceandindier- ence probability, researchers have found that a subject’s utility function is influenced by the chosen method. The behavioral experiment conducted for this work confirms this observation. I find that elicited function parameters depend on the specific method. However, I do not find evidence for other reseachers’ claim that the certain equivalence method results in more risk-seeking behavior. Inaddition, this workinvestigates theconsistencyofsubjects’responsestoapproximationmethods with ones elicited by the two established methods. Studies reviewed for this dissertation omitted this comparison. This work finds that function parameters derived by approximation methods overestimate parameters elicited via established methods in most cases. xvii An alternative method is the elicitation of utility via preferences between uncertain gambles. This approach has been shown to be more consistent and more predictive of actual choice behavior sinceitwasfoundtobelesssusceptibletobias. Becausepreferencemethodsrelyonalargenumber ofobservedchoices,thesecondresearchareaofthisworkisthederivationofalgorithmstoovercome this disadvantage. The algorithms converge in less than twenty assessment rounds to a rational decision-maker’s utility function. Additionally, they provide lotteries to test adherence to induced utility functions under incentives. Under incentives a rational decision-maker chooses the lottery that maximizes their likelihood of meeting or exceeding a target. The target’s probability distribution was shown to be the utility function induced into the probability maximizing decision-maker. By using the theoretically in- ducedutilityasafunctiontobeelicited,thepresentedalgorithmsgenerateaseriesoflotteries. The work pertaining to the third research area uses these lotteries and the utility functions to generate theminabehavioralexperiment. Subjectswereaskedtochooseororderlotteriesundercertainand uncertain targets, where the target distributions were the utility functions to derive the lotteries. I find that in most cases humans prefer the lottery that maximizes their probability of achieving or exceeding their target. Because of probability maximizing behavior and the specific design of the presented lotteries, most subjects exhibited an induced utility function consistent with theoretical results. An analysis of third factors influencing probability maximizing behavior under incentives found no significant impact of age, gender, target generating function, or whether the target was certain or uncertain. Time spent on the survey was the most significant influence on consistency with normative behavior. Subjects who spent less time on the survey had a lower consistency with probability maximization under targets, which decreased significantly once the task was switched from choosing to ordering lotteries. The task had no significant influence on choice behavior for subjects who spent more time on answering the experiment’s questions. xviii Preface Before you lies my dissertation “Utility Functions Induced by Certain and Uncertain Incentive Schemes.” IwrotethisthesistopartiallyfulfilltherequirementsformyDoctorofPhilosophydegree at the University of Southern California (USC). The choice of topic was motivated by my interest in the intersection between normative decision theory and behavioral practice. In particular, it was motivated by: • Standard economic and decision analytic theory prescribe the utility functions induced by certain types of incentives. These prescriptions, however, depend on several assumptions. The aim of this study was to test the degree to which some of these assumptions hold true for human behavior. • Althoughtheoreticalresearchspecifieshowriskattitudechangesunderincentives,elicitingthe form of the induced utility function is challenging. Past experiments used observed choices, fitted a pre-determined utility function, and argued on the basis of the fitted parameter. This approach appears intuitive, but does not include the opportunity to determine whether subjects decide on the basis of the theoretically induced utility function. • The main purpose of incentives is to encourage a subject to choose the same alternative the incentive imposing party prefers. I want to investigate if decision-making behavior be- comesmoreconsistentwithnormativedecision-makingunderincentives, thusbecomingmore xix predictable. Thistopiccameaboutafterworkingonarangeofdierentresearchprojectsduringmyacademic career at USC, including decision analysis for designing large-scale engineering systems, options valuation, and value of information about demand uncertainty, and forecasting methods. I found incentives play a role in all of these domains. For example, when designing large-scale systems, incentives in the form of requirements and deadlines are often used to keep the project on track. In forecasting competitions, scoring rules that rank human experts are used as incentives to ensure truthful reporting and revelation of private information. Given their prevalent role, I want to investigate how they aect decision-making behavior of human subjects. Mydeepestgratitudegoesouttoallthepeopleinmyacademicandpersonallifewhosupported me on this journey. On the academic side, my adviser Professor Ali E. Abbas supported me in my pursuits at USC since my first day. He motivated me to work on topics that seemed daunting, held me to high standards, and thereby helped me to grow. Also, I would like to thank the members of my committee. Professor Sze-Chuan Suen had a big influence on my work through her expertise in modeling and her guidance on analyzing experimental data sets. Professor Shinyi Wu’s counsel in designing and executing the behavioral experiment in this work was invaluable. In my private life, my wife Shira helped me stay the course, supported me by proof-reading, and encouraged me to transform convoluted thoughts into simple concepts. I am also very grateful to my mother and late grandfather who emboldened me to pursue this endeavor. Maximilian Zellner Los Angeles, 08. April 2022 xx Chapter 1 Scope and Approach 1.1 Research Goals and Questions Themainquestionguidingthisresearchis: “How do incentives aect a person’s decision-making behavior?”. However, this question is large and ambiguous, and multiple research disciplines have investigated similar questions, all arriving at dierent conclusions. Instead, the subset of this question examined in this dissertation are: 1. “Are the results of common utility elicitation methods consistent with each other when applied with human decision-makers?” 2. “Does a human decision-maker’s utility function change in accordance to normative decision theory when certain and uncertain incentives are imposed?” 3. “Are there other factors determining this change other than the imposed incentive?” In question 1, I explore whether dierences exist in the assessed parameters of utility functions depending on the used elicitation method. The study considers the widely used certain equivalence and indierence probability methods, as well as approximation methods devised for exponential decision-makers. The answer serves two purposes. First, it sheds light on whether these methods 1 Table 1.1: Overview of methods and their purpose Method Purpose Literature review - Puts research in wider context - Informs the use of other methods Agent-based simulation - Shows whether research questions can be answered via preference orderings - Motivational example to use lottery design algorithms Elicitation algorithm Creates lotteries to assess induced utility functions via preference statements Behavioral experiment - Provides data to assess consistency of utility elicitation methods - Provides data about human decision-makers choice between lotteries under incentives - Provides data to investigate other factors aecting decision-making under incentive canbeusedinterchangeablyornot. Second,itprovidesinsightintowhetherthesemethodsmightbe useful when eliciting utility functions under incentives. Answering question 2 matters because it is a central tenet of decision analysis and principal-agent theory. Question 3 will investigate whether there are other factors explaining the existence or absence of a changing utility function in the context of the behavioral experiment. Demographic factors, including age, gender, and educational background, were previously found to be predictors of choice behavior under no incentives. 1.2 Research Approach and Methodology Thisdissertationemploysacombinationofliteraturesurvey,agent-basedsimulation,algorithms designed by me to devise lotteries for utility elicitation, and a behavioral experiment to answer the research questions. Table 1.1 summarizes each method and their purpose in this work. The purpose of the literature review is to put this dissertation in a wider academic context. Disagreements exist in academic literature on how incentives shape decision-making behavior. By discussing what other researchers have found I put the methods used in this dissertation on a more 2 sound basis. The agent-based simulation model is designed to determine whether one can deduce the in- duced utility function by analyzing the choices between randomly generated lotteries made by the decision-maker. Itallowsmetounambiguouslydeterminetheinducedutilityfunctionbyanalyzing agents’preferencesbetweenlotteriesaslongasthenumberofobservationsinbigenough. However, this number of required observations might be too big for an implementation in a behavioral ex- periment. This requirement for many observations to uniquely identify the induced utility function also serves as a motivational example for algorithmic procedures to design the lotteries presented to decision-makers. These algorithms are a unique contribution and take the theoretically induced utility function as target generating function and produce a sequence of lotteries, which serves two purposes. First, if the subject adheres to normative decision theory, their choice between lotteries allows to unambiguously determine their utility functions with the least amount of observations. This makes preference statements a viable alternative to indierence methods for the elicitation of utilityfunctions. Second,bycreatinglotteriesbasedonthenormativeinducedutilityfunctions,one can investigate whether the human decision-makers decide on the basis of the same function. The behavioral experiment’s purpose is to determine whether human responses to indierence and ap- proximation utility elicitation methods are consistent, and whether human decision-makers decide normatively under targets/incentive schemes. 1.3 Structure of Dissertation The remainder of this dissertation consists of five chapters. The literature review (i) evaluates findings on incentives discussed in literature by performing both an automated search and classical survey. Next, the results of a behavioral experiment comparing common indierence and approxi- mation methods (ii) are discussed. It compares the similarity of parameters derived by indierence 3 andapproximationmethodsforfivedierentfunctionsusingmultiplefittingapproaches. Anagent- based simulation model then investigates whether induced utility functions can be deduced from observed preference orderings between lotteries (iii). Its findings are used to motivate and inform the creation of algorithms that design lotteries to elicit utilities via preference statements. I de- signed and built three algorithms that are applied to create lotteries under certain (fixed) and uncertain (exponential and logarithmic) targets to be presented to human decision-makers. The resultsofabehavioralexperimentinvestigatinghumandecision-makingbehaviorundercertainand uncertain targets (iv) using these lotteries are discussed next. If the subjects adhere to the pref- erence sequence produced by the algorithms, they operate under the theoretically induced utility function. By using these lotteries we can therefore show whether the subjects make decisions in accordance to normative decision theory under incentives. Theworkconcludeswithasummaryofthefindingsandadiscussionofotherresearchdirections. 4 Chapter 2 Literature Review on Decision-Making Under Incentives Chapter Overview Thissectiondiscussespreviousresearchondecision-makingunderincentivesacrossabreadthof academicdisciplinesincludingeconomics,decisionanalysis,andbehavioralpsychology. Thesefields employ a wide array of methodologies, including purely mathematical deliberations, behavioral experiments, and computer simulations. Their often disparate findings, for example the role of extrinsic and intrinsic motivation in incentive design or the eects of incentives of risk taking, are discussed. I also provide background on using computer simulations to study behavioral or sociological observations. This chapter concludes by placing the research at the intersection of purely mathematical deliberations, behavioral experiments, and computer simulations, arguing that these methods can complement each other to study decision-making under incentives. 5 2.1 Introduction The goals of this literature review are: • Identification of academic disciplines researching incentives and key issues • Surveyance of approaches and methods proposed by dierent disciplines and how the disci- plines relate to each other in the context of incentives • Identification of research gap and placement of this work In this chapter, I will describe the literature review process and identify focus areas of research publications on decision-making under incentives. The second section elaborates on the methods and results of incentive research in various academic disciplines. Section three reviews methods of utility elicitation and simulation. The chapter concludes with a comparison and summary of previous research on decision-making under incentives and the placement of this work in the larger context of the field. 2.2 Literature Review Process Figure 2.1 visualizes the literature review process. To obtain an initial overview of research into incentive eects on decision-making, I first determine search terms and conduct an algorithmic literature search using Google Scholar and a self-coded R-package. These results and relevant papers are then reviewed and summarized. 2.3 Results of Algorithmic Literature Search The terms used in this initial search were “Incentive eects on decision-making” and “Decision- making under incentives” . The results allowed me to determine the main research foci by a 6 Determine search terms Algorithmic search and frequency analysis Identification of relevant papers Read and review Review referenced publications Summarize findings Figure 2.1: Literature review process conducting frequency analysis of both publication titles and abstracts. Journal or outlet names were also retrieved to identify the academic fields investigating these topics. Figures 2.2 and 2.3 show the results of the Google Scholar search for “Decision-making un- der incentives” for both publication title and abstract. The most frequent terms in the title are “financial,” “eects,” “information,” and “performance.” “Information,” “conditions,” “risk,” “in- vestment,” and “process” appear most frequently in the abstract. One can partially explain the frequency of these terms by looking at the methods used in those publications. To answer incentive related questions, the majority uses principal-agent frameworks, whose underlying concept is an information dierential between principal and agent (Fudenberg and Tirole 1991; Gibbons 1998; Laont and Martimort 2002; Prendergast 1999). Figures 2.4 and 2.5 perform a similar word cloud representation of the results produced for the search “Incentive eects on decision-making.” The most frequent terms in the titles are “per- 7 financial effects information performance based economic risk health experimental management theory uncertainty role public care choice control economics investment organizational contracts design impact policy analysis quality research approach compensation cost evidence institutions mechanisms pay regulation environmental managerial model system systems tax time behavior conservation effect monetary organizations political supply aversion contract loss market service team behavioral corporate empirical firm investigation optimal participation physicians policies process review risks social study task versus Figure 2.2: Title wordcloud for search "Decision-making under incentives" information risk conditions process investment based theory optimal individual performance uncertainty 1 influence model policy time financial costs analysis choice cost firm decisionmaking managers public authority clinical effect level makers 2 economic social system behavior control agents decentralized impact market rules tax agent effort physicians contract effects research study task function government people political quality role subjects affect capital individuals management power private production 3 5 benefits current experiment local maker results assumption firms fixed health journal paper planning result structure studies systems behavioral business care compensation form game makes outcome utility 4 ability allocation approach change efficient factors found human low offer participants participation policies principal real rule scheme short strategic tend term agency al centralized china circumstances context economics environment expected feedback focus free judgment lead nature outcomes patients price processes provide rational report review subject choices collective consequences contrast criteria data delegation difficult directly executive experiments increase increasing levels mechanism monetary note play pressure prior project projects rate requires resource schemes strategy tasks treatment values view article attention autonomy cognitive communication coordination corporate court employee experimental framework future improve informs involved involvement key land law limited literature lower manager managerial markets money organizational page payments preferences prospect provided related relative service single support terms action assume benefit demand depends effective employees environmental evaluation examine hand hospital hypothesis induce issues manager's mechanisms medical medicine observed organization organizations patient profit proposed resources roles sample services simple tion total university wealth Figure 2.3: Abstract wordcloud for search "Decision-making under incentives" 8 formance (based),” “compensation,” “analysis,” “evidence,” “risk,” and “management.” For the abstract the common terms are “performance (based),” “model,” “compensation,” “risk,” and “information.” Comparing the results obtained for both terms, one can conclude that the main in- terests of matching publications concern the role of information, performance, (performance based) compensation, and risk. performance based compensation evidence analysis model management risk experimental pay contracts control stock structure system systems choice financial information investment theory impact mechanisms role social tax corporate insurance review capital executive strategic cognitive design economic game health incentives managerial managers policies research reward approach firms knowledge monetary motivation public schemes structures task agency aversion behavioral care cost development efficiency effort empirical equity influence learning mechanism options organizational salience study behavior chinese compatible human income monitoring option plans policy processing programs project Figure 2.4: Title wordcloud for search "Incentive eects on decision-making" 2.4 Research Observations on Decision-Making under Incentives Several academic disciplines, including economics, operations research, psychology, and com- puter science, study incentives and how they shape human decision-making. The resulting research can be classified as normative or descriptive depending on their goals and assumptions. The most commonly used normative model in economics to discuss incentives is the principal- agent framework (Laont and Martimort 2002; Prendergast 1999). In its most basic form, the principal cannot directly observe the actions being taken by the agent; only the realized output. 9 performance model based risk compensation information study incentives process research 1 pay 2 choice analysis influence management managers system investment paper results individual page studies systems behavior financial tax time al control literature options 4 5 economic firms schemes 3 data design equity examine firm stock processes reward role task university business effort impact income game positive potential subjects type evidence increase mechanisms option plans policy project relative social structure theory worker capital contracts cost employee experimental government negative organizational play related rewards section set significant treatment uncertainty 6 9 china empirical executive factors level managerial motivation people probability structures costs decisionmaking measures monetary plan program provide repeated review salience short strategic strong team trade utility alternative change characteristics contract examined experiment extent incentiveeffects increases individuals interaction manager motivational optimal participation power price scheme school selection sharing stimuli theoretical 2000 agents authority conditions due economics effective employees expected experience found future investigate land magnitude political quality rates term terms action adverse article assumptions choices context corporate focus joint learning monitoring ownership period policies real relevant requires researchers response rules size strategy variables 1995 2007 affect alignment assume benefits citation cognitive contexts differences differential download evaluation experiments function increased labor lead main makers measure mechanism opportunity outcomes previous programs purpose rate recent setting similar status view workers 0 absence addition agent attention aversion central compatible complex credit development discussed efficiency environmental examines explore federal firm's form functions implementation impulsive influenced influencing issue levels low manager's national owners perceived reflects rights simple specific technology tests theincentive tion types variable versus Figure 2.5: Abstract wordcloud for search "Incentive eects on decision-making" Because of this lack of observability, the principal designs the incentive such that it uses the observable output as the sole measure of goal attainment. To agree to such a contract, the agent requires the payment of a risk premium because this contractual agreement transfers uncertainty about goal attainment to herself. The goal of the principal-agent problem is to determine the incentivethatmakestheagentagreetothecontractwhilesimultaneouslymaximizingtheprincipal’s expected utility (Laont and Martimort 2002). The field of organization science has adopted a principal-agent view of incentives to study organizational structure and dynamics, deducing organizational designs aimed at ecient and fast decision-making (Eisenhardt and Bourgeois 1988; Eisenhardt 1989; Eisenhardt and Zbaracki 1992). These pursuits of organizational scientists have not produced quantifiable and generalizable results that also include behavioral decision theoretic insights. The second normative field of decision theory studying decision-making under incentives is decision analysis. Instead of interpreting the design of an incentive scheme as an optimization 10 problem, research in decision analysis focuses on an agent’s utility function induced by an outside incentive scheme (Abbas and Matheson 2005, 2009; Abbas, Matheson, and Bordley 2009; Glover 2012; Holmstrøm 1999). It has been shown that if the divisions of a firm can be structured such that their performance is independent, there exists an eective utility function under a generalized target-basedincentive(Abbas, Matheson, andBordley2009). Thismeansthatifacompanywishes to induce a certain utility function to align decision-making, it has to ensure that the performance lotteries of the departments are independent from each other. The same research also found that under most linear target-based incentive schemes the induced utility function is S-shaped, mean- ing that decision-makers are risk seeking below and risk averse above the target level. Abbas, Matheson, and Bordley (2009) further observed that risk aversion increases with wealth and that dierent divisions might value portfolios dierently under these linear incentive schemes. Using the decision analytic concept of utility functions, Pennings and Smidts (2003) empirically inves- tigated the relation between chosen organizational structure and subjects’ utility function. They found that although the global shape of the utility function influences the organizational behavior, the decision-maker’s risk attitude does not aect the choice of production systems (Pennings and Smidts 2003). Comparingthenormativeresearchfieldsyieldssimilaritiesanddierences,bothofwhichcanbe found in their assumptions. The most prominent dierence is the standard principal-agent model’s assumptionoftheprincipal’sriskneutralityandtheagent’sriskaversion. Decisionanalyticmodels in comparison do not make this assumption regarding risk attitude. Instead, they appear to be more flexible in their choice of risk aversion and aim to determine the appropriate incentive to induce the principal’s utility function in the agent. The main similarity between the normative areas is their assumption that an extrinsic reward is sucient to produce the desired action if it is big enough. 11 Descriptive research using behavioral data have presented both supporting and conflicting ev- idence for principal-agent theory, sometimes resulting in modified models to accommodate these findings. For example, pay for performance incentive schemes were shown to increase labor output (Banker, Lee, and Potter 1996; Groves et al. 1994; Lazear 2000). It was also shown that piece rate compensation, a specific form of performance pay, was associated with employees working harder (Foster and Rosenzweig 1994). A similar improvement of agent performance was observed when studying incentives in tournaments (Ehrenberg and Bognanno 1990; Knoeber 1989; Knoeber and Thurman 1994). In contrast, research also exists about the negative eects of incentives. For example, free- riding was observed in teams when the reward was tied to group performance instead of individual performance (Gaynor and Pauly 1990). The phenomenon of free-riding was also investigated in the context of prosocial behavior, showing the perverse eect of high-contributors being rewarded for theircontribution. Inthis case, rewardofcontributinggroupmembers reducedtheirangertowards non-contributorsanddiminishedthepressureonnon-compliantmemberstocontribute(Fusterand Meier 2010). Furthermore, it was shown that choosing very strong incentives, i.e. a high monetary reward, can “crowd-out” intrinsic employee motivation (Cappelen and Bjørn-Atle Reme 2015; Frey and Jegen 2002; Gneezy and Rustichini 2000; Kreps 1997). A poorly designed incentive scheme can also result in incentive distortions (Gibbons 1998; Jensen 2003; Kerr 1975). Under such conditions, the incentive results in the agent choosing an optimal action for herself, which does not maximize the principal’s expected utility. An example fromDenmarkhasshownthatpoorlydesignedauctionsofpublicconstructioncontractsresultedin cost-overruns and delays, because they incentivized untruthful behavior of the contractors. These negative consequences could no longer be observed if the contract specified penalties for overruns (Flyvberg, Bruzelius, and Rothengatter 2003; Flyvberg, Holm, and Buhl 2007; Sanderson 2012). 12 An alternative method, other than penalties, to prevent incentive distortions is the use of implicit incentives. This form of incentives is supposed to supplement explicit incentives and prevent unde- siredactionsbyrewardingadherencetopolicyguidelines,providingjobguarantees,jobenrichment, and supplying status rewards such as promotions (Baker, Gibbons, and Murphy 1994; Besley and Ghatak 2008; Gibbs et al. 2009; Glover 2012; Holmstrøm 1999). Working in an environment that caters to an agent’s preferences, i.e. her intrinsic motivators, was also shown to reduce the need for strong external incentives (Prendergast 2008). Criticismofdecisionanalyticmodelscomesfrombehavioraldecisionscientistsandpsychologists. Their main concern lies with the assumption of unbounded rationality made by normative decision science (Simon 1979). Experiments have shown that human subjects make decisions inconsistent withnormativedecisiontheory, whichledbehavioraldecisionsciencetodevelopconceptsofitsown todescribehumandecision-making. Themostprominentofthosearesatisficing(Simon1956,1959), heuristics (Gigerenzer and Gaissmaier 2011; Gigerenzer, Todd, and Group 1999; Schwenk 1984), andprospecttheory(KahnemanandTversky1979; TverskyandKahneman1992). Withrespectto goal setting, satisficing perceives a target as a set of constraints that need to be satisfied. Following this logic, the human decision-maker chooses the actions that satisfies those constraints the best (Simon, 1964). Proponents of the heuristics view on decision-making assume that because humans are not capable of including all information and possible outcomes in the evaluation of a complex decisionproblem,theyresorttorulesofthumb(GigerenzerandGaissmaier2011;Gigerenzer,Todd, and Group 1999). 2.5 Review of Elicitation and Simulation Methods This research applies utility elicitation and simulation methods to answer its questions. In the following sections, I will discuss elicitation methods, including their process and potential 13 limitations, modelling methods and how they can be used to simulate and study human decision- making behavior. 2.5.1 Review of Utility Elicitation Methods Depending on the academic field, the term utility has dierent meanings. In the field of eco- nomics,utilityisdefinedasamonetaryvalue(forexampleFudenbergandTirole(1991); Kahneman andTversky(1979); LaontandMartimort(2002); TverskyandKahneman(1992)). Contrastingly, the field of decision analysis defines utility as an indierence probability. The probability of receiv- ing the best outcome makes the decision-maker indierent between keeping an uncertain deal with outcomesAandCorsellingitforacertainoutcomeB,wherethepreferenceorderingofoutcomesis A>B>C. Assuming that the utility function over all outcomes is twice-dierentiable, the decision- maker’s risk aversion is then given by the negative fraction of its second over its first derivative (Howard and Abbas 2015; Neumann and Morgenstern 1944). The decision-analytic definition of utility and risk aversion will be used throughout this dissertation. Using the probabilistic deal outlined in the previous paragraph, four main methods exist to elicit utility functions: • Certain equivalence: This method elicits an indierence value for B given A, C, and proba- bility p. The subject is asked for the value of B that makes them indierent between selling the uncertain deal with prospects A and C with probabilities p and 1-p respectively. At indierence, value B is the personal indierence selling price and the certain equivalent of the deal. • Probability equivalence: Under this scenario, the indierence probability p is elicited for given levels of A, B, and C. Similar to the certain equivalence method, it is assumed that the decision-maker possesses the uncertain deal. They are asked which probability p of prospect 14 A would make them indierent between keeping the deal or selling it for B. At indierence, probability p is then the utility of the uncertain lottery. This method is also referred to indierence probability method. • Gain equivalence: The level of the most preferred outcome A is elicited for given values of B, C, and probability p. • Loss equivalence: The level of the least preferred outcome C is elicited for given values of A, B, and probability p (Hershey, Kunreuther, and Schoemaker 1982). Criticismoftheseelicitationmethodsstemfrominconsistentresultsproducedduringbehavioral experiments. Applyingdierentmethods,behavioralresearchhasshownthatsignificantdierences exist in the resulting utility function (Hershey, Kunreuther, and Schoemaker 1982; Hershey and Schoemaker1985). AccordingtoHershey,Kunreuther,andSchoemaker(1982),theanalysteliciting the utility function must consider five forms of biases introduced by the chosen elicitation method, where bias is described as behavior violating the axioms of rational decision-making (see Howard (1992)). Hershey, Kunreuther, and Schoemaker (1982) describe the five biases as response mode bias, biases induced by the probability and outcome levels, aspiration level eects, inertia eects, andcontexteects. Responsemodebiasoccursinbehavioralexperimentswhentheutilityfunction derived by the certain equivalent method does not match the function produced by probability equivalence (Hershey and Schoemaker 1985). In particular, Hershey and Schoemaker (1985) found thatcertaintyequivalencemethodsyieldgreaterrisk-seekingthanprobabilityequivalencemethods. Inertia eects can be observed when comparing the utility functions derived by asking whether the participant wants to assume or transfer risk away, meaning selling or buying an uncertain lottery. Furthermore, the analyst should choose the elicitation lotteries such that subjects feel a familiarity withtheprospectstopreventcontextbias(HersheyandSchoemaker1985). Anexampletoprevent contextbiaswasshownbyPenningsandSmidts(2003)whenanalyzingtheeectofriskaversionon 15 the choice of production systems. Because the experiment subjects were hog farmers, the authors chose the performance lottery and the outcomes were prices of hog futures, although they only used the certain equivalence method to derive the farmer’s utility functions (Pennings and Smidts 2003). The biases presented by Hershey, Kunreuther, and Schoemaker (1982) and Hershey and Schoe- maker(1985)aresimilartothebiasesfoundinprobabilityencoding(Abbasetal. 2008;Spetzlerand Holstein 1975). This makes intuitive sense because utility is defined as an indierence probability. 2.5.2 Review of Simulation Methods to Model Human Behavior Dierentwaystodistinguishsimulationmodelsarealongtime(discreteversuscontinuoustime) oralongthelevelofdetail(BorshchevandGrigoryev2013). Distinguishingsimulationmodelsalong their level of detail, one can choose between system dynamics, which models changes in the system stateusingdierentialequations(Forrester2013;Sterman2000),discrete-eventsimulation(Rossetti 2015), and agent-based simulation, also called microsimulation (Borshchev and Grigoryev 2013). In contrast to system dynamics models and discrete-event simulation, agent-based simulation applies a bottom-up procedure, meaning that agent behavior is determined at the smallest unit of the system without specifying overarching system boundaries. The agents are then interacting with each other over time, and the behavior of the overall system is shaped by the behavior of its most basic units. In comparison to the other two types of simulation modeling, the agent-based approach holds several advantages. It is scalable because agents can be added easily. And it does not require many assumptions regarding overall system behavior because it emanates over time from the agents’ interactions (Macal and North 2010). For example, a system dynamics model requires the definition of a structural model first, which can constrain the type of system behaviors one can see during the simulation. 16 Becauseofitsflexibilityandtheabundanceofsimulationsoftwareavailabletoconstructmodels, agent-based simulation has been used to study phenomena in diverse fields such as multicriteria decision-making (Bishop, Stock, and Williams 2009), design of computer clusters (Zhang and Xu 2017), social conflict (Lemos 2018), human behavior in systems (Bonabeau 2002), design of re- silient systems (Colson, Nehrir, and Gunderson 2011), water conservation (Ng et al. 2011), and organization science (Anussornnitisarn, Nof, and Etzion 2005; Fioretti 2013; Rivkin and Siggelkow 2003; Siggelkow and Levinthal 2003). Davis, Eisenhardt, and Bingham (2007) present a process of how simulation approaches can be used in organization science to derive novel theories, which are otherwise mathematically intractable or too costly to study using behavioral experiments. The proposedprocessstartswiththecleardefinitionofaresearchquestion,theidentificationofasimple theory, and the choice of an appropriate simulation approach. One then chooses the computational representationandverifiesitbeforeconductingexperimentstoderivenoveltheories. Thesetheories then have to be validated using empirical data (Davis, Eisenhardt, and Bingham 2007). 2.6 Research Placement In surveying the existing research on the eects of incentives on decision-making, it is clear that tension exists between the fields of normative and descriptive theories, and that the eects of incentives on decision-making can be explained from many dierent angles. Oneoftheseanglesisthedistinctionofextrinsicandintrinsicmotivation,whichisoftenreferred towhendiscussingstrongandweakincentivesinprincipal-agentmodels. Fromtheliteraturereview onecandeducethatacommonlyheldviewamongresearchersisthatifanemployeedoesnotpossess sucientintrinsicmotivation,acompanyneedstoimplementstrongincentivestoincreaseextrinsic motivation. At the same time, the company could save on incentive expenditure if the employee possesses sucient intrinsic motivation. This dierentiation has led to recommendations to adjust 17 hiringprocessestorecruitemployeeswithhighintrinsicmotivation, andtoincreasecommunication about corporate values and mission statements. Asecondangleexploredbyresearchisthedierentialofriskaversionbetweenactors. Following principal-agent theory, to align decision-making a more risk averse agent has to be compensated for taking on the risk of not achieving a certain performance. In decision analysis, the imposition of a performance target induces a utility function that is independent from the agent’s, allowing for the alignment of decision-making behavior. It is important to note that the term utility has two dierentmeaningsineconomicsanddecisionanalysis. Economicsdefinesutilityasamonetaryvalue to the decision-maker, whereas decision analysis assumes utility to be the indierence probability between an uncertain lottery and a certain deal. The third angle is how incentives can and should be used in social structures. For example, decision theory research has shown that incentives will not result in expected utility maximiza- tion if the performance lotteries of agents are correlated because of the organizational structure. Also, some research in principal-agent theory argues that the way target achievement is measured has to be aligned with the chosen organizational structure because the structure can obscure the contribution of one single agent to the goal. The research in this dissertation adopts the second angle, using normative decision-analytical concepts of induced utility functions, and contrasts them with behavioral data. This includes a behavioralexperimenttoassesstheconsistencyofhumandecision-makers’responsesunderdierent elicitationmethods. Thesameexperimentprovidesdataonwhethersubjectsactnormativelyunder imposed target schemes, supporting or refuting whether they act under the normatively induced utility function. 18 Chapter 3 Comparison of Utility Elicitation Methods Chapter Overview The most common methods to determine a person’s utility function are indierence methods. This chapter presents the results of a behavioral study eliciting 12 subjects’ utility functions by certain equivalence and indierence probability methods, and the responses to two approximation methods developed for exponential utility functions. Curve fitting approaches are applied to de- termine the parameters of five types of utility functions and to identify the best fit. There are four main findings from this behavioral experiment. First, approximation methods perform poorly at predicting the parameter(s) of the utility function obtained by fitting to individual or all ob- servations simultaneously. Second, fitting to all observations simultaneously should be preferred overfittingtoindividualassessmentsandthenaggregatingtheobtainedvaluesbecauseofthelatter method’sincreasedsusceptibilitytooutliers. Third, noparameterobtainedbyfittingtoanindivid- ual assessment’s response was reliably close the the fitted value for the exponential and logarithmic utility function. Fourth, the standard deviation of individual assessment parameters was lower us- 19 ing the certain equivalence method compared to indierence probability assuming exponential and logarithmic utility functions. This suggests that subjects were more consistent under the certain equivalence method. 3.1 Introduction The goal of this chapter is to answer the research question “Are the results of common utility elicitation methods consistent with each other when applied with human decision-makers?”.This question is important because previous research has shown that indierence probability and cer- tain equivalence methods introduce inherent bias into the assessment process, potentially yielding dierent results (Farquhar 1984; Hershey and Schoemaker 1985). Abbas et al. (2008) compare the results of two probability encoding methods along monotonicity, accuracy, and precision metrics. They find that the fixed variable method, which elicited a percentile of the variable’s distribution, was superior. Assuming a utility function normalized to the interval 0 and 1, the fixed variable methodcorresponds to theindierence probabilityelicitationmethod. The researchinthis chapter seeks to validate these claims and to augment them with a comparison to approximation methods. To contrast theory and behavioral reality, and to determine whether this chapter’s research question is true or not, a behavioral experiment was designed and deployed online via Qualtrics. In total, 12 (8 male, 4 female) graduate students at USC’s Industrial and Systems Engineering department participated in this study. I first captured demographic factors to enable an analysis of third factors influencing the participants’ responses. Because the population was very homoge- neous (similar educational background and very narrow age range) this analysis is omitted. Next, participants were randomly assigned to either indierence probability (IP) or certain equivalence (CE) methods. After performing the CE or IP methods, subjects answered two questions that correspond to the approximation methods M 1 and M 2 . In comparison to the CE and IP methods, 20 theseapproximationmethodsusetheresponsestodeterminetheutilityfunctionparameterviaone indierence question. Multiple forms of utility functions were then fit to the responses and insights were derived by comparing the fitting results along several metrics. The following section will provide an overview of assessment methods and background on the utility functions that are fitted to the assessments. Results are then discussed in section two, which is split into subsections on general curve-fitting, results on one-parametric utility functions (exponential and logarithmic), and results pertaining to two-parametric utility functions (linear risk tolerance and logistic). The chapter concludes with a discussion of findings, limitations, and further research directions. 3.2 Overview of Assessment Methods and Background The goal of this section is to provide an overview and background on elicitation methods and utility functions. Utility elicitation methods are being discussed in the first subsection. The second subsection contains background on the utility functions and how the experiment’s observation can be used to derive their parameters. 3.2.1 Indierence Utility Elicitation Methods The most common ways to elicit utility functions are certain equivalence and indierence prob- abilitymethods. Empiricalresearchhasshownthataperson’sriskaversioncandierdependingon theusedelicitationmethod(Farquhar1984;HersheyandSchoemaker1985). IncontrasttoHershey and Schoemaker (1985), a between-subject design was chosen and subjects were randomly assigned to one of the two elicitation methods. This design keeps the experiment duration short, but limits thecomparabilityofassessedparametersbecauseitassumesthattherearenosignificantdierences between subjects. To mitigate the risk of subjects not understanding the elicitation method, videos 21 were developed and used to explain the methods and their implementation in detail. In the experiment, both utility elicitation methods assess five points. The certain equivalence method elicits the certain amounts that make a decision-maker indierent between receiving an amount or selling their deal with probabilities of 10%, 25%, 50%, 75%, 90% of receiving $100. The reverseistruefortheindierencemethod,whichasksthedecision-makertoidentifytheprobability makes them indierent between keeping or selling for $10, $25, $50, $75, $90 respectively. Figures 3.1 and 3.2 showcase how these assessments are implemented in Qualtrics. Figure 3.1: Example: Visualization of certain equivalence method Because the experiment is conducted online, it was not possible to ask probing or clarifying questions to remove any inconsistent assessments. To mitigate this issue, the study provided participants the opportunity after each assessment to review their answers and change them if necessary. After completing the assessments using indierence methods, subjects were presented with two approximation methods, M 1 and M 2 . Instead of using several assessments, these approximation methods relied on only one assessment to derive the parameter of the decision-maker’s utility func- 22 Figure 3.2: Example: Visualization of indierence probability method tion. Method M 1 asks the decision-maker which amount R would make them indierent between receivingnothingorplayingagamblethatpaysRor≠ R 2 withequalprobability. Thesameparame- terRiselicitedinmethodM 2 ,butthepayosoftheuncertaindealare Rand≠ Rwithprobabilities 3 4 and 1 4 respectively. Figures 3.3 and 3.4 show the implementation of the approximation methods during the behavioral experiment. 3.2.2 Utility Functions and Methods to Determine Parameters Table 3.1 summarizes the utility functions evaluated in this chapter. They can be distinguished along the number of parameters that are assessed. It is assumed that all are normalized to an interval between zero and one to maintain the interpretability of utility as preference probability. This assumption is omitted from table 3.1 for the sake of brevity and readability. Please refer to appendix I.I for the normalized forms and the respective inverse/certain equivalent of these functions. 23 Figure 3.3: Visualization of approximation method M1 Figure 3.4: Visualization of approximation method M2 24 Table 3.1: Assessed utility functions Function name Functional form Determined parameters Linear U(x)= x x ú Exponential U(x)=1≠ e ≠ x/fl fl Logarithmic U(x)=ln(x+w) w Linear Risk Tolerance U(x)=(fl +÷x ) 1≠ 1/÷ fl,÷ Logistic U(x)= L 1+e ≠ k(x≠ x 0 ) k,x 0 The approach to analyze responses is contingent on the utility function to be evaluated: • One-parametric utility functions: Exponential and logarithmic utility functions are one- parametric because they feature either risk tolerance or wealth as function parameters. To analyze the data, a curve is fit to each single assessment of a subject, yielding five parameter values. Next, a utility function of the same form is fit to all five points simultaneously (see appendix I.I). The approximation methods provide two more values for the parameter. As- suming an exponential utility function, the outcome chosen by the subject when faced with the deals of M 1 and M 2 yields their risk tolerance (Howard and Abbas 2015). In case of the logarithmic utility function, the values provided in M 1 and M 2 provide the exact wealth and a lower bound on wealth respectively (see appendix I.II). • Two-parametric utility functions: The two-parametric functions are linear risk tolerance and logistic,whichrequiretheevaluationoftwoparameterseach. Becausefittingatwoparametric function to a single observation is not possible due to a lack of degrees of freedom, this particularanalysisisomitted. Instead,thesefunctionsarefittoallfivepointssimultaneously. Because the certain amounts in the approximation methods are equal, one can assume that the expected utilities of the deals should also be equal. Therefore, the parameters obtained by curve fitting are compared to the ones that establish equality between the expected utility of the deals in M 1 and M 2 . 25 The following section presents the results and insights of these fitting procedures. 3.3 Results of Behavioral Experiment This section covers the results of this behavioral experiment and includes three subsections. The first discusses results of fitting several dierent utility functions to the observations and their monotonicity. The second discusses findings regarding one-parametric utility functions. The third delves into findings regarding two-parametric functions. 3.3.1 Results of Curve-Fitting Procedures and Monotonicity of Assessments Evaluating the monotonicity of the points serves two purposes: 1. Ifthepointsonthex,U(x)planearenotmonotonouslyincreasing,thedecision-makerviolates the axioms of rational decision-making. They would prefer a deal with lower to a deal with higher probability of obtaining the same positive outcome. 2. All utility functions in the curve fitting procedure are monotonously increasing. The fraction ofmonotoneassessmentscan(partially)explainhighfittingerrorsasmeasuredbythesquared error. The subjects’ responses and the squared errors of the fitted utility functions are separated by elicitationmethod. Table3.2and3.3showtheresultsofsubjectsassignedtothecertainequivalence and indierence probability methods respectively. Figures 3.5 and 3.6 plot the utility functions fit to each subject’s assessments, which are symbolized by red circles. From the tables, one can make observations regarding subjects’ risk neutrality, the relationship between curve fitting error and observations’ montonocity, as well as the relation between the number of function parameters and the function’s fitting error. 26 Risk neutrality implies that the squared error of the linear utility function is lowest. Apart from looking at the fitting error, one can see that subjects 4 and 8 assigned the expectation of the presented deals as their certain equivalent, making them risk neutral by definition. Comparing groups along their number of risk neutral subjects, we find two risk neutral subjects in the certain equivalence and none in the indierence probability group. One can also observe that the highest fitting error occurred for subjects whose rank correlation coecient was less than 1. This result makes intuitive sense. The exception was subject 2, whose Spearman coecient was 1, but whose fitted logistic squared error was highest compared to other subjects in their group. This observation can be explained by looking at subject 2’s individual responses. For probabilities less than or equal to 50%, they value the uncertain deal less than its expectation, implying risk aversion and a concave utility function. For higher probabilities, the subject becomes risk neutral. A logistic utility function is S-shaped, i.e. first convex and then concave, meaning that it does not fit well to such assessments. Although the indierence probability group had a higher number of non-monotonous responses at fewer subjects, one cannot say with certainty that the elicitation method is responsible for an increaseinnon-monotonousassessments. ThereweretwosubjectswithaSpearmanrankcorrelation coecient less than 1 in the indierence probability group (fraction 40%), and one in the certain equivalence group (fraction: 1/7). This might suggest that assessments produced using the certain equivalence method are more monotone, i.e. consistent, compared to the indierence probability method. Additional research is needed to conclusively determine the validity of this hypothesis. Withtwo-parametricutilityfunctionpossessingmoredegreesoffreedomtofittheobservations, they feature a lower fitting error than one-parametric functions as long as montonicity of the as- sessments is given. Ignoring subjects whose best-fitting function is linear, this assumptions holds true. The exception is the function fit to subject 10’s responses. This subject’s best fitting utility 27 Table 3.2: Individual responses to certain equivalence questions Subjects 2 4 7 8 9 11 12 CE for p success 10% [$] 5 10 7 10 10 20 5 CE for p success 25% [$] 10 25 25 25 20 25 15 CE for p success 50% [$] 20 50 45 50 45 20 40 CE for p success 75% [$] 75 75 65 75 55 30 65 CE for p success 90% [$] 90 90 80 90 80 50 86 Linear sq. error 1150.0 0.0 234.0 0.0 550.0 4625.0 341.0 Exponential sq. error 513.8 0.0 91.9 0.0 167.9 660.2 13.3 Logarithmic sq. error 441.6 3.4 101.7 3.4 204.0 1178.9 8.9 Linear Risk sq. error 367.7 0.0 110.2 0.0 236.3 2387.6 6.4 Logistic sq. error 635.2 0.5 24.0 0.5 62.3 200.8 29.5 Spearman Coecient 1.0 1.0 1.0 1.0 1.0 0.8 1.0 function is logarithmic, i.e. one-parametric. However, this person’s assessments were monotonously decreasing with a Spearman rank correlation coecient of -1. For subjects 6 and 11, whose assess- ments’ montonicity was less than 1, the assumption that two-parametric utility function fit better than one-parametric ones still held. 28 Table 3.3: Individual responses to indierence probability questions Subjects 1 3 5 6 10 Indierence p at $10 [%] 12 20 24 10 80 Indierence p at $25 [%] 27 35 30 25 73 Indierence p at $50 [%] 55 60 51 50 65 Indierence p at $50 [%] 80 85 80 100 61 Indierence p at $90 [%] 90 100 100 100 58 Linear sq. error 0.006 0.050 0.035 0.072 0.865 Exponential sq. error 0.001 0.007 0.022 0.054 0.508 Logarithmic sq. error 0.001 0.007 0.022 0.056 0.368 Linear Risk sq. error 0.001 0.007 0.018 0.057 0.492 Logistic sq. error 0.001 0.009 0.024 0.025 0.518 Spearman Coecient 1.000 1.000 1.000 0.975 -1.000 3.3.2 Results of Fitting One - Parametric Utility Functions This section discusses results of fitting one-parametric utility functions to subjects’ responses. Both exponential and logarithmic utility functions are fit to each assessment point separately to obtainthefunctionparameters,andseveralsummaryparametersarecalculated. Then,thesevalues are compared to the parameters obtained by fitting these functions jointly to all points, and to the parameters obtained via two approximation methods (M 1 and M 2 ). 3.3.2.1 ComparisonofElicitationMethodsAssuminganExponentialUtilityFunction The analysis fit an exponential utility function to each subject’s assessments, regardless of whether their best fitting utility function was exponential or not. Tables 3.4 and 3.5 contain the results for the certain equivalence and indierence probability groups. One can group the observa- tions into four main areas of comparison: values obtained by fitting all observations simultaneously versus fitted to individual points, values obtained by fitting all observations simultaneously versus 29 0 20 40 60 80 100 0.0 0.2 0.4 0.6 0.8 1.0 Subject: 1 0 20 40 60 80 100 0.0 0.2 0.4 0.6 0.8 1.0 Subject: 2 0 20 40 60 80 100 0.0 0.2 0.4 0.6 0.8 1.0 Subject: 3 0 20 40 60 80 100 0.0 0.2 0.4 0.6 0.8 1.0 Subject: 4 0 20 40 60 80 100 0.0 0.2 0.4 0.6 0.8 1.0 Subject: 5 0 20 40 60 80 100 0.0 0.2 0.4 0.6 0.8 1.0 Subject: 6 0 0 Exponential Logarithmic Linear Risk Tolerance Logistic Figure 3.5: Subjects’ 1-6 responses and fit curves 30 0 20 40 60 80 100 0.0 0.2 0.4 0.6 0.8 1.0 Subject: 7 0 20 40 60 80 100 0.0 0.2 0.4 0.6 0.8 1.0 Subject: 8 0 20 40 60 80 100 0.0 0.2 0.4 0.6 0.8 1.0 Subject: 9 0 20 40 60 80 100 0.0 0.2 0.4 0.6 0.8 1.0 Subject: 10 0 20 40 60 80 100 0.0 0.2 0.4 0.6 0.8 1.0 Subject: 11 0 20 40 60 80 100 0.0 0.2 0.4 0.6 0.8 1.0 Subject: 12 0 0 Exponential Logarithmic Linear Risk Tolerance Logistic Figure 3.6: Subjects’ 7-12 responses and fit curves 31 approximation methods, approximation methods M 1 versus M 2 , and certain equivalence versus indierence probability methods. Comparing the value when fitting to all observations simultaneously with the minimum and maximum values of fitting to individual points, one finds that the former falls consistently within the range established by the minimum and maximum of the latter only when one includes risk- neutral assessments/infinite values. In case these assessments are excluded, the fitted value falls within this interval 80% of times under both elicitation methods. Assessing whether the mean or median of the individually fitted points is closer to the value obtained by fitting to all, one can see thatundertheindierenceprobabilitymethod, 20%offittedvaluesareclosesttothemean, 60%to themedian,andin20%ofcasesthesemeasuresarethesame. Thereverseistrueforpointsassessed bythecertainequivalencemethod. 60%ofthetimethefittedvalueisclosertothemeanthantothe median,andin40%ofcasesitisclosertothemedian. Comparingtheparametersofeachindividual assessmentwiththeparameterderivedbyfittingtoallofarespondent,thereisnoarticulartrendin the indierence probability (IP) group. Not one particular assessment consistently produced a risk tolerance that was close to the fitted composite parameter across subjects. A similar observation holds true for subjects in the certain equivalence (CE) group. Checking whether the values obtained by the approximation method are similar to the fitted- to-all parameter, one finds that in 3/5 (IP) and 2/7 (CE) cases, the fitted parameter was lower than any of the ones produced by the approximation methods. In 2/5 (IP) and 2/7 (CE) of cases the fitted values were higher, and only 1/7 (CE) were within the range if one ignores 2 risk neutral cases. Mathematically, the risk tolerance parameter produced by M2 is slightly lower than the one obtained by M1 for an exponential decision-maker. Judging from the observations, however, the opposite is true for the indierence probability group, except for subject 5 whose choice is the 32 possible minimum set by the survey. For the certain equivalence group, the results are mixed, with 3/7 (M1 < M2), 2/7 (M1 > M2), and 2/7 set to the survey maximum or minimum. Because these approximations methods were designed for exponential decision-makers, it is reasonable to narrow this comparison to only the subjects whose best fitting utility function was exponential. The decision-makers who meet this criterion are subjects 1 and 3. In both cases, the parameters obtainedbyfittingtoallobservations,aswellasthemeanandmedianoftheindividualassessments’ parameters, were lower than the minimum of the parameters obtained by using the approximation methods. When comparing the standard deviation of the parameters obtained by fitting to observations individuallybetweengroups,wefindthatit’smeanacrosssubjectsis52.21and150.7forthecertain equivalence and the indierence probability groups (excluding risk neutrality). This means that the individual assessments and parameters obtained by the certain equivalence method vary less, and are therefore more consistent, than the values elicited by the indierence probability method. Hershey and Schoemaker (1985) found that the certain equivalence method resulted in higher risk seeking behavior compared to the indierence probability methods. Following this particular research, one would expect a higher risk tolerance in subjects who were assigned to the certain equivalence method in this experiment. Taking the mean of the fitted risk tolerance, of the mean and median of the individual assessments, and of the mean of the approximation methods, the observation made by Hershey and Schoemaker (1985) cannot be replicated. In fact, the opposite, where subjects assigned to the certain equivalence group had a lower risk tolerance, was found to be the case. 33 Table 3.4: Risk tolerance fit (Method: Certain Equivalence) Subjects 2 4 7 8 9 11 12 fl 1 58.52 Œ 120.80ŒŒ -66.25 58.52 fl 2 37.85ŒŒ Œ 168.66 Œ 72.35 fl 3 30.48 Œ 248.33 Œ 248.33 30.48 121.63 fl 4 ŒŒ 106.69 Œ 56.49 22.16 106.69 fl 5 ŒŒ 66.25 Œ 66.25 22.76 136.64 Mean 42.28 Œ 135.52 Œ 134.93 2.28 99.17 Median 37.85 Œ 113.75 Œ 117.46 22.46 106.69 Max 58.52 Œ 248.33 Œ 248.33 30.48 136.64 Min 30.48 Œ 66.25 Œ 56.49 -66.25 58.52 St.dev. 14.54 Œ 78.68 Œ 91.04 45.85 32.93 Range 28.04 Œ 182.08 Œ 191.84 96.73 78.12 fl fitted 85.30 Œ 150.81 Œ 89.64 24.63 107.74 fl M1 500 1000 50 0 50 200 20 fl M2 200 1000 60 0 101 248 9 Mean fl M1 ,fl M2 350.00 1000.00 55.00 0.00 75.50 224.00 14.50 Spearman Coecient 1.00 1.00 1.00 1.00 1.00 0.82 1.00 34 Table 3.5: Risk tolerance fit (Method: Indierence probability) Subjects 1 3 5 6 10 fl 1 237.30 54.25 40.25 Œ 6.21 fl 2 476.61 99.93 194.70 Œ 19.32 fl 3 249.16 123.32 1249.83 Œ 80.77 fl 4 177.87 82.56 177.87 1.81 -72.39 fl 5 Œ 2.20 2.20 2.20 -18.47 Mean 285.24 72.45 332.97 2.01 3.09 Median 243.23 82.56 177.87 2.01 6.21 Max 476.61 123.32 1249.83 2.20 80.77 Min 177.87 2.20 2.20 1.81 -72.39 St.dev. 131.34 46.67 519.35 0.28 55.83 Range 298.75 121.11 1247.63 0.39 153.16 fl fitted 273.16 91.37 170.35 150.73 8.58 fl M1 299 357 0 40 200 fl M2 598 386 0 101 300 Mean fl M1 ,fl M2 448.50 371.50 0.00 70.50 250.00 Spearman Coecient 1.00 1.00 1.00 0.97 -1.00 35 3.3.2.2 Comparison of Elicitation Methods Assuming a Logarithmic Utility Function A similar analysis to the one assuming an exponential utility function was performed for the logarithmic functional form. Tables 3.6 and 3.7 contain the results for the certain equivalence and indierence probability groups. The observations are again grouped into four main areas of com- parison: values obtained by fitting to all observations simultaneously versus fitted to individual points, values obtained by fitting to all observations simultaneously versus approximation meth- ods, approximation method M 1 versus M 2 , and certain equivalence versus indierence probability method. Assessing whether the parameter value when fitting to all observations simultaneously falls within the range of the minimum and maximum values obtained by fitting to individual points, one finds that former falls within the later only when one includes risk-neutral assessments/infinite values. This observation is identical to the one assuming an exponential utility function. Similarly, neither the mean nor the median of fitting to individual points are indicative of the parameter producedwhenfittingtoallassessments. Forpointsassessedbytheindierenceprobabilitymethod, 40%offittedvaluesareclosesttothemean, 40%tothemedian, andin10%ofcasesthesemeasures are the same. This distribution is dierent for the responses under the certain equivalence method. 5/7 of times the fitted value is closer to the mean than to the median, and in 2/7 median and mean are the same. Assessing whether one particular assessment yields a parameter that is close to the fit-to-all parameter, one finds that this is not true for the indierence probability group. In the certain equivalence group, the individual points of assessments 1 and 4 are close to fitted value several times. In contrast to the exponential case, the approximation methods do not yield similar values for the wealth parameter when solved analytically. Instead, method M 1 yields the decision-maker’s 36 exact wealth, while methodM 2 produces a lower bound on the same parameter (see appendix I.II). Investigating whether the fit-to-all parameter is greater or equal to the lower bound shows that it is not in most cases. Excluding risk neutral subjects, the fitted value is greater than the lower bound in 1/5 (CE) and 2/5 (IP) cases. In the remaining cases, the fitted value is less than the lower bound. At the same time, the fitted parameter is not close to the value obtained byM 1 ,with the exception of subject 9. Comparingthevaluesobtainedbyapproximationmethods,onewouldexpectthevalueobtained by M 1 to be consistently higher than the lower bound. However, this is not the case throughout and is contingent on the elicitation method. It is higher for four out of seven subjects assigned to the certain equivalence method (includingrisk neutral subjects), but only forone out offive sujects belonging to the indierence probability group. The mean standard deviation of individual wealth parameters across subjects is 41.98 and 139.28 for the certain equivalence and indierence probability groups (excluding risk neutrality). This means that observations obtained under the certain equivalence method are more consistent with each other than ones elicited using indierence probability. 3.3.3 Results of Fitting Two - Parametric Utility Functions This section discusses insights from fitting two-parametric utility functions. The derived func- tion parameters are compared to the values that establish equality between the expected utility of deals/lotteries of the approximation methods. 37 Table 3.6: Wealth fit (Method:Certain Equivalence) Subjects 2 4 7 8 9 11 12 w 1 34.03 Œ 90.84ŒŒ Œ 34.03 w 2 15.13ŒŒ Œ 132.53 Œ 42.41 w 3 6.67 Œ 202.50 Œ 202.50 6.67 80.00 w 4 ŒŒ 58.14 Œ 16.56 0.89 58.14 w 5 ŒŒ 18.81 Œ 18.81 0.10 80.55 Mean 18.61 Œ 92.57 Œ 92.60 2.55 59.03 Median 15.13 Œ 74.49 Œ 75.67 0.89 58.14 Max 34.03 Œ 202.50 Œ 202.50 6.67 80.55 Min 6.67 Œ 18.81 Œ 16.56 0.10 34.03 St.dev. 14.01 Œ 78.98 Œ 91.10 3.59 21.24 Range 27.37 Œ 183.69 Œ 185.94 6.57 46.52 w fitted 35.57 1000.00 116.60 1000.00 55.83 1.05 62.97 w M1 500 1000 50 0 50 200 20 w lb 200 1000 60 0 101 248 9 Spearman Coecient 1.00 1.00 1.00 1.00 1.00 0.82 1.00 3.3.3.1 Comparison of Elicitation Methods Assuming aLinearRisk Tolerance Utility Function Anadvantageoffittingatwo-parametricutilityfunctiontoanindividual’sassessmentisalower fittingerrorduetomoreflexibility. However,Ifindthatonehastoconsiderlimitationswhenfitting a linear risk tolerance utility function to the responses of the approximation methods because the deal outcomes constrict the choice of fitted parameters. More specifically, if ÷ is even,fl>÷x because x can be less than zero, which would result in a negative root (see appendix I.III). Table 3.8 visualizes the parameters produced by equating the approximation methods and the curve fitting procedure. One can observe that in 8/12 cases, the parameter fl that minimizes the dierence between expected utilities of gambles in method M 1 and M 2 is at its imposed minimum offl>÷x ,where x is the chosen indierence amount in M 2 . This is not true for the remaining 38 Table 3.7: Wealth fit (Method:Indierence Probability) Subjects 1 3 5 6 10 w 1 203.64 28.38 16.97 Œ 0.00 w 2 436.34 64.39 156.33 Œ 0.64 w 3 201.68 78.41 1200.33 Œ 38.58 w 4 123.57 33.52 123.57 0.00 Œ w 5 Œ 0.00 0.00 0.00 Œ Mean 241.31 40.94 299.44 0.00 13.07 Median 202.66 33.52 123.57 0.00 0.64 Max 436.34 78.41 1200.33 0.00 38.58 Min 123.57 0.00 0.00 0.00 0.00 St.dev. 135.26 30.99 508.05 0.00 22.09 Range 312.76 78.41 1200.33 0.00 38.58 w fitted 229.12 51.10 120.35 122.71 0.20 w M1 299 357 0 40 200 w lb 598 386 0 101 300 Spearman Coecient 1.00 1.00 1.00 0.97 -1.00 four cases because the best fitting parameter ÷ is uneven, meaning that we do not need to impose this artificial minimum on parameter fl . Omitting subjects 4 and 8, who are risk neutral, fl is consistently higher for the approximation method than for the general curve fitting procedure. The summary measures of mean and median support this claim. While the mean fl is higher for the curve fitting procedure because of the extreme values produced by fitting to the risk neutral assessments of subjects 4 and 8, the median is significantly lower than the one produced by the approximation methods. The median ÷ using the approximation and curve fitting methods is the same. 3.3.3.2 Comparison of Elicitation Methods Assuming a Logistic Utility Function Table 3.9 summarizes the results for each subject. In 8/12 cases, the function’s midpoint parameter x 0 is higher when using the approximation method. For the remaining cases, where this 39 Table 3.8: Comparison parameters derived by approximation and curve fitting - Linear Risk Tol- erance Approximation method Curve fitting Subject x M1 x M2 fl M1,M2 ÷ M1,M2 fl÷ 1 299.0 598.0 1196.0 2.0 190.0 2.0 2 500.0 200.0 1896.0 3.0 0.0 3.0 3 357.0 386.0 772.0 2.0 22.5 2.0 4 1000.0 1000.0 2000.0 2.0 10000.0 20.0 5 0.0 0.0 1.0 3.0 0.0 5.0 6 40.0 101.0 202.0 2.0 97.5 2.0 7 50.0 60.0 120.0 2.0 87.5 2.0 8 0.0 0.0 1.0 3.0 10000.0 20.0 9 50.0 101.0 202.0 2.0 32.5 2.0 10 200.0 300.0 600.0 2.0 0.0 2.0 11 200.0 248.0 496.0 2.0 0.0 2.0 12 20.0 9.0 134.0 11.0 7.5 3.0 Mean 226.3 250.2 635.0 3.0 1703.1 5.4 Median 125.0 150.5 349.0 2.0 27.5 2.0 Max 1000.0 1000.0 2000.0 11.0 10000.0 20.0 Min 0.0 0.0 1.0 2.0 0.0 2.0 St.dev. 292.2 297.1 707.3 2.6 3875.9 6.9 Range 1000.0 1000.0 1999.0 9.0 10000.0 18.0 40 Table 3.9: Comparison parameters derived by approximation and curve fitting - Logistic Utility Function Approximation method Curve fitting Subject x M1 x M2 k M1,M2 x 0,M1,M2 kx 0 1 299.00 598.00 0.09 59.00 0.02 30.00 2 500.00 200.00 0.11 59.00 0.02 0.00 3 357.00 386.00 0.10 60.00 0.03 17.50 4 1000.00 1000.00 0.09 60.00 0.01 50.00 5 0.00 0.00 0.01 1.00 0.02 15.00 6 40.00 101.00 0.24 24.00 0.06 42.50 7 50.00 60.00 0.28 84.00 0.04 37.50 8 0.00 0.00 0.01 1.00 0.01 50.00 9 50.00 101.00 0.20 29.00 0.04 32.50 10 200.00 300.00 0.10 58.00 0.20 0.00 11 200.00 248.00 0.11 60.00 0.16 27.50 12 20.00 9.00 0.18 23.00 0.02 0.00 Mean 226.33 250.25 0.13 43.17 0.05 25.21 Median 125.00 150.50 0.10 58.50 0.03 28.75 Max 1000.00 1000.00 0.28 84.00 0.20 50.00 Min 0.00 0.00 0.01 1.00 0.01 0.00 St.dev. 292.19 297.06 0.08 26.58 0.06 18.66 Range 1000.00 1000.00 0.27 83.00 0.19 50.00 parameter was lower for the approximation methods, the elicited x M1 and x M2 were set to their minimal values by the subject. A similar observation can be made for the steepness parameter k. For most cases it was higher when using the approximation methods. There was only one case in which it was lower, other than for x M1 and x M2 being set to their minimums or the individual assessments being monotonously decreasing (Subject 10). There is no detectable dierence in the fitted steepness parameter k between the elicitation method groups. The summary measures mean, median, and standard deviation of the function parameters were consistently higher for the approximation methods compared to the values obtained by fitting to the observations under certain equivalence and indierence probability methods. 41 3.4 Conclusion In investigating whether human decision-maker’s responses to dierent utility elicitation meth- ods are consistent with each other, existing research has shown that there are dierences between the established certain equivalence and indierence probability methods. This research aimed to validate these results for multiple types of utility functions, and extend them to approximation methods that elicit the function parameter via a single indierence assessment. The methodology leveragedabehavioralexperimentthateliciteddemographicfactors,assignedsubjectstoeithercer- tain equivalence or indierence probability methods, and concluded by asking subjects to answer two questions for the approximation methods M 1 and M 2 .Theresultswerederivedbyfittingmul- tiple types of utility functions to each subject’s assessment individually and simultaneously to all, and by comparing the derived values to the respective parameters obtained via the approximation method. The findings can be summarized as: • There are significant dierence between function parameters depending on whether we use certain equivalence or indierence probability methods. Both the median and mean risk tolerance and wealth were lower for the subjects assigned to the certain equivalence group. ThissupportsthefindingsderivedbyHersheyandSchoemaker(1985)thatelicitedparameter depends on the applied elicitation method. However, the results do not support the claim by the authors that the certain equivalence method produces more risk-seeking behavior than the indierence probability method. • Theparametersdeterminedbyapproximationmethodsoverestimatetheparametersobtained by curve fitting. I used lotteries to approximate the risk tolerance of exponential decision- makers and extended their use to other utility functions. However, the prospect of losing a 42 monetary amount imposes restrictions on the parameters to be fitted. This implies the need to devise approximation lotteries for utility functions other than exponential. • Decision-maker’s are not consistent between the two approximation methods assuming they follow an exponential utility function. The risk tolerance parameters of the approximation methods should be almost identical. Instead, they are (very) dierent. • Fitting all assessments simultaneously is preferred to using approximation methods, despite the higher assessment eort. The former approach appears to more robust to outliers, while the latter produces highly variable responses depending on the presented gamble. To build on the findings from this research, there are three main research directions worth exploring. First,thesubjectpoolshouldbeexpandedtobemorenumerousandmoreheterogeneous. This would ensure that the results are generalizable to the overall population. The second research direction is to investigate the response to indierence methods using a within-subject experiment design, similar to Hershey and Schoemaker (1985). Using such a design, the subject would reveal their utility function under both methods, allowing one to investigate the similarity of the derived functions and parameters. Thirdly, one might extend the experiment to have the subject choose between lotteries multiple times to observe the predictive power of the assessed function. 43 Chapter 4 Lottery Design Algorithms to Elicit Utility Functions via Preference Statements Chapter Overview In this this chapter I will discuss the applicability of preference statements to determine a person’sutilityfunction. Anagent-basedsimulationmodelisdevisedtoshowthatusingasubject’s choicebetweenrandomlygeneratedlotteriesmightrequiremanyobservationsbeforetheboundson the subject’s fitted utility function are suciently tight. The lottery design algorithms developed in this chapter reduce the number of necessary assessments. They elicit step, exponential, and logistic utility function parameters and converge to close bounds in less than 20 iterations at most. I will discuss application areas of these algorithms, including utility elicitation with and without incentives. 44 4.1 Introduction In an extensive survey of utility assessment methods, Farquhar (1984) distinguishes four methodological categories: preference comparison, probability equivalence, value equivalence, and certainty equivalence methods. The author further categorizes them into standard-gamble and paired-gamble methods. When discussing preference comparison methods with paired gambles, the author highlights the method’s advantages to be elicitation simplicity and bias reduction (see Hershey and Schoemaker (1985) for bias introduced by probability and certainty equivalence methods), but cautions that the method’s applications are mostly consistency checks and multi- attribute independence tests (see for example Keeney and Raia (1993)). Some researchers use preferencecomparisonmethods, mostlywitheven-chancedgambles, toconstructanorderedmetric scale for utilities (Debreau 1959; McClelland and Coombs 1975; Pfanzagl 1959; Suppes and Winet 1955). Others use linear and non-linear programming approaches to estimate an admissible utility function by fitting it to the observed data (Bradley and Frey 1975; Suppes and Walsh 1959). A comparison of utility assessment methods according to choice predictability was performed by Daniels and Keller (1992). They compare indierence judgments, i.e. probability or certainty equivalence methods, with choice-based/preference comparison procedures when eliciting exponen- tialutilityfunctions. Theauthorsfoundchoice-basedprocedurestobemoreprecisewhenpredicting choicebehavior. However,theresearchersconstrictedtheutilityvaluetobeobtainedbyeachchoice and the subjects had to perform more than twenty iterations to yield reasonable bounds on the exponential utility function (Daniels and Keller 1992). The authors highlight that using choice- based procedures to elicit utility function requires a large amount of choice data, but that this disadvantage might be outweighed by higher confidence and consistency across judgments, as well as higher replicability. 45 Therequirementofmoreobservationsbychoice-based,orpreferencecomparison,methodscould suggest that these approaches are less suitable to elicit utilities in behavioral experiments where data collection is costly and time-intensive. The goal of any elicitation is for the subjects to answer as few questions as possible to obtain tight enough lower and upper bounds on their personal utility function. This means that if one were to use a preference comparison method, the lotteries presented to a human subject in a behavioral experiment have to be designed such that the lower andupperboundsconvergeecientlytowardsthedecision-maker’srealriskaversion. Theresulting assessment would be easier cognitively and reduce ambiguity compared to indierence methods. This trade-o between lower cognitive load and higher predictive power of preference statement methods on the one hand, and a large number of observations on the other, motivate the design of the algorithms presented in this chapter. In comparison to the method devised by Abbas (2004), the presented algorithms do not rely on entropy to decide on the next question to ask, and ask for preference between uncertain lotteries instead of between a certain prospect and an uncertain deal. The algorithm presented by Chajewska, Koller, and Parr (2000) diers from the presented approach in that the former assumes the utility function to be a prior distribution that is updated using Bayes theorem after a choice is observed. Additionally, the questions asked by Chajewska, Koller, and Parr (2000) are similar to the ones of Abbas (2004), asking for preference between a certain amount and an uncertain deal. The goals of this chapter are to: 1. Demonstrate how preference statements can be used to determine (induced) utility function through an agent-based simulation model. 2. Presentthethreealgorithmsdevelopedtoelicitstep,exponentialandlogisticutilityfunctions. 3. Showcase the convergence of these algorithms. 4. Discuss the application of these algorithms in utility elicitation with and without incentives. 46 Table 4.1: Overview of notation and its meaning Notation Meaning T Fixed target amount (step utility function) “ Risk aversion coecient of an exponential utility function t Interval of the risk aversion coecient at iteration t defined by the lower bound “ l t and upper bound “ u t . x 0 Midpoint of the logistic utility function k Steepness of the logistic utility function L Maximum of logistic utility function, assumed to be 1. [X t 0 ;K t ] Interval of the parameters x 0 and k at iteration t. Both are defined by upper and lower bounds: X t 0 =[x l 0 ,x u 0 ] and K t =[k l ,k u ] L i Lottery iœ [1,3] x i Non-zero outcome of lottery i p i Probability of x i to occur x 0 Minimum outcome, assumed to be 0. x ú Maximum outcome c Correction term used in the algorithm for the logistic utility function as a function of the initial correction term c 0 , the rate of adjustment r c , and the number n k of previous adjustments of parameter k. This chapter first used an example of an agent-based simulation model (ABM) to highlight the use of preference statements for utility elicitation. It shows how the principal’s utility function can be used as a target generating function such that a fully rational agent maximizes the principal’s expected utility. The model also highlights the need for a systematic lottery design if the choices between them were used to elicit the agent’s induced utility function. The second section unpacks the algorithmic methods to design lotteries for step, exponential, and logarithmic utility functions. This chapter concludes with a discussion of how these algorithms are used to elicit utility functions with and without incentives, their limitations and future work. Both the model and the algorithm rely on mathematical notation, which is summarized in table 4.1. 47 4.2 Motivational Example for Decision-Making under Indepen- dent Target Schemes The goals of this motivational example are: 1. To present an alternative approach to impose expected utility maximizing incentives. Previ- ousworkbyotherauthorshasrequiredtheimpositionoftheaspirationequivalent(Abbasand Matheson 2005; Abbas, Matheson, and Bordley 2009) of each performance lottery to ensure expectedutilitymaximizationoftheprincipal. Instead,theapproachusedinthissectionuses the principal’s utility function as the distribution of an uncertain target. If the agent chooses the lottery that maximizes their probability of meeting or exceeding the target, they auto- matically maximize the principal’s expected utility. In contrast to the aspiration equivalent method, this approach does not require knowledge about the lotteries at the agent’s disposal. Calculating the aspiration equivalent can become dicult or impossible if the lotteries are plentiful or if the principal does not know the performance lotteries available to the agent. Furthermore, the presented approach eliminates the incentive to the agent to lie about their lotteriesbecauseknowledgeoftheexactlotteriestodeterminethetargetisnolongerrequired. 2. To present ways utility functions induced in the agent can be reconstructed from their prefer- ence between lotteries. This will also highlight the necessity to carefully curate the presented lotteries in order to elicit the induced utility function via preference statements between lotteries. The following sections will discuss the agent based simulation model and results. The moti- vational example concludes with the reconstruction of the induced utility function via preference orderings, limitations, and how the results aect the elicitation of induced utility functions in 48 behavioral settings. 4.2.1 Assumptions of the Motivational Example The main assumptions of the motivational example are : • Actors of the model are only motivated by extrinsic motivation. Meaning, they decide solely on the basis of the extrinsically imposed incentive. • Actors are fully rational, utility and probability maximizing. • There are no limits to their ability to perform mathematical calculations. • Utility functions are normalized to an interval of zero and one, thereby being comparable to cumulative probability functions. • Payos under every target scheme are constant, meaning that they do not depend on the degree of target attainment. Once the target is achieved or exceeded, the actor receives the same payo regardless of the extent of target exceedance. Actors are not penalized, meaning they do not incur personal losses. • Actors do not deceive. The following definitions and explanations apply: • There are two types of actors in the model, termed principal and agent used to describe business titles of manager and employee. The principal takes on a managing role of the agent, who works towards the goal set by the principal by choosing between performance lotteries. • A performance lottery is an uncertain gamble characterized by outcomes and probabilities. • An incentive is a scheme used to entice the agent to act on the principal’s behalf. The incentivesinthischaptercanbedistinguishedbetweencertainanduncertaintargets. Certain 49 targets are fixed and need to be achieved or exceeded. Uncertain incentive are probability distributions, and are either normalized exponential and logistic functions. 4.2.2 Setup and Structure of Agent-Based Simulation Model (ABM) The two types of agents in this model can be characterized by attributes and methods. In addition, the environment contains methods and attributes that cannot be assigned to either agent type, but are necessary for the simulation. Table 4.2 summarizes the attributes and methods of each component. An agent of type principal is characterized by a name, their personal utility function and its parameters, as well as the lotteries that are passed into it through the environment. The lottery attributeisnotrequired,althoughithelpstoquantifyapotentialvaluegapthattheprincipalincurs iftheemployeechoosesalotteryontheirbehalf. Anon-zerovaluegapoccursiftheemployeechooses a lottery that is not the preferred choice by the principal. The principal possesses two methods. One is the ability to create an incentive scheme and to assign them to an agent. They can choose between fixed or uncertain incentives. If they choose a fixed target, they have to provide the target amount. In case of an uncertain target, they can choose between a normalized exponential target distribution or a normalized logistic distribution. The second method is to make a fictitious choice between the lotteries presented to the agent, meaning the principal determines the lottery they would prefer if they got to choose instead of the agent. The agent is characterized by the attributes name, the lotteries available to them, and the incentive/target scheme they are choosing under. This last attribute is passed into them by the principal. Their only method is to make a choice between the lotteries during every iteration of the simulation. The assumption is that they decide perfectly rationally, meaning that they maximize their probability of achieving or exceeding the target and do not pursue any other motivation. 50 Table 4.2: Components of the ABM and their attributes and methods Component Attributes Methods Principal - Name - Parameters of own utility function - Available lotteries - Create incentive scheme - Determine own preferred choice Agent - Name - Available lotteries - Imposed incentive scheme Choose lottery that maximizes target attainment probability Environment Generate lotteries The environment generates the lotteries in a two-step procedure. First, the outcomes of two binary lotteries are created by sampling the best prospect from the interval (x 0 ,x ú ]. The worst prospect is automatically set to x 0 . Then, the probability of the best outcome is sampled from a beta distribution. The probability of the worst outcome occurring is calculated as one minus the sample probability. A top-level view of how the dierent components of the model interact during the simulation is provided in the pseudocode outlined in algorithm 1. Algorithm 1: High level view of ABM simulation 1 Initiate agents and environment 2 Agent of type principal assigns target scheme to employee 3 while iÆ simRuns do 4 Environment creates lotteries 5 Agent of type employee chooses preferred lottery 6 Agent of type principal determines own preferred lottery 7 end 51 Table 4.3: Parameter values used in ABM simulation Simulation parameter Value General parameters Number of simulation iterations 500 Parameters of agent type principal Parameter “ for principal with exponential utility function -0.1 Parameters x 0 ,k for principal with logistic utility function 50, 0.15 Fixed target parameter 30 Exponential target parameter “ -0.1 Logistic target parameters x 0 ,k 50, 0.15 Parameters of simulation environment Number of lotteries per iteration 2 Number of lottery outcomes 2 Minimum outcome per lottery 0 Maximum outcome per lottery 100 Shape parameters of beta sample distribution 2,2 4.2.3 Agent-Based Simulation Results The results of the agent-based simulation are obtained using the parameter settings given in table 4.3. The simulation contains two principals with an exponential and logistic utility function each. If they impose an uncertain target, they choose a distribution equal to their own utility function. In addition, the simulation contains three agents. One operates under a fixed target, while the others decide under exponential and logistic uncertain targets. The first subsection discusses the value gap each principal incurs if they had each of the agents choose on their behalf. The second section discusses results of reconstructing the agents’ induced utility function via their choices between lotteries. 4.2.3.1 Value Gap Analysis This section discusses the value or utility gap the principals incur over the course of the simu- lation if they chose to impose dierent target schemes. A principal incurs a value gap if the agent that is deciding on their behalf chooses a lottery that does not maximize their expected utility. 52 The principal follows either an exponential or logistic utility function, and can choose to impose their own utility function as a target distribution or another type. Figures 4.1 and 4.2 visualize the cumulative expected utility gap for the exponential and logistic principal had they imposed dierent incentive schemes. Onecanobservetwopoints. First,thegapofexpectedutilityremainsflatduringthesimulation if the principals impose their own utility function as the target distribution. Second, imposing a logistic target instead of a fixed target results in a lower expected utility gap over the long run. An explanation for this observation is the shape of the logistic utility function. It is convex below the sigmoid midpoint x 0 , meaning that it has the same curvature as the principal’s risk seeking exponential utility function. However, the fixed target remains the same across all iterations. If one changed it to be the aspiration equivalent of the available lotteries (see Abbas and Matheson (2005)), the expected utility gap would remain zero as well. 0 100 200 300 400 500 0 2 4 6 8 Cumulative Expected Utility Gap under di↵erent incentive schemes for an exponential decision-maker Iteration Cumulative Expected Utility Gap Exponential Goal Logistic Goal Fixed Goal Figure 4.1: Principal’s utility gap introduced by dierent target schemes 53 0 100 200 300 400 500 0 2 4 6 8 10 12 14 Cumulative Expected Utility Gap under di↵erent incentive schemes for a logistic decision-maker Iteration Cumulative Expected Utility Gap Exponential Goal Logistic Goal Fixed Goal Figure 4.2: Principal’s utility gap introduced by dierent target schemes 4.2.3.2 Reconstruction of Utility Functions via Preference Statements between Lot- teries This section discusses the reconstruction of the agents’ utility function induced by the im- posed target schemes. The function parameters are found by minimizing the choice error between theoretical and observed choices. The results of fitting the fixed target to the observed choices can be seen in figure 4.3. The approximated lower bound is 30 and the upper bounds is 30, compared to the real target of 30. Fittinganexponentialutilityfunctiontothechoicesoftheagentundertheexponentialincentive produced the lower and upper bound on the risk aversion coecient shown in figure 4.4. Their numeric values are -0.1 and -0.1 respectively. These are the risk aversion coecients that minimize the classificiation error between the agent’s observed and the probability maximizing choices. The principal’s real risk aversion coecient is -0.1 and is contained within the bounds. The utility curves from the lower and upper bounds, as well as of the principal’s utility function, are shown in 54 0 20 40 60 80 100 0.0 0.1 0.2 0.3 0.4 0.5 Classification Error for Fixed Targets Target Classification Error Error Lower bound Upper bound Figure 4.3: Lower and Upper Bound Approximation of Step-Utility Function -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.00 0.10 0.20 0.30 Classification Error for Exponential Targets Risk aversion coecient Classification Error Misclassified choices Lower Bound Upper Bound Figure 4.4: Lower and Upper Bound Approximation of Exponential Utility Function 55 0 20 40 60 80 100 0.0 0.2 0.4 0.6 0.8 1.0 Outcome Utility Lower Bound Upper Bound Principal Risk Aversion Figure 4.5: Lower and Upper Bounds on Utility Function figure 4.5 Similarly, a normalized logistic utility function is fit to the observed choices made by the agent underthelogistictargetscheme. Comparedtostepandexponentialtargets, thegoalinthissection is to find two function parameters (x 0 and k) instead of one. Onecanseefromfigure4.6thattheclassificationerrorisminimalatapproximately50and0.15. Using the all possible parameter settings that minimize the classification error, one can determine the upper and lower bounds on the utility function. These bounds are given in figure 4.7, showing that the lower and upper bound are so close together they are almost non-distinguishable as two lines. 4.2.4 Summary of Results and Implications The motivational example highlighted the need of many simulation runs to obtain reasonably tight bounds on the utility curves if one were to use preference statements. This means that if utility elicitation via binary preference statements is to be a viable alternative to other methods, 56 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 0 25 50 75 100 Parameter x0 Parameter k 0.0 0.1 0.2 0.3 Classification Error Classification Error for values of x0 and k Figure 4.6: Heatmap of classification error for dierent parameter setting of X0 and k 0.00 0.25 0.50 0.75 1.00 0 25 50 75 100 Outcome Utility Figure 4.7: Permissible logistic utility functions 57 it has to converge quickly to tight bounds. If successful, this procedure can be used in behavioral experiments to elicit utility functions induced by incentive schemes. The next section will discuss the algorithmic approach developed to curate lotteries for preference assessments, thereby reducing the number of required observations. 4.3 LotteryDesignAlgorithmstoElicitUtilityFunctionsviaPref- erence Statements The following algorithms were built to design lotteries that elicit a decision-maker’s utility function via preference statements. The first subsection provides the thought process behind the algorithms, followed by underlying assumptions, and the algorithms for step, exponential, and logistic utility functions. Each of the following sections contain the algorithm’s pseudocode and an application example. The chapter concludes with a determination about whether the produced lotteries and the normative choice between them converge to one specific family of utility function. 4.3.1 Rationale of Utility-Elicitation Algorithms Usingpreferencestatementstoelicitutilitymeansthatfunctionsarefittothepreferenceorders of lotteries. Figures 4.8 and 4.9 demonstrate how subjects are asked for their preference ordering of binary lotteries. From the responses one can learn whetherEU GambleI >EU GambleII in the case of two lotteries, or whetherEU GambleI >EU GambleII >EU GambleIII in the case of three, assuming the decision-maker maximizes their expected utility. The choice of whether to use two or three gambles per algorithm iteration depends on the number of parameters of the utility function to be estimated. If there is only one parameter to be estimated, such as in the case of step or exponential utility functions, using two gambles is sucient. With two parameters, three lotteries are required to ensure enough degrees of freedom 58 for the estimation. Figure 4.8: Preference ordering of two lotteries Figure 4.9: Preference ordering of three lotteries Algorithm 2 provides a top-level view on the elicitation of utility function parameters via pref- erence statements. It first initializes the bounds on the parameters to be elicited. For example, one has to specify the range of the fixed target T in case of a step-utility function, the risk aversion coecient “ for the normalized exponential function, and parameters x 0 (midpoint of the sigmoid) and k (steepness of the curve) for the logistic utility function. From these bounds the algorithm 59 determines a (set of) specific parameter(s), which is used to determine two or three lotteries. The lotteries are generated such that their expected utilities are equal given the specific parameter(s). Meaning, a decision-maker would be indierent between the lotteries if the parameters of their utility function are the same as the ones used to generate them. In the next step, the preference ordering made by the decision-maker is observed. With any preference ordering other than indif- ference, the bounds on the parameters are adjusted and the algorithm repeats the generation of parameters and lotteries. The algorithm stops once the decision-maker signals indierence because this means the correct parameters were found. It is advisable to stop the algorithm after a certain amount of iterations, even if indierence has not been achieved. Algorithm 2: Top-level view of elicitation algorithms Input : Maximum number of iterations Output: Utility function parameters and used lotteries 1 Initiate parameter bounds 2 while iÆ maxIterations | Subject is not indierent btw. lotteries do 3 Determine parameter from parameter bounds 4 Generate lotteries 5 Observe preference ordering 6 Update parameter bounds 7 end 4.3.2 Assumptions of Algorithms The developed algorithms assume that the decision-maker follows either a step, normalized exponential or logistic utility function. Furthermore, it assumes that the decision-maker is fully 60 rational and maximizes expected utility when making a decision, and that they also do not make inconsistent assessments due to their full rationality. In the context of researching whether the decision-maker adheres to normative decision-making under incentives the assumptions are reasonable because: 1. If one uses a utility function as the function to generate an uncertain target, the decision- maker should decide on the basis of the same utility function. 2. The algorithms create a sequence of lotteries and choices that are consistent with normative decision theory. If the decision-maker deviates from this sequence, they violate normative decision theory or have a dierent type of utility function. 4.3.3 Algorithm for Step Utility Functions and Example of Application The goal of the step-utility algorithm is to find parameter T at which the utility jumps from zero to one. In contrast to the subsequent algorithms, this one stops after a certain number of iterations (in this case 30). This is because the decision-maker’s choice of indierence between the lotteries can also mean that both nonzero outcomes of the lotteries are below the target. If this is the case, the expected utility of each lottery is zero, resulting in indierence. Algorithm 3 shows the pseudo-code for eliciting a step utility function via preference statement. The example eliciting a decision-maker’s step utility function assumes that the real function jumpsfromzerotooneatthefixedtarget’svalueof65. Figure4.10showshowthelowerandupper boundsonthestep-utilityfunctiongetsmallerforselectnumberofalgorithmiterations. Fromfigure 4.11 one can see that the size of the gap between upper and lower bound is strictly decreasing. It gets smaller very quickly at first and then approaches zero. In this particular example, the gap is less than 1% (visualized as a red horizontal line) after 7 iterations. 61 0 20 40 60 80 100 0.0 0.2 0.4 0.6 0.8 1.0 Outcome x Utility Real Target Lower bound it. 3 Upper bound it. 3 Lower bound it. 5 Upper bound it. 5 Lower bound it. 10 Upper bound it. 10 Figure 4.10: Convergence of step-utility algorithm 62 Algorithm 3: Algorithm to elicit step utility functions Input : Initial lower and upper bound on fixed target T: T L ,T U Output: Updated T L ,T U , series of elicitation lotteries 1 while T U ≠ T L Ø Â do 2 Choose the probabilities p 1 and p 2 of receiving the nonzero outcome of lotteries L 1 and L 2 such that p 1 ”=p 2 and p 1 ,p 2 œ (0,1]. 3 Determine nonzero outcomes x 1 ,x 2 of lotteries L 1 ,L 2 at iteration i from the upper and lower bounds on the fixed target T. 4 if p 1 x 1 . 8 end 9 Update bounds on parameter T after observing choice between lotteries. 10 if p 1 L 2 then 12 T U =x 1 ,T L =x 2 13 else if L 1 =L 2 then 14 T L =max(x 1 ,x 2 )=x 1 15 else 16 T U =x 2 17 end 18 else 19 if L 1 >L 2 then 20 T U =x 1 21 else if L 1 =L 2 then 22 T L =max(x 1 ,x 2 )=x 2 23 else 24 T U =x 2 ,T L =x 1 25 end 26 end 27 end 63 0 5 10 15 20 25 30 0.0 0.2 0.4 0.6 0.8 1.0 Iteration Gap size in fraction of original interval Figure 4.11: Convergence of step-utility function algorithm 4.3.4 Algorithm for Exponential Utility Functions and Example of Application Algorithm4showsthepseudo-codeassuminganormalizedexponentialutilityfunction. Theex- ampleshowcasingitsapplicationassumesadecision-makerfollowinganexponentialutilityfunction with “ =≠ 0.015 and x 0 =0, x ú = 100. The lower and upper bound on the risk aversion coecient are initialized to=[ ≠ 0.1,0.1]. The algorithm then attempts to find the decision-maker’s risk aversion coecient by searching in the space created by the bounds. 64 Algorithm 4: Algorithm to elicit exponential utility functions Input : Initial lower and upper bound on exponential target parameter : “ l ,“ u Output: Updated “ l ,“ u , series of elicitation lotteries 1 Initiate parameters at iteration t=0 2 0 =[l,u],where l and u denote the lower and upper bounds on the risk aversion coecient “ 3 x 0 and x ú , which denote the minimum and maximum of the lottery outcomes. 4 while “ U ≠ “ L Ø Â or L 1 =L 2 do 5 Initiate the risk aversion coecient at iteration t at which the choice of the decision-maker will be evaluated: 6 “ t = “ u t ≠ “ l t≠ 1 2 +“ l t≠ 1 +‘, 7 where ‘ is a very small amount if “ u t ≠ “ l t≠ 1 2 +“ l t≠ 1 =0. 8 Choose the following parameters for L 1 = at random: 9 x 1 œ [x 0 ,x ú ], which is the best outcome of the binary lottery L 1 10 p 1 œ (0,1], which is the probability of outcome x 1 occurring 11 Choose the parameters for L 2 = such that: 12 p 2 œ [max( p 1 (U(x 1 )≠ U(x 0 )) 1≠ U(x 0 ) ,0);1]. This parameter can be chosen at random from the defined interval. 13 x 2 =≠ 1 “ t ln Ë 1≠ Ë p 1 p 2 U(x 1 )+(1≠ p 1 p 2 )U(x 0 ) ÈÈ . This parameter is contingent on the choice of p 2 . 14 Observe the decision-maker’s choice between L 1 and L 2 and update the interval t in the following way: 15 if if L 1 >L 2 and p 1 >p 2 then 16 t =[“ t ,“ u t≠ 1 ] 17 else if L 1 >L 2 and p 1 p 2 then 22 t =[“ l t≠ 1 ,“ t ] 23 end Figures 4.12 and 4.13 shows the convergence of the bounds to the decision-maker’s real utility 65 2 4 6 8 10 -0.10 -0.05 0.00 0.05 0.10 Iteration Risk Aversion Coecient Assumed Risk Aversion Lower Bound Upper Bound Real Risk Aversion Figure 4.12: Convergence of bounds towards real risk aversion function. The gap between upper and lower bound decreases monotonically and is less than 1% after 8 iterations in this example. To determine the bounds, the lotteries summarized in table 4.4 were generated. 4.3.5 Algorithm for Logistic Utility Functions and Example of Application This section presents an algorithm to determine the two parameters of a decision-maker’s nor- malized utility function. In comparison to the step or normalized exponential utility functions, the algorithm no longer searches along a line but a grid. Solely updating the lower and upper bounds on the parameters of the utility function will result in the algorithm getting stuck because of the shape of the decision frontier. Therefore, the algorithm is expanded with a correction term that adjusts the bounds on one parameter if the other one is being changed. It determines the parameters over several iterations, during which the decision-maker is asked to order three lotteries according to his or her preferences. 66 2 4 6 8 10 0.0 0.2 0.4 0.6 0.8 1.0 Iteration Gap size in fraction of original interval Figure 4.13: Bounds on Risk Aversion Coecient at dierent iterations of algorithm Table 4.4: Summary of algorithm assuming normalized exponential utility function Iteration Proposed““ L “ U p 1 p 2 x 1 x 2 Choice 1 0.000 -0.100 0.100 0.85 0.75 75 85.000 2 2 -0.050 -0.100 0.000 0.54 0.90 75 65.095 2 3 -0.025 -0.050 0.000 0.63 1.00 25 17.453 2 4 -0.012 -0.025 0.000 0.90 0.50 75 106.726 2 5 -0.019 -0.025 -0.012 0.07 0.55 45 8.312 2 6 -0.016 -0.019 -0.012 1.00 0.05 65 229.738 1 7 -0.014 -0.016 -0.012 0.76 0.70 55 58.209 2 8 -0.015 -0.016 -0.014 0.87 0.80 95 99.315 2 9 -0.015 -0.016 -0.015 0.44 0.35 85 96.237 1 10 -0.015 -0.015 -0.015 0.57 0.40 65 80.636 1 67 Because the logistic utility function given by equation (6.30) has two parameters, having a subject choose between two lotteries does not provide enough degrees of freedom to estimate the parameters. Therefore, algorithm 5 presents the subject with three lotteries and asks to order them according to their preference. The lotteries are given by L 1 =, L 2 =< p 2 ,x 2 ;(1≠ p 2 ),0 >, and L 3 =,where p 1 at random: 6 x 1 œ [x 0 ,x ú ], which is the best outcome of the binary lottery L 1 7 p 1 œ (0,1], which is the probability of outcome x 1 occurring 8 Choose the parameters for L 2 = such that: 9 p 2 œ [max( p 1 (U(x 1 )≠ U(x 0 )) 1≠ U(x 0 ) ,0);1]. This parameter can be chosen at random from the defined interval. 10 x 2 =≠ 1 “ t ln Ë 1≠ Ë p 1 p 2 U(x 1 )+(1≠ p 1 p 2 )U(x 0 ) ÈÈ . 11 Observe the decision-maker’s choice between L 1 and L 2 and update the interval t in the following way: 12 if L 1 >L 2 >L 3 then 13 [X t 0 ;K t ]=[x t≠ 1 0 ,x u 0 ;k l ,k u ] 14 else if L 3 >L 2 >L 1 then 15 [X t 0 ;K t ]=[x l 0 ,x t≠ 1 0 ;k l ,k u ] 16 else if L 2 >L 1 >L 3 or L 2 >L 3 >L 1 then 17 [X t 0 ;K t ]=[x l 0 (1≠ c),x u 0 ;k t≠ 1 ,k u ] 18 else if L 1 >L 3 >L 2 or L 3 >L 1 >L 2 then 19 [X t 0 ;K t ]=[x l 0 ,x u 0 (1+c);k l ,k t≠ 1 ] 20 end Table 4.5 summarizes the parameters used in the example showcasing the application of algo- rithm 5. Figure 4.14 shows how the algorithm converges on the parameters of the decision-maker’s utility function and table 4.6 summarizes the lotteries used during a select number of iterations. 69 0 10 20 30 40 50 0 20 40 60 80 100 Iteration Parameter x0 Value of suggested x0 Real x0 0 10 20 30 40 50 0.0 0.2 0.4 0.6 0.8 1.0 Iteration Parameter k Value of suggested k Real k Figure 4.14: Convergence of algorithm to parameters 70 Table 4.5: Parameter settings logistic utility elicitation example X 0 0 K 0 Max. logistic function Decision-maker’s x 0 Decision-maker’s k c 0 r c Value [0,100] [0,1] 1 50 0.15 0.1 0.2 Table 4.6: Summary of algorithm assuming normalized logistic utility function Iteration x 0 kp 1 p 2 p 3 x 1 x 2 x 3 1 50.00 0.50 0.09 0.84 0.91 87.00 45.76 45.58 12 50.07 0.16 0.19 0.68 0.94 89.00 44.00 41.28 24 49.56 0.15 0.16 0.25 0.40 87.00 53.28 46.87 36 49.98 0.15 0.21 0.41 0.95 81.00 50.19 41.54 50 49.98 0.15 0.14 0.45 0.99 100.00 44.69 37.99 Figure 4.15 shows the utility functions resulting from the proposed parameters during the first, second, and fifth iteration of the algorithm and the decision-maker’s utility function. It also plots the nonzero outcomes of the lotteries L 1 , L 2 , and L 3 respectively, which would make the decision- maker indierent if his or her utility function had the same parameters as the proposed function. 4.3.6 Testing for Uniqueness of the Obtained Solution Having devised algorithms that converge on a rational decision-maker’s utility function, the question now becomes whether the used lotteries and observed choices uniquely identify the func- tions assumed by the algorithms. This section checks whether the choices made by the decision- maker under the assumption of a certain functional form are unique to the function, or whether we could have observed the same choice behavior if the decision-maker had another utility function. To this end, step, linear, normalized exponential, logistic, and logarithmic functions are fit to the choices made when performing the three algorithms. If the algorithms devise lotteries that are unique to decision-makers with the assumed utility functions, one expects that the classification error between observation and fitted choice is minimal if the fitted and assumed utility functions 71 0 20 40 60 80 100 0.0 0.2 0.4 0.6 0.8 1.0 Outcome Utility 0 20 40 60 80 100 0.0 0.2 0.4 0.6 0.8 1.0 Outcome Utility 0 20 40 60 80 100 0.0 0.2 0.4 0.6 0.8 1.0 Outcome Utility 0 0 Proposed function Real function X1 X2 X3 Figure 4.15: Proposed and real logistic utility function including outcomes of lotteries 72 Table 4.7: Classification errors between fitted induced utility and target generating function using lottery generating algorithms Target generating functions Fitted function Step Exponential: Risk seeking Exponential: Risk averse Logistic Linear 0.67 0.30 0.40 0.78 Step 0.00 0.40 0.40 0.54 Exponential 0.67 0.00 0.00 0.54 Logarithmic 0.67 0.40 0.20 0.70 Logistic 0.60 0.30 0.20 0.00 are the same. Table 4.7 shows the minimum classification errors when fitting dierent utility functions to the lotteries and choices observed under the application examples of algorithms 3 - 5. One can observe that the classification errors are at zero when the type of fitted and lottery generating function agree. Additionally, the normalized logarithmic utility function can yield a similar preference order (withexceptions)thanthatunderthenormalizedexponentialifthedecision-makerwasriskaverse. 4.3.7 Application Areas, Conclusion, and Limitations Preference methods are not commonly used in utility elicitation. Although they are cognitively easy and supposedly less biased than indierence methods, they require greater amounts of obser- vations. This makes them less suitable for behavioral experiments because additional observations require time and money. The algorithmic approaches developed in this chapter reduce the number of iterations necessary to elicit the parameters of step, normalized exponential, and normalized lo- gisticutilityfunctions. Byusingthesenovelalgorithms, whichadjusttheboundsoftheparameters upon making an observation and create new lotteries, the assumed function converges quickly to a fully rational decision-maker’s utility function. The proposed algorithms can be used in two ways. One is to use them to elicit a person’s 73 utility function under no incentive. In this case, one has to assume a functional form, present the lotteriestothesubject,recordthechoice,anddeterminethenextlotteriesbasedonthisobservation. However, this assumes a specific functional form and does not allow for inconsistent assessments. The second area where the algorithms can be used is studying utility functions under incentives. Assumingacertainoruncertainincentivescheme,onecanusethealgorithmstogenerateasequence oflotteriesanddeterminetherationalchoicebeforepresentingthelotteriestosubjects. Comparing the rational with the observed choice, one can determine whether the subject adheres to normative decision-making behavior under incentives. This is how chapter 5 of this dissertation applies the designed algorithms. The proposed algorithms feature some limitations based on their assumptions. The limitations include: • The decision-maker follows the underlying function used to generate the presented lotteries. The proposed algorithms assume a functional form of the decision-maker’s utility function. This is a reasonable assumption to make when studying decision-making under incentives because the elicited function should converge to the function used to generate the uncertain target. • In the case of the normalized logistic utility function, the best outcome of the first lottery on which the subsequent calculations are performed has to be greater than the decision-maker’s x 0 . Otherwise, the preference ordering is reversed. 74 Chapter 5 Human Decision-Making Under Certain and Uncertain Incentives Chapter Overview This chapter investigates the adherence of human decision-makers to normative behavior under certain and uncertain incentive schemes. 12 subjects were asked to choose between or order lotter- ies according to their preference under dierent incentive scenarios. For the purpose of this study, normative decision-making behavior is defined as choosing the lottery that maximizes the proba- bility of achieving or exceeding a target. The lotteries presented to them were designed so that fitting a function to the observations produces a unique utility function as long as subjects decide normatively. One finding is that adherence to normative behavior decreases with the complexity of the incentive scheme. On average, the bounds on the induced utility functions became larger as the incentives became uncertain and the task was changed from choosing to ordering lotteries. Another observation pertaining to third factors is that the subject pool can be divided into two groups whose composition does not change with the imposed scheme. One group spent less time on the survey, had a higher share of assessments consistent with their personal utility function, 75 a lower degree of normative decision-making under incentives, and a steeper decline in normative behavioroncethetargetbecamemorecomplex. Theothergroupspentmoretime,wasmoreconsis- tent with probability maximization instead of with maximizing their own personal expected utility, and their degree of consistency with normative behavior stayed relatively high and constant across all schemes. There was no significant dierence between the groups along demographic factors, including age, gender, and educational background. 5.1 Introduction The following research questions were explored: 1. “Does a human decision-maker’s utility function change in accordance to normative decision theory when certain and uncertain incentives are imposed?” 2. “Are there other factors determining the change described in question 1, other than the im- posed incentive?” 3. “Are preference statements sucient to elicit induced utility functions?” Question 1 looks at normative decision-making behavior under incentives and real-life observa- tions. Scientificpublicationsresearchingincentivesusingpurelytheoreticalandbehavioralmethods have found that decision-makers’ utility functions change when incentives are imposed. Behavioral research mostly discusses changes in the convexity of utility curves rather than measuring the func- tions explicitly (for an example see Pennings and Smidts (2003)). In contrast, normative decision theory research on incentives describes the mathematical form of the induced utility function un- der the imposition of specific incentives (see Abbas and Matheson (2005), Abbas, Matheson, and Bordley (2009), Abbas and Matheson (2009)). Because of the discrepancy between normative and behavioral research, this experiment sets out to elicit the mathematical form of utility functions 76 induced by incentives using preference statements. Question 2 controls for potential third factors when investigating a change between personal and induced utility functions. For example, previous research on decision-making without incen- tives found female subjects to be more risk averse than male participants (Powell and Ansic 1997; Sapienza, Zingales, and Maestripieri 2009). The eects of educational attainment and age on risk aversion are mixed, finding evidence for both increasing and decreasing influences (Jung 2015; Wang and Hanna 1998; Morin and Suarez 1983; Mata et al. 2011). Question 2 determines whether these observations extend to incentives and sheds some light on the predictability of choice under incentives if certain demographic factors were known. Question 3 concerns a specific utility elicitation method. In his survey of these methods, Far- quhar (1984) finds that using preference statements is cognitively easier and introduces less bias. This particular method is chosen under the assumption that imposing incentives when eliciting utility functions adds another level of complexity, thereby increasing cognitive load on the subject. Tocompensateforthemethod’sdisadvantageofrequiringlotsofpreferencestatementstoconverge, we employ an algorithm that assumes the normative functional form and produces the lotteries for theassessment(seechapter4). Thatway, onecandeterminetheperson’sutilityfunctionunderthe targetandcompareittotheinducedutilityfunctionproposedbynormativetheory. Theadherence to normative behavior can then be used as a validation of the algorithms proposed in chapter 4. Themethodologychosentoanswerthesequestionswasabehavioralexperiment,conductedfrom NovembertoDecember2021viaQualtricswith12graduatestudents(8maleand4female)inUSC’s Industrial and Systems Engineering department. Details on its design and data transformation methods can be found in the appendix. The remainder of this chapter discusses the determination of the subjects’ personal utility function, introduces terminology and assumptions used throughout this chapter, and defines measures to aggregate subjects’ responses. I then discuss the experiment 77 results, first elaborating on the basis of individual target schemes, and concluding analysis of third factors. I end with a summary of findings. 5.2 DeterminationofPersonalUtilityFunction, Terminology, and Assumptions 5.2.1 Elicitation of Personal Utility Functions Before they were asked to choose or order lotteries under incentives, subjects were randomly assignedtoeithercertainequivalenceorindierenceprobabilitymethodstodeterminetheirprivate utility functions. The goal of this elicitation was to control for a subject’s personal utility function whenanalyzingtheirchoicesunderincentives. Table6.1inappendixI.Iprovidesanoverviewofthe applied functions and their respective inverses, which are used to determine the certain equivalent of an uncertain deal. Note that all functions are normalized to the interval [x 0 ,x ú ]=[0,100] to maintain the interpretability of utility as preference probability. Thefunctionfeaturingthelowestsquarederrorwasdeterminedtobeadecision-maker’spersonal utilityfunction. Tables3.2and3.3inchapter3showtheresponsesandfittingresultsforallsubjects in the certain equivalence and indierence probability group. 5.2.2 Terminology and Assumptions The definitions and assumptions holding throughout this chapter are: • A target is fixed or certain if there is only one target outcome. • A binary target can take on two possible values, each with a non-zero probability. Similarly, a target can have three or four outcomes. • Target functions are the cumulative distribution function of the target. 78 • An “averse” exponential target function is a normalized exponential utility function with risk aversion parameter“> 0. The function is termed “seeking” if“< 0. • “Logistic” targets are when the function to generate the target follows the functional form given by equation (5.3). The functions used to generate the target are given by equations (5.1) - (5.3). Parameter T denotes the fixed target, x an outcome, “ the risk aversion coecient of an exponential function, L,x 0 ,k the maximum value, midpoint, and steepness of the logistic function, and x 0 and x ú are the worst and best outcomes the functions are normalized over. S(x)= Y _____] _____[ 1, ifx>=T 0, ifx<T (5.1) F(x)= (1≠ e ≠ “x )≠ (1≠ e ≠ “x 0 ) (1≠ e ≠ “x ú )≠ (1≠ e ≠ “x 0 ) (5.2) H(x)= U(x)≠ U(x 0 ) U(x ú )≠ U(x 0 ) (5.3) U(x)= L 1+e ≠ k(x≠ x 0 ) (5.4) The algorithms proposed in chapter 4 generated the lotteries presented to the subjects. Tables 6.2 - 6.9 in the appendix section 6.3 summarize the lotteries and the associated targets. Note that the lotteries used under binary targets are the same lotteries as under the fixed/certain target scheme. Table 5.1 summarizes the stages of the experiment, their associated target details, and in- 79 Table 5.1: Summary of tasks Stage Number targets Number assessments Alternatives (incl. indi.) Target function Target function parameters Task 11 8 3 S(x) T = 65 Choose 2 2 8 3 None None Choose 3 3 10 3 F(x) “ =0.015 Choose 4 3 10 3 F(x) “ =≠ 0.015 Choose 5 4 16 4 H(x) x 0 = 50 k=0.15 Order structions. Column 2 presents the number of target outcomes under which subjects either choose or order (column 6) the lotteries presented to them. Individual target outcomes are derived by discretizing the target function, whose steps to correspond to the best outcome of every lottery presented during an assessment (assuming that the probability of achieving a target with a zero outcome is zero). The number of lotteries presented to the subjects during every assessment are given in column 4. 5.2.3 Background on Calculated Measures Several measures based on observed and normative choices, as well as on choices based on the elicited personal utility function, summarize the subjects’ responses for further analysis. One can distinguishbetweenconsistency,value,probabilitygapmeasures,andmeasuresderivedfromfitting the induced utility function to the observations. Consistency Measures Consistency measures indicate how often choices between lotteries are the same under two dif- ferentassumptionsasafractionofthenumberofassessments. Thethreeconsistencymeasuresused 80 in the summary tables below are “Consistency with personal utility,” “Consistency with probabil- ity maximization,” and “Consistency between normative and personal utility.” “Consistency with personal utility” measures how often the decision-maker chooses lotteries under the incentive that they would have chosen under their personal utility function. “Consistency with probability maxi- mization” measures how often the observed choices are consistent with probability maximization, i.e. normative decision-making behavior under incentives. And, “Consistency between normative and personal utility” assesses how often normative and choice under the personal utility function are identical. This indicator matters because, if it is high, determining whether the observed choice is due to personal preference or the imposed incentive becomes dicult. I calculate consistency of choice under two dierent scenarios S 1 and S 2 according to (5.5), where n denotes the number of assessments under a particular incentive. The scenarios describe whether we take the observed choices, the choices the decision-maker would have made if they decided based only on the fitted utility function, or the normative choices under the respective target to calculate the consistency. Consistency = q i a i n , (5.5) where a i = Y _____] _____[ 1 L S 1 i =L S 2 i 0 L S 1 i ”=L S 2 i . (5.6) Value Gap Measures Value gap measures identify the loss or gain in value that the subject incurs by choosing a 81 lottery. To determine this value, the analysis assumes that participants follow their personal utility function determined at the beginning of the experiment. First, the dierence between the expected utility of the chosen lottery, minus the expected utility of the lottery that should have been chosen under the personal utility function or normative behavior, is determined. In case of an observed indierence between gambles that does not align with behavior under personal utility or normative behavior, the gap is calculated as the mean expected utility of the available lotteries, minus the normative or personal utility choice. The value gap is then calculated by using the expected utility gap as the input argument to the inverse of the subject’s personal utility function. Under logistic target schemes, where the experiment asks subjects to order lotteries according to their preference, the value gap is determined by taking the inverse of the expected utility dierence between the observed preferred lotteries (in first place of the ordering) and the most preferred lotteries under normative or personal utility functions. Probability Gap Measures The probability gap is the dierence between the probability of meeting or exceeding the target under the normative choice minus the same probability under the observed choice. Because the normativechoiceisalwaysthelotterywiththehighesttargetachievementprobability,thismeasure iseitherzero,whennormativeandobservedchoicealign,orpositive. Underlogistictargetschemes, theprobabilitygapisdeterminedbythetargetprobabilitydierencebetweentheobservedpreferred lotteries and the preferred lotteries under normative or personal utility functions. Fitting Results 82 Fitting results summarize the best fitted parameter, the number of possible solutions, and the fitting error. The best fitted parameter minimizes the classification error between observed and theoretical choices. I assumed that the fitted curve belongs to the same functional family as the target function. If there are multiple optimal solutions, the best fitted parameter is determined to be the value that is closest to the normative value. The fitting error is the fraction of lottery choices that are inconsistent with the fitted induced utility function. 5.3 Experiment Results This section discusses gathered responses and and provides an analysis of results. Each sub- section is dedicated to one particular incentive or target scheme and contains a table with the measures defined for each subject. It also features a visualization of the personal, fitted induced, and normative induced utility functions. Please refer to appendix III.V for individual responses for every assessment and incentive scheme. 5.3.1 Results of Choice Under Fixed Targets Thissectiondiscussesthesubjects’responsesunderafixed/certaintargetscheme. Thefollowing observations can be made from table 5.2: • 75% of subjects were perfectly consistent with probability maximizing behavior under the fixed target. • There were no participants who exclusively maximized their personal expected utility. The fraction of observations maximizing personal expected utility was lower than 50% per person. • In less than 40% of all cases normative and best personal alternative were identical. The implications of these findings on the incurred value loss and gain are twofold. First, people 83 whowerelessconsistentwiththeirownutilityfunctionincurredabiggermeanvaluelosscompared to subjects who mostly chose alternatives that maximized their personal expected utility. Second, subjectswhoseobservedchoicesweremoreconsistentwiththeirpersonalutilityfunction, i.e.hada low personal value loss, were less consistent with probability maximization under the fixed target. When fitting the induced utility function, only one optimal solution existed for 11 of 12 sub- jects. This optimal solution was close or equal to the normative target value of $65. Figures 5.1 and 5.2 visualize this observation: the fitted and normative induced utility functions are hardly distinguishable from each other. However, 60 optimal solutions exist for Subject 2, who exhibited a very low degree of probability maximizing behavior. The range of the best fitting target for Subject 2 is big, and the normative target is contained therein. A consequence of this big range of best fitting parameters is Subject 2’s high error rate. The fitting error, which is the fraction of misclassified preferred lotteries compared to the observed preferences, is 50% for Subject 2. The error rate for all other subjects, for whom only one optimal solution existed, was much lower. Toobservewhethersubjectswereawareofthepresentedtargetinformation, twolotteriesunder dierent fixed target settings for identical lotteries were added after the eight initial assessments. The target amounts were chosen such that a probability maximizing subject would switch their preference compared to their previous choice under the original target. The lotteries presented to the subjects in assessment rounds 2 and 3 were chosen for this comparison. Looking at the individual choices in table 6.10 and 6.12 in appendix III.V, one can see that, under the original target, 23 out of 24 observed choices were consistent with probability maximizing behavior. After imposing two dierent fixed targets, we expected to see a switch of preferred lotteries. This change occurred in 20 out of 24 cases, indicating that subjects were aware of the target information and maximized their target achievement probability. However, Subject 2 was an outlier with 50% of probabilitymaximizingbehaviorduringtheinitialassessmentsand0%underthevalidationtargets. 84 Table 5.2: Aggregation of responses under fixed target Subject 1 2 3 4 5 6 7 8 9 10 11 12 Comparison observation to personal utility Consistency with pers. utility 0.75 0.62 0.38 0.38 0.50 0.75 0.38 0.38 0.38 0.25 0.25 0.38 Mean value gap -0.58 -0.06 -7.29 -8.67 -6.76 -1.52 -9.57 -8.67 -9.13 -1.10 -12.37 -6.58 Median value gap 0.00 0.00 -1.98 -2.50 -0.47 0.00 -3.64 -2.50 -3.37 -0.07 -10.43 -1.35 Max value gap 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 Min value gap -2.79 -0.45 -20.79 -25.98 -20.66 -7.14 -28.34 -25.98 -26.74 -4.96 -25.13 -19.00 St. dev. value gap 1.10 0.16 9.33 11.43 9.47 2.86 12.29 11.43 11.65 1.76 9.79 8.63 Comparison observation to prob. maximization Consistency with prob. maxim. 0.62 0.25 1.00 1.00 0.75 0.62 1.00 1.00 1.00 1.00 1.00 1.00 Mean value gap 7.61 5.58 0.00 0.00 0.18 9.64 0.00 0.00 0.00 0.00 0.00 0.00 Median value gap 0.00 0.34 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 Max value gap 24.16 16.43 0.00 0.00 1.47 32.97 0.00 0.00 0.00 0.00 0.00 0.00 Min value gap 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 St. dev. value gap 11.09 7.68 0.00 0.00 0.52 14.90 0.00 0.00 0.00 0.00 0.00 0.00 Mean prob. gap 0.07 0.18 0.00 0.00 0.01 0.07 0.00 0.00 0.00 0.00 0.00 0.00 Max prob. gap 0.55 0.65 0.00 0.00 0.10 0.55 0.00 0.00 0.00 0.00 0.00 0.00 Min prob. gap 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 Comparison personal utility to prob. maximization Consistency btw. normative and personal utility 0.38 0.38 0.38 0.38 0.38 0.38 0.38 0.38 0.38 0.25 0.25 0.38 Fitting results Best fitted fixed target 64.5 64.5 65 65 65 64.5 65 65 65 65 65 65 Maximum best fitted fixed target 64.5 75 65 65 65 64.5 65 65 65 65 65 65 Minimum best fitted fixed target 64.5 0 65 65 65 64.5 65 65 65 65 65 65 Number of optimal solutions 1 60 1 1 1 1 1 1 1 1 1 1 Fitting error 0.12 0.5 0 0 0.25 0.12 0 0 0 0 0 0 85 5.3.2 Results of Choice Under Binary Targets Thissectiondiscussessubjects’choicessubjectsunderbinarytargets, i.e.uncertaintargetswith twooutcomes. Identicallotteriesasunderthefixed/certaintargetsituationswereusedtoeliminate the possible influence of lotteries on choice behavior. The main purpose was not to derive the subjects’adherencetoaparticularinducedutilityfunction, buttoseewhetherthemereimposition of an uncertain target made them less or more consistent with probability maximization compared to fixed/certain targets. Because the lotteries were not designed in a way that would produce a particular induced utility function if the subject decided normatively, a visual representation of personal and induced utility functions per subject is omitted. With respect to consistency with probability or expected utility maximizing behavior, table 5.3 shows the following: • 79.25%ofobservedchoicesmaximizedtheprobabilityofachievingorexceedingtheuncertain target across all subjects. This constitutes a decrease of over 5% compared to probability maximizing choices under the fixed target. • On average, 47% of observed choices were consistent with maximizing subjects’ expected utility. This is a 2% increase from the fixed target case. The fraction of choices maximizing personal expected utility ranged from 25% to 88% of per subject. • A similar fraction of choices compared to the fixed target was consistent between probability and expected utility maximization. In 75% of cases, subjects were perfectly consistent with probability maximization under a fixed target. This fraction more than halved under the binary target. For almost 50% of the remaining subjects the fraction of probability maximizing behavior declined, and stayed the same for one subject. Subject 2, whose share of probability maximizing choices was particularly low under the 86 0 20 40 60 80 100 0.0 0.2 0.4 0.6 0.8 1.0 Subject: 1 0 20 40 60 80 100 0.0 0.2 0.4 0.6 0.8 1.0 Subject: 2 0 20 40 60 80 100 0.0 0.2 0.4 0.6 0.8 1.0 Subject: 3 0 20 40 60 80 100 0.0 0.2 0.4 0.6 0.8 1.0 Subject: 4 0 20 40 60 80 100 0.0 0.2 0.4 0.6 0.8 1.0 Subject: 5 0 20 40 60 80 100 0.0 0.2 0.4 0.6 0.8 1.0 Subject: 6 0 0 Personal U-Function Induced (fitted) Induced (normative) Figure 5.1: Personal, induced (fitted), induced (normative) utility functions under fixed target (Subjects 1-6) 87 0 20 40 60 80 100 0.0 0.2 0.4 0.6 0.8 1.0 Subject: 7 0 20 40 60 80 100 0.0 0.2 0.4 0.6 0.8 1.0 Subject: 8 0 20 40 60 80 100 0.0 0.2 0.4 0.6 0.8 1.0 Subject: 9 0 20 40 60 80 100 0.0 0.2 0.4 0.6 0.8 1.0 Subject: 10 0 20 40 60 80 100 0.0 0.2 0.4 0.6 0.8 1.0 Subject: 11 0 20 40 60 80 100 0.0 0.2 0.4 0.6 0.8 1.0 Subject: 12 0 0 Personal U-Function Induced (fitted) Induced (normative) Figure 5.2: Personal, induced (fitted), induced (normative) utility functions under fixed target (Subjects 7-12) 88 fixed target, increased by a factor of three under the binary target. One can conclude that making the target uncertain while leaving the lotteries identical slightly decreased the subjects’ adherence to normative probability maximization. The eect of less probability and more expected utility maximizing behavior can also be seen in the incurred value gaps. Because subjects’ observed choices were more consistent with maximizing their own expected utility compared to under the fixed target, their mean incurred value gap was lower. Also, because only 1/3 of subjects were perfectly maximizing their success probability, a value gap of zero between observed and normative choice was only observed in 4 subjects. In comparison to the fixed target, where only value gains by violating probability maximization could be observed, we also see a value loss under this target. This loss occurs when the lottery that maximizesasubject’sprobabilityofmeetingorexceedingthetargetalsomaximizesexpectedutility, but was not chosen as their preferred choice. This finding implies that, under the fixed target, subjects violated normative probability maximizing behavior. Under the binary target, however, both normative probability maximization and expected utility maximization were violated. 89 Table 5.3: Aggregation of responses under binary target Subject 1 2 3 4 5 6 7 8 9 10 11 12 Comparison observation to personal utility Consistency with pers. utility 0.62 0.62 0.62 0.38 0.88 0.38 0.25 0.38 0.38 0.25 0.38 0.50 Mean value gap -1.27 -0.59 -1.37 -8.26 -0.24 -8.86 -10.54 -8.26 -8.88 -0.51 -11.00 -5.08 Median value gap 0.00 0.00 0.00 -2.50 0.00 0.00 -6.45 -2.50 -3.37 -0.07 -10.30 -0.56 Max value gap 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 Min value gap -8.35 -3.56 -6.97 -25.98 -1.96 -32.97 -28.34 -25.98 -26.74 -1.83 -25.13 -19.00 St. dev. value gap 2.93 1.23 2.44 11.33 0.69 14.97 11.68 11.33 11.58 0.82 10.70 8.60 Comparison observation to prob. maximization Consistency with prob. maxim. 0.50 0.75 0.75 1.00 0.25 0.62 0.88 1.00 1.00 1.00 0.88 0.88 Mean value gap 6.38 4.10 5.19 0.00 5.93 3.00 -1.07 0.00 0.00 0.00 1.31 0.67 Median value gap 0.00 0.00 0.00 0.00 1.20 0.00 0.00 0.00 0.00 0.00 0.00 0.00 Max value gap 24.16 16.43 20.79 0.00 20.66 16.84 0.00 0.00 0.00 0.00 10.45 5.38 Min value gap 0.00 0.00 0.00 0.00 -1.96 0.00 -8.57 0.00 0.00 0.00 0.00 0.00 St. dev. value gap 11.00 7.59 9.61 0.00 9.33 6.13 3.03 0.00 0.00 0.00 3.69 1.90 Mean prob. gap 0.10 0.02 0.02 0.00 0.17 0.05 0.01 0.00 0.00 0.00 0.00 0.04 Max prob. gap 0.55 0.10 0.10 0.00 0.55 0.29 0.10 0.00 0.00 0.00 0.03 0.29 Min prob. gap 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 Comparison personal utility to prob. maximization Consistency btw. normative and personal utility 0.38 0.38 0.38 0.38 0.38 0.38 0.38 0.38 0.38 0.25 0.25 0.38 90 5.3.3 Results of Choice Under Averse Targets Table 5.4 summarizes the observed choices under an uncertain target generated from a normal- ized exponential cumulative distribution with parameter “ =0.015, discretized to three outcomes. Looking at the consistency measures, one can observe: • Four out of twelve subjects behaved perfectly rationally under this target, meaning that all their choices maximized their probability of success. This is the same number of subjects as under the binary target, and half the number of subjects under the fixed target scheme. • Average consistency between observed choice and normative probability maximization across all subjects is around 80%. This is similar to the consistency under the binary target. • The average consistency between observed choice and choice under the assumed personal utility function across all subjects is 57.5%. This is considerably higher than under the fixed and binary target. Thecomparativelyhighconsistencyofobservationswithchoicesassumingpersonalutilityfunc- tions and normative probability maximization can be explained by the high share of lotteries that would have been chosen under both assumptions. This average overlap of 67.5% is almost double than under fixed and binary targets. As a result, we also see more value losses when comparing observations to the normative choice, because both expected utility and probability maximization principles are violated. Because normative probability maximizing behavior is less frequent than than under the fixed target case, fitting the induced utility function has a higher error rate on average and a greater number of optimal solutions. The space of possible optimal solutions sometimes does not even contain the parameter of the normative induced utility function (compare to Subject 1-4s’ fitted parameter values). Additionally, Subjects 1 and 2 exhibit risk seeking behavior (“< 0)despite 91 their own personal risk neutrality or aversion, and the risk averse behavior the target was supposed to induce. In contrast, Subject 3 exhibited more risk averse behavior than warranted by either their personal utility function or the incentive scheme. For Subjects 5 and 6, the bounds on the risk aversion parameter produced by the optimal solutions was large compared to other subjects with more normative choice behavior. Theseresultsarevisualizedinfigures5.3and5.4. ForSubjects7-12,theboundsontheinduced utility function are (almost) identical to the normative induced utility function. For Subjects 1-5, either of the bounds constituted the best fitting induced utility, and for Subject 6, both best fit and normative functions lie between rhe upper and lower bound. 92 Table 5.4: Aggregation of responses under exponential (averse) target Subject 1 2 3 4 5 6 7 8 9 10 11 12 Comparison observation to personal utility Consistency with pers. utility 0.80 0.30 0.20 0.80 0.60 0.50 0.60 0.70 0.70 0.30 0.50 0.90 Mean value gap -1.12 -0.32 -0.72 -0.18 -0.24 -3.30 -0.97 -0.30 -0.58 -0.21 -6.39 -0.01 Median value gap 0.00 -0.06 -0.34 0.00 0.00 -1.91 0.00 0.00 0.00 -0.11 -1.58 0.00 Max value gap 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 Min value gap -10.20 -1.62 -2.70 -1.50 -1.91 -9.21 -3.23 -1.50 -2.79 -0.81 -18.27 -0.10 St. dev. value gap 3.21 0.52 0.94 0.47 0.60 3.75 1.34 0.56 1.02 0.28 7.76 0.03 Comparison observation to prob. maximization Consistency with prob. maxim. 0.50 0.60 0.40 0.90 0.70 0.70 1.00 1.00 0.90 0.90 1.00 1.00 Mean value gap -0.93 -0.28 -0.69 0.12 -0.24 -0.49 0.00 0.00 0.13 -0.08 0.00 0.00 Median value gap 0.00 0.00 -0.27 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 Max value gap 0.89 0.00 0.00 1.19 0.00 8.09 0.00 0.00 1.27 0.00 0.00 0.00 Min value gap -10.20 -1.62 -2.70 0.00 -1.91 -7.66 0.00 0.00 0.00 -0.81 0.00 0.00 St. dev. value gap 3.30 0.53 0.96 0.38 0.61 4.08 0.00 0.00 0.40 0.26 0.00 0.00 Mean prob. gap 0.01 0.01 0.01 0.00 0.01 0.00 0.00 0.00 0.00 0.00 0.00 0.00 Max prob. gap 0.04 0.04 0.04 0.01 0.04 0.02 0.00 0.00 0.01 0.04 0.00 0.00 Min prob. gap 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 Comparison personal utility to prob. maximization Consistency btw. normative and personal utility 0.70 0.70 0.80 0.70 0.90 0.60 0.60 0.70 0.60 0.40 0.50 0.90 Fitting results Best fitted “ -0.002 -0.002 0.029 0.014 0.016 0.015 0.015 0.015 0.015 0.015 0.015 0.015 Maximum best fitted “ -0.002 -0.002 0.07 0.014 0.07 0.028 0.015 0.015 0.015 0.015 0.015 0.015 Minimum best fitted “ -1 -1 0.029 0.012 0.016 0.008 0.015 0.015 0.015 0.015 0.015 0.015 Number of optimal solutions 999 999 42 3 47 9 1 1 1 1 1 1 Fitting error 0.1 0 0.3 0 0.2 0.3 0 0 0.1 0.1 0 0 93 5.3.4 Results of Choice Under Seeking Targets Table 5.5 summarizes the subjects’ observed choices under an exponential target with lotteries that were generated using a risk aversion coecent “ =≠ 0.015. One can observe that: • Five out of twelve subjects behave perfectly rationally by maximizing their probability of achieving or exceeding the uncertain target. On average, probability maximizing behavior is slightly higher than under binary or averse targets. • Substantiallyhigherfractionofconsistencybetweenobservedandexpectedutilitymaximizing choice, resulted in smaller personal value losses compared to fixed, binary, and averse targets. • A similar share of overlap existed between lotteries preferred under expected utility and probability maximization compared to the averse target. Incomparisontoprevioustargetschemes,thefittingerrorishigherfortheseekingtarget. Also, there is no case in which only one optimal solution exists, even for subjects who chose perfectly rationally. This observation may indicate that the number of choices presented to the subjects was too low. Figures 5.5 and 5.6 visualize the findings on the fitting of induced utility functions. In the case of Subjects 1 and 5, there exists a wide band of possible induced utility functions spanning risk averse and risk seeking behaviors. Additionally, the normative utility function is not contained within this band for those subjects. A similar observation holds true for Subject 2, although their bounds contain the normative function. Subjects 3 and 6 exhibit risk averse behavior, implying that the normative function is not contained within the large bounds on the fitted utility function. In contrast, Subjects 4, and 7-12 exhibit a high degree of probability maximizing behavior, leading to tight bounds on the observed induced utility function that contain the normative function. 94 0 20 40 60 80 100 0.0 0.2 0.4 0.6 0.8 1.0 Subject: 1 0 20 40 60 80 100 0.0 0.2 0.4 0.6 0.8 1.0 Subject: 2 0 20 40 60 80 100 0.0 0.2 0.4 0.6 0.8 1.0 Subject: 3 0 20 40 60 80 100 0.0 0.2 0.4 0.6 0.8 1.0 Subject: 4 0 20 40 60 80 100 0.0 0.2 0.4 0.6 0.8 1.0 Subject: 5 0 20 40 60 80 100 0.0 0.2 0.4 0.6 0.8 1.0 Subject: 6 0 0 Personal U-Function Induced (fitted) Induced (normative) Lower bound Upper bound Figure 5.3: Personal, induced (fitted), induced (normative) utility functions under averse target (Subjects 1-6) 95 0 20 40 60 80 100 0.0 0.2 0.4 0.6 0.8 1.0 Subject: 7 0 20 40 60 80 100 0.0 0.2 0.4 0.6 0.8 1.0 Subject: 8 0 20 40 60 80 100 0.0 0.2 0.4 0.6 0.8 1.0 Subject: 9 0 20 40 60 80 100 0.0 0.2 0.4 0.6 0.8 1.0 Subject: 10 0 20 40 60 80 100 0.0 0.2 0.4 0.6 0.8 1.0 Subject: 11 0 20 40 60 80 100 0.0 0.2 0.4 0.6 0.8 1.0 Subject: 12 0 0 Personal U-Function Induced (fitted) Induced (normative) Lower bound Upper bound Figure 5.4: Personal, induced (fitted), induced (normative) utility functions under averse target (Subjects 7-12) 96 Table 5.5: Aggregation of responses under exponential (seeking) target Subject 1 2 3 4 5 6 7 8 9 10 11 12 Comparison observation to personal utility Consistency with pers. utility 0.80 0.60 0.70 0.70 0.70 0.70 0.60 0.70 0.70 0.70 0.70 0.60 Mean value gap -0.86 -0.44 -0.83 -0.98 -2.16 -0.84 -1.79 -0.98 -1.64 -0.09 -5.03 -0.92 Median value gap 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 Max value gap 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 Min value gap -8.56 -2.41 -4.68 -4.65 -14.38 -5.12 -7.55 -4.65 -8.01 -0.47 -20.60 -4.31 St. dev. value gap 2.71 0.87 1.72 1.74 4.83 1.83 2.65 1.74 2.88 0.17 8.50 1.62 Comparison observation to prob. maximization Consistency with prob. maxim. 0.60 0.70 0.60 1.00 0.50 0.70 0.90 1.00 1.00 0.90 1.00 1.00 Mean value gap 0.24 0.17 0.39 0.00 -1.36 1.13 -0.24 0.00 0.00 0.05 0.00 0.00 Median value gap 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 Max value gap 5.25 3.00 5.90 0.00 3.91 10.17 0.00 0.00 0.00 0.51 0.00 0.00 Min value gap -8.56 -1.66 -3.40 0.00 -14.38 -5.12 -2.39 0.00 0.00 0.00 0.00 0.00 St. dev. value gap 3.65 1.14 2.30 0.00 5.41 4.16 0.76 0.00 0.00 0.16 0.00 0.00 Mean prob. gap 0.00 0.00 0.01 0.00 0.01 0.00 0.00 0.00 0.00 0.00 0.00 0.00 Max prob. gap 0.01 0.01 0.02 0.00 0.06 0.02 0.02 0.00 0.00 0.01 0.00 0.00 Min prob. gap 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 Comparison personal utility to prob. maximization Consistency btw. normative and personal utility 0.70 0.60 0.70 0.70 0.60 0.70 0.70 0.70 0.70 0.60 0.70 0.60 Fitting results Best fitted “ -0.012 -0.015 0.017 -0.015 -0.012 0.017 -0.015 -0.015 -0.015 -0.015 -0.015 -0.015 Maximum best fitted “ 0.016 0.016 1 -0.014 0.016 1 -0.014 -0.014 -0.014 -0.014 -0.014 -0.014 Minimum best fitted “ -0.012 -0.015 0.017 -0.015 -0.012 0.017 -0.015 -0.015 -0.015 -0.015 -0.015 -0.015 Number of optimal solutions 28 30 984 2 28 984 2 2 2 2 2 2 Fitting error 0.2 0.3 0.2 0 0.2 0.2 0.1 0 0 0.1 0 0 97 5.3.5 Results of Choice Under Logistic Targets The final incentive scheme that subjects were presented with used three lotteries generated under the assumption of a logistic normative utility function. In contrast to the previous schemes, these lotteries were ordered according to subjects’ preferences. Table 5.6 summarizes the subjects’ choices. Regarding the choice consistency measures one can observe: • 45% average consistency between choice and personal expected utility maximization across subjects. This figure is significantly lower than the one exhibited under both exponential target schemes. • 78% average consistency between choice and normative probability maximizing behavior. This is approximately 2% lower than under the other schemes with uncertain targets. • 54%averageconsistencybetweenchoiceunderpersonalexpectedutilityandprobabilitymax- imizing. This value was approximately 13% higher under exponential targets. From the curve fitting results, one can observe that the number of optimal solutions is higher than for the other incentive schemes, even for subjects with perfectly normative choice behavior. To reduce the number of possible optimal solutions, the number of assessments was increased from 10 to 16. This proved insucient to produce one single optimal solution for fully rational decision-makers. Figures 5.7 and 5.8 show that utility functions fit to Subjects 4, 7, 9, 10, and 12 were identical to the normative utility function. Subject 2 and 8 had an identical midpoint parameter x 0 as the normative function, but not the same steepness parameter k as the normative function. Both function parameters of the remaining subjects diverged from the normative function. 98 0 20 40 60 80 100 0.0 0.2 0.4 0.6 0.8 1.0 Subject: 1 0 20 40 60 80 100 0.0 0.2 0.4 0.6 0.8 1.0 Subject: 2 0 20 40 60 80 100 0.0 0.2 0.4 0.6 0.8 1.0 Subject: 3 0 20 40 60 80 100 0.0 0.2 0.4 0.6 0.8 1.0 Subject: 4 0 20 40 60 80 100 0.0 0.2 0.4 0.6 0.8 1.0 Subject: 5 0 20 40 60 80 100 0.0 0.2 0.4 0.6 0.8 1.0 Subject: 6 0 0 Personal U-Function Induced (fitted) Induced (normative) Figure 5.5: Personal, induced (fitted), induced (normative) utility functions under seeking target (Subjects 1-6) 99 0 20 40 60 80 100 0.0 0.2 0.4 0.6 0.8 1.0 Subject: 7 0 20 40 60 80 100 0.0 0.2 0.4 0.6 0.8 1.0 Subject: 8 0 20 40 60 80 100 0.0 0.2 0.4 0.6 0.8 1.0 Subject: 9 0 20 40 60 80 100 0.0 0.2 0.4 0.6 0.8 1.0 Subject: 10 0 20 40 60 80 100 0.0 0.2 0.4 0.6 0.8 1.0 Subject: 11 0 20 40 60 80 100 0.0 0.2 0.4 0.6 0.8 1.0 Subject: 12 0 0 Personal U-Function Induced (fitted) Induced (normative) Lower bound Upper bound Figure 5.6: Personal, induced (fitted), induced (normative) utility functions under seeking target (Subjects 7-12) 100 Table 5.6: Aggregation of responses under logistic target Subject 1 2 3 4 5 6 7 8 9 10 11 12 Comparison observation to personal utility Consistency with pers. utility 0.62 0.50 0.06 0.56 0.25 0.44 0.56 0.56 0.56 0.31 0.44 0.56 Mean value gap -0.95 -0.40 -8.67 -1.54 -6.86 -7.86 -1.77 -1.08 -2.08 -0.25 -9.74 -1.23 Median value gap 0.00 0.00 -6.38 0.00 -7.49 -3.52 0.00 0.00 0.00 -0.12 -4.96 0.00 Max value gap 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 Min value gap -13.07 -3.49 -23.33 -11.50 -20.93 -22.32 -13.67 -11.50 -13.35 -1.02 -28.79 -8.70 St. dev. value gap 3.25 0.95 7.33 3.29 6.97 8.61 3.66 2.91 3.75 0.33 10.87 2.37 Comparison observation to prob. maximization Consistency with prob. maxim. 0.56 0.38 0.12 0.94 0.31 0.44 1.00 0.88 1.00 1.00 0.69 1.00 Mean value gap 0.31 0.51 -6.89 -0.46 -5.63 -6.11 0.00 0.00 0.00 0.00 0.13 0.00 Median value gap 0.00 0.00 -4.93 0.00 -2.50 0.00 0.00 0.00 0.00 0.00 0.00 0.00 Max value gap 11.15 6.93 0.00 0.00 1.58 4.78 0.00 0.00 0.00 0.00 22.25 0.00 Min value gap -13.07 -0.24 -22.18 -7.40 -19.68 -22.32 0.00 0.00 0.00 0.00 -28.79 0.00 St. dev. value gap 4.62 1.74 7.72 1.85 7.11 8.58 0.00 0.00 0.00 0.00 13.17 0.00 Mean prob. gap 0.04 0.02 0.11 0.00 0.13 0.05 0.00 0.00 0.00 0.00 0.01 0.00 Max prob. gap 0.34 0.18 0.34 0.00 0.37 0.34 0.00 0.00 0.00 0.00 0.09 0.00 Min prob. gap 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 Comparison personal utility to prob. maximization Consistency btw. normative and personal utility 0.62 0.50 0.50 0.62 0.56 0.62 0.56 0.62 0.56 0.31 0.44 0.56 Fitting results Best fitted parameter x 0 45 50 72.5 50 37.5 45 50 50 50 50 60 50 Best fitted parameter k 0.08 0.05 0.12 0.15 0.08 0.11 0.15 0.1 0.15 0.15 0.1 0.15 Number of optimal solutions 5 51 3 3 12 9 3 7 3 3 7 3 Fitting error 0.19 0.34 0.69 0.06 0.56 0.44 0 0.06 0 0 0.25 0 101 5.3.6 Summary Analysis of Third Factors and Subject Feedback A subject’s adherence to normative behavior under incentives could potentially be explained by third factors. These factors include demographics and the time participants spent on the survey. Table 5.7 summarizes the consistency of each subject with probability maximizing behavior, their demographicbackground,andthetimetheyspentfinishingtheexperiment. Themethodsemployed in this section are k-means clustering and testing of the hypotheses: • H 1 : The type of target scheme has no significant influence on normative probability maxi- mizing behavior. • H 2 : Thedemographicfactorsageandgenderofasubjectdonothaveanysignificantinfluence on normative probability maximizing behavior under specific target schemes . • H 3 : The time spent on the survey/ experiment does not have a significant influence on normative probability maximizing behavior. Performing an ANOVA analysis using these three hypothesss yields the results summarized in tables 5.8 - 5.10. One can observe that: • WhentestingforhypothesisH 1 , onlytheintercepthasasucientlylowp-value. Thep-value for all other target schemes is much higher. Therefore, H 1 cannot be rejected. • Age and gender do not have a significant influence on consistency under dierent types of targets. Note that the subjects were very similar in age and educational background. • Thetime asubjectspentonansweringthe surveyhas asignificantinfluenceontheadherence of their responses with normative decision-making under targets at the 5% confidence level. 102 0 20 40 60 80 100 0.0 0.2 0.4 0.6 0.8 1.0 Subject: 1 0 20 40 60 80 100 0.0 0.2 0.4 0.6 0.8 1.0 Subject: 2 0 20 40 60 80 100 0.0 0.2 0.4 0.6 0.8 1.0 Subject: 3 0 20 40 60 80 100 0.0 0.2 0.4 0.6 0.8 1.0 Subject: 4 0 20 40 60 80 100 0.0 0.2 0.4 0.6 0.8 1.0 Subject: 5 0 20 40 60 80 100 0.0 0.2 0.4 0.6 0.8 1.0 Subject: 6 0 0 Personal U-Function Induced (fitted) Induced (normative) Figure 5.7: Personal, induced (fitted), induced (normative) utility functions under logistic target (Subjects 1-6) 103 0 20 40 60 80 100 0.0 0.2 0.4 0.6 0.8 1.0 Subject: 7 0 20 40 60 80 100 0.0 0.2 0.4 0.6 0.8 1.0 Subject: 8 0 20 40 60 80 100 0.0 0.2 0.4 0.6 0.8 1.0 Subject: 9 0 20 40 60 80 100 0.0 0.2 0.4 0.6 0.8 1.0 Subject: 10 0 20 40 60 80 100 0.0 0.2 0.4 0.6 0.8 1.0 Subject: 11 0 20 40 60 80 100 0.0 0.2 0.4 0.6 0.8 1.0 Subject: 12 0 0 Personal U-Function Induced (fitted) Induced (normative) Figure 5.8: Personal, induced (fitted), induced (normative) utility functions under logistic target (Subjects 7-12) 104 Table 5.7: Summary of consistency with normative behavior under dierent target schemes and demographic data per subject Subject 1 2 3 4 5 6 7 8 9 10 11 12 Consistency with normative behavior under target Fixed 0.62 0.25 1.00 1.00 0.75 0.62 1.00 1.00 1.00 1.00 1.00 1.00 Binary 0.50 0.75 0.75 1.00 0.25 0.62 0.88 1.00 1.00 1.00 0.88 0.88 Exponential (averse) 0.50 0.60 0.40 0.90 0.70 0.70 1.00 1.00 0.90 0.90 1.00 1.00 Exponential (seeking) 0.60 0.70 0.60 1.00 0.50 0.70 0.90 1.00 1.00 0.90 1.00 1.00 Logistic 0.56 0.38 0.12 0.94 0.31 0.44 1.00 0.88 1.00 1.00 0.69 1.00 Demographic information Age 28 26 24 26 25 27 24 24 26 31 24 25 Gender Male Male Male Male Male Female Female Female Male Male Female Male Years work experience 022 010 3 0 410 0 Probability Theory Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Decision Theory Yes Yes Yes Yes Yes Yes Yes No Yes Yes No No Total time spent on survey in seconds 1323 2164 2166 3807 752 3184 3005 71544 3411 4593 3603 3321 105 Table5.8: Coecient-levelestimatesformodelestimatingthevariationinadherencetoprobability maximizing behavior (hypothesis H 1 ) Predictor Estimate Sq. Error t p (Intercept) 0.79 0.071 11.22 0.000 TargetTypeExponential (averse) 0.01 0.100 0.08 0.940 TargetTypeExponential (seeking) 0.03 0.100 0.33 0.746 TargetTypeFixed 0.06 0.100 0.61 0.545 TargetTypeLogistic -0.10 0.100 -0.99 0.325 Kk-means clustering groups subjects along the dimensions of time spent on the survey and consistency with normative behavior. The goal is to identify whether separable groups exist and whether the composition of these groups change with the imposed incentives. First, clusterings are performed for all five incentive schemes. Then, I determine the specific subject within each cluster before calculating the average consistency with probability and expected utility maximizing behavior within the clusters. This analysis omits subject 8 because they spent a very long time on the survey compared to other subjects (see table 5.7), which would skew the analysis results. Figures 5.9 an 5.10 show the results of the k-means clustering. The members of each group under each incentive are summarized by table 5.11. One can see that there are two distinct groups for all five target schemes and that the members of the groups are identical. Determining the average consistency with probability and expected utility maximization per group per target scheme, as well as the average time it took each group’s members to complete the experiment yields table 5.12. One can make three main observations. First, Group 2, whose members spent more time on the survey, exhibited significantly higher normative behavior under targets than members of Group 1. Second, this fraction did not decline once the target schemes were made more complex by adding uncertainty and multiple target outcomes. Third, the fraction of expected utility maximization of subjects in Group 1 was not consistently higher or lower than the fraction of probability maximizing behavior under incentives. This may imply a somewhat 106 Table5.9: Coecient-levelestimatesformodelestimatingthevariationinadherencetoprobability maximizing behavior (hypothesis H 2 ) Predictor Estimate Sq. Error t p (Intercept) 0.70 0.430 1.63 0.108 TargetTypeExponential (averse) 0.01 0.099 0.08 0.940 TargetTypeExponential (seeking) 0.03 0.099 0.33 0.744 TargetTypeFixed 0.06 0.099 0.61 0.542 TargetTypeLogistic -0.10 0.099 -1.00 0.322 Age 0.01 0.017 0.39 0.700 GenderMale -0.12 0.072 -1.66 0.103 Table5.10: Coecient-levelestimatesformodelestimatingthevariationinadherencetoprobability maximizing behavior (hypothesis H 3 ) Predictor Estimate Sq. Error t p (Intercept) 0.764 0.033 22.88 0.00 TimeSpent 0.000 0.000 2.10 0.04 Table 5.11: Members of groups per target scheme according to k-means clustering (k = 2, outlier subject 8 removed) Target Scheme Group 1 Group 2 Fixed 1,2,3,5 4,6,7,9,10,11,12 Binary 1,2,3,5 4,6,7,9,10,11,12 Exponential (averse) 1,2,3,5 4,6,7,9,10,11,12 Exponential (seeking) 1,2,3,5 4,6,7,9,10,11,12 Logistic 1,2,3,5 4,6,7,9,10,11,12 107 -2000 -1000 0 1000 -2 -1 0 1 2 Cluster target type: Fixed -2000 -1000 0 1000 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 Cluster target type: Binary -2000 -1000 0 1000 -1 0 1 2 Cluster target type: Exponential (averse) Figure 5.9: Consistency vs. Time clusters of subjects under fixed, binary, and averse targets 108 -2000 -1000 0 1000 -1.0 0.0 1.0 2.0 Cluster target type: Exponential (seeking) -2000 -1000 0 1000 -2 0 1 2 3 Cluster target type: Logistic Figure 5.10: Consistency vs. Time of subjects under seeking and logistic target 109 random choice behavior under targets for this group, instead of either maximizing expected utility orsuccessprobability. Anotherpossibleimplicationisthatthemechanismusedintheexperimentto incentivize truthful reporting and probability maximizing behavior was not strong enough. Figure 5.11 visualizes the consistency with probability maximizing behavior and its variability between target schemes and identified groups. I perform two more ANOVA analyses after filtering the data for group composition. These test for the following hypotheses: • Givenasubject’sgroup, thecertainoruncertainnatureofatargetdoesnothaveasignificant influence on the subject’s consistency with probability maximizing behavior. • Given a subject’s group, the nature of the task (choice between lotteries vs. ordering them) does not have a significant influence on the subject’s consistency with probability maximizing behavior. The results of these tests are shown in tables 5.13 and 5.14. They demonstrate that if a subject spent less time on the survey, consistency was no for Group 2. A subject’s ability to understand the instructions and determine the normative choice may be a factor explaining their consistency with probability maximizing behavior. Table 5.15 summarizes the subjects’ feedback after the conclusion of the experiment. Responses were encoded using a 5-point Likert scale, where 5 is agreement and 1 disagreement with a statement. An average score of 4.75 indicates that most subjects agreed with statement 1 and 2 about the clarity of the instructions and tasks. There was not a big dierence in these questions’ scores between the Group 1andGroup2. Thissuggeststhatunclearinstructionsandtaskscanlikelyberuledoutasareason for choices inconsistent with normative behavior under incentives. Members belonging to Group spent more time on the survey and were more consistent with normative behavior under incentives. On average, they reported that the experiment was mentally exhausting more often, as indicated 110 Table 5.12: Consistency with probability and expected utility maximization per group under dif- ferent target schemes Group 1 Group 2 Fixed target Avg. consistency with prob. maximization 0.655 0.946 Avg. consistency with expected util. maximization 0.562 0.396 Binary target Avg. consistency with prob. maximization 0.562 0.894 Avg. consistency with expected util. maximization 0.685 0.36 Exponential (averse) target Avg. consistency with prob. maximization 0.55 0.914 Avg. consistency with expected util. maximization 0.475 0.614 Exponential (seeking) target Avg. consistency with prob. maximization 0.6 0.929 Avg. consistency with expected util. maximization 0.7 0.671 Logistic target Avg. consistency with prob. maximization 0.342 0.85 Avg. consistency with expected util. maximization 0.358 0.49 Average time spent on survey in sec. 1601 3561 0.25 0.50 0.75 1.00 Fixed Binary Averse Seeking Logistic Target type Consistency with prob. max. Group Group 1 Group 2 Figure 5.11: Consistency with prob.max. along groups 111 Table 5.13: Group 1: Coecient-level estimates for certainty and task type Predictor Estimate Sq. Error t p (Intercept) 0.65 0.097 6.75 0.000 certaintyTypeUncertain -0.08 0.112 -0.75 0.463 TaskTypeOrder -0.23 0.112 -2.04 0.057 Table 5.14: Group 2: Coecient-level estimates for certainty and task type Predictor Estimate Sq. Error t p (Intercept) 0.95 0.048 19.69 0.000 certaintyTypeUncertain -0.03 0.056 -0.52 0.605 TaskTypeOrder -0.05 0.056 -0.98 0.335 by the higher average Likert score. This may indicate that they put in more eort to determine the success probability of each of the presented lottery. Because all participants were ISE graduate students, each possessed the theoretical knowledge of calculate the best lottery under every target. 5.4 Conclusion This part of my dissertation investigated whether human decision-makers’ utility functions under targets/incentives change in accordance to normative decision theory. In addition, I aimed to determine whether this change can be explained by third factors concerning the decision-maker, including age, gender, and educational background. The third goal was to validate whether the algorithms proposed in chapter 4 are sucient to elicit induced utility via preference statements. Findings confirm that subjects maximized the probability of achieving or exceeding their target and mostly acted normatively. Therefore, the induced utility functions were consistent with theo- retical research. This finding, however, depended on the type of target. Where there was only one possible target, i.e. it was certain, subjects chose most consistently. The adherence to probability maximization decreased when subjects were asked to choose between two lotteries or indierence 112 Table 5.15: Aggregate and groupwise feedback Across all subjects Across group 1 Across group 2 Question Mean St.dev. Mean St.dev. Mean St.dev. The questions were explained clearly 4.75 0.45 4.25 0.50 5.00 0.00 The visual materials supported me in answering the questions 4.75 0.62 4.25 0.96 5.00 0.00 The survey length was appropriate 3.17 1.53 3.00 1.41 3.25 1.67 I did not understand what was expected of me during the survey 1.75 1.22 2.00 0.82 1.62 1.41 I had diculty reading the supporting data or material 1.58 1.16 1.50 0.58 1.62 1.41 Answering some questions was mentally exhausting 3.00 1.76 2.50 1.29 3.25 1.98 betweenthemwhenthetargetwasuncertain. Afurtherdecreasewasdetectedwhenaddingathird lottery and asking the participants to order them according to their preference. This has several implications for the practical use of incentives. First, a certain target should be imposed whenever possible. However, to ensure that subjects choose such that someone else’s utilityismaximized,thisfixedtargethastobetheaspirationequivalentofalottery(seeAbbasand Matheson (2005)). A second implication is that having decision-makers order lotteries according to their preference should be avoided in favor of them choosing their preferred alternative under the same target. Assessing the influence of demographics on decision-making, standard demographic factors, including age and gender, did not have significant influence on adherence to normative behavior. Instead, the time a subject spent answering the questions determined the degree of consistency with normative behavior. Less time and a change of task decreased consistency and may suggest 113 that the incentive for truthful reporting and eort was not strong enough for some participants. Thelotteriesusedinthisexperimentweregeneratedbyalgorithmsoutlinedinchapter4. Given the experiment’s results on decision-making behavior are consistent with normative theory, one could conclude that the algorithms and the use of preference statements is a strong method to assess induced utility functions. 114 Chapter 6 Conclusion This dissertation explored utility functions induced by certain and uncertain incentive schemes usinganalyticalandempiricalmethods. Thefactorsmotivatingtheinvestigationofthistopicarose from surveying related literature. First, this survey did not find behavioral data or studies that could validate or disprove theoretical findings on utility functions induced by incentives. Second, utility elicitation methods were not discussed under the assumption of imposed incentives. Lastly, third factors, including age and gender, are discussed when researching their eect on subjects’ risk aversion. However, the discussion of third factors was not found to be prevalent when investigating utilityfunctionsinducedbyincentives. Thesemotivatingfactorsresultedinthreeresearchquestions answered by this work: 1. “Are the results of common utility elicitation methods consistent with each other when applied with human decision-makers?” 2. “Does a human decision-maker’s utility function change in accordance to normative decision theory when certain and uncertain incentives are imposed?” 3. “Are there other factors determining this change other than the imposed incentive?” 115 Question1hasbeenansweredbyseveralresearchersforthecaseofindierencemethods. These methodsconsistofcertainequivalenceandindierenceprobabilitymethods,andwerefoundtoyield dierent results (Hershey and Schoemaker 1985). The research conducted for this dissertation con- tributes to this question in three ways. It validated previous findings, and included approximation methodsintheevaluations. Also,moreformsofutilityfunctionsanddierentanalyticalprocedures were used than in previous studies. Question 2 was never answered before by a behavioral experiment, although theoretical results on utility functions induced by incentives exist. Answering this question produced several research contributions. First, algorithms to elicit the utility function via preference statements were devel- oped. With previous findings on inconsistencies between results elicited by indierence methods, and the added complexity of imposed incentives, I considered preference statements as a good al- ternative. These algorithms, which design the lotteries based on a subject’s response and converge quicklytotheirtrueutilityfunction,didnotexistbefore. Second,theelicitationofutilityfunctions inducedbyseveraltypesofcertainanduncertainincentivesvalidatedtheoreticalfindings. Previous behavioralstudiesassessedthechangesinriskattitudeonceanincentivewasimposed(Prendergast 1999, 2008; Pennings and Smidts 2003), but did not determine the specific form of the resulting utility function. Question 3 set out to explore the influence of third factors on probability maximizing behavior and utility functions under incentives. It contributes to research by finding that demographic factors that were determined to impact subjects’ risk aversion, no longer do so once incentives were imposed. Furthermore, it provides observations supporting statements on the required strength of incentives (Gneezy and Rustichini 2000). The remainder of this chapter is structured as follows. Subsection 1 addresses the first research question by summarizing findings on consistency between human decision-makers’ utility functions 116 elicited by dierent elicitation methods. The next subsection recapitulates observations regard- ing research questions 2 and 3. The chapter concludes by discussing future research directions originating from this work. 6.1 Consistency of Human Decision-Makers’ Utility Assessments Across Elicitation Methods In creating a behavioral experiment to assess whether the responses to indierence and ap- proximation methods were consistent, I was able to determine that the individual points assessed during the indierence method diverged widely from each other on the basis of the fitted curve parameter(s). This is significant, because it implies that the methods cannot be used interchange- ably. Becauseoftheindividualpoints’divergence,onecanassumethatfittingcurvestoallassessed pointssimultaneouslyismorerobustandsusceptibletooutliers. Aggregationbytakingthemedian, which is also robust to outlier assessments, did not consistently produce parameters that were close or equal to the ones produced by curve fitting to all points simultaneously or the approximation methods. Theindierencemethodasubjectwasassignedtoinfluencedthestandarddeviationoftheutility function parameters when fitting to individual points. Under the certain equivalence method, the standarddeviationoftheindividualparameterswassubstantiallylowerthanundertheindierence probability method. This supports findings made by Hershey and Schoemaker (1985) that elicited utility diers depending on the used method. However, my findings deviated from their claim that the certain equivalence method produces more risk seeking behavior. Instead I observed less risk seeking compared to the indierence probability group. Note that Hershey and Schoemaker (1985) uses a within-subject design while this research employed a between-subject design. 117 Approximationmethodsthatelicitasubject’sutilityfunctionparameterviaasingleassessment did not yield parameters that were close to ones elicited by indierence methods. This observation was true for cases in which parameters obtained from fitting to individual assessments were aggre- gated using mean and median, and in which parameters were obtained by fitting to all assessments of the respective indierence method. The value obtained by fitting to all was outside of the range produced by the approximation methods most of the time. Because of the chosen probabilities and outcomes of the approximation methods, they also imposed constraints on the function parameters tobefitforlogarithmicandlinearrisktoleranceutilityfunctions. Thisrequiresthatapproximation methods are tailored to a specific form of utility function. There are two limitations to these findings. First, in comparison to Hershey and Schoemaker (1985) this experiment used a between-subject instead of a within-subject design. However, the study population had a similar educational background (all were graduate students of USC’s ISE department), and a narrow age range. The second limitation is due to the mode the experiment was delivered to subjects. Because of an online format, it was not possible for both subject and experimenter to ask clarifying questions. The individual points were elicited via direct assessment instead of via a series of questions, which is aligned to similar studies using in-person experiments. 6.2 NormativeBehaviorofHumanDecision-MakersUnderIncen- tives Human decision-makers’ consistency with normative behavior under certain and uncertain in- dependent targets was assessed using preferences between lotteries. The research found overall consistencywithnormative behavior, meaningthatsubjectschosethe lotterythatmaximizedtheir probability of meeting or exceeding the imposed target. As the incentives and tasks became more 118 complex, the number of inconsistencies with this behavior increased. Inanalyzingtheresponses,Ilookedtoisolatethereasonfordecliningconsistencywithnormative behavior and did not find a significant influence of demographic factors (age, gender, personal risk aversion) on consistency with normative behavior. Interestingly, the time a subject spent on the survey was found to be a predictor of the number of assessments consistent with probability maximization. Using k-means clustering, two groups were formed that diered in the amount of time they spent on the experiment. Their composition did not change across the imposed targets, implying a subject’s adherence to normative choice behavior did not change dramatically between dierent schemes. One group’s consistency with normative behavior declined substantially when the imposed targets became more complex and the other’s remained constant at a high level. Even adherence to probability maximization under the certain target, which was the simplest imposed incentive, was lower for the group that spent less time. This may indicate that the incentives paid to ensure truthful reporting and to put forth the eort to determine probability maximizing lotteries was not powerful enough. Second, anecdotal evidence relayed after subjects finished the experiment suggests that they were uncertain whether the experiment aimed at their intuitive choice or the analytic solution to it in a few cases. Given that all subjects had knowledge of probability theory, they had the background to determine the normative choice. The instructions of the experiment provided background on the target schemes, but deliberately did not specify whether to use intuition or their knowledge of probability theory. Subjects’ responses to clarifying questions during the experiment and to general feedback questions at conclusion suggest that they understood the instructions and that these were clear. Because indierence methods were found to introduce bias and produce diering results (Far- quhar 1984; Hershey and Schoemaker 1985), three algorithms were developed to elicit utility func- tions for step, exponential, and logistic decision-makers via preference comparisons. Preference 119 methods were found to be less susceptible to bias and more predictive of actual choice behavior (Daniels and Keller 1992). These developed algorithms converge in few iterations, and were used to generate lotteries beforehand assuming a probability maximizing subject under the respective incentive. Fitting a curve to a subject’s responses to questions of which of these lottery they pre- ferred would produce a function consistent with the theoretically induced utility function. This design meant that the number of assessments was pre-determined, which led to wide bounds on the induced utility function for subjects with inconsistent assessments. 6.3 Future Research Directions Possible directions for future research pertain to utility elicitation methods without incentives, thealgorithmicproceduretoelicitutilityviapreferencemethods,andutilityelicitationunderdier- ent types of incentives. The evaluation of utility elicitation methods can be expanded by adopting a within-subject design. Furthermore, preference and approximation methods refined to specific forms of utility functions can be added to the comparison. Incorporating choices between lotteries after a subject’s utility function has been assessed by all methods would permit the comparison of methods along choice predictability. The presented algorithms were designed with the experiment to elicit induced utility functions in mind. The experiment’s purpose was to investigate human decision-makers’ adherence to nor- mative behavior under incentives, meaning that the algorithms generated lotteries for a normative decision-makerundertheassumptionofapre-determinedutilityfunction. Therearethreeresearch directions to make this approach more viable for situations in which knowledge about which func- tional form to assume is not given. First, one could include preliminary assessments that narrow down the function to be used from a set of functions. These preliminary assessments can include questions to test adherence to the delta property or whether the utility function is S-shaped or not. 120 Second, algorithms to determine other functional forms need to be devised. Third, inconsistent assessments of subjects need to be accommodated. All incentive schemes in this dissertation were independent, meaning that only the subject’s choice of lottery influenced target attainment. Future research could investigate the adherence to normative behavior under competitive and/or collaborative incentives. When making a decision under collaborative or competitive incentives, the subject has to consider the choices made by the people they collaborate or compete with. Additionally, one could expand the incentive scheme by specifying outcomes for both the decision-maker and a third party on whose behalf they make a choice between lotteries. This would enable an investigation into whether and how decision-makers make a tradeo between personal and third-party gain, and what the resulting risk attitude will be. 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Given these assessments, one can determine Â by minimizing the squared error between fitted and observed quantities. Min Â N ÿ i ! ˆ U(x i )≠ U(x i ) " 2 , (6.1) where ˆ U(x i ) denotes the fitted and U(x i ) the observed indierence probability. The amount x i is the certain amount provided to the subject in exchange for the uncertain deal. For the certain equivalent the function to be minimized is 132 Table 6.1: Overview of utility functions fitted to elicited statements Function name Functional form U(x) Inverse function U ≠ 1 (x) Linear x x ú U(x)x ú Exponential (1≠ e ≠ “x )≠ (1≠ e ≠ “x 0 ) (1≠ e ≠ “x ú )≠ (1≠ e ≠ “x 0 ) ≠ 1 “ ln(e ≠ “x 0 ≠ U(x)[e ≠ “x 0 ≠ e ≠ “x ú ]) Logarithmic ln(x+w)≠ ln(x 0 +w) ln(x ú +w)≠ ln(x 0 +w) e U(x)[ln(x ú +w)≠ ln(x 0 +w)]+ln(x 0 +w) ≠ w Linear Risk Tolerance (fl +÷x ) 1≠ 1 ÷ ≠ (fl +÷x 0 ) 1≠ ÷ ≠ 1 (fl +÷x ú ) 1≠ 1 ÷ ≠ (fl +÷x 0 ) 1≠ 1 ÷ 1≠ 1 ÷ Ò U(x)(fl +÷x ú ) 1≠ 1 ÷ +(fl +÷x 0 ) 1≠ 1 ÷ (1≠ U(x))≠ fl ÷ Logistic L 1≠ e ≠ k(x≠ x 0 ) ≠ L 1≠ e ≠ k(x 0 ≠ x 0 ) L 1≠ e ≠ k(x ú ≠ x 0 ) ≠ L 1≠ e ≠ k(x 0 ≠ x 0 ) x 0 ≠ ln ! L U(x) L 1+e ≠ k(x ú ≠ x 0 ) +(1≠ U(x)) L 1+e ≠ k(x 0 ≠ x 0 ) ≠ 1 " k Min Â N ÿ i ! ˆ x i ≠ ˜ x i " 2 , (6.2) where ˆ x i denotes the fitted and ˜ x i the observed certain equivalent. Given the data obtained from the behavioral experiment, this curve fitting procedure is applied to the answers provided by each participant and for each type of utility function. The choice of fittingproceduredependsonwhichutilityelicitationmethodthesubjectwasassignedto. Table6.1 summarizes the utility functions fit to the observations and their inverses, i.e. certain equivalents. Appendix I.II - Determination of wealth parameter using approximation meth- ods M 1 and M 2 The logarithmic utility function is given by U(x)=ln(x+w) (6.3) wherewdenotesthedecision-maker’sinitialwealth. InapproximationmethodM 1 thesubjectis 133 askedwhichamountRwouldmakethemindierentbetweennotreceivinganythingoranuncertain deal of receiving R or losing R/2 with equal probability. Therefore, ln(0+w)= ln(R+w) 2 + ln ! ≠ R 2 +w " 2 (6.4) Then, ln(w 2 )=ln(R+w)+ln ! ≠ R 2 +w " (6.5) ln(w 2 )=ln ! ≠ R 2 2 + Rw 2 +w 2 " (6.6) which yields R =w (6.7) As a result of equation (6.7), a fully rational decision-maker following a logarithmic utility functionwouldprovidehisinitialwealthlevelwhenpromptedbyapproximationmethodM 1 . Using method M 2 , one obtains ln(0+w)= 3ln(R+w) 4 + ln(≠ R+w) 4 (6.8) Inspecting equation (6.8), one finds thatR>w and w =Œ at equality. Therefore, method M 2 yields a lower bound on the decision-maker’s initial wealth. 134 Appendix I.III - Determination of parameters using approximation methodsM 1 and M 2 for linear risk tolerance The linear risk tolerance utility function is given by equation (6.9). U(x)=(fl +÷x ) 1≠ 1 ÷ (6.9) Given presumed equality between the lotteries in M 1 and M 2 since their certain equivalent is zero, one can state 1 2 (fl +÷x ) 1≠ 1 ÷ + 1 2 (fl ≠ ÷ x 2 ) 1≠ 1 ÷ = 3 4 (fl +÷x ) 1≠ 1 ÷ + 1 4 (fl ≠ ÷x ) 1≠ 1 ÷ . (6.10) Equation (6.10) can be simplified to 2 1 2 (fl ≠ ÷ x 2 ) 1≠ 1 ÷ ≠ (fl +÷x ) 1≠ 1 ÷ =(fl ≠ ÷x ) 1≠ 1 ÷ . (6.11) Equation (6.11) cannot be solved analytically for the two functional parameters. Using the experimentresponses, wedeterminetheseparametersbyminimzingthesquareddierencebetween theleftandright-handsideofequation (6.10). Onecanobservefromthetermsthoughthatfl>÷x whenx< 0 and÷> 0 is even. Otherwise, the term within the even root would be negative. Appendix II - Chapter 3 ABM and algorithms Appendix II.I - Using utility functions as target generating functions I define a lottery L =,where p i and x i denote discrete probabilities and outcomes respectively. However, a lottery can also be continuous, which I assume for the 135 followingequations. Adecisionmakersubscribingtotheaxiomsofrationaldecisionmakingchooses the lottery that maximizes his or her expected utility, i.e. max m ⁄ +Œ ≠Œ p m U(x m )dx (6.12) Now, suppose that we give the agent an incentive that pays a fixed amount if a random target is achieved or exceeded, i.e. target T Æ x. This means, that they no longer care about the utility of outcome x i , but rather its probability of being greater or equal to the uncertain target T.We also tell the decision maker the distribution F from which the target is going to be sampled when determining whether he or she receives the payo. Therefore, a rational decision maker would choose the lottery that maximizes the probability of attaining or exceeding the target, i.e. max m ⁄ +Œ ≠Œ p m F(T Æ x m )dx (6.13) Function (6.13) implies that if we take the principal’s utility function as the incentive scheme’s cumulative distribution function, the agent automatically chooses the lottery that maximizes the principal’s expected utility. The advantages of using the utility function as a function to generate uncertain targets has the following advantages: 1. Eliminates the need to determine the aspiration equivalent for every available lottery. The uncertain target applies to all lotteries and results in expected utility maximization for the principal. 2. Agent does not benefit from lying about his or her performance lotteries and principal does not need knowledge about the lotteries in the first place. The disadvantages of the approach are that the principal’s utility function has to be normalized 136 to a interval between zero and one. Appendix II.II - Background on incentive schemes with certain target Under an incentive scheme with a certain target, the decision maker knows the exact target against which his or her performance will be measured. Upon achieving this fixed amount, he or she will receive a specific amount, independent of by how much the target has been exceeded. The induced utility function is therefore a step-function that jumps from zero to one at the fixed target. U(x)= Y _____] _____[ 1 if xØ T 0 ifx<T . (6.14) Figure6.1showsthedierentsteputilityfunctionsinducedbydierenttargetvalues T. Because thedeterminationoftheriskaversionfunctionrequirestheutilityfunctiontobetwicedierentiable, one cannot determine the risk aversion function in this particular case. 0 20 40 60 80 100 0.0 0.2 0.4 0.6 0.8 1.0 Outcome x Utility Target: 30 Target: 50 Target: 80 Figure 6.1: Step utility function with dierent targets 137 Appendix II.III - Background on normalized exponential utility functions The first distribution that will be used as an uncertain target is the normalized exponential utility function. Usually, the exponential utility function does not need to be restricted to the interval of zero and one. I choose this interval such that the utility remains interpretable as an indierence probability. The normalized exponential utility function is given by U(x)= (1≠ e ≠ “x )≠ (1≠ e ≠ “x min ) (1≠ e ≠ “x max )≠ (1≠ e ≠ “x min ) . (6.15) Using the von Neumann - Morgenstern definition of risk aversion, which is given by “ (x)=≠ U ÕÕ (x) U Õ (x) , (6.16) the risk aversion function of the normalized exponential utility function is given by “ (x)=“. (6.17) Asaresultof (6.17), theriskaversionfunctionisconstantandtakesonthevalueofthedecision maker’s risk aversion coecient. Appendix II.IV - Background on normalized logistic utility function The normalized logistic function has the form U(x)= L 1+e ≠ k(x≠ x 0 ) ≠ L 1+e ≠ k(x 0 ≠ x 0 ) L 1+e ≠ k(x ú ≠ x 0 ) ≠ L 1+e ≠ k(x 0 ≠ x 0 ) (6.18) ,whereL=1denotesthemaximumoftheutilitycurve,k thesteepness, andx 0 isthemidpoint of the sigmoid. 138 Figure 6.2 visualizes the eects of parameters k and x 0 on the shape of the utility function. Increasing parameter k increases convexity of the utility curve below x 0 and concavity above. Therefore, a decision-maker who follows this utility curve is risk-seeking below x 0 and risk-averse above it. At the same time altering target x 0 results in a dierent inflection point at which the utility function changes from convex to concave, giving it its characteristic S-shape. One can solve equation (6.18) for the certain equivalent by taking the inverse. The certain equivalent Â x can be calculated by Â x =x 0 ≠ ln 1 L U(x) ≠ 1 2 k . (6.19) Dierentiating the logistic utility function, the risk aversion function can be written as “ (x)= k(1≠ e ≠ 2k(x≠ x 0 ) ) (1+e ≠ k(x≠ x 0 ) ) 2 . (6.20) As one can see from figure 6.3 the risk aversion function converges to parameter k as x goes to infinity and to≠ k as x goes to negative infinity. “ (x)= k(1≠ e ≠ 2k(x≠ x 0 ) ) (1+e ≠ k(x≠ x 0 ) ) 2 = k(1≠ 1 e 2k(x≠ x 0 ) ) [1+ 1 e k(x≠ x ) ] 2 (6.21) lim xæŒ k(1≠ 1 e 2k(x≠ x 0 ) ) [1+ 1 e k(x≠ x ) ] 2 =k (6.22) lim xæ≠Œ k(1≠ 1 e 2k(x≠ x 0 ) ) [1+ 1 e k(x≠ x 0 ) ] 2 =≠ k (6.23) 139 0 20 40 60 80 100 0.0 0.2 0.4 0.6 0.8 1.0 Outcome Utility k =0.15 k =0.2 k =0.25 x 0 0 20 40 60 80 100 0.0 0.2 0.4 0.6 0.8 1.0 Outcome Utility k =0.15 k =0.2 k =0.25 x 0 Figure 6.2: Logistic utility function for dierent parameters x0 and k 140 0 20 40 60 80 100 -0.2 -0.1 0.0 0.1 0.2 Outcome Risk aversion k =0.1 k =0.2 x 0 Figure 6.3: Risk aversion function for dierent values of parameter k 141 Appendix II.V - Proof of algorithm to elicit exponential utility functions I assume that the decision maker follows a normalized exponential utility function of the form U N (x)= U(x)≠ U(x 0 ) U(x ú )≠ U(x 0 ) , (6.24) where x 0 =0 and U(x)=1≠ e ≠ “x . The algorithm presented below works on the assumption that the decision maker chooses be- tweenlotteriesL 1 =,L 2 =, orindierencebetweenthem. Assuming a certain value for risk aversion coecient “ we can determine outcome x of lottery L 2 such that lottery L 1 is preferred over lottery L 2 . p 1 U(x 1 )+(1≠ p 1 )U(0)>p 2 U(x 2 )+(1≠ p 2 )U(0) (6.25) p 1 U(x 1 )≠ U(x 0 ) U(x ú )≠ U(x 0 ) +(1≠ p 1 ) U(0)≠ U(x 0 ) U(x ú )≠ U(x 0 ) > p 2 U(x 2 )≠ U(x 0 ) U(x ú )≠ U(x 0 ) +(1≠ p 2 ) U(0)≠ U(x 0 ) U(x ú )≠ U(x 0 ) (6.26) The second terms on each side of (6.26) reduce to zero because of x 0 =0 and both sides can be multiplied by U(x ú )≠ U(x 0 ) to produce (6.27) p 1 (U(x 1 )≠ U(x 0 ))>p 2 (U(x 2 )≠ U(x 0 )) (6.27) For L 1 to be preferred over L 2 for a given risk aversion coecient “ , outcome x 2 has to be x 2 Ø≠ 1 “ ln 5 1≠ 5 p 1 p 2 U(x 1 )+(1≠ p 1 p 2 )U(x 0 ) 66 (6.28) 142 To design a feasible lottery L 2 that is consistent with this preference ordering, probability p 2 has to meet following condition: p 2 œ [max( p 1 (U(x 1 )≠ U(x 0 )) 1≠ U(x 0 ) ,0);1] (6.29) I impose that any probability has to be between 0 and 1. Furthermore, probability p 2 > p 1 (U(x 1 )≠ U(x 0 )) 1≠ U(x 0 ) because the term within the natural logarithm would be less than or equal to 0 otherwise. Appendix II.VI - Proof of algorithm to elicit logistic utility func- tions I assume the following utility function with two parameters U(x)= L 1+e ≠ k(x≠ x 0 ) (6.30) where L denotes the maximum of the function, x 0 the sigmoid midpoint and k the logistic growth rate or steepness of the curve. Similarly to the section above, I assume that the utility curve is normalized such that U(x ú )=1 and U(x 0 )=0. For given parameters x 0 and k, Lottery L 1 = is preferred over lottery L 2 = if x 2 >x 0 ≠ ln( L p 1 p 2 U(x 1 )+U(x 0 )(1≠ p 1 p 2 ) ≠ 1) k (6.31) Solving for x 2 for a given set of parameters x 0 and k. 143 p 1 (U(x 1 )≠ U(x 0 ))>p 2 (U(x 2 )≠ U(x 0 )) (6.32) p 1 p 2 U(x 1 )+(1≠ p 1 p 2 )U(x 0 )>U(x 2 ) (6.33) L 1+e ≠ k(x 2 ≠ x 0 ) < p 1 p 2 U(x 1 )+(1≠ p 1 p 2 )U(x 0 ) (6.34) e ≠ k(x 2 ≠ x 0 ) < L p 1 p 2 U(x 1 )+(1≠ p 1 p 2 )U(x 0 ) ≠ 1 (6.35) ≠ k(x 2 ≠ x 0 )<ln C L p 1 p 2 U(x 1 )+(1≠ p 1 p 2 )U(x 0 ) ≠ 1 D (6.36) Solving for x 2 such that lottery L 1 is preferred over L 2 : x 2 >x 0 ≠ ln 5 L p 1 p 2 U(x 1 )+(1≠ p 1 p 2 )U(x 0 ) ≠ 1 6 k (6.37) One can see from equation (6.31) that p 1 ln( p 2 (1≠ e ≠ kx 2 )≠ p 1 (1≠ e ≠ kx 1 ) p 1 e ≠ kx 2(1≠ e ≠ kx 1)≠ p 2 e ≠ kx 1(1≠ e ≠ kx 2) ) k (6.38) Finding parameter x 0 for a given set of parameters p 1 , p 2 , x 1 , x 2 , x 0 =0, and L=1, such that lottery L 1 is preferred over lottery L 2 . 144 p 1 (U(x 1 )≠ U(x 0 ))>p 2 (U(x 2 )≠ U(x 0 )) (6.39) p 1 p 2 > U(x 2 )≠ U(x 0 ) U(x 1 )≠ U(x 0 ) (6.40) With x 0 =0, p 1 p 2 > 5 L 1+e ≠ k(x 2 ≠ x 0 ) ≠ L 1+e kx 0 65 L 1+e ≠ k(x 1 ≠ x 0 ) ≠ L 1+e kx 0 6 ≠ 1 (6.41) Assuming that L=1 p 1 p 2 > e kx 0 ≠ e ≠ k(x 2 ≠ x 0 ) 1+e ≠ k(x 2 ≠ x 0 ) 1+e ≠ k(x 1 ≠ x 0 ) e kx 0 ≠ e ≠ k(x 1 ≠ x 0 ) (6.42) p 1 p 2 > 1≠ e ≠ kx 2 1+e ≠ k(x 2 ≠ x 0 ) 1+e ≠ k(x 1 ≠ x 0 ) 1≠ e ≠ kx 1 (6.43) Rearranging for x 0 yields e kx 0 Ë p 1 e ≠ kx 2 (1≠ e ≠ kx 1 )≠ p 2 e ≠ kx 1 (1≠ e ≠ kx 2 ) È >p 2 (1≠ e ≠ kx 2 )≠ p 1 (1≠ e ≠ kx 1 ) (6.44) kx 0 >ln C p 2 (1≠ e ≠ kx 2 )≠ p 1 (1≠ e ≠ kx 1 ) p 1 e ≠ kx 2 (1≠ e ≠ kx 1 )≠ p 2 e ≠ kx 1 (1≠ e ≠ kx 2 ) D (6.45) x 0 > ln Ë p 2 (1≠ e ≠ kx 2 )≠ p 1 (1≠ e ≠ kx 1 ) p 1 e ≠ kx 2(1≠ e ≠ kx 1)≠ p 2 e ≠ kx 1(1≠ e ≠ kx 2) È k (6.46) 145 Appendix III - Chapter 4: Behavioral experiment to assess decision-making behavior under incentives Appendix III.I - Experiment setup The behavioral experiment was set up to be taken in Qualtrics. Subjects were selected from a pool of graduate students at the ISE department of USC. Links to the experiment were sent to them via an email distribution list. Every participant went through the process visualized by 6.4. After consent was given, demographic factors (age, gender, educational attainment, work experience, and character traits) and personal risk attitude were assessed using a collection of methods. Next, every subject was presented with choice situations under fixed, binary, exponential (averse and seeking), and logistic targets. Once subjects finished the experiment participation was compensated. Compensation consisted of two components. The first component of 15 dollars was certain, meaning that every participant received it. 10 dollars were paid based on a random mechanism. This mechanism randomly determined one of the subject’s choices and generated a randomnumberbetween0and1. Thevariableamountwasdisbursediftherandomnumberwasless than or equal the probability of achieving or exceeding the target with the selected gamble/lottery. Thepurposeofthismechanismwastoincentivizetruthfulreporting. Subjectscouldmaximizetheir chances of obtaining the higher payout if they chose lotteries that maximized their probability of target achievement. Appendix III.II - Ten-item inventory for Big 5 character trait elicitation This is the ten-item inventory developed by Gosling, Rentfrow, and Swann (2003). The items elicit to which degree one agrees with the following statements: 146 Informed consent Elicitation of demographic factors Assessment of risk aversion via indifference probability Assessment of risk aversion via certain equivalence Risk tolerance approximation Choice under fixed target Choice under uncertain target with two outcomes Choice under uncertain target with three outcomes Choice under uncertain target with four outcomes Conclusion and determination of payoff Figure 6.4: Process flow of the experiment 147 1. I see myself as extroverted, enthusiastic 2. I see myself as critical, quarrelsome 3. I see myself as dependable, self-disciplined 4. I see myself as anxious, easily upset 5. I see myself as open to new experiences, complex 6. I see myself as reserved, quiet 7. I see myself as sympathetic, warm 8. I see myself as disorganized, careless 9. I see myself as calm, emotionally stable 10. I see myself as conventional, uncreative According to how much one agrees with each of the ten statements, the following score is assigned to item: • 1: Disagree strongly • 2: Disagree moderately • 3: Disagree a little • 4: Neither agree or disagree • 5: Agree a little • 6: Agree moderately • 7: Agree strongly To determine the score for each of the big five character traits, sum over the degrees assigned to the associated question by the subject (“R” denotes reverse-scored items): • Extroversion: 1, 6R • Agreeableness: 2R, 7 148 Table 6.2: Lotteries to elicit step utility function Lottery 1 Lottery 2 Assessment p 1 x 1 p 2 x 2 p 1 x 1 p 2 x 2 Certain target: T = 65 1 0.20 50.00 0.80 0 0.90 25.00 0.10 0 2 0.75 75.00 0.25 0 0.80 63.00 0.20 0 3 0.55 69.00 0.45 0 0.40 72.00 0.60 0 4 0.65 66.00 0.35 0 0.70 64.00 0.30 0 5 0.15 64.00 0.85 0 0.10 65.00 0.90 0 6 0.55 65.00 0.45 0 0.95 64.90 0.05 0 7 0.60 64.99 0.40 0 1.00 64.97 0.00 0 8 0.30 65.00 0.70 0 0.20 65.00 0.80 0 Certain target: T = 60 9 0.75 75.00 0.25 0 0.80 63.00 0.20 0 Certain target: T = 70 10 0.55 69.00 0.45 0 0.40 72.00 0.60 0 • Conscientiousness: 3, 8R • Emotional Stability: 4R, 9 • Openness to Experiences: 5, 10R Appendix III.III - Lotteries presented to experiment participants Tosummarizethelotteriestoelicittheinducedutilityfunctionunderalogistictargetconcisely, we abbreviate the following table. Note that for each lottery the worst outcome is zero and the corresponding probability is 1≠ p best . Appendix III.IV - Fitting procedure for utility functions under incentives Theprocesstofittheinducedutilityfunctionsistwo-pronged. First, Iconverttheparticipants’ answersintobinaryoutcomesusingone-hotencoding, meaningthatchoicebecomesavariablewith 1 if chosen and 0 otherwise. Then, I derive a function that generates a preference encoding for each particular incentive scheme. The parameter(s) of the incentive scheme (step, exponential, logistic) 149 Table 6.3: Lotteries to elicit exponential utility function (risk averse) Lottery 1 Lottery 2 Assessment p 1 x 1 p 2 x 2 p 1 x 1 p 2 x 2 1 0.20 55 0.80 0 0.60 19 0.40 0 2 0.60 55 0.40 0 0.70 26 0.30 0 3 0.70 85 0.30 0 0.90 50 0.10 0 4 0.40 55 0.60 0 0.50 41 0.50 0 5 0.20 35 0.80 0 0.65 8 0.35 0 6 0.80 25 0.20 0 0.70 30 0.30 0 7 0.35 25 0.65 0 0.60 14 0.40 0 8 0.19 35 0.81 0 0.78 7 0.22 0 9 0.90 35 0.10 0 0.70 50 0.30 0 10 0.50 55 0.50 0 0.65 33 0.35 0 Table 6.4: Lotteries to elicit exponential utility function (risk seeking) Lottery 1 Lottery 2 Assessment p 1 x 1 p 2 x 2 p 1 x 1 p 2 x 2 1 0.20 75 0.80 0 0.15 90 0.85 0 2 0.30 35 0.70 0 0.50 28 0.50 0 3 0.70 55 0.30 0 0.05 100 0.95 0 4 0.30 65 0.70 0 0.55 42 0.45 0 5 1.00 65 0.00 0 0.55 88 0.45 0 6 0.75 45 0.25 0 0.35 74 0.65 0 7 0.90 55 0.10 0 0.65 69 0.35 0 8 0.40 25 0.60 0 0.70 13 0.30 0 9 1.00 55 0.00 0 0.65 72 0.35 0 10 0.90 75 0.10 0 0.10 100 0.90 0 150 Table 6.5: Lotteries to elicit logistic utility function Lottery 1 Lottery 2 Lottery 3 Assessment p best x best p best x best p best x best 1 0.50 83 0.60 52 0.70 52 2 0.10 84 0.30 53 0.50 50 3 0.40 87 0.40 26 0.50 26 4 0.35 97 0.45 25 0.50 19 5 0.20 86 0.20 39 0.40 26 6 0.30 91 0.60 71 0.90 65 7 0.50 93 0.70 41 0.70 39 8 0.30 94 0.90 55 0.95 53 9 0.05 55 0.20 40 0.90 35 10 0.10 89 0.10 58 0.25 43 11 0.30 85 0.30 53 0.65 44 12 0.30 91 0.40 55 0.60 44 13 0.20 90 0.50 48 0.80 44 14 0.40 85 0.45 50 0.65 49 15 0.40 85 0.65 54 0.85 50 16 0.20 95 0.40 50 0.75 42 Table 6.6: Binary target infromation f(x L1 =T) f(x L2 =T) x L1 x L2 0.50 0.50 25.00 50.00 0.25 0.75 63.00 75.00 0.20 0.80 69.00 72.00 0.50 0.50 64.00 66.00 0.50 0.50 64.00 65.00 0.50 0.50 64.90 65.00 0.50 0.50 64.97 64.99 0.50 0.50 50.00 65.00 151 Table 6.7: Averse target information x L1 F(x L1 )Ø Tx L2 F(x L2 )Ø T 55 0.70 19 0.300 55 0.70 26 0.400 85 0.95 50 0.700 55 0.70 41 0.600 35 0.50 8 0.150 25 0.40 30 0.475 25 0.40 14 0.250 35 0.55 7 0.150 35 0.50 50 0.700 55 0.70 33 0.500 Table 6.8: Seeking target information x L1 F(x L1 )Ø Tx L2 F(x L2 )Ø T 75 0.60 90 0.85 35 0.20 28 0.15 55 0.35 100 1.00 65 0.50 42 0.25 65 0.50 88 0.80 45 0.30 74 0.60 55 0.35 69 0.50 25 0.15 13 0.06 55 0.35 72 0.50 75 0.60 100 1.00 152 Table 6.9: Logistic target infromation x L1 F(x L1 )Ø Tx L2 F(x L2 )Ø Tx L3 F(x L3 )Ø T 83 0.95 52 0.60 52 0.600 84 0.95 53 0.60 50 0.500 87 0.95 26 0.03 26 0.030 97 0.97 25 0.03 19 0.010 86 0.95 39 0.15 26 0.025 91 0.98 71 0.95 65 0.900 93 0.95 41 0.20 39 0.150 94 0.95 55 0.70 53 0.600 55 0.65 40 0.20 35 0.100 89 0.95 58 0.75 43 0.250 85 0.95 53 0.60 44 0.300 91 0.95 55 0.65 44 0.300 90 0.95 48 0.45 44 0.300 85 0.95 50 0.50 49 0.450 85 0.95 54 0.65 50 0.500 95 0.95 50 0.55 42 0.250 are then found by minimizing the squared dierence between observed and theoretical preference ordering (see equations (6.48) and (6.47) ). The generating function G(L i ) takes the lotteries L used in assessment i and generates an encodedpreferenceorderingundertheassumptionofanincentivescheme. Incaseofone-parametric utility functions, function (6.47) is used to generate a preference ordering under the assumption of the parameter Â to be fitted. It determines the expected utility for the parameter by assuming the subject operates under the induced utility function. G(L i )= Y __________] __________[ 1,0,0 if EU 1 >EU 2 0,1,0 if EU 1 <EU 2 0,0,1 if EU 1 =EU 2 (6.47) The parameter(s) of the induced utility functions are then found by minimizing the squared 153 classification error, which is the dierence between the vector generated by G(L i ) and the observed order x i . Grid-search is used to find the parameter Â of the assumed utility function instead of conventional optimization because the non-dierentiability of the objective. Min Â ÿ i,j ! G(L i )≠ x i " 2 (6.48) Appendix III.V - Individual responses under dierent imposed target schemes Thissectioncontainstheresponsesof12subjectsunderdierenttargets. Thefollowingnumeric values encode a choice between lotteries: • 0: Indierence between lotteries 1 and 2 • 1: Lottery 1 is preferred choice • 2: Lottery 2 is preferred choice • 3: Lottery 3 is preferred choice (only applicable under logistic target) • 4: Indierence between lotteries 1,2, and 3 (only applicable under logistic target) For example, one subject’s response under the logistic target could be 3214. This sequence implies that they prefer lottery 3 over 2 over 1 over indierence. We label an observed preference orderingasinconsistentifindierenceispreferredoversomealternatives, butnotoverothers. Such anorderingviolatestheaxiomsofrationaldecisionmaking. Anexampleofaninconsistentordering is 3241, which implies that they prefer lottery 3 over 2 over indierence over 1. 154 Table 6.10: Responses per subject to assessments 1-4 under fixed target 1 2 3 4 5 6 7 8 9 10 11 12 Assessment 1 Choice 2 2 0 0 0 2 0 0 0 0 0 0 Personal U choice 2 2 2 2 2 2 2 2 2 2 2 2 Normative choice 0 0 0 0 0 0 0 0 0 0 0 0 Personal value gap 0.00 0.00 -12.84 -12.50 -11.89 0.00 -12.64 -12.50 -12.92 -4.96 -17.60 -12.01 Normative value gap 12.65 11.11 0.00 0.00 0.00 11.24 0.00 0.00 0.00 0.00 0.00 0.00 P Norm,Choice 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 Assessment 2 Choice 1 1 1 1 1 1 1 1 1 1 1 1 Personal U choice 1 1 1 1 1 1 1 1 1 2 2 1 Normative choice 1 1 1 1 1 1 1 1 1 1 1 1 Personal value gap 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 -0.03 -10.21 0.00 Normative value gap 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 P Norm,Choice 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 Assessment 3 Choice 1 0 1 1 1 1 1 1 1 1 1 1 Personal U choice 1 1 1 1 1 1 1 1 1 1 1 1 Normative choice 1 1 1 1 1 1 1 1 1 1 1 1 Personal value gap 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 Normative value gap 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 P Norm,Choice 0.00 0.15 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 Assessment 4 Choice 1 0 1 1 1 1 1 1 1 1 1 1 Personal U choice 2 2 2 2 2 2 2 2 2 2 2 2 Normative choice 1 1 1 1 1 1 1 1 1 1 1 1 Personal value gap -1.83 -0.45 -1.68 -1.90 -0.93 -4.98 -2.95 -1.90 -2.81 -0.06 -10.40 -1.12 Normative value gap 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 P Norm,Choice 0.00 0.65 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 155 Table 6.11: Responses per subject to assessments 5-8 under fixed target 1 2 3 4 5 6 7 8 9 10 11 12 Assessment 5 Choice 2 0 2 2 1 2 2 2 2 2 2 2 Personal U choice 1 1 1 1 1 1 1 1 1 1 1 1 Normative choice 2 2 2 2 2 2 2 2 2 2 2 2 Personal value gap -2.79 0.00 -2.28 -3.10 0.00 -7.14 -4.33 -3.10 -3.92 -0.07 -10.45 -1.58 Normative value gap 0.00 0.69 0.00 0.00 1.47 0.00 0.00 0.00 0.00 0.00 0.00 0.00 P Norm,Choice 0.00 0.10 0.00 0.00 0.10 0.00 0.00 0.00 0.00 0.00 0.00 0.00 Assessment 6 Choice 2 2 1 1 1 2 1 1 1 1 1 1 Personal U choice 2 2 2 2 2 2 2 2 2 2 2 2 Normative choice 1 1 1 1 1 1 1 1 1 1 1 1 Personal value gap 0.00 0.00 -20.74 -25.91 -20.60 0.00 -28.28 -25.91 -26.69 -1.83 -25.13 -18.95 Normative value gap 24.10 16.38 0.00 0.00 0.00 32.92 0.00 0.00 0.00 0.00 0.00 0.00 P Norm,Choice 0.55 0.55 0.00 0.00 0.00 0.55 0.00 0.00 0.00 0.00 0.00 0.00 Assessment 7 Choice 2 2 0 0 1 2 0 0 0 0 0 0 Personal U choice 2 2 2 2 2 2 2 2 2 2 2 2 Normative choice 0 0 0 0 0 0 0 0 0 0 0 0 Personal value gap 0.00 0.00 -20.79 -25.98 -20.66 0.00 -28.34 -25.98 -26.74 -1.83 -25.13 -19.00 Normative value gap 24.16 16.43 0.00 0.00 0.00 32.97 0.00 0.00 0.00 0.00 0.00 0.00 P Norm,Choice 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 Assessment 8 Choice 1 1 1 1 1 1 1 1 1 1 1 1 Personal U choice 1 1 1 1 1 1 1 1 1 1 1 1 Normative choice 1 1 1 1 1 1 1 1 1 1 1 1 Personal value gap 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 Normative value gap 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 P Norm,Choice 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 156 Table 6.12: Responses per subject to validation questions under step targets 1 2 3 4 5 6 7 8 9 10 11 12 Step validation question 1 Choice 2 1 2 2 1 2 2 2 2 2 2 2 Personal U choice 1 1 1 1 1 1 1 1 1 2 2 1 Normative choice 2 2 2 2 2 2 2 2 2 2 2 2 Personal value gap -4.21 0.00 -1.98 -5.85 0.00 -8.25 -5.38 -5.85 -3.87 0.00 0.00 -1.62 Normative value gap 0.00 0.55 0.00 0.00 1.96 0.00 0.00 0.00 0.00 0.00 0.00 0.00 P Norm,Choice 0.00 0.05 0.00 0.00 0.05 0.00 0.00 0.00 0.00 0.00 0.00 0.00 Step validation question 2 Choice 1 1 2 2 2 2 2 2 2 2 2 2 Personal U choice 1 1 1 1 1 1 1 1 1 1 1 1 Normative choice 2 2 2 2 2 2 2 2 2 2 2 2 Personal value gap 0.00 0.00 -6.97 -9.15 -5.71 -16.84 -11.86 -9.15 -10.92 -0.27 -17.13 -5.38 Normative value gap 8.35 3.56 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 P Norm,Choice 0.40 0.40 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 157 Table 6.13: Responses per subject to assessments 1-4 under binary target 1 2 3 4 5 6 7 8 9 10 11 12 Assessment 1 Choice 2 2 2 2 2 2 2 2 2 2 2 2 Personal U choice 2 2 2 2 2 2 2 2 2 2 2 2 Normative choice 2 2 2 2 2 2 2 2 2 2 2 2 Personal value gap 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 Normative value gap 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 P Norm,Choice 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 Assessment 2 Choice 0 1 1 1 2 1 1 1 1 1 1 1 Personal U choice 1 1 1 1 1 1 1 1 1 2 2 1 Normative choice 1 1 1 1 1 1 1 1 1 1 1 1 Personal value gap 0.00 0.00 0.00 0.00 -1.96 0.00 0.00 0.00 0.00 -0.03 -10.21 0.00 Normative value gap 0.00 0.00 0.00 0.00 -1.96 0.00 0.00 0.00 0.00 0.00 0.00 0.00 P Norm,Choice 0.55 0.00 0.00 0.00 0.55 0.00 0.00 0.00 0.00 0.00 0.00 0.00 Assessment 3 Choice 2 2 2 2 1 1 2 2 2 2 2 1 Personal U choice 1 1 1 1 1 1 1 1 1 1 1 1 Normative choice 2 2 2 2 2 2 2 2 2 2 2 2 Personal value gap -8.35 -3.56 -6.97 -9.15 0.00 0.00 -11.86 -9.15 -10.92 -0.27 -17.13 0.00 Normative value gap 0.00 0.00 0.00 0.00 5.71 16.84 0.00 0.00 0.00 0.00 0.00 5.38 P Norm,Choice 0.00 0.00 0.00 0.00 0.29 0.29 0.00 0.00 0.00 0.00 0.00 0.29 Assessment 4 Choice 1 1 1 1 2 1 1 1 1 1 1 1 Personal U choice 2 2 2 2 2 2 2 2 2 2 2 2 Normative choice 1 1 1 1 1 1 1 1 1 1 1 1 Personal value gap -1.83 -0.45 -1.68 -1.90 0.00 -4.98 -2.95 -1.90 -2.81 -0.06 -10.40 -1.12 Normative value gap 0.00 0.00 0.00 0.00 0.93 0.00 0.00 0.00 0.00 0.00 0.00 0.00 P Norm,Choice 0.00 0.00 0.00 0.00 0.30 0.00 0.00 0.00 0.00 0.00 0.00 0.00 158 Table 6.14: Responses per subject to assessments 5-8 under binary target 1 2 3 4 5 6 7 8 9 10 11 12 Assessment 5 Choice 1 2 2 2 1 0 2 2 2 2 1 2 Personal U choice 1 1 1 1 1 1 1 1 1 1 1 1 Normative choice 2 2 2 2 2 2 2 2 2 2 2 2 Personal value gap 0.00 -0.69 -2.28 -3.10 0.00 0.00 -4.33 -3.10 -3.92 -0.07 0.00 -1.58 Normative value gap 2.79 0.00 0.00 0.00 1.47 7.14 0.00 0.00 0.00 0.00 10.45 0.00 P Norm,Choice 0.03 0.00 0.00 0.00 0.03 0.03 0.00 0.00 0.00 0.00 0.03 0.00 Assessment 6 Choice 2 2 2 1 2 1 1 1 1 1 1 1 Personal U choice 2 2 2 2 2 2 2 2 2 2 2 2 Normative choice 1 1 1 1 1 1 1 1 1 1 1 1 Personal value gap 0.00 0.00 0.00 -25.91 0.00 -32.92 -28.28 -25.91 -26.69 -1.83 -25.13 -18.95 Normative value gap 24.10 16.38 20.74 0.00 20.60 0.00 0.00 0.00 0.00 0.00 0.00 0.00 P Norm,Choice 0.08 0.08 0.08 0.00 0.08 0.00 0.00 0.00 0.00 0.00 0.00 0.00 Assessment 7 Choice 2 2 2 1 2 1 1 1 1 1 1 1 Personal U choice 2 2 2 2 2 2 2 2 2 2 2 2 Normative choice 1 1 1 1 1 1 1 1 1 1 1 1 Personal value gap 0.00 0.00 0.00 -25.98 0.00 -32.97 -28.34 -25.98 -26.74 -1.83 -25.13 -19.00 Normative value gap 24.16 16.43 20.79 0.00 20.66 0.00 0.00 0.00 0.00 0.00 0.00 0.00 P Norm,Choice 0.10 0.10 0.10 0.00 0.10 0.00 0.00 0.00 0.00 0.00 0.00 0.00 Assessment 8 Choice 1 1 1 1 1 0 2 1 1 1 1 1 Personal U choice 1 1 1 1 1 1 1 1 1 1 1 1 Normative choice 1 1 1 1 1 1 1 1 1 1 1 1 Personal value gap 0.00 0.00 0.00 0.00 0.00 0.00 -8.57 0.00 0.00 0.00 0.00 0.00 Normative value gap 0.00 0.00 0.00 0.00 0.00 0.00 -8.57 0.00 0.00 0.00 0.00 0.00 P Norm,Choice 0.00 0.00 0.00 0.00 0.00 0.10 0.10 0.00 0.00 0.00 0.00 0.00 159 Table 6.15: Responses per subject to assessments 1-4 under averse target 1 2 3 4 5 6 7 8 9 10 11 12 Assessment 1 Choice 1 1 1 2 2 2 2 2 2 1 2 2 Personal U choice 2 2 2 2 2 1 1 2 1 2 1 2 Normative choice 2 2 2 2 2 2 2 2 2 2 2 2 Personal value gap -1.05 -1.62 -2.05 0.00 0.00 -9.21 -2.09 0.00 -1.02 -0.81 -13.19 0.00 Normative value gap -1.05 -1.62 -2.05 0.00 0.00 0.00 0.00 0.00 0.00 -0.81 0.00 0.00 P Norm,Choice 0.04 0.04 0.04 0.00 0.00 0.00 0.00 0.00 0.00 0.04 0.00 0.00 Assessment 2 Choice 1 1 1 1 1 1 1 1 1 1 1 1 Personal U choice 1 1 1 1 1 1 1 1 1 2 1 1 Normative choice 1 1 1 1 1 1 1 1 1 1 1 1 Personal value gap 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 -0.01 0.00 0.00 Normative value gap 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 P Norm,Choice 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 Assessment 3 Choice 2 1 1 1 1 1 1 1 1 1 1 1 Personal U choice 1 1 1 1 1 1 1 1 1 2 2 1 Normative choice 1 1 1 1 1 1 1 1 1 1 1 1 Personal value gap -10.20 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 -0.22 -18.27 0.00 Normative value gap -10.20 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 P Norm,Choice 0.03 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 Assessment 4 Choice 1 1 2 2 2 1 2 2 2 2 2 2 Personal U choice 1 2 1 1 1 1 1 1 1 2 2 2 Normative choice 2 2 2 2 2 2 2 2 2 2 2 2 Personal value gap 0.00 -0.06 -0.01 -1.50 -0.07 0.00 -2.92 -1.50 -1.99 0.00 0.00 0.00 Normative value gap 0.89 -0.06 0.00 0.00 0.00 8.09 0.00 0.00 0.00 0.00 0.00 0.00 P Norm,Choice 0.02 0.02 0.00 0.00 0.00 0.02 0.00 0.00 0.00 0.00 0.00 0.00 160 Table 6.16: Responses per subject to assessments 5-8 under averse target 1 2 3 4 5 6 7 8 9 10 11 12 Assessment 5 Choice 1 1 2 1 2 2 1 1 1 1 1 1 Personal U choice 1 2 1 1 1 1 1 1 1 2 1 2 Normative choice 1 1 1 1 1 1 1 1 1 1 1 1 Personal value gap 0.00 -0.31 -0.84 0.00 -0.00 -7.66 0.00 0.00 0.00 -0.60 0.00 -0.10 Normative value gap 0.00 0.00 -0.84 0.00 -0.00 -7.66 0.00 0.00 0.00 0.00 0.00 0.00 P Norm,Choice 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 Assessment 6 Choice 2 2 1 2 2 1 2 2 2 2 2 2 Personal U choice 2 1 2 2 2 2 2 2 2 1 2 2 Normative choice 2 2 2 2 2 2 2 2 2 2 2 2 Personal value gap 0.00 -0.02 -0.40 0.00 0.00 -5.29 0.00 0.00 0.00 -0.09 0.00 0.00 Normative value gap 0.00 0.00 -0.40 0.00 0.00 -5.29 0.00 0.00 0.00 0.00 0.00 0.00 P Norm,Choice 0.00 0.00 0.01 0.00 0.00 0.01 0.00 0.00 0.00 0.00 0.00 0.00 Assessment 7 Choice 1 1 1 2 2 2 2 2 1 2 2 2 Personal U choice 1 2 2 1 2 1 1 1 1 2 1 2 Normative choice 2 2 2 2 2 2 2 2 2 2 2 2 Personal value gap 0.00 -0.35 -0.13 -0.35 0.00 -3.81 -1.51 -0.35 0.00 0.00 -13.31 0.00 Normative value gap 0.17 -0.35 -0.13 0.00 0.00 0.00 0.00 0.00 1.27 0.00 0.00 0.00 P Norm,Choice 0.01 0.01 0.01 0.00 0.00 0.00 0.00 0.00 0.01 0.00 0.00 0.00 Assessment 8 Choice 1 1 2 1 2 2 2 2 2 2 2 2 Personal U choice 1 2 1 1 2 1 1 1 1 2 1 2 Normative choice 2 2 2 2 2 2 2 2 2 2 2 2 Personal value gap 0.00 -0.74 -0.27 0.00 0.00 -7.01 -3.23 -1.19 -2.79 0.00 -16.00 0.00 Normative value gap 0.85 -0.74 0.00 1.19 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 P Norm,Choice 0.01 0.01 0.00 0.01 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 161 Table 6.17: Responses per subject to assessments 9-10 under averse target 1 2 3 4 5 6 7 8 9 10 11 12 Assessment 9 Choice 2 2 1 2 1 2 2 2 2 2 2 2 Personal U choice 2 1 2 2 2 2 2 2 2 1 1 2 Normative choice 2 2 2 2 2 2 2 2 2 2 2 2 Personal value gap 0.00 -0.05 -0.79 0.00 -0.46 0.00 0.00 0.00 0.00 -0.24 -3.17 0.00 Normative value gap 0.00 0.00 -0.79 0.00 -0.46 0.00 0.00 0.00 0.00 0.00 0.00 0.00 P Norm,Choice 0.00 0.00 0.04 0.00 0.04 0.00 0.00 0.00 0.00 0.00 0.00 0.00 Assessment 10 Choice 1 1 2 1 2 1 1 1 1 1 1 1 Personal U choice 1 1 1 1 1 1 1 1 1 2 1 1 Normative choice 1 1 1 1 1 1 1 1 1 1 1 1 Personal value gap 0.00 0.00 -2.70 0.00 -1.91 0.00 0.00 0.00 0.00 -0.13 0.00 0.00 Normative value gap 0.00 0.00 -2.70 0.00 -1.91 0.00 0.00 0.00 0.00 0.00 0.00 0.00 P Norm,Choice 0.00 0.00 0.03 0.00 0.03 0.00 0.00 0.00 0.00 0.00 0.00 0.00 162 Table 6.18: Responses per subject to assessments 1-4 under seeking target 1 2 3 4 5 6 7 8 9 10 11 12 Assessment 1 Choice 0 0 1 2 1 0 2 2 2 2 2 2 Personal U choice 1 1 1 1 1 1 1 1 1 1 1 1 Normative choice 2 2 2 2 2 2 2 2 2 2 2 2 Personal value gap 0.00 0.00 0.00 -1.50 0.00 0.00 -3.34 -1.50 -3.16 -0.06 -10.46 -1.03 Normative value gap 1.62 0.40 1.66 0.00 0.80 6.26 0.00 0.00 0.00 0.00 0.00 0.00 P Norm,Choice 0.01 0.01 0.01 0.00 0.01 0.01 0.00 0.00 0.00 0.00 0.00 0.00 Assessment 2 Choice 2 0 1 2 2 2 2 2 2 2 2 2 Personal U choice 2 2 2 2 2 2 2 2 2 2 2 2 Normative choice 2 2 2 2 2 2 2 2 2 2 2 2 Personal value gap 0.00 -1.66 -3.40 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 Normative value gap 0.00 -1.66 -3.40 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 P Norm,Choice 0.00 0.01 0.01 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 Assessment 3 Choice 1 1 1 1 1 1 1 1 1 1 1 1 Personal U choice 1 1 1 1 1 1 1 1 1 1 1 1 Normative choice 1 1 1 1 1 1 1 1 1 1 1 1 Personal value gap 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 Normative value gap 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 P Norm,Choice 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 Assessment 4 Choice 2 1 1 1 2 1 1 1 1 1 1 1 Personal U choice 2 2 2 2 2 2 2 2 2 2 2 2 Normative choice 1 1 1 1 1 1 1 1 1 1 1 1 Personal value gap 0.00 -2.41 -4.68 -3.60 0.00 -3.31 -4.59 -3.60 -5.23 -0.47 -19.24 -3.53 Normative value gap 4.09 0.00 0.00 0.00 3.11 0.00 0.00 0.00 0.00 0.00 0.00 0.00 P Norm,Choice 0.01 0.00 0.00 0.00 0.01 0.00 0.00 0.00 0.00 0.00 0.00 0.00 163 Table 6.19: Responses per subject to assessments 5-8 under averse target 1 2 3 4 5 6 7 8 9 10 11 12 Assessment 5 Choice 1 1 1 1 2 1 1 1 1 1 1 1 Personal U choice 1 1 1 1 1 1 1 1 1 1 1 1 Normative choice 1 1 1 1 1 1 1 1 1 1 1 1 Personal value gap 0.00 0.00 0.00 0.00 -14.38 0.00 0.00 0.00 0.00 0.00 0.00 0.00 Normative value gap 0.00 0.00 0.00 0.00 -14.38 0.00 0.00 0.00 0.00 0.00 0.00 0.00 P Norm,Choice 0.00 0.00 0.00 0.00 0.06 0.00 0.00 0.00 0.00 0.00 0.00 0.00 Assessment 6 Choice 2 1 1 1 1 1 1 1 1 1 1 1 Personal U choice 1 1 1 1 1 1 1 1 1 1 1 1 Normative choice 1 1 1 1 1 1 1 1 1 1 1 1 Personal value gap -8.56 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 Normative value gap -8.56 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 P Norm,Choice 0.01 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 Assessment 7 Choice 1 1 1 2 1 1 2 2 2 1 2 2 Personal U choice 1 1 1 1 1 1 1 1 1 1 1 1 Normative choice 2 2 2 2 2 2 2 2 2 2 2 2 Personal value gap 0.00 0.00 0.00 -4.65 0.00 0.00 -7.55 -4.65 -8.01 0.00 -20.60 -4.31 Normative value gap 5.25 3.00 5.90 0.00 3.91 10.17 0.00 0.00 0.00 0.51 0.00 0.00 P Norm,Choice 0.01 0.01 0.01 0.00 0.01 0.01 0.00 0.00 0.00 0.01 0.00 0.00 Assessment 8 Choice 1 1 2 1 1 2 2 1 1 1 1 1 Personal U choice 1 2 1 1 2 1 1 1 1 2 1 2 Normative choice 1 1 1 1 1 1 1 1 1 1 1 1 Personal value gap 0.00 -0.30 -0.27 0.00 -0.13 -5.12 -2.39 0.00 0.00 -0.34 0.00 -0.32 Normative value gap 0.00 0.00 -0.27 0.00 0.00 -5.12 -2.39 0.00 0.00 0.00 0.00 0.00 P Norm,Choice 0.00 0.00 0.02 0.00 0.00 0.02 0.02 0.00 0.00 0.00 0.00 0.00 164 Table 6.20: Responses per subject to assessments 9-10 under averse target 1 2 3 4 5 6 7 8 9 10 11 12 Assessment 9 Choice 1 1 1 1 2 1 1 1 1 1 1 1 Personal U choice 1 1 1 1 1 1 1 1 1 1 1 1 Normative choice 1 1 1 1 1 1 1 1 1 1 1 1 Personal value gap 0.00 0.00 0.00 0.00 -7.07 0.00 0.00 0.00 0.00 0.00 0.00 0.00 Normative value gap 0.00 0.00 0.00 0.00 -7.07 0.00 0.00 0.00 0.00 0.00 0.00 0.00 P Norm,Choice 0.00 0.00 0.00 0.00 0.03 0.00 0.00 0.00 0.00 0.00 0.00 0.00 Assessment 10 Choice 1 1 1 1 1 1 1 1 1 1 1 1 Personal U choice 1 1 1 1 1 1 1 1 1 1 1 1 Normative choice 1 1 1 1 1 1 1 1 1 1 1 1 Personal value gap 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 Normative value gap 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 P Norm,Choice 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 165 Table 6.21: Responses per subject to assessments 1-4 under logistic target 1 2 3 4 5 6 7 8 9 10 11 12 Assessment 1 Choice 1234 3124 1234 1324 3124 1324 1324 1324 1324 1324 1324 1324 Personal U choice 1324 3124 3124 1324 1324 1324 1324 1324 1324 3214 3214 1324 Normative choice 1324 1324 1324 1324 1324 1324 1324 1324 1324 1324 1324 1324 Personal value gap 0.00 0.00 -0.49 0.00 -0.56 0.00 0.00 0.00 0.00 -0.28 -18.68 0.00 Normative value gap 0.00 0.12 0.00 0.00 -0.56 0.00 0.00 0.00 0.00 0.00 0.00 0.00 P Norm,Choice 0.00 0.06 0.00 0.00 0.06 0.00 0.00 0.00 0.00 0.00 0.00 0.00 Assessment 2 Choice 3214 3241 1324 3214 1324 3214 3214 3214 3214 3214 3214 3214 Personal U choice 3214 3214 3214 3214 3214 3214 3214 3214 3214 3214 3214 3214 Normative choice 3214 3214 3214 3214 3214 3214 3214 3214 3214 3214 3214 3214 Personal value gap 0.00 0.00 -14.91 0.00 -13.39 0.00 0.00 0.00 0.00 0.00 0.00 0.00 Normative value gap 0.00 0.00 -14.91 0.00 -13.39 0.00 0.00 0.00 0.00 0.00 0.00 0.00 P Norm,Choice 0.00 0.00 0.15 0.00 0.15 0.00 0.00 0.00 0.00 0.00 0.00 0.00 Assessment 3 Choice 1324 1243 1324 1324 3214 1324 1324 1324 1324 1324 1324 1324 Personal U choice 1324 1324 1324 1324 1324 1324 1324 1324 1324 3124 1324 1324 Normative choice 1324 1324 1324 1324 1324 1324 1324 1324 1324 1324 1324 1324 Personal value gap 0.00 0.00 0.00 0.00 -12.35 0.00 0.00 0.00 0.00 -0.00 0.00 0.00 Normative value gap 0.00 0.00 0.00 0.00 -12.35 0.00 0.00 0.00 0.00 0.00 0.00 0.00 P Norm,Choice 0.00 0.00 0.00 0.00 0.36 0.00 0.00 0.00 0.00 0.00 0.00 0.00 Assessment 4 Choice 1234 1234 2134 1234 2134 1234 1234 1234 1234 1234 1234 1234 Personal U choice 1234 1234 1234 1234 1234 1234 1234 1234 1234 3214 1234 1234 Normative choice 1234 1234 1234 1234 1234 1234 1234 1234 1234 1234 1234 1234 Personal value gap 0.00 0.00 -11.81 0.00 -12.80 0.00 0.00 0.00 0.00 -0.02 0.00 0.00 Normative value gap 0.00 0.00 -11.81 0.00 -12.80 0.00 0.00 0.00 0.00 0.00 0.00 0.00 P Norm,Choice 0.00 0.00 0.33 0.00 0.33 0.00 0.00 0.00 0.00 0.00 0.00 0.00 166 Table 6.22: Responses per subject to assessments 5-8 under logistic target 1 2 3 4 5 6 7 8 9 10 11 12 Assessment 5 Choice 1234 3124 2134 1234 1234 3214 1234 1234 1234 1234 1234 1234 Personal U choice 1324 1324 1324 1324 1324 1324 1324 1324 1324 3124 1324 1324 Normative choice 1234 1234 1234 1234 1234 1234 1234 1234 1234 1234 1234 1234 Personal value gap 0.00 -0.24 -4.93 0.00 0.00 -15.46 0.00 0.00 0.00 -0.22 0.00 0.00 Normative value gap 0.00 -0.24 -4.93 0.00 0.00 -15.46 0.00 0.00 0.00 0.00 0.00 0.00 P Norm,Choice 0.00 0.18 0.16 0.00 0.00 0.18 0.00 0.00 0.00 0.00 0.00 0.00 Assessment 6 Choice 3214 3214 2134 3214 3214 3214 3214 3214 3214 3214 3214 3214 Personal U choice 3214 3214 3214 3214 3214 3214 3214 3214 3214 3214 3214 3214 Normative choice 3214 3214 3214 3214 3214 3214 3214 3214 3214 3214 3214 3214 Personal value gap 0.00 0.00 -13.14 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 Normative value gap 0.00 0.00 -13.14 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 P Norm,Choice 0.00 0.00 0.24 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 Assessment 7 Choice 2134 1324 2134 1234 3214 2314 1234 1234 1234 1234 1234 1234 Personal U choice 1234 1234 1234 1234 1234 1234 1234 1234 1234 2314 2314 1234 Normative choice 1234 1234 1234 1234 1234 1234 1234 1234 1234 1234 1234 1234 Personal value gap -13.07 0.00 -6.26 0.00 -8.73 -22.32 0.00 0.00 0.00 -0.19 -16.02 0.00 Normative value gap -13.07 0.00 -6.26 0.00 -8.73 -22.32 0.00 0.00 0.00 0.00 0.00 0.00 P Norm,Choice 0.34 0.00 0.34 0.00 0.37 0.34 0.00 0.00 0.00 0.00 0.00 0.00 Assessment 8 Choice 2314 3214 1324 2314 1324 2314 2314 2314 2314 2314 2314 2314 Personal U choice 3214 3214 3214 3214 3214 3214 3214 3214 3214 3214 3214 3214 Normative choice 2314 2314 2314 2314 2314 2314 2314 2314 2314 2314 2314 2314 Personal value gap -0.93 0.00 -23.33 -0.85 -20.93 -1.20 -1.19 -0.85 -1.31 -0.06 -9.92 -0.68 Normative value gap 0.00 0.24 -22.03 0.00 -19.68 0.00 0.00 0.00 0.00 0.00 0.00 0.00 P Norm,Choice 0.00 0.06 0.34 0.00 0.34 0.00 0.00 0.00 0.00 0.00 0.00 0.00 167 Table 6.23: Responses per subject to assessments 9-12 under logistic target 1 2 3 4 5 6 7 8 9 10 11 12 Assessment 9 Choice 3214 3421 2134 3214 3214 3214 3214 3214 3214 3214 3214 3214 Personal U choice 3214 3214 3214 3214 3214 3214 3214 3214 3214 3214 3214 3214 Normative choice 3214 3214 3214 3214 3214 3214 3214 3214 3214 3214 3214 3214 Personal value gap 0.00 0.00 -22.18 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 Normative value gap 0.00 0.00 -22.18 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 P Norm,Choice 0.00 0.00 0.05 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 Assessment 10 Choice 3124 3124 1234 1234 3124 3124 1234 1324 1234 1234 1234 1234 Personal U choice 3124 3124 3124 3124 3124 3124 3124 3124 3124 3124 3124 3124 Normative choice 1234 1234 1234 1234 1234 1234 1234 1234 1234 1234 1234 1234 Personal value gap 0.00 0.00 -2.93 -1.85 0.00 0.00 -3.63 -1.85 -4.01 -0.22 -16.21 -1.99 Normative value gap 2.36 1.11 0.00 0.00 1.58 4.78 0.00 0.00 0.00 0.00 0.00 0.00 P Norm,Choice 0.03 0.03 0.00 0.00 0.03 0.03 0.00 0.00 0.00 0.00 0.00 0.00 Assessment 11 Choice 3124 1324 1234 1324 2134 2314 1324 1324 1324 1324 1324 1324 Personal U choice 3124 3124 3124 3124 3124 3124 3124 3124 3124 3124 3124 3124 Normative choice 1324 1324 1324 1324 1324 1324 1324 1324 1324 1324 1324 1324 Personal value gap 0.00 -3.49 -6.31 -3.10 -9.84 -18.44 -6.53 -3.10 -7.68 -0.89 -22.63 -4.59 Normative value gap 4.54 0.00 0.00 0.00 -4.45 -14.23 0.00 0.00 0.00 0.00 0.00 0.00 P Norm,Choice 0.09 0.00 0.00 0.00 0.10 0.10 0.00 0.00 0.00 0.00 0.00 0.00 Assessment 12 Choice 1324 1324 1324 1234 3214 2134 1234 1234 1234 1234 2134 1234 Personal U choice 3124 3124 3124 1324 3124 3124 3124 1324 3124 3214 3214 3124 Normative choice 1234 1234 1234 1234 1234 1234 1234 1234 1234 1234 1234 1234 Personal value gap -1.17 -1.67 -3.74 0.00 0.00 -5.84 -2.54 0.00 -4.24 -0.62 -17.79 -2.36 Normative value gap 0.00 0.00 0.00 0.00 1.40 -4.44 0.00 0.00 0.00 0.00 14.18 0.00 P Norm,Choice 0.00 0.00 0.00 0.00 0.10 0.03 0.00 0.00 0.00 0.00 0.03 0.00 168 Table 6.24: Responses per subject to assessments 13-16 under logistic target 1 2 3 4 5 6 7 8 9 10 11 12 Assessment 13 Choice 3214 3214 3124 3214 1324 2314 3214 3214 3214 3214 1234 3214 Personal U choice 3214 3214 3214 3214 3214 3214 3214 3214 3214 3214 3214 3214 Normative choice 3214 3214 3214 3214 3214 3214 3214 3214 3214 3214 3214 3214 Personal value gap 0.00 0.00 0.00 0.00 -16.01 -16.98 0.00 0.00 0.00 0.00 -28.79 0.00 Normative value gap 0.00 0.00 0.00 0.00 -16.01 -16.98 0.00 0.00 0.00 0.00 -28.79 0.00 P Norm,Choice 0.00 0.00 0.00 0.00 0.05 0.01 0.00 0.00 0.00 0.00 0.05 0.00 Assessment 14 Choice 3124 1324 2134 1324 2314 2314 1324 1324 1324 1324 3214 1324 Personal U choice 1324 3124 3124 1324 3124 1324 3124 1324 3124 3214 3214 3124 Normative choice 1324 1324 1324 1324 1324 1324 1324 1324 1324 1324 1324 1324 Personal value gap -0.06 -0.96 -7.63 0.00 -6.25 -16.34 -0.81 0.00 -2.68 -0.44 0.00 -1.44 Normative value gap -0.06 0.00 -4.93 0.00 -5.12 -16.34 0.00 0.00 0.00 0.00 20.26 0.00 P Norm,Choice 0.09 0.00 0.15 0.00 0.15 0.15 0.00 0.00 0.00 0.00 0.09 0.00 Assessment 15 Choice 3214 3214 2314 2314 3214 2134 3214 3124 3214 3214 1234 3214 Personal U choice 3214 3214 3214 3214 3214 3214 3214 3214 3214 3214 3214 3214 Normative choice 3214 3214 3214 3214 3214 3214 3214 3214 3214 3214 3214 3214 Personal value gap 0.00 0.00 -6.45 -7.40 0.00 -12.81 0.00 0.00 0.00 0.00 -25.84 0.00 Normative value gap 0.00 0.00 -6.45 -7.40 0.00 -12.81 0.00 0.00 0.00 0.00 -25.84 0.00 P Norm,Choice 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.04 0.00 Assessment 16 Choice 3214 3214 1324 2134 2134 2314 2134 2134 2134 2134 3214 2134 Personal U choice 3214 3214 3214 3214 3214 3214 3214 3214 3214 3214 3214 3214 Normative choice 2134 2134 2134 2134 2134 2134 2134 2134 2134 2134 2134 2134 Personal value gap 0.00 0.00 -14.54 -11.50 -8.94 -16.33 -13.67 -11.50 -13.35 -1.02 0.00 -8.70 Normative value gap 11.15 6.93 -3.66 0.00 0.00 0.00 0.00 0.00 0.00 0.00 22.25 0.00 P Norm,Choice 0.03 0.03 0.03 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.03 0.00 169 Appendix III.VI - Visualizations used during the behavioral experiment Figure 6.5: Screenshot: Introductory video 170 Figure 6.6: Screenshot: Age Demographics Figure 6.7: Screenshot: Gender Demographics 171 Figure 6.8: Screenshot: Work Experience Demographics Figure 6.9: Screenshot: Education Demographics 172 Figure 6.10: Screenshot: Decision Theory Demographics Figure 6.11: Screenshot: Probability Theory Demographics 173 Figure 6.12: Screenshot: Big Five Character Traits Question 1 Figure 6.13: Screenshot: Big Five Character Traits Question 2 174 Figure 6.14: Screenshot: Big Five Character Traits Question 3 Figure 6.15: Screenshot: Big Five Character Traits Question 4 175 Figure 6.16: Screenshot: Big Five Character Traits Question 5 Figure 6.17: Screenshot: Big Five Character Traits Question 6 176 Figure 6.18: Screenshot: Big Five Character Traits Question 7 Figure 6.19: Screenshot: Big Five Character Traits Question 8 177 Figure 6.20: Screenshot: Big Five Character Traits Question 9 Figure 6.21: Screenshot: Big Five Character Traits Question 10 178 Figure 6.22: Screenshot: Indierence Probability Method Explanatory Video Screen 1 179 Figure 6.23: Screenshot: Indierence Probability Method Explanatory Video Screen 2 180 Figure 6.24: Screenshot: Indierence Probability Method Explanatory Video Screen 3 181 Figure 6.25: Screenshot: Indierence Probability Method Assessment 1 Figure 6.26: Screenshot: Indierence Probability Method Assessment 1 Validation 182 Figure 6.27: Screenshot: Indierence Probability Method Assessment 2 Figure 6.28: Screenshot: Indierence Probability Method Assessment 2 Validation 183 Figure 6.29: Screenshot: Indierence Probability Method Assessment 3 Figure 6.30: Screenshot: Indierence Probability Method Assessment 3 Validation 184 Figure 6.31: Screenshot: Indierence Probability Method Assessment 4 Figure 6.32: Screenshot: Indierence Probability Method Assessment 4 Validation 185 Figure 6.33: Screenshot: Indierence Probability Method Assessment 5 Figure 6.34: Screenshot: Indierence Probability Method Assessment 5 Validation 186 Figure 6.35: Screenshot: Certain Equivalence Method Explanatory Video Screen 1 Figure 6.36: Screenshot: Certain Equivalence Method Explanatory Video Screen 2 187 Figure 6.37: Screenshot: Certain Equivalence Method Explanatory Video Screen 3 Figure 6.38: Screenshot: Certain Equivalence Method Explanatory Video Screen 4 188 Figure 6.39: Screenshot: Certain Equivalence Method Assessment 1 Figure 6.40: Screenshot: Certain Equivalence Method Assessment 1 Validation 189 Figure 6.41: Screenshot: Certain Equivalence Method Assessment 2 Figure 6.42: Screenshot: Certain Equivalence Method Assessment 2 Validation 190 Figure 6.43: Screenshot: Certain Equivalence Method Assessment 3 Figure 6.44: Screenshot: Certain Equivalence Method Assessment 3 Validation 191 Figure 6.45: Screenshot: Certain Equivalence Method Assessment 4 Figure 6.46: Screenshot: Certain Equivalence Method Assessment 4 Validation 192 Figure 6.47: Screenshot: Certain Equivalence Method Assessment 5 Figure 6.48: Screenshot: Certain Equivalence Method Assessment 5 Validation 193 Figure 6.49: Screenshot: Risk Tolerance Assessment Introductory Video Figure 6.50: Screenshot: Risk Tolerance Assessment Introductory Video Validation 194 Figure 6.51: Screenshot: Risk Tolerance Assessment 1 Figure 6.52: Screenshot: Risk Tolerance Assessment 1 Validation 195 Figure 6.53: Screenshot: Risk Tolerance Assessment 2 Figure 6.54: Screenshot: Risk Tolerance Assessment 2 Validation 196 Figure 6.55: Screenshot: Introduction to Decision Making under Targets Section 197 Figure 6.56: Screenshot: Introduction to Decision Making under Targets Section Validation 198 Figure 6.57: Screenshot: Introduction to decision making under fixed targets 199 Figure 6.58: Screenshot: Introduction to decision making under fixed targets validation 200 Figure 6.59: Screenshot: Decision making under fixed targets assessment 1 201 Figure 6.60: Screenshot: Decision making under fixed targets assessment 2 202 Figure 6.61: Screenshot: Decision making under fixed targets assessment 3 203 Figure 6.62: Screenshot: Decision making under fixed targets assessment 4 204 Figure 6.63: Screenshot: Decision making under fixed targets assessment 5 205 Figure 6.64: Screenshot: Decision making under fixed targets assessment 6 206 Figure 6.65: Screenshot: Decision making under fixed targets assessment 7 207 Figure 6.66: Screenshot: Decision making under fixed targets assessment 8 208 Figure 6.67: Screenshot: Decision making under fixed targets validation choice 1 209 Figure 6.68: Screenshot: Decision making under fixed targets validation choice 2 210 Figure 6.69: Screenshot: Introduction to decision making under targets with two outcomes Figure 6.70: Screenshot: Introduction to decision making under targets with two outcomes valida- tion 211 Figure 6.71: Screenshot: Decision making under targets with two outcomes assessment 1 212 Figure 6.72: Screenshot: Decision making under targets with two outcomes assessment 2 213 Figure 6.73: Screenshot: Decision making under targets with two outcomes assessment 3 214 Figure 6.74: Screenshot: Decision making under targets with two outcomes assessment 4 215 Figure 6.75: Screenshot: Decision making under targets with two outcomes assessment 5 216 Figure 6.76: Screenshot: Decision making under targets with two outcomes assessment 6 217 Figure 6.77: Screenshot: Decision making under targets with two outcomes assessment 7 218 Figure 6.78: Screenshot: Decision making under targets with two outcomes assessment 8 219 Figure 6.79: Screenshot: Introduction to decision making under targets with three outcomes 220 Figure 6.80: Screenshot: Introduction to decision making under targets with three outcomes vali- dation 221 Figure 6.81: Screenshot: Decision making under targets with three outcomes assessment 1 - averse 222 Figure 6.82: Screenshot: Decision making under targets with three outcomes assessment 2 - averse 223 Figure 6.83: Screenshot: Decision making under targets with three outcomes assessment 3 - averse 224 Figure 6.84: Screenshot: Decision making under targets with three outcomes assessment 4 - averse 225 Figure 6.85: Screenshot: Decision making under targets with three outcomes assessment 5 - averse 226 Figure 6.86: Screenshot: Decision making under targets with three outcomes assessment 6 - averse 227 Figure 6.87: Screenshot: Decision making under targets with three outcomes assessment 7 - averse 228 Figure 6.88: Screenshot: Decision making under targets with three outcomes assessment 8 - averse 229 Figure 6.89: Screenshot: Decision making under targets with three outcomes assessment 9 - averse 230 Figure6.90: Screenshot: Decisionmakingundertargetswiththreeoutcomesassessment10-averse 231 Figure6.91: Screenshot: Decisionmakingundertargetswiththreeoutcomesassessment1-seeking 232 Figure6.92: Screenshot: Decisionmakingundertargetswiththreeoutcomesassessment2-seeking 233 Figure6.93: Screenshot: Decisionmakingundertargetswiththreeoutcomesassessment3-seeking 234 Figure6.94: Screenshot: Decisionmakingundertargetswiththreeoutcomesassessment4-seeking 235 Figure6.95: Screenshot: Decisionmakingundertargetswiththreeoutcomesassessment5-seeking 236 Figure6.96: Screenshot: Decisionmakingundertargetswiththreeoutcomesassessment6-seeking 237 Figure6.97: Screenshot: Decisionmakingundertargetswiththreeoutcomesassessment7-seeking 238 Figure6.98: Screenshot: Decisionmakingundertargetswiththreeoutcomesassessment8-seeking 239 Figure6.99: Screenshot: Decisionmakingundertargetswiththreeoutcomesassessment9-seeking 240 Figure 6.100: Screenshot: Decision making under targets with three outcomes assessment 10 - seeking 241 Figure 6.101: Screenshot: Introduction to decision making under targets with four outcomes 242 Figure 6.102: Screenshot: Introduction to decision making under targets with four outcomes vali- dation 243 Figure 6.103: Screenshot: Decision making under targets with four outcomes assessment 1 244 Figure 6.104: Screenshot: Decision making under targets with four outcomes assessment 2 245 Figure 6.105: Screenshot: Decision making under targets with four outcomes assessment 3 246 Figure 6.106: Screenshot: Decision making under targets with four outcomes assessment 4 247 Figure 6.107: Screenshot: Decision making under targets with four outcomes assessment 5 248 Figure 6.108: Screenshot: Decision making under targets with four outcomes assessment 6 249 Figure 6.109: Screenshot: Decision making under targets with four outcomes assessment 7 250 Figure 6.110: Screenshot: Decision making under targets with four outcomes assessment 8 251 Figure 6.111: Screenshot: Decision making under targets with four outcomes assessment 9 252 Figure 6.112: Screenshot: Decision making under targets with four outcomes assessment 10 253 Figure 6.113: Screenshot: Decision making under targets with four outcomes assessment 11 254 Figure 6.114: Screenshot: Decision making under targets with four outcomes assessment 12 255 Figure 6.115: Screenshot: Decision making under targets with four outcomes assessment 13 256 Figure 6.116: Screenshot: Decision making under targets with four outcomes assessment 14 257 Figure 6.117: Screenshot: Decision making under targets with four outcomes assessment 15 258 Figure 6.118: Screenshot: Decision making under targets with four outcomes assessment 16 259 Figure 6.119: Screenshot: Elicitation of email address 260 Figure 6.120: Screenshot: Elicitation of survey feedback 261

Abstract (if available)

Abstract Utility functions encode a decision-maker's preference under uncertainty. This research investigates human decision-maker's utility functions under certain and uncertain incentive schemes. Previous decision analysis research has focused primarily on deriving the form of a fully rational decision-maker's utility function under a particular incentive or target. This work focuses on three distinct areas of utility research with human subjects: the consistency of human decision-makers' assessed utilities under common elicitation methods without incentives, the applicability of preference methods, and humans' adherence to normative probability maximizing behavior and induced utility theory under incentives.
When comparing the most common utility elicitation methods, certain equivalence and indifference probability, researchers have found that a subject's utility function is influenced by the chosen method. The behavioral experiment conducted for this work confirms this observation. I find that elicited function parameters depend on the specific method. However, I do not find evidence for other reseachers' claim that the certain equivalence method results in more risk-seeking behavior. In addition, this work investigates the consistency of subjects' responses to approximation methods with ones elicited by the two established methods. Studies reviewed for this dissertation omitted this comparison. This work finds that function parameters derived by approximation methods overestimate parameters elicited via established methods in most cases.
An alternative method is the elicitation of utility via preferences between uncertain gambles. This approach has been shown to be more consistent and more predictive of actual choice behavior since it was found to be less susceptible to bias. Because preference methods rely on a large number of observed choices, the second research area of this work is the derivation of algorithms to overcome this disadvantage. The algorithms converge in less than twenty assessment rounds to a rational decision-maker's utility function. Additionally, they provide lotteries to test adherence to induced utility functions under incentives.
Under incentives a rational decision-maker chooses the lottery that maximizes their likelihood of meeting or exceeding a target. The target's probability distribution was shown to be the utility function induced into the probability maximizing decision-maker. By using the theoretically induced utility as a function to be elicited, the presented algorithms generate a series of lotteries. The work pertaining to the third research area uses these lotteries and the utility functions to generate them in a behavioral experiment. Subjects were asked to choose or order lotteries under certain and uncertain targets, where the target distributions were the utility functions to derive the lotteries. I find that in most cases humans prefer the lottery that maximizes their probability of achieving or exceeding their target. Because of probability maximizing behavior and the specific design of the presented lotteries, most subjects exhibited an induced utility function consistent with theoretical results. An analysis of third factors influencing probability maximizing behavior under incentives found no significant impact of age, gender, target generating function, or whether the target was certain or uncertain. Time spent on the survey was the most significant influence on consistency with normative behavior. Subjects who spent less time on the survey had a lower consistency with probability maximization under targets, which decreased significantly once the task was switched from choosing to ordering lotteries. The task had no significant influence on choice behavior for subjects who spent more time on answering the experiment's questions.

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Asset Metadata

Creator Zellner, Maximilian (author)

Core Title Utility functions induced by certain and uncertain incentive schemes

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Industrial and Systems Engineering

Degree Conferral Date 2022-05

Publication Date 04/25/2022

Defense Date 04/08/2022

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag decision analysis,decisions under incentives,incentives,OAI-PMH Harvest,utility,utility functions

Format application/pdf (imt)

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Abbas, Ali (committee chair), Suen, Sze-Chuan (committee member), Wu, Shinyi (committee member)

Creator Email mzellner@usc.edu,zellnermaximilian@gmail.com

Permanent Link (DOI) https://doi.org/10.25549/usctheses-oUC111117823

Unique identifier UC111117823

Document Type Dissertation

Format application/pdf (imt)

Rights Zellner, Maximilian

Type texts

Source 20220427-usctheses-batch-933 (batch), University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

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