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Computational model of human behavior in security games with varying number of targets
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Computational model of human behavior in security games with varying number of targets
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COMPUTATIONAL MODEL OF HUMAN BEHAVIOR IN SECURITY GAMES WITH VARYING NUMBER OF TARGETS by Mohit Goenka A Thesis Presented to the FACULTY OF THE USC VITERBI SCHOOL OF ENGINEERING UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree MASTER OF SCIENCE (COMPUTER SCIENCE) May 2011 Copyright 2011 Mohit Goenka ii Acknowledgements Successful inception of a project involves interests and efforts of many people so it becomes obligatory on our part to extend our thanks to them. It gives me immense pleasure to express my deepest sense of gratitude and sincere thanks to Dr. Milind Tambe, Computer Science and Industrial and Systems Engineering Departments, University of Southern California, for his revered guidance throughout the research work, which made this task a pleasant job and to facilitate me with the departmental resources that proved to be very essential in building the foundations of the project. His noteworthy dictum helped me regain my focus as and where required. I have furthermore to thank, Dr. Richard John, Psychology Department, University of Southern California, for his continued support throughout my work towards the thesis. I am also grateful to Dr. Rajiv Maheswaran, Computer Science Department and Information Sciences Institute, University of Southern California, for his appreciation and his encouraging attitude that brought me to complete the project. I would like to express our sincere thanks to Rong Yang, James Pita, Manish Jain and all the other members of Teamcore Research Group, whose advice and support enabled me to adopt a constructive and organized approach towards my research work. There is many more populace I am entitled to thank, for their mere presence helped me in outlining requirements which provided me with valuable sample data sets for relevant analysis. iii Table of Contents Acknowledgements ii List of Tables v List of Figures xi Abbreviations xiv Abstract xv Chapter One: Introduction 1 Chapter Two: Stackelberg Games 4 Chapter Three: Related Work 6 Prospect Theory 6 PT-Attract 7 COBRA 8 DOBSS 9 QRE 9 Chapter Four: Payoff Structure Classification 11 Chapter Five: Experimental Setup 13 Chapter Six: Experimental Results 17 Results Based on Clusters 17 Results Based on Total Number of Gates 22 Results Based on Game Models 27 Results Based on Time Taken by Participants 34 Rankings of Game Models 39 Chapter Seven: Statistical Analysis 41 Cluster 1 41 Cluster 2 44 Cluster 3 46 Cluster 4 49 Chapter Eight: Summary 52 Bibliography 54 iv Appendices Appendix A: Payoff Structures for 3 Gate Settings 57 Appendix B: Payoff Structures for 6 Gate Settings 58 Appendix C: Payoff Structures for 9 Gate Settings 59 Appendix D: Payoff Structures for 12 Gate Settings 60 Appendix E: Payoff Structures for 15 Gate Settings 62 Appendix F: Defender Strategies for 3 Gate Settings 64 Appendix G: Defender Strategies for 6 Gate Settings 66 Appendix H: Defender Strategies for 9 Gate Settings 68 Appendix I: Defender Strategies for 12 Gate Settings 70 Appendix J: Defender Strategies for 15 Gate Settings 73 Appendix K: Participants’ Choices for Cluster 1 76 Appendix L: Participants’ Choices for Cluster 2 81 Appendix M: Participants’ Choices for Cluster 3 86 Appendix N: Participants’ Choices for Cluster 4 91 Appendix O: Average Response Times (Uncapped) 96 Appendix P: Average Response Times (Capped) 100 v List of Tables Table 1: Sample Stackelberg game 4 Table 2: Count of randomly generated payoff structures 11 Table 3: Money paid to the participants 16 Table 4: Average Expected Utilities of Game Models for Cluster 1 18 Table 5: Average Expected Utilities of Game Models for Cluster 2 19 Table 6: Average Expected Utilities of Game Models for Cluster 3 20 Table 7: Average Expected Utilities of Game Models for Cluster 4 21 Table 8: Average Expected Utilities of Game Models for 3 Gates 22 Table 9: Average Expected Utilities of Game Models for 6 Gates 23 Table 10: Average Expected Utilities of Game Models for 9 Gates 24 Table 11: Average Expected Utilities of Game Models for 12 Gates 25 Table 12: Average Expected Utilities of Game Models for 15 Gates 26 Table 13: Average Expected Utilities for PT Model 27 Table 14: Average Expected Utilities for PT-Attract Model 28 Table 15: Average Expected Utilities for COBRA (α = 0.15) Model 29 Table 16: Average Expected Utilities for COBRA (α = 0.5) Model 30 Table 17: Average Expected Utilities for DOBSS Model 31 Table 18: Average Expected Utilities for QRE (λ = 0.45) Model 32 Table 19: Average Expected Utilities for QRE (λ = 0.76) Model 33 Table 20: Time Weighted Average Expected Utilities against No. of Gates for Cluster 1 35 vi Table 21: Time Weighted Average Expected Utilities against No. of Gates for Cluster 2 36 Table 22: Time Weighted Average Expected Utilities against No. of Gates for Cluster 3 37 Table 23: Time Weighted Average Expected Utilities against No. of Gates for Cluster 4 38 Table 24: Game Model Point Allocation 39 Table 25: Game Model Point Allocation for Time Weighted Average Expected Utilities 40 Table 26: p-values for 3 Gate settings in Cluster 1 41 Table 27: p-values for 6 Gate settings in Cluster 1 42 Table 28: p-values for 9 Gate settings in Cluster 1 42 Table 29: p-values for 12 Gate settings in Cluster 1 43 Table 30: p-values for 15 Gate settings in Cluster 1 43 Table 31: p-values for 3 Gate settings in Cluster 2 44 Table 32: p-values for 6 Gate settings in Cluster 2 44 Table 33: p-values for 9 Gate settings in Cluster 2 45 Table 34: p-values for 12 Gate settings in Cluster 2 45 Table 35: p-values for 15 Gate settings in Cluster 2 46 Table 36: p-values for 3 Gate settings in Cluster 3 46 Table 37: p-values for 6 Gate settings in Cluster 3 47 Table 38: p-values for 9 Gate settings in Cluster 3 47 Table 39: p-values for 12 Gate settings in Cluster 3 48 Table 40: p-values for 15 Gate settings in Cluster 3 48 vii Table 41: p-values for 3 Gate settings in Cluster 4 49 Table 42: p-values for 6 Gate settings in Cluster 4 49 Table 43: p-values for 9 Gate settings in Cluster 4 50 Table 44: p-values for 12 Gate settings in Cluster 4 50 Table 45: p-values for 15 Gate settings in Cluster 4 51 Table 46: Payoff Structure for 3 Gate Settings in Cluster 1 57 Table 47: Payoff Structure for 3 Gate Settings in Cluster 2 57 Table 48: Payoff Structure for 3 Gate Settings in Cluster 3 57 Table 49: Payoff Structure for 3 Gate Settings in Cluster 4 57 Table 50: Payoff Structure for 6 Gate Settings in Cluster 1 58 Table 51: Payoff Structure for 6 Gate Settings in Cluster 2 58 Table 52: Payoff Structure for 6 Gate Settings in Cluster 3 58 Table 53: Payoff Structure for 6 Gate Settings in Cluster 4 58 Table 54: Payoff Structure for 9 Gate Settings in Cluster 1 59 Table 55: Payoff Structure for 9 Gate Settings in Cluster 2 59 Table 56: Payoff Structure for 9 Gate Settings in Cluster 3 59 Table 57: Payoff Structure for 9 Gate Settings in Cluster 4 59 Table 58: Payoff Structure for 12 Gate Settings in Cluster 1 60 Table 59: Payoff Structure for 12 Gate Settings in Cluster 2 60 Table 60: Payoff Structure for 12 Gate Settings in Cluster 3 60 Table 61: Payoff Structure for 12 Gate Settings in Cluster 4 61 Table 62: Payoff Structure for 15 Gate Settings in Cluster 1 62 viii Table 63: Payoff Structure for 15 Gate Settings in Cluster 2 62 Table 64: Payoff Structure for 15 Gate Settings in Cluster 3 63 Table 65: Payoff Structure for 15 Gate Settings in Cluster 4 63 Table 66: Defender Strategies for 3 Gate Settings in Cluster 1 64 Table 67: Defender Strategies for 3 Gate Settings in Cluster 2 64 Table 68: Defender Strategies for 3 Gate Settings in Cluster 3 64 Table 69: Defender Strategies for 3 Gate Settings in Cluster 4 65 Table 70: Defender Strategies for 6 Gate Settings in Cluster 1 66 Table 71: Defender Strategies for 6 Gate Settings in Cluster 2 66 Table 72: Defender Strategies for 6 Gate Settings in Cluster 3 66 Table 73: Defender Strategies for 6 Gate Settings in Cluster 4 67 Table 74: Defender Strategies for 9 Gate Settings in Cluster 1 68 Table 75: Defender Strategies for 9 Gate Settings in Cluster 2 68 Table 76: Defender Strategies for 9 Gate Settings in Cluster 3 68 Table 77: Defender Strategies for 9 Gate Settings in Cluster 4 69 Table 78: Defender Strategies for 12 Gate Settings in Cluster 1 70 Table 79: Defender Strategies for 12 Gate Settings in Cluster 2 70 Table 80: Defender Strategies for 12 Gate Settings in Cluster 3 71 Table 81: Defender Strategies for 12 Gate Settings in Cluster 4 72 Table 82: Defender Strategies for 15 Gate Settings in Cluster 1 73 Table 83: Defender Strategies for 15 Gate Settings in Cluster 2 73 Table 84: Defender Strategies for 15 Gate Settings in Cluster 3 74 ix Table 85: Defender Strategies for 15 Gate Settings in Cluster 4 75 Table 86: Participants’ Choices (in fraction) for 3 Gate Settings in Cluster 1 76 Table 87: Participants’ Choices (in fraction) for 6 Gate Settings in Cluster 1 77 Table 88: Participants’ Choices (in fraction) for 9 Gate Settings in Cluster 1 78 Table 89: Participants’ Choices (in fraction) for 12 Gate Settings in Cluster 1 79 Table 90: Participants’ Choices (in fraction) for 15 Gate Settings in Cluster 1 80 Table 91: Participants’ Choices (in fraction) for 3 Gate Settings in Cluster 2 81 Table 92: Participants’ Choices (in fraction) for 6 Gate Settings in Cluster 2 82 Table 93: Participants’ Choices (in fraction) for 9 Gate Settings in Cluster 2 83 Table 94: Participants’ Choices (in fraction) for 12 Gate Settings in Cluster 2 84 Table 95: Participants’ Choices (in fraction) for 15 Gate Settings in Cluster 2 85 Table 96: Participants’ Choices (in fraction) for 3 Gate Settings in Cluster 3 86 Table 97: Participants’ Choices (in fraction) for 6 Gate Settings in Cluster 3 87 Table 98: Participants’ Choices (in fraction) for 9 Gate Settings in Cluster 3 88 Table 99: Participants’ Choices (in fraction) for 12 Gate Settings in Cluster 3 89 Table 100: Participants’ Choices (in fraction) for 15 Gate Settings in Cluster 3 90 Table 101: Participants’ Choices (in fraction) for 3 Gate Settings in Cluster 4 91 Table 102: Participants’ Choices (in fraction) for 6 Gate Settings in Cluster 4 92 Table 103: Participants’ Choices (in fraction) for 9 Gate Settings in Cluster 4 93 Table 104: Participants’ Choices (in fraction) for 12 Gate Settings in Cluster 4 94 Table 105: Participants’ Choices (in fraction) for 15 Gate Settings in Cluster 4 95 x Table 106: Average Response Times (Uncapped) for Cluster 1 96 Table 107: Average Response Times (Uncapped) for Cluster 2 97 Table 108: Average Response Times (Uncapped) for Cluster 3 98 Table 109: Average Response Times (Uncapped) for Cluster 4 99 Table 110: Average Response Times (Capped) for Cluster 1 100 Table 111: Average Response Times (Capped) for Cluster 2 101 Table 112: Average Response Times (Capped) for Cluster 3 102 Table 113: Average Response Times (Capped) for Cluster 4 103 xi List of Figures Figure 1: Empirical Function for Prospect Theory 7 Figure 2: The Guards and The Treasure game interface 13 Figure 3: Average Expected Utilities against No. of Gates for Cluster 1 18 Figure 4: Average Expected Utilities against No. of Gates for Cluster 2 19 Figure 5: Average Expected Utilities against No. of Gates for Cluster 3 20 Figure 6: Average Expected Utilities against No. of Gates for Cluster 4 21 Figure 7: Average Expected Utilities against Various Clusters for 3 Gates 22 Figure 8: Average Expected Utilities against Various Clusters for 6 Gates 23 Figure 9: Average Expected Utilities against Various Clusters for 9 Gates 24 Figure 10: Average Expected Utilities against Various Clusters for 12 Gates 25 Figure 11: Average Expected Utilities against Various Clusters for 15 Gates 26 Figure 12: Average Expected Utilities for PT Model 27 Figure 13: Average Expected Utilities for PT-Attract Model 28 Figure 14: Average Expected Utilities for COBRA (α = 0.15) Model 29 Figure 15: Average Expected Utilities for COBRA (α = 0.5) Model 30 Figure 16: Average Expected Utilities for DOBSS Model 31 Figure 17: Average Expected Utilities for QRE (λ = 0.45) Model 32 Figure 18: Average Expected Utilities for QRE (λ = 0.76) Model 33 Figure 19: Time Weighted Average Expected Utilities against No. of Gates for Cluster 1 35 Figure 20: Time Weighted Average Expected Utilities against No. of Gates for Cluster 2 36 xii Figure 21: Time Weighted Average Expected Utilities against No. of Gates for Cluster 3 37 Figure 22: Time Weighted Average Expected Utilities against No. of Gates for Cluster 4 38 Figure 23: Participants’ Choices against Gate Number for 3 Gate Settings in Cluster 1 76 Figure 24: Participants’ Choices against Gate Number for 6 Gate Settings in Cluster 1 77 Figure 25: Participants’ Choices against Gate Number for 9 Gate Settings in Cluster 1 78 Figure 26: Participants’ Choices against Gate Number for 12 Gate Settings in Cluster 1 79 Figure 27: Participants’ Choices against Gate Number for 15 Gate Settings in Cluster 1 80 Figure 28: Participants’ Choices against Gate Number for 3 Gate Settings in Cluster 2 81 Figure 29: Participants’ Choices against Gate Number for 6 Gate Settings in Cluster 2 82 Figure 30: Participants’ Choices against Gate Number for 9 Gate Settings in Cluster 2 83 Figure 31: Participants’ Choices against Gate Number for 12 Gate Settings in Cluster 2 84 Figure 32: Participants’ Choices against Gate Number for 15 Gate Settings in Cluster 2 85 Figure 33: Participants’ Choices against Gate Number for 3 Gate Settings in Cluster 3 86 Figure 34: Participants’ Choices against Gate Number for 6 Gate Settings in Cluster 3 87 Figure 35: Participants’ Choices against Gate Number for 9 Gate Settings in Cluster 3 88 xiii Figure 36: Participants’ Choices against Gate Number for 12 Gate Settings in Cluster 3 89 Figure 37: Participants’ Choices against Gate Number for 15 Gate Settings in Cluster 3 90 Figure 38: Participants’ Choices against Gate Number for 3 Gate Settings in Cluster 4 91 Figure 39: Participants’ Choices against Gate Number for 6 Gate Settings in Cluster 4 92 Figure 40: Participants’ Choices against Gate Number for 9 Gate Settings in Cluster 4 93 Figure 41: Participants’ Choices against Gate Number for 12 Gate Settings in Cluster 4 94 Figure 42: Participants’ Choices against Gate Number for 15 Gate Settings in Cluster 4 95 Figure 43: Average Response Times (Uncapped) against Number of Gates for Cluster 1 96 Figure 44: Average Response Times (Uncapped) against Number of Gates for Cluster 2 97 Figure 45: Average Response Times (Uncapped) against Number of Gates for Cluster 3 98 Figure 46: Average Response Times (Uncapped) against Number of Gates for Cluster 4 99 Figure 47: Average Response Times (Capped) against Number of Gates for Cluster 1 100 Figure 48: Average Response Times (Capped) against Number of Gates for Cluster 2 101 Figure 49: Average Response Times (Capped) against Number of Gates for Cluster 3 102 Figure 50: Average Response Times (Capped) against Number of Gates for Cluster 4 103 xiv Abbreviations ARMOR: Assistant for Randomized Monitoring Over Routes COBRA: Combined OBservability and Rationality Assumption DOBSS: Decomposed Optimal Bayesian Stackelberg Solver FAMS: Federal Air Marshal Service GUARDS: Game theoretic Unpredictable And Randomly Deployed Security IRIS: Intelligent Randomization In Scheduling LAX: Los Angeles International Airport MILP: Mixed Integer Linear Program MLE: Maximum Likelihood Estimator No.: Number PT: Prospect Theory QRE: Quantal Response Equilibrium TSA: Transportation Security Administration xv Abstract Security is one of the biggest concerns all around the world. There are only a limited number of resources that can be allocated in security coverage. Terrorists can exploit any pattern of monitoring deployed by the security personnel. It becomes important to make the security pattern unpredictable and randomized. In such a scenario, the security forces can be randomized using algorithms based on Stackelberg games. Stackelberg games have recently gained significant importance in deployment for real world security. Game-theoretic techniques make a standard assumption that adversaries’ actions are perfectly rational. It is a challenging task to account for human behavior in such circumstances. What becomes more challenging in applying game-theoretic techniques to real-world security problems is the standard assumption that the adversary is perfectly rational in responding to security forces’ strategy, which can be unrealistic for human adversaries. Different models in the form of PT, PT-Attract, COBRA, DOBSS and QRE have already been proposed to address the scenario in settings with fixed number of targets. My work focuses on the evaluation of these models when the number of targets is varied, giving rise to an entirely new problem set. 1 Chapter One: Introduction In Stackelberg games, one player, the leader, commits to a strategy publicly before the remaining players, the followers, make their decision [7]. These types of commitments are necessary for security agents in a number of domains, pertaining to attacker-defender scenarios [1, 2, 11, 15] and Stackelberg games are well-suited to appropriately model these commitments [14, 16]. For example, in airport settings there may be eight terminals serving passengers, as at LAX, but only five bomb sniffing canine units to patrol the terminals. In such a scenario, the canine units are the first one to decide on randomizing their patrolling over the eight terminals. Meanwhile, adversaries may conduct surveillance to act according to this committed strategy of the canine units. It is a well-known fact that game-theoretic approaches make an assumption of perfect rationality which induces errors in the prediction of human behavior for multi-agent decision making problems [3, 5]. Various models are being developed and studied in order to account for the variations in human behavior from the initial assumption of perfect rationality. Behavioral game theory and cognitive science are both fully devoted to this domain. The multi-agent systems community has shown growing interest in adopting these models for decision-making and providing advice to human decision- makers [6, 26]. There has been profound work in improving the computational models of human behavior, especially in the field of security games [27]. Stackelberg games have been able to handle these needs in the best way possible [14, 16, 17, 18]. 2 The challenges in moving beyond rationality have been very significant. There is very little consensus on what model suits which domain, and if one model can outperform all the others. Thus, it becomes a very important research question to analyze all these models and determine which is more suited to one setting or the other. Another important problem that needs to be addressed in this respect is to compute the best strategy based on complex mathematical equations. This is because the calculation of mixed strategy equilibriums can involve matrices of very large sizes making the calculations difficult to handle. In this context, some of the models have proven themselves to be more effective than the others. Game-theoretic models are now being used for analyzing real-world security resource allocation problems [8, 20]. These models provide enough complexity to the system so as to not allow attackers to find any patterns in the allocation of security. ARMOR [16], IRIS [22] and GUARDS [19] are the best examples of the systems that use this approach for allocation of security in the real-world domain. In the past, such systems were designed under the assumption that the attackers are perfectly rational and would only work on maximizing their own benefit, from the system. Such a system would work best against a very intelligent attacker who can calculate his own rewards in the best way possible. However, this is not true in all the cases. The attackers may be prone to human errors that the system may not be robust enough to handle. There is a need to design a system that can not only account for the human errors but also exploit it in the best way possible. 3 Beside addressing the issue of perfect rationality and determining the best model that can account for irrational human behavior, this work addresses the different domains individually. ARMOR uses DOBSS for randomizing the checkpoints on the roadways entering the airport and canine patrol routes within the airport terminals at Los Angeles International airport. IRIS is used in limited international sectors by FAMS. GUARDS is being used by TSA for scheduling airport security operations. It is noteworthy that in general, these domains have specific number of targets that the adversaries can attack. My work further explains the effectiveness of these models and analyzes their performance in the cases where the attackers are encountered with settings pertaining to varying number of targets. My analysis has been carried out based on the data collected via the online game The Guards and The Treasure designed to simulate different security scenarios. The payoff structures used in the games are based on classification techniques that ascertain separation of models from each-other spanning the game space as widely as possible. The organization of this report is as follows: Chapter Two defines Stackelberg games in detail. In Chapter Three, I will discuss various game models that have been taken into consideration for this work. Chapter Four details the payoff structure classification. Chapter Five and Chapter Six deal with experimental setup and the corresponding results. Chapter Seven gives a statistical analysis of the results and Chapter Eight is the summary of my work. 4 Chapter Two: Stackelberg Games In a Stackelberg game, a leader first commits to a strategy followed by the follower, who can selfishly optimize his reward based on the action chosen by the leader. It is noteworthy, that in some game settings, the leader actually has an advantage of making the first move. Considering a simple example of Stackelberg game, the payoff structure may be shown as in Table 1. x y a 2, 1 4, 0 b 1, 0 3, 2 Table 1: Sample Stackelberg game In this case, the leader’s utilities are represented in the rows corresponding to a and b while the follower’s utilities are represented in the columns corresponding to x and y. The pure-strategy Nash equilibrium for this game is denoted by the action <a, x>. When the leader plays a, the follower plays x. This gives the leader a payoff of 2. In this case, the strategy b is strictly dominated by a. However, when we view this game as a Stackelberg game, the leader has a choice of committing to strategy b. In this case, the follower would respond by choosing strategy y, as this would give follower a reward of 2. Thus, committing to a strategy actually increases the reward of leader from 2 to 3. In addition to this, the leader can actually choose to play a mixed strategy of playing a and b with probabilities 0.5 each. In this case, the leader would fetch a utility of 3.5. When expanding such settings to security games, the assumption of rationality implies that the defender makes its decision assuming that if it chooses b, the attacker would 5 respond by choosing d. Such an assumption induces errors when the adversary happens to be irrational in its behavior. Another aspect of the game is that the follower may not be able to observe the applied strategy before making its choice. If the defender can determine the attackers’ responses to such mixed-strategies, it still remains a hard question to design optimal strategy against such adversaries. 6 Chapter Three: Related Work This work is motivated by the various algorithms developed to compute optimal defender strategies in Stackelberg games. These algorithms have been designed taking into consideration the different aspects of human decision making such as risk/loss aversion, non-linear weighing of probabilities and bounded rationality. Prospect Theory Prospect Theory (PT) describes a simple form of game involving alternatives that involve risk where the probabilities are known. The model tries to model real-life choices rather than optimal decisions. This theory describes the decision making process as maximization of the prospect which in general, has close proximity to what is referred to as expected utility by other models. The theory describes how individuals evaluate potential losses and gains. The decision-making is based on the editing of the choices to determine what humans equate as similar outcomes followed by the evaluation of the choices obtained through the calculation of utility value. The general principles of probability do not hold well as the payoff values and the assigned probabilities are perceived lower than what they actually signify. An empirical form of the function is shown in Figure 1. 7 Figure 1: Empirical function for Prospect Theory Any outcome which is lower than the reference point is considered to be a loss, while the one higher than the reference point is considered to be gain. There are many different forms of the value function which are reflexive and loss-averse [10]. PT-Attract PT-Attract is a model proposed by Yang et al [27], which is a linear program based on PT, designed to improve the PT model by introducing new parameters addressing more number of constraints. The imprecision for human adversaries to make the optimal choices can affect the expected outcome of a defender strategy based on the PT model. These losses can be fatal when considering real-world security domains. In order to compensate for these losses, the PT-Attract model introduces a new constraint that 8 accounts for all the Ɛ-optimal strategies corresponding to any particular choice of the defender. It turns out that the defender reward as obtained by the PT model is smaller than the defender reward for the Ɛ-optimal adversarial strategies. COBRA COBRA is one of the leading works to account for human behavior [17]. This model is supported by experimental evidence over the human subjects. Whether or not this model is as fast as some of the other models is questionable. This model defines the adversaries’ behavior from two different prospects of bounded rationality on computing the optimal strategy and anchoring bias caused by the attackers’ inability to get the precise observation of the mixed strategy that the defender is using for the purpose of security allocation. The calculations here are based on Mixed Integer Linear Program (MILP) with variables to account for selections of Ɛ-optimal adversarial strategies as in the case of PT-Attract model. The major problem with this model is that finding an optimal solution in a Bayesian Stackelberg game is NP-hard [4], thus COBRA happens to be a MILP facing an NP-hard problem.An Alpha (α) parameter is varied to account for the limited number of observations that a human adversary may be able to use. The value α can vary from 0 to 1 to denote the ignorance prior while 1 - α would denote the occurrences viewed by the adversaries. Thus, the value α = 0 would mean that the adversaries have fully observed the system while α = 1 would mean that the adversaries 9 are completely ignorant of the system. In my experiments, the α parameter has been set to 0.15 and 0.5. DOBSS DOBSS happens to be the one of the most efficient general Stackelberg solvers [14]. It is in currently being used in the ARMOR software that is deployed for security scheduling at the Los Angeles International airport. It directly uses the Bayesian representations for calculations and thus, does not require the normal-form Bayesian game to go through the Harsanyi transformation [9]. This gives a very significant speedup over the other multiple linear program methods [4]. It uses the property that instead of carrying out the Harsanyi transformation, the same result can be achieved by evaluating the payoffs against all the game matrices one-by-one and then carrying out a weighted sum. QRE Quantal Response Equilibrium (QRE) acts as a base for many different types of studies [21, 25]. It states that the response of individuals in Stackelberg games is quantal, and the chance of making an error increases as the cost of such an error decreases [13]. Thus, it does consider that very costly errors are unlikely to occur. It is noteworthy that QRE, in general, can return results that are significantly different than Nash equilibrium. The equilibrium condition is based on the realization of the beliefs of the opponent player. In the base model, every strategy is used with nonzero probability. In modeling the error of 10 the players, lambda (λ) parameter is used. λ = 0 denotes that the players can play any pure strategy with equal probability while λ = ∞ denotes that the quantal response is same as the best response. λ is very sensitive when it comes to the payoff strategies thus, a very crucial step is to determine the exact value of λ for deploying into the QRE model. My experiments are based on λ values of 0.45 and 0.76, pre-determined by Maximum Likelihood Estimator (MLE) used in Yang et al [27]. 11 Chapter Four: Payoff Structure Classification In order to justify the results, it becomes critically important to account for the entire state-space of the payoff structures. It is also important that the strategies generated from different models should be separated from each other. In order to span the entire state space, a very large number of payoff structures are randomly generated, as in Table 2. No. of gates No. of randomly generated payoff structures 3 1000 6 1000 9 1000 12 100 15 100 Table 2: Count of randomly generated payoff structures The range of the rewards and the penalties are the same for both the defender as well as the attacker. The rewards in each of these cases vary from 1 to 10, while the penalties vary from -1 to -10. This is consistent with the previous work carried out in this domain [17, 27]. In order to separate the different models from each other, the randomly generated payoff structures are divided into four different groups using k-means clustering. There are 8 different features extracted in the same way as in Yang et al [27]. These features denote the adversaries’ utilities, the defenders’ utilities and the level of gain of one player over the other. 12 The difference of mixed strategies is calculated using Kullback-Leibler divergence [12], which gives a non-symmetric measure of the difference between the various payoff structures. Being a premetric, it ascertains that the distance between any two payoff structure is greater than or equal to zero and generates a topology on the space of generalized distributions. Thereafter, from each group one payoff structure is selected such that it is closest to the group center. 13 Chapter Five: Experimental Setup The tests on human participants were conducted via an online game The Guards and The Treasure designed specifically for this purpose. The interface of the game is shown in Figure 2. Figure 2: The Guards and The Treasure game interface 14 The experiments were designed to study the human behavior and their perception of the different payoff structures. This is to account for the standard perfect rationality assumption that the various game models would otherwise make. The role of leader is played by the guard while the role of follower is played by the attacker. The mixed strategy is presented to the attackers in the form of probabilities. For clarity we present both the probability of the presence of a guard as well as the probability of the absence of the guard. The leader strategies are decided on the basis of the various game models discussed earlier. Each participant plays 35 rounds in all. Each of the rounds corresponds to one of the 5 gate settings and one of the 7 guard allocation strategies generated by the different game models. The various gate settings are as follows: 3 gates and 1 guard 6 gates and 2 guards 9 gates and 3 guards 12 gates and 4 guards 15 gates and 7 guards The various guard allocation strategies are as follows: PT PT-Attract COBRA (α = 0.15) 15 COBRA (α = 0.5) DOBSS QRE (λ = 0.45) QRE (λ = 0.76) In each of the rounds, the participant gets to attack one of the gates. In each of the rounds, exactly one-third of the gates are protected by the guards based on the displayed probability distribution, marked as Probability of Guard in Figure 2. An attack is considered to be successful if the attacked gate was not being protected by a guard. An attack is considered to be a failure if the attacked gate was being protected by a guard. For a successful attack the player receives points corresponding to Your Rewards in Figure 2, while for a failure the player loses points corresponding to Your Penalties in Figure 2. As described in Chapter Four: Payoff Structure Classification, there are four different payoff structures corresponding to four different groups decided using k-means clustering. Each of the participant plays only of these groups, but plays against all the payoff structures of the group. The order of the games is randomized to ensure that there is no learning and that any bias due to ordering of the games can be removed. Also, the intermediate results of the rounds are not presented to the participants. This is to ensure that their choices are not affected by the points they have already earned. 16 Each subject begins with base money of $8.00. Thereafter, for every point scored, the participant wins $0.10, while for every point lost, the participant loses $0.10. The monetary data is included in Table 3. Cluster No. of Participants Highest Lowest Average 1 21 $22.20 $5.80 $12.03 2 25 $23.00 $12.50 $17.65 3 23 $22.70 $12.20 $17.79 4 22 $25.30 $11.80 $19.38 Total 91 $25.30 $5.80 $16.81 Table 3: Money paid to the participants In the experimental setup, it was also ensured that the data is only recorded once the participant has played all the rounds of the cluster, in which he is participating. 17 Chapter Six: Experimental Results In total, 91 participants played the game. The participation based on cluster is given in Table 3. In order to evaluate the performances of different game models, defenders’ utilities are calculated based on the participants’ choices. These calculations are based on the following formula: ( ) ∑( ( )) where, ( ) denotes the average expected utility of the defender against number of gates in the specified cluster, is the total number of participants, who played that cluster, are the number of different gates chosen by the participant, ( ) is the defender’s expected utility for gate given as the mixed strategy Results Based on Clusters The average expected utilities against no. of gates for various clusters are illustrated in Tables 4 to 7 and Figures 3 to 6. Table 4 and Figure 3 show the average expected utilities of various game models for cluster 1. For 3 gates we see that the average expected utility for QRE (λ = 0.45) is higher than all the other game models. This denotes that its performance was better than the 18 other models. Similarly, PT-Attract dominates the 6 gates setting. PT-Attract and COBRA (α = 0.5) are closely contested in 9 gates setting. PT-Attract and PT models are leader in 12 gates setting and 15 gates setting respectively. Game Model 3 Gates 6 Gates 9 Gates 12 Gates 15 Gates PT -4.35 -1.122 -0.194 -1.949 0.29 PT-Attract -4.57 -1.094 0.296 -0.03 -1.061 COBRA (Alpha = 0.15) -4.113 -1.137 -0.651 -1.559 -0.162 COBRA (Alpha =0.5) -3.965 -1.491 0.284 -1.801 -0.551 DOBSS -4.408 -2.267 -0.342 -1.72 -0.3 QRE (Lambda = 0.45) -3.511 -1.243 -0.32 -1.886 -1.561 QRE (Lambda = 0.76) -3.696 -1.274 -0.999 -2.756 -0.663 Table 4: Average Expected Utilities of Game Models for Cluster 1 Figure 3: Average Expected Utilities against No. of Gates for Cluster 1 Table 5 and Figure 4 show the average expected utilities of various game models for cluster 2. For 3 gates we see that the highest average expected utility is fetched by PT- 19 Attract. PT-Attract and QRE (λ = 0.76) are closely contested in 6 game setting. PT model then dominates the games for 9 gates, 12 gates and 15 gates. Game Model 3 Gates 6 Gates 9 Gates 12 Gates 15 Gates PT -0.405 -1.946 1.337 1.042 0.83 PT-Attract -0.295 -0.036 -1.031 -1.084 -0.685 COBRA (Alpha = 0.15) -0.476 -5.28 -0.768 -1.68 -0.263 COBRA (Alpha =0.5) -0.593 -4.186 -0.546 -1.536 -0.385 DOBSS -0.574 -3.225 -2.397 0.1 0.183 QRE (Lambda = 0.45) -0.91 -0.242 -1.58 -1.215 -1.044 QRE (Lambda = 0.76) -0.814 0.069 -2.167 -1.515 -0.339 Table 5: Average Expected Utilities of Game Models for Cluster 2 Figure 4: Average Expected Utilities against No. of Gates for Cluster 2 Cluster 3, illustrated in Table 6 and Figure 5, has mixed results. PT dominates 3 gates setting. COBRA (α = 0.5) turns out to be the best for 6 gates setting. DOBSS leads in case of 9 gates and 12 gates. PT-Attract is the best for 15 gates. 20 Game Model 3 Gates 6 Gates 9 Gates 12 Gates 15 Gates PT -2.523 -2.283 -2.713 -2.479 -3.341 PT-Attract -3.536 -0.311 -1.918 -2.911 -1.374 COBRA (Alpha = 0.15) -5.333 -1.149 -3.339 -2.606 -2.006 COBRA (Alpha =0.5) -5.64 -0.193 -3.39 -2.754 -3.953 DOBSS -4.462 -0.699 -1.618 -2.224 -2.446 QRE (Lambda = 0.45) -3.909 -0.932 -2.123 -2.578 -1.712 QRE (Lambda = 0.76) -4.213 -0.903 -2.594 -2.425 -2.021 Table 6: Average Expected Utilities of Game Models for Cluster 3 Figure 5: Average Expected Utilities against No. of Gates for Cluster 3 In Cluster 4, illustrated by Table 7 and Figure 6, all the models seem to perform really well for 3 gates by fetching positive average expected utilities. In other game sets, DOBSS performs the best for 6 gates and 9 gates, 12 gates are dominated by PT-Attract and COBRA (α = 0.5) is the only model to fetch positive rewards in 15 gates setting. 21 Game Model 3 Gates 6 Gates 9 Gates 12 Gates 15 Gates PT 1.093 -2.341 -3.541 -2.956 -0.906 PT-Attract 1.412 -2.54 -2.99 -2.391 -1.062 COBRA (Alpha = 0.15) 1.999 -1.844 -1.31 -2.967 -0.224 COBRA (Alpha =0.5) 2.134 -2.091 -2.017 -4.419 0.104 DOBSS 2.198 -3.025 -2.697 -3.848 -0.586 QRE (Lambda = 0.45) 1.68 -3.164 -1.857 -2.858 -0.515 QRE (Lambda = 0.76) 1.433 -3.359 -2.344 -3.328 -0.241 Table 7: Average Expected Utilities of Game Models for Cluster 4 Figure 6: Average Expected Utilities against No. of Gates for Cluster 4 22 Results Based on Total Number of Gates The average expected utilities against various clusters for different number of total gates are illustrated in Tables 8 to 12 and Figures 7 to 11. Game Model Cluster 1 Cluster 2 Cluster 3 Cluster 4 PT -4.35 -0.405 -2.523 1.093 PT-Attract -4.57 -0.295 -3.536 1.412 COBRA (Alpha = 0.15) -4.113 -0.476 -5.333 1.999 COBRA (Alpha =0.5) -3.965 -0.593 -5.64 2.134 DOBSS -4.408 -0.574 -4.462 2.198 QRE (Lambda = 0.45) -3.511 -0.91 -3.909 1.68 QRE (Lambda = 0.76) -3.696 -0.814 -4.213 1.433 Table 8: Average Expected Utilities of Game Models for 3 Gates Figure 7: Average Expected Utilities against Various Clusters for 3 Gates 23 Table 8 and Figure 7 illustrate the comparative performance of models for 3 gates. QRE (λ = 0.45) dominates cluster 1. PT-Attract is best in cluster 2. PT leads in cluster 3 while DOBSS is best for cluster 4. This mixed performance shows the effect of irrational human behavior in various clusters. Game Model Cluster 1 Cluster 2 Cluster 3 Cluster 4 PT -1.122 -1.946 -2.283 -2.341 PT-Attract -1.094 -0.036 -0.311 -2.54 COBRA (Alpha = 0.15) -1.137 -5.28 -1.149 -1.844 COBRA (Alpha =0.5) -1.491 -4.186 -0.193 -2.091 DOBSS -2.267 -3.225 -0.699 -3.025 QRE (Lambda = 0.45) -1.243 -0.242 -0.932 -3.164 QRE (Lambda = 0.76) -1.274 0.069 -0.903 -3.359 Table 9: Average Expected Utilities of Game Models for 6 Gates Figure 8: Average Expected Utilities against Various Clusters for 6 Gates 24 Table 9 and Figure 8 illustrate the comparative performance of models for 6 gates. PT- Attract dominates cluster 1. QRE (λ = 0.76) is best in cluster 2. COBRA (α = 0.5) leads in cluster 3 while COBRA (α = 0.15) is best for cluster 4. Again, mixed performance is obtained for 6 gate settings. Game Model Cluster 1 Cluster 2 Cluster 3 Cluster 4 PT -0.194 1.337 -2.713 -3.541 PT-Attract 0.296 -1.031 -1.918 -2.99 COBRA (Alpha = 0.15) -0.651 -0.768 -3.339 -1.31 COBRA (Alpha =0.5) 0.284 -0.546 -3.39 -2.017 DOBSS -0.342 -2.397 -1.618 -2.697 QRE (Lambda = 0.45) -0.32 -1.58 -2.123 -1.857 QRE (Lambda = 0.76) -0.999 -2.167 -2.594 -2.344 Table 10: Average Expected Utilities of Game Models for 9 Gates Figure 9: Average Expected Utilities against Various Clusters for 9 Gates 25 For 9 gates, as shown in Table 10 and Figure 9, PT-Attract leads cluster 1, PT is best in cluster 2, DOBSS is the first one in cluster 3 and COBRA (α = 0.15) has the highest average expected utility in cluster 4. Game Model Cluster 1 Cluster 2 Cluster 3 Cluster 4 PT -1.949 1.042 -2.479 -2.956 PT-Attract -0.03 -1.084 -2.911 -2.391 COBRA (Alpha = 0.15) -1.559 -1.68 -2.606 -2.967 COBRA (Alpha =0.5) -1.801 -1.536 -2.754 -4.419 DOBSS -1.72 0.1 -2.224 -3.848 QRE (Lambda = 0.45) -1.886 -1.215 -2.578 -2.858 QRE (Lambda = 0.76) -2.756 -1.515 -2.425 -3.328 Table 11: Average Expected Utilities of Game Models for 12 Gates Figure 10: Average Expected Utilities against Various Clusters for 12 Gates In 12 gates setting, as shown in Table 11 and Figure 10, PT-Attract leads cluster 1 and cluster 4, PT is best in cluster 2 and DOBSS is the first one in cluster 3. 26 Game Model Cluster 1 Cluster 2 Cluster 3 Cluster 4 PT 0.29 0.83 -3.341 -0.906 PT-Attract -1.061 -0.685 -1.374 -1.062 COBRA (Alpha = 0.15) -0.162 -0.263 -2.006 -0.224 COBRA (Alpha =0.5) -0.551 -0.385 -3.953 0.104 DOBSS -0.3 0.183 -2.446 -0.586 QRE (Lambda = 0.45) -1.561 -1.044 -1.712 -0.515 QRE (Lambda = 0.76) -0.663 -0.339 -2.021 -0.241 Table 12: Average Expected Utilities of Game Models for 15 Gates Figure 11: Average Expected Utilities against Various Clusters for 15 Gates For 15 gates setting, PT leads cluster 1 and cluster 2. PT-Attract turns out to be the best for cluster 3 and COBRA (α = 0.15) dominates cluster 4. The inconsistency in model performance across settings with same number of gates, incline towards the deduction that participants are unable to process all the given information in the best possible rational manner. 27 Results Based on Game Models On the basis of the different game models, the average expected utilities against varying gates and clusters are illustrated in Tables 13 to 19 and Figures 12 to 18. 3 Gates 6 Gates 9 Gates 12 Gates 15 Gates Cluster 1 -4.35 -1.122 -0.194 -1.949 0.29 Cluster 2 -0.405 -1.946 1.337 1.042 0.83 Cluster 3 -2.523 -2.283 -2.713 -2.479 -3.341 Cluster 4 1.093 -2.341 -3.541 -2.956 -0.906 Table 13: Average Expected Utilities for PT Model Figure 12: Average Expected Utilities for PT Model 28 3 Gates 6 Gates 9 Gates 12 Gates 15 Gates Cluster 1 -4.57 -1.094 0.296 -0.03 -1.061 Cluster 2 -0.295 -0.036 -1.031 -1.084 -0.685 Cluster 3 -3.536 -0.311 -1.918 -2.911 -1.374 Cluster 4 1.412 -2.54 -2.99 -2.391 -1.062 Table 14: Average Expected Utilities for PT-Attract Model Figure 13: Average Expected Utilities for PT-Attract Model 29 3 Gates 6 Gates 9 Gates 12 Gates 15 Gates Cluster 1 -4.113 -1.137 -0.651 -1.559 -0.162 Cluster 2 -0.476 -5.28 -0.768 -1.68 -0.263 Cluster 3 -5.333 -1.149 -3.339 -2.606 -2.006 Cluster 4 1.999 -1.844 -1.31 -2.967 -0.224 Table 15: Average Expected Utilities for COBRA (α = 0.15) Model Figure 14: Average Expected Utilities for COBRA (α = 0.15) Model 30 3 Gates 6 Gates 9 Gates 12 Gates 15 Gates Cluster 1 -3.965 -1.491 0.284 -1.801 -0.551 Cluster 2 -0.593 -4.186 -0.546 -1.536 -0.385 Cluster 3 -5.64 -0.193 -3.39 -2.754 -3.953 Cluster 4 2.134 -2.091 -2.017 -4.419 0.104 Table 16: Average Expected Utilities for COBRA (α = 0.5) Model Figure 15: Average Expected Utilities for COBRA (α = 0.5) Model 31 3 Gates 6 Gates 9 Gates 12 Gates 15 Gates Cluster 1 -4.408 -2.267 -0.342 -1.72 -0.3 Cluster 2 -0.574 -3.225 -2.397 0.1 0.183 Cluster 3 -4.462 -0.699 -1.618 -2.224 -2.446 Cluster 4 2.198 -3.025 -2.697 -3.848 -0.586 Table 17: Average Expected Utilities for DOBSS Model Figure 16: Average Expected Utilities for DOBSS Model 32 3 Gates 6 Gates 9 Gates 12 Gates 15 Gates Cluster 1 -3.511 -1.243 -0.32 -1.886 -1.561 Cluster 2 -0.91 -0.242 -1.58 -1.215 -1.044 Cluster 3 -3.909 -0.932 -2.123 -2.578 -1.712 Cluster 4 1.68 -3.164 -1.857 -2.858 -0.515 Table 18: Average Expected Utilities for QRE (λ = 0.45) Model Figure 17: Average Expected Utilities for QRE (λ = 0.45) Model 33 3 Gates 6 Gates 9 Gates 12 Gates 15 Gates Cluster 1 -3.696 -1.274 -0.999 -2.756 -0.663 Cluster 2 -0.814 0.069 -2.167 -1.515 -0.339 Cluster 3 -4.213 -0.903 -2.594 -2.425 -2.021 Cluster 4 1.433 -3.359 -2.344 -3.328 -0.241 Table 19: Average Expected Utilities for QRE (λ = 0.76) Model Figure 18: Average Expected Utilities for QRE (λ = 0.76) Model 34 Results Based on Time Taken by Participants When considering the response of participants in an experiment with varying number of gates, it becomes critically important that the participants pay importance to the data presented to them. This is necessary to eradicate conditions where a participant may make choices based on instinct rather than analysis of the defenders’ strategy. To assign more credits to participants who paid attention to such details, I carried out another calculation of average expected utilities weighted on the basis of the time taken by the participants. This calculation is based on the response time capped at 120 seconds, i.e. if any participant took more than 120 seconds to analyze a payoff structure; his response time was recorded as 120 seconds only. This was done to ensure that one or more participants do not get a very heavy weightage over other participants. The calculation of time based average expected utility is carried out as follows: ( ) ∑( ( )) where, ( ) denotes the time based average expected utility of the defender corresponding to the given number of gates in the specified cluster, is the total time spent by the participants, who played that cluster, are the number of different gates chosen by the participant, ( ) is the defender’s expected time based utility for gate given as the mixed strategy 35 The time weighted average expected utilities against no. of gates for various clusters are illustrated in Tables 20 to 23 and Figures 19 to 22. Game Model 3 Gates 6 Gates 9 Gates 12 Gates 15 Gates PT -4.877 -1.135 -0.651 -1.961 0.033 PT-Attract -4.102 -0.69 0.364 0.299 -1.141 COBRA (Alpha = 0.15) -4.337 -1.82 -0.693 -1.122 -0.292 COBRA (Alpha =0.5) -4.155 -0.903 0.092 -0.755 -0.92 DOBSS -4.38 -2.069 0.075 -1.916 0.476 QRE (Lambda = 0.45) -3.305 -1.083 -0.44 -1.952 -1.322 QRE (Lambda = 0.76) -3.835 -1.114 -0.756 -3.014 -0.21 Table 20: Time Weighted Average Expected Utilities of Game Models for Cluster 1 Figure 19: Time Weighted Average Expected Utilities against No. of Gates for Cluster 1 36 Game Model 3 Gates 6 Gates 9 Gates 12 Gates 15 Gates PT -0.331 -1.975 0.132 1.006 0.57 PT-Attract -1.098 0.014 -0.798 -1.395 -0.411 COBRA (Alpha = 0.15) -0.585 -4.908 -0.64 -1.949 -0.363 COBRA (Alpha =0.5) -0.499 -3.427 -0.5 -2.172 -0.189 DOBSS -0.933 -2.61 -2.529 0.444 0.867 QRE (Lambda = 0.45) -0.915 -0.285 -1.607 -1.229 -0.802 QRE (Lambda = 0.76) -0.822 -0.017 -2.129 -1.817 -0.351 Table 21: Time Weighted Average Expected Utilities of Game Models for Cluster 2 Figure 20: Time Weighted Average Expected Utilities against No. of Gates for Cluster 2 37 Game Model 3 Gates 6 Gates 9 Gates 12 Gates 15 Gates PT -1.514 -1.968 -2.46 -2.938 -3.003 PT-Attract -2.889 -0.272 -2.194 -2.952 -1.118 COBRA (Alpha = 0.15) -5.215 -1.681 -3.783 -2.538 -1.963 COBRA (Alpha =0.5) -5.28 0.023 -3.548 -2.617 -3.905 DOBSS -2.908 -1.025 -1.808 -1.183 -2.378 QRE (Lambda = 0.45) -4.229 -0.907 -2.074 -2.465 -1.671 QRE (Lambda = 0.76) -2.621 -0.82 -2.52 -1.764 -2.238 Table 22: Time Weighted Average Expected Utilities of Game Models for Cluster 3 Figure 21: Time Weighted Average Expected Utilities against No. of Gates for Cluster 3 38 Game Model 3 Gates 6 Gates 9 Gates 12 Gates 15 Gates PT 1.013 -1.738 -3.667 -2.624 -0.902 PT-Attract 1.291 -2.465 -3.298 -2.144 -1.441 COBRA (Alpha = 0.15) 2.165 -1.703 -2.681 -3.023 -0.576 COBRA (Alpha =0.5) 2.485 -1.493 -1.668 -5.065 0.505 DOBSS 1.296 -2.823 -2.557 -2.944 -0.558 QRE (Lambda = 0.45) 2.044 -3.386 -2.015 -2.992 -1.313 QRE (Lambda = 0.76) 1.828 -3.613 -2.367 -3.66 -0.263 Table 23: Time Weighted Average Expected Utilities of Game Models for Cluster 4 Figure 22: Time Weighted Average Expected Utilities against No. of Gates for Cluster 4 39 Rankings of Game Models From the experiments, there does not seem to be a distinct game model that would be superior to all the other models in every case. Thus, to rank the game models, a different ranking scheme is deployed. A table is designed to count the number of times a game model is ranked 1, considering all the game settings, then the number of times it is ranked 1 or 2, the number of times it is ranked 1, 2 or 3 and so on. Each position gives 1 point to the game model. Thus, the overall ranking can then be calculated based on this point system. The points allocated are shown in Table 24. Rank 1 1 or 2 1, 2 or 3 1, 2, 3 or 4 1, 2, 3, 4 or 5 1, 2, 3, 4, 5 or 6 Total PT 5 7 11 12 14 17 66 PT-Attract 6 10 11 13 14 17 71 COBRA (Alpha = 0.15) 2 5 11 13 14 18 63 COBRA (Alpha =0.5) 2 8 9 11 13 17 60 DOBSS 4 4 8 9 16 18 59 QRE (Lambda = 0.45) 0 3 6 12 15 16 52 QRE (Lambda = 0.76) 1 3 4 10 14 17 49 Table 24: Game Model Point Allocation Based on this ranking system, PT-Attract Model outperforms all the other game models. On similar lines, when Time Weighted Average Utilities are considered the point allocation obtained is as shown in Table 25. 40 Rank 1 1 or 2 1, 2 or 3 1, 2, 3 or 4 1, 2, 3, 4 or 5 1, 2, 3, 4, 5 or 6 Total PT 4 7 8 10 14 16 59 PT-Attract 6 7 10 13 14 17 67 COBRA (Alpha = 0.15) 0 2 6 9 12 17 46 COBRA (Alpha =0.5) 5 11 12 13 15 17 73 DOBSS 4 4 7 10 15 18 58 QRE (Lambda = 0.45) 1 4 9 12 15 18 59 QRE (Lambda = 0.76) 0 5 8 13 15 17 58 Table 25: Game Model Point Allocation for Time Weighted Average Expected Utilities Based on this point allocation, COBRA (α = 0.5) outperforms the other models, however, PT-Attract follows as a close second. From both the obtained tables, it can be derived that PT-Attract and COBRA (α = 0.5) outperform all the other models in this set of experiments. 41 Chapter Seven: Statistical Analysis In order to carry out the statistical analysis over the data set, a robust method is needed. This is because of the non-normal distribution of the data. I chose to run Yuen’s test for comparing trimmed means [28]. For my tests, the standard 20% trimmed mean was used. A trimmed mean refers to a situation where a certain proportion of the largest and smallest sample points are removed and the remaining sample points are averaged. This is typically done to help reduce variance in data collections that may have extreme outliers that can skew data sets [23, 24]. This method has been considered due to its use in some of the previous work [18]. Cluster 1 The statistical probability parameters (p-value) for various game models against each other in cluster 1 are shown in Tables 26 to 30. PT PT- Attract COBRA (α = 0.15) COBRA (α = 0.5) DOBSS QRE (λ = 0.45) QRE (λ = 0.76) PT 1 0.6225 0.0962 0.1654 0.4407 0.0016 0.0215 PT-Attract 0.6225 1 0.0268 0.0489 0.1572 0.0004 0.0054 COBRA (α = 0.15) 0.0962 0.0268 1 0.0004 0 0 0.0423 COBRA (α = 0.5) 0.1654 0.0489 0.0004 1 0 0 0.0104 DOBSS 0.4407 0.1572 0 0 1 0 0.0005 QRE (λ = 0.45) 0.0016 0.0004 0 0 0 1 0.0094 QRE (λ = 0.76) 0.0215 0.0054 0.0423 0.0104 0.0005 0.0094 1 Table 26: p-values for 3 Gate settings in Cluster 1 42 PT PT- Attract COBRA (α = 0.15) COBRA (α = 0.5) DOBSS QRE (λ = 0.45) QRE (λ = 0.76) PT 1 0.2233 0.0003 0.1576 0.5006 0.918 0.5075 PT-Attract 0.2233 1 0.2351 0.5868 0.2245 0.2064 0.3092 COBRA (α = 0.15) 0.0003 0.2351 1 0.8687 0.0696 0 0 COBRA (α = 0.5) 0.1576 0.5868 0.8687 1 0.1449 0.1542 0.206 DOBSS 0.5006 0.2245 0.0696 0.1449 1 0.4736 0.3898 QRE (λ = 0.45) 0.918 0.2064 0 0.1542 0.4736 1 0.2686 QRE (λ = 0.76) 0.5075 0.3092 0 0.206 0.3898 0.2686 1 Table 27: p-values for 6 Gate settings in Cluster 1 PT PT- Attract COBRA (α = 0.15) COBRA (α = 0.5) DOBSS QRE (λ = 0.45) QRE (λ = 0.76) PT 1 0.3417 0 0.0003 0.0029 0.0079 0 PT-Attract 0.3417 1 0 0.2736 0.0013 0.0047 0 COBRA (α = 0.15) 0 0 1 0 0.9699 0.014 0.8738 COBRA (α = 0.5) 0.0003 0.2736 0 1 0.0003 0.0002 0 DOBSS 0.0029 0.0013 0.9699 0.0003 1 0.1535 0.9027 QRE (λ = 0.45) 0.0079 0.0047 0.014 0.0002 0.1535 1 0.052 QRE (λ = 0.76) 0 0 0.8738 0 0.9027 0.052 1 Table 28: p-values for 9 Gate settings in Cluster 1 43 PT PT- Attract COBRA (α = 0.15) COBRA (α = 0.5) DOBSS QRE (λ = 0.45) QRE (λ = 0.76) PT 1 0.0131 0.8214 0.187 0.187 0.187 0.0414 PT-Attract 0.0131 1 0.1722 0 0 0 0 COBRA (α = 0.15) 0.8214 0.1722 1 0.2994 0.2994 0.2944 0.1078 COBRA (α = 0.5) 0.187 0 0.2994 1 0 0 0.1098 DOBSS 0.187 0 0.2994 0 1 0 0.1098 QRE (λ = 0.45) 0.187 0 0.2944 0 0 1 0.1098 QRE (λ = 0.76) 0.0414 0 0.1078 0.1098 0.1098 0.1098 1 Table 29: p-values for 12 Gate settings in Cluster 1 PT PT- Attract COBRA (α = 0.15) COBRA (α = 0.5) DOBSS QRE (λ = 0.45) QRE (λ = 0.76) PT 1 0.0467 0.8915 0.512 0.4957 0.1525 0.4116 PT-Attract 0.0467 1 0.0057 0.0235 0.1293 0.9235 0.043 COBRA (α = 0.15) 0.8915 0.0057 1 0.006 0.3729 0.1061 0.0209 COBRA (α = 0.5) 0.512 0.0235 0.006 1 0.8053 0.2079 0.5527 DOBSS 0.4957 0.1293 0.3729 0.8053 1 0.3249 0.9875 QRE (λ = 0.45) 0.1525 0.9235 0.1061 0.2079 0.3249 1 0.2639 QRE (λ = 0.76) 0.4116 0.043 0.0209 0.5527 0.9875 0.2639 1 Table 30: p-values for 15 Gate settings in Cluster 1 44 Cluster 2 The statistical probability parameters (p-value) for various game models against each other in cluster 2 are shown in Tables 31 to 35. PT PT- Attract COBRA (α = 0.15) COBRA (α = 0.5) DOBSS QRE (λ = 0.45) QRE (λ = 0.76) PT 1 0.4592 0.8231 0.6536 0.9574 0.1744 0.238 PT-Attract 0.4592 1 0.1867 0.1219 0.2582 0.0157 0.0243 COBRA (α = 0.15) 0.8231 0.1867 1 0.0213 0.3114 0 0 COBRA (α = 0.5) 0.6536 0.1219 0.0213 1 0.0348 0 0 DOBSS 0.9574 0.2582 0.3114 0.0348 1 0 0 QRE (λ = 0.45) 0.1744 0.0157 0 0 0 1 0 QRE (λ = 0.76) 0.238 0.0243 0 0 0 0 1 Table 31: p-values for 3 Gate settings in Cluster 2 PT PT- Attract COBRA (α = 0.15) COBRA (α = 0.5) DOBSS QRE (λ = 0.45) QRE (λ = 0.76) PT 1 0.0209 0.0025 0.0041 0.0876 0.2459 0.125 PT-Attract 0.0209 1 0.0004 0.0003 0.0086 0 0 COBRA (α = 0.15) 0.0025 0.0004 1 0.514 0.125 0.0009 0.0007 COBRA (α = 0.5) 0.0041 0.0003 0.514 1 0.3061 0.0011 0.0008 DOBSS 0.0876 0.0086 0.125 0.3061 1 0.0271 0.0192 QRE (λ = 0.45) 0.2459 0 0.0009 0.0011 0.0271 1 0.0001 QRE (λ = 0.76) 0.125 0 0.0007 0.0008 0.0192 0.0001 1 Table 32: p-values for 6 Gate settings in Cluster 2 45 PT PT- Attract COBRA (α = 0.15) COBRA (α = 0.5) DOBSS QRE (λ = 0.45) QRE (λ = 0.76) PT 1 0 0 0 0 0 0 PT-Attract 0 1 0 0 0.0755 0.0507 0.0249 COBRA (α = 0.15) 0 0 1 0 0.0347 0.0118 0.007 COBRA (α = 0.5) 0 0 0 1 0.0099 0.0012 0.001 DOBSS 0 0.0755 0.0347 0.0099 1 0.5268 0.8044 QRE (λ = 0.45) 0 0.0507 0.0118 0.0012 0.5268 1 0.6116 QRE (λ = 0.76) 0 0.0249 0.007 0.001 0.8044 0.6116 1 Table 33: p-values for 9 Gate settings in Cluster 2 PT PT- Attract COBRA (α = 0.15) COBRA (α = 0.5) DOBSS QRE (λ = 0.45) QRE (λ = 0.76) PT 1 0.0064 0.0076 0.0011 0.0051 0.0006 0.001 PT-Attract 0.0064 1 0.9348 0 0 0 0 COBRA (α = 0.15) 0.0076 0.9348 1 0.0571 0.8371 0.0114 0.0421 COBRA (α = 0.5) 0.0011 0 0.0571 1 0 0 0.0001 DOBSS 0.0051 0 0.8371 0 1 0 0 QRE (λ = 0.45) 0.0006 0 0.0114 0 0 1 0 QRE (λ = 0.76) 0.001 0 0.0421 0.0001 0 0 1 Table 34: p-values for 12 Gate settings in Cluster 2 46 PT PT- Attract COBRA (α = 0.15) COBRA (α = 0.5) DOBSS QRE (λ = 0.45) QRE (λ = 0.76) PT 1 0.0227 0 0 0.8253 0.0053 0 PT-Attract 0.0227 1 0.9865 0.5453 0.5967 0.2236 0.8147 COBRA (α = 0.15) 0 0.9865 1 0 0.5849 0.1482 0.1408 COBRA (α = 0.5) 0 0.5453 0 1 0.7142 0.0718 0.0001 DOBSS 0.8253 0.5967 0.5849 0.7142 1 0.2715 0.5337 QRE (λ = 0.45) 0.0053 0.2236 0.1482 0.0718 0.2715 1 0.2004 QRE (λ = 0.76) 0 0.8147 0.1408 0.0001 0.5337 0.2004 1 Table 35: p-values for 15 Gate settings in Cluster 2 Cluster 3 The statistical probability parameters (p-value) for various game models against each other in cluster 3 are shown in Tables 36 to 40. PT PT- Attract COBRA (α = 0.15) COBRA (α = 0.5) DOBSS QRE (λ = 0.45) QRE (λ = 0.76) PT 1 0.2873 0.0319 0.0377 0.2037 0.2345 0.1579 PT-Attract 0.2873 1 0.0591 0.0975 0.5339 0.9172 0.5182 COBRA (α = 0.15) 0.0319 0.0591 1 0.7999 0.5924 0.0322 0.146 COBRA (α = 0.5) 0.0377 0.0975 0.7999 1 0.5072 0.0808 0.1851 DOBSS 0.2037 0.5339 0.5924 0.5072 1 0.5329 0.7543 QRE (λ = 0.45) 0.2345 0.9172 0.0322 0.0808 0.5329 1 0.4044 QRE (λ = 0.76) 0.1579 0.5182 0.146 0.1851 0.7543 0.4044 1 Table 36: p-values for 3 Gate settings in Cluster 3 47 PT PT- Attract COBRA (α = 0.15) COBRA (α = 0.5) DOBSS QRE (λ = 0.45) QRE (λ = 0.76) PT 1 0.0006 0.0363 0 0.0058 0.0065 0.0089 PT-Attract 0.0006 1 0.5601 0.0771 0.7168 0.0376 0.0553 COBRA (α = 0.15) 0.0363 0.5601 1 0.137 0.7861 0.7458 0.7208 COBRA (α = 0.5) 0 0.0771 0.137 1 0.118 0.0003 0.0005 DOBSS 0.0058 0.7168 0.7861 0.118 1 0.3775 0.3714 QRE (λ = 0.45) 0.0065 0.0376 0.7458 0.0003 0.3775 1 0.9049 QRE (λ = 0.76) 0.0089 0.0553 0.7208 0.0005 0.3714 0.9049 1 Table 37: p-values for 6 Gate settings in Cluster 3 PT PT- Attract COBRA (α = 0.15) COBRA (α = 0.5) DOBSS QRE (λ = 0.45) QRE (λ = 0.76) PT 1 0.0852 0.0238 0.2964 0.3587 0.2489 0.9153 PT-Attract 0.0852 1 0.0077 0.0528 0.8435 0.2677 0.104 COBRA (α = 0.15) 0.0238 0.0077 1 0.8758 0.0875 0.0098 0.0752 COBRA (α = 0.5) 0.2964 0.0528 0.8758 1 0.1672 0.1583 0.3518 DOBSS 0.3587 0.8435 0.0875 0.1672 1 0.5929 0.3615 QRE (λ = 0.45) 0.2489 0.2677 0.0098 0.1583 0.5929 1 0.3671 QRE (λ = 0.76) 0.9153 0.104 0.0752 0.3518 0.3615 0.3671 1 Table 38: p-values for 9 Gate settings in Cluster 3 48 PT PT- Attract COBRA (α = 0.15) COBRA (α = 0.5) DOBSS QRE (λ = 0.45) QRE (λ = 0.76) PT 1 0.3712 0.8314 0.756 0.8033 0.9002 0.8535 PT-Attract 0.3712 1 0.2687 0.2496 0.0477 0.0706 0.064 COBRA (α = 0.15) 0.8314 0.2687 1 0.8537 0.2994 0.4389 0.3899 COBRA (α = 0.5) 0.756 0.2496 0.8537 1 0.0285 0.0814 0.0945 DOBSS 0.8033 0.0477 0.2994 0.0285 1 0.5996 0.8256 QRE (λ = 0.45) 0.9002 0.0706 0.4389 0.0814 0.5996 1 0.8277 QRE (λ = 0.76) 0.8535 0.064 0.3899 0.0945 0.8256 0.8277 1 Table 39: p-values for 12 Gate settings in Cluster 3 PT PT- Attract COBRA (α = 0.15) COBRA (α = 0.5) DOBSS QRE (λ = 0.45) QRE (λ = 0.76) PT 1 0.0002 0.005 0.4105 0.04 0 0.0002 PT-Attract 0.0002 1 0.4761 0.0001 0.2256 0.4728 0.2194 COBRA (α = 0.15) 0.005 0.4761 1 0.0026 0.5992 0.7619 0.7551 COBRA (α = 0.5) 0.4105 0.0001 0.0026 1 0.0205 0 0.0003 DOBSS 0.04 0.2256 0.5992 0.0205 1 0.34 0.7371 QRE (λ = 0.45) 0 0.4728 0.7619 0 0.34 1 0.2799 QRE (λ = 0.76) 0.0002 0.2194 0.7551 0.0003 0.7371 0.2799 1 Table 40: p-values for 15 Gate settings in Cluster 3 49 Cluster 4 The statistical probability parameters (p-value) for various game models against each other in cluster 4 are shown in Tables 41 to 45. PT PT- Attract COBRA (α = 0.15) COBRA (α = 0.5) DOBSS QRE (λ = 0.45) QRE (λ = 0.76) PT 1 0.5852 0.0577 0.0363 0.0524 0.1727 0.4684 PT-Attract 0.5852 1 0.1119 0.0646 0.1031 0.3786 0.8508 COBRA (α = 0.15) 0.0577 0.1119 1 0 0.7015 0 0.1415 COBRA (α = 0.5) 0.0363 0.0646 0 1 0.7566 0 0.0791 DOBSS 0.0524 0.1031 0.7015 0.7566 1 0.0515 0.1306 QRE (λ = 0.45) 0.1727 0.3786 0 0 0.0515 1 0.492 QRE (λ = 0.76) 0.4684 0.8508 0.1415 0.0791 0.1306 0.492 1 Table 41: p-values for 3 Gate settings in Cluster 4 PT PT- Attract COBRA (α = 0.15) COBRA (α = 0.5) DOBSS QRE (λ = 0.45) QRE (λ = 0.76) PT 1 0.934 0.0694 0.2381 0.76 0.2963 0.2311 PT-Attract 0.934 1 0.02 0.0913 0.693 0.0667 0.1191 COBRA (α = 0.15) 0.0694 0.02 1 0.4484 0.3063 0.0038 0.0048 COBRA (α = 0.5) 0.2381 0.0913 0.4484 1 0.6068 0.0169 0.0203 DOBSS 0.76 0.693 0.3063 0.6068 1 0.3304 0.259 QRE (λ = 0.45) 0.2963 0.0667 0.0038 0.0169 0.3304 1 0.6771 QRE (λ = 0.76) 0.2311 0.1191 0.0048 0.0203 0.259 0.6771 1 Table 42: p-values for 6 Gate settings in Cluster 4 50 PT PT- Attract COBRA (α = 0.15) COBRA (α = 0.5) DOBSS QRE (λ = 0.45) QRE (λ = 0.76) PT 1 0.0407 0 0 0 0 0 PT-Attract 0.0407 1 0.0011 0.0315 0.2394 0.0015 0.0034 COBRA (α = 0.15) 0 0.0011 1 0.0163 0.0037 0.232 0.1388 COBRA (α = 0.5) 0 0.0315 0.0163 1 0.0868 0 0.0021 DOBSS 0 0.2394 0.0037 0.0868 1 0.0004 0.0021 QRE (λ = 0.45) 0 0.0015 0.232 0 0.0004 1 0.2363 QRE (λ = 0.76) 0 0.0034 0.1388 0.0021 0.0021 0.2363 1 Table 43: p-values for 9 Gate settings in Cluster 4 PT PT- Attract COBRA (α = 0.15) COBRA (α = 0.5) DOBSS QRE (λ = 0.45) QRE (λ = 0.76) PT 1 0.733 0.7725 0.1651 0.1895 0.7197 0.9357 PT-Attract 0.733 1 0.9094 0.0752 0.0937 0.9342 0.889 COBRA (α = 0.15) 0.7725 0.9094 1 0.24 0.2539 0.8818 0.8492 COBRA (α = 0.5) 0.1651 0.0752 0.24 1 0.9808 0.0536 0.2609 DOBSS 0.1895 0.0937 0.2539 0.9808 1 0.0713 0.2793 QRE (λ = 0.45) 0.7197 0.9342 0.8818 0.0536 0.0713 1 0.9078 QRE (λ = 0.76) 0.9357 0.889 0.8492 0.2609 0.2793 0.9078 1 Table 44: p-values for 12 Gate settings in Cluster 4 51 PT PT- Attract COBRA (α = 0.15) COBRA (α = 0.5) DOBSS QRE (λ = 0.45) QRE (λ = 0.76) PT 1 0.0521 0 0 0.1005 0.0198 0 PT-Attract 0.0521 1 0.0155 0.002 0.6278 0.3629 0.0961 COBRA (α = 0.15) 0 0.0155 1 0.0552 0.3767 0.446 0.1636 COBRA (α = 0.5) 0 0.002 0.0552 1 0.1355 0.1268 0.0163 DOBSS 0.1005 0.6278 0.3767 0.1355 1 0.8081 0.643 QRE (λ = 0.45) 0.0198 0.3629 0.446 0.1268 0.8081 1 0.8274 QRE (λ = 0.76) 0 0.0961 0.1636 0.0163 0.643 0.8274 1 Table 45: p-values for 15 Gate settings in Cluster 4 52 Chapter Eight: Summary This work expands the domain of game-theoretic techniques as a solution to security allocation problems. Most of the previous work has been done on specific security settings with pre-defined number of targets. With varying number of targets, the game settings can also be expanded over multiple security domains at the same time. Among all the evaluated game models including PT, PT-Attract, COBRA, DOBSS and QRE; PT-ATTRACT and COBRA (α = 0.5) seem to perform better than the others. The results, however, are very close to determine any particular game model to be dominating all the others. Unlike the other experiments based on Stackelberg Games, it has been remarkably surprising to see the below par performance of QRE game model. Also, the PT model seems to perform much better than our expectations. PT-Attract has been one of the superior models in the previous game settings as well, so we see no surprise in its performance [27]. DOBSS evolved as one of the superior models when initially proposed, however, its performance has not been comparable to the other recent models, which the case in these is set of experiments as well [16]. One of the major lessons learnt from this work has been the acknowledgement of human behavior and its influence on the obtained results. The performance seems to vary across settings with same number of gates. People tend to choose gates that have lower rewards but higher probability of guard over those with higher rewards and lower probability of 53 guard. This may refer to their inability to fetch and process all the information that is provided to them. Considering that there has been use of models that do not even account for the irrational human behavior, it becomes critically important that this be taken into account for making any significant decisions. 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Gate 1 Gate 2 Gate 3 Your Rewards 8 6 3 Your Penalties -4 -8 -2 Guards' Rewards 8 5 1 Guards' Penalties -5 -9 -7 Table 46: Payoff Structure for 3 Gate Settings in Cluster 1 Gate 1 Gate 2 Gate 3 Your Rewards 7 6 8 Your Penalties -5 -1 -5 Guards' Rewards 8 7 3 Guards' Penalties -4 -4 -9 Table 47: Payoff Structure for 3 Gate Settings in Cluster 2 Gate 1 Gate 2 Gate 3 Your Rewards 7 3 8 Your Penalties -6 -3 -3 Guards' Rewards 3 6 7 Guards' Penalties -5 -10 -2 Table 48: Payoff Structure for 3 Gate Settings in Cluster 3 Gate 1 Gate 2 Gate 3 Your Rewards 2 7 6 Your Penalties -4 -3 -3 Guards' Rewards 1 8 8 Guards' Penalties -2 -1 -3 Table 49: Payoff Structure for 3 Gate Settings in Cluster 4 58 Appendix B: Payoff Structures for 6 Gate Settings The payoff structures for 6 gate settings are given in the Tables 50 to 53. Gate 1 Gate 2 Gate 3 Gate 4 Gate 5 Gate 6 Your Rewards 4 7 2 2 3 1 Your Penalties -5 -3 -5 -2 -2 -2 Guards' Rewards 2 2 5 8 3 1 Guards' Penalties -3 -5 -7 -6 -1 -4 Table 50: Payoff Structure for 6 Gate Settings in Cluster 1 Gate 1 Gate 2 Gate 3 Gate 4 Gate 5 Gate 6 Your Rewards 10 6 3 10 4 9 Your Penalties -3 -8 -2 -7 -7 -8 Guards' Rewards 4 5 5 9 5 9 Guards' Penalties -6 -1 -9 -2 -10 -6 Table 51: Payoff Structure for 6 Gate Settings in Cluster 2 Gate 1 Gate 2 Gate 3 Gate 4 Gate 5 Gate 6 Your Rewards 7 9 10 6 2 6 Your Penalties -6 -6 -10 -5 -2 -1 Guards' Rewards 8 9 3 8 10 7 Guards' Penalties -9 -9 -2 -5 -4 -1 Table 52: Payoff Structure for 6 Gate Settings in Cluster 3 Gate 1 Gate 2 Gate 3 Gate 4 Gate 5 Gate 6 Your Rewards 3 3 10 6 8 8 Your Penalties -1 -8 -6 -9 -4 -1 Guards' Rewards 4 3 6 3 4 7 Guards' Penalties -9 -1 -7 -4 -3 -9 Table 53: Payoff Structure for 6 Gate Settings in Cluster 4 59 Appendix C: Payoff Structures for 9 Gate Settings The payoff structures for 9 gate settings are given in the Tables 54 to 57. Gate 1 Gate 2 Gate 3 Gate 4 Gate 5 Gate 6 Gate 7 Gate 8 Gate 9 Your Rewards 9 8 7 3 7 7 7 1 10 Your Penalties -4 -6 -10 -3 -2 -8 -1 -1 -5 Guards' Rewards 8 2 8 6 8 10 4 5 4 Guards' Penalties -1 -4 -4 -5 -9 -3 -8 -6 -3 Table 54: Payoff Structure for 9 Gate Settings in Cluster 1 Gate 1 Gate 2 Gate 3 Gate 4 Gate 5 Gate 6 Gate 7 Gate 8 Gate 9 Your Rewards 2 6 5 7 8 4 9 4 3 Your Penalties -4 -8 -6 -10 -8 -3 -2 -8 -3 Guards' Rewards 7 7 1 3 4 9 6 10 2 Guards' Penalties -5 -2 -10 -2 -9 -8 -10 -5 -6 Table 55: Payoff Structure for 9 Gate Settings in Cluster 2 Gate 1 Gate 2 Gate 3 Gate 4 Gate 5 Gate 6 Gate 7 Gate 8 Gate 9 Your Rewards 2 10 6 8 9 4 2 9 8 Your Penalties -9 -2 -1 -7 -9 -6 -3 -8 -2 Guards' Rewards 4 1 8 2 10 2 9 9 8 Guards' Penalties -7 -8 -3 -2 -7 -8 -4 -6 -7 Table 56: Payoff Structure for 9 Gate Settings in Cluster 3 Gate 1 Gate 2 Gate 3 Gate 4 Gate 5 Gate 6 Gate 7 Gate 8 Gate 9 Your Rewards 4 2 1 9 7 10 6 1 7 Your Penalties -9 -4 -2 -9 -2 -8 -5 -6 -4 Guards' Rewards 5 2 10 7 1 6 5 9 6 Guards' Penalties -6 -10 -3 -3 -7 -2 -10 -3 -1 Table 57: Payoff Structure for 9 Gate Settings in Cluster 4 60 Appendix D: Payoff Structures for 12 Gate Settings The payoff structures for 12 gate settings are given in the Tables 58 to 61. Gat e 1 Gat e 2 Gat e 3 Gat e 4 Gat e 5 Gat e 6 Gat e 7 Gat e 8 Gat e 9 Gat e 10 Gat e 11 Gat e 12 Your Rewards 6 3 10 9 4 6 3 9 6 7 2 10 Your Penalties -1 -8 -5 -4 -10 -4 -6 -10 -3 -1 -2 -9 Guards' Rewards 8 1 6 1 6 4 2 5 10 1 3 7 Guards' Penalties -1 -7 -3 -2 -9 -1 -8 -9 -2 -8 -2 -7 Table 58: Payoff Structure for 12 Gate Settings in Cluster 1 Gat e 1 Gat e 2 Gat e 3 Gat e 4 Gat e 5 Gat e 6 Gat e 7 Gat e 8 Gat e 9 Gat e 10 Gat e 11 Gat e 12 Your Rewards 3 3 7 3 4 9 7 5 9 4 10 2 Your Penalties -2 -10 -8 -4 -10 -1 -7 -6 -8 -9 -3 -1 Guards' Rewards 1 2 2 7 4 6 3 9 2 10 1 2 Guards' Penalties -2 -2 -9 -5 -9 -9 -10 -8 -8 -4 -3 -1 Table 59: Payoff Structure for 12 Gate Settings in Cluster 2 Gat e 1 Gat e 2 Gat e 3 Gat e 4 Gat e 5 Gat e 6 Gat e 7 Gat e 8 Gat e 9 Gat e 10 Gat e 11 Gat e 12 Your Rewards 8 6 5 9 2 3 6 10 4 6 4 1 Your Penalties -9 -2 -5 -3 -5 -6 -10 -8 -1 -8 -9 -2 Guards' Rewards 1 7 1 3 7 3 2 7 9 8 3 6 Guards' Penalties -7 -3 -8 -10 -5 -6 -4 -6 -6 -3 -4 -2 Table 60: Payoff Structure for 12 Gate Settings in Cluster 3 61 Gat e 1 Gat e 2 Gat e 3 Gat e 4 Gat e 5 Gat e 6 Gat e 7 Gat e 8 Gat e 9 Gat e 10 Gat e 11 Gat e 12 Your Rewards 7 8 9 5 3 1 7 7 8 8 5 2 Your Penalties -4 -8 -9 -8 -1 -1 -2 -10 -10 -3 -2 -5 Guards' Rewards 1 8 9 9 4 2 8 7 10 5 1 2 Guards' Penalties -6 -5 -6 -5 -1 -8 -1 -8 -4 -5 -10 -2 Table 61: Payoff Structure for 12 Gate Settings in Cluster 4 62 Appendix E: Payoff Structures for 15 Gate Settings The payoff structures for 15 gate settings are given in the Tables 62 to 65. Gate 1 Gate 2 Gate 3 Gate 4 Gate 5 Gate 6 Gate 7 Gate 8 Your Rewards 7 1 4 7 5 7 3 6 Your Penalties -2 -2 -3 -10 -6 -3 -3 -6 Guards' Rewards 3 2 2 10 10 6 10 1 Guards' Penalties -2 -6 -5 -7 -4 -7 -9 -2 Gate 9 Gate 10 Gate 11 Gate 12 Gate 13 Gate 14 Gate 15 Your Rewards 9 10 2 5 3 3 6 Your Penalties -4 -10 -1 -4 -1 -2 -5 Guards' Rewards 8 10 1 7 5 1 9 Guards' Penalties -10 -1 -7 -5 -2 -8 -3 Table 62: Payoff Structure for 15 Gate Settings in Cluster 1 Gate 1 Gate 2 Gate 3 Gate 4 Gate 5 Gate 6 Gate 7 Gate 8 Your Rewards 1 9 4 5 7 4 1 4 Your Penalties -2 -9 -3 -4 -1 -2 -3 -6 Guards' Rewards 2 3 3 9 6 9 10 8 Guards' Penalties -5 -4 -9 -10 -4 -4 -10 -5 Gate 9 Gate 10 Gate 11 Gate 12 Gate 13 Gate 14 Gate 15 Your Rewards 3 7 4 2 8 3 7 Your Penalties -8 -2 -9 -3 -4 -1 -9 Guards' Rewards 1 1 9 2 7 7 9 Guards' Penalties -3 -8 -2 -10 -4 -8 -9 Table 63: Payoff Structure for 15 Gate Settings in Cluster 2 63 Gate 1 Gate 2 Gate 3 Gate 4 Gate 5 Gate 6 Gate 7 Gate 8 Your Rewards 4 4 3 1 7 6 10 5 Your Penalties -10 -4 -1 -9 -10 -7 -8 -4 Guards' Rewards 2 6 5 9 8 10 1 6 Guards' Penalties -7 -6 -6 -8 -8 -2 -5 -5 Gate 9 Gate 10 Gate 11 Gate 12 Gate 13 Gate 14 Gate 15 Your Rewards 1 7 10 2 1 6 8 Your Penalties -4 -6 -5 -5 -8 -8 -8 Guards' Rewards 7 8 3 5 4 6 1 Guards' Penalties -5 -9 -6 -5 -4 -4 -4 Table 64: Payoff Structure for 15 Gate Settings in Cluster 3 Gate 1 Gate 2 Gate 3 Gate 4 Gate 5 Gate 6 Gate 7 Gate 8 Your Rewards 7 3 3 8 10 8 1 2 Your Penalties -3 -1 -2 -10 -1 -3 -5 -9 Guards' Rewards 3 4 1 9 7 5 8 7 Guards' Penalties -3 -1 -5 -1 -7 -9 -5 -8 Gate 9 Gate 10 Gate 11 Gate 12 Gate 13 Gate 14 Gate 15 Your Rewards 4 10 1 9 8 6 6 Your Penalties -9 -10 -2 -3 -5 -9 -5 Guards' Rewards 7 3 4 9 8 4 10 Guards' Penalties -1 -4 -9 -6 -8 -8 -8 Table 65: Payoff Structure for 15 Gate Settings in Cluster 4 64 Appendix F: Defender Strategies for 3 Gate Settings The defender strategies for 3 gate settings are given in the tables 66 to 69. Game Model Gate 1 Gate 2 Gate 3 PT 0.44 0.19 0.37 PT-Attract 0.43 0.19 0.38 COBRA (α = 0.15) 0.38 0.32 0.3 COBRA (α = 0.5) 0.42 0.31 0.27 DOBSS 0.5 0.29 0.21 QRE (λ = 0.45) 0.34 0.39 0.27 QRE (λ = 0.76) 0.35 0.37 0.27 Table 66: Defender Strategies for 3 Gate Settings in Cluster 1 Game Model Gate 1 Gate 2 Gate 3 PT 0.22 0.51 0.27 PT-Attract 0.23 0.49 0.28 COBRA (α = 0.15) 0.3 0.34 0.36 COBRA (α = 0.5) 0.28 0.33 0.39 DOBSS 0.29 0.36 0.35 QRE (λ = 0.45) 0.25 0.28 0.46 QRE (λ = 0.76) 0.26 0.29 0.45 Table 67: Defender Strategies for 3 Gate Settings in Cluster 2 Game Model Gate 1 Gate 2 Gate 3 PT 0.27 0.22 0.5 PT-Attract 0.28 0.24 0.49 COBRA (α = 0.15) 0.42 0.19 0.4 COBRA (α = 0.5) 0.48 0.09 0.44 DOBSS 0.36 0.12 0.52 QRE (λ = 0.45) 0.38 0.35 0.27 QRE (λ = 0.76) 0.4 0.29 0.31 Table 68: Defender Strategies for 3 Gate Settings in Cluster 3 65 Game Model Gate 1 Gate 2 Gate 3 PT 0.1 0.47 0.43 PT-Attract 0.11 0.45 0.44 COBRA (α = 0.15) 0.04 0.43 0.53 COBRA (α = 0.5) 0 0.45 0.55 DOBSS 0.03 0.52 0.46 QRE (λ = 0.45) 0.14 0.39 0.47 QRE (λ = 0.76) 0.15 0.39 0.46 Table 69: Defender Strategies for 3 Gate Settings in Cluster 4 66 Appendix G: Defender Strategies for 6 Gate Settings The defender strategies for 6 gate settings are given in the tables 70 to 73. Game Model Gate 1 Gate 2 Gate 3 Gate 4 Gate 5 Gate 6 PT 0.25 0.54 0.13 0.37 0.46 0.24 PT-Attract 0.28 0.58 0.16 0.43 0.24 0.31 COBRA (α = 0.15) 0.43 0.65 0.22 0.4 0.14 0.16 COBRA (α = 0.5) 0.49 0.66 0.16 0.4 0.15 0.14 DOBSS 0.36 0.62 0.18 0.31 0.45 0.08 QRE (λ = 0.45) 0.31 0.58 0.41 0.5 0.06 0.14 QRE (λ = 0.76) 0.34 0.59 0.36 0.42 0.11 0.18 Table 70: Defender Strategies for 6 Gate Settings in Cluster 1 Game Model Gate 1 Gate 2 Gate 3 Gate 4 Gate 5 Gate 6 PT 0.6 0.2 0.4 0.36 0.15 0.29 PT-Attract 0.63 0.22 0.46 0.21 0.17 0.31 COBRA (α = 0.15) 0.76 0.25 0.11 0.33 0.05 0.5 COBRA (α = 0.5) 0.66 0.28 0.18 0.29 0.16 0.43 DOBSS 0.59 0.27 0.14 0.45 0.16 0.39 QRE (λ = 0.45) 0.56 0.12 0.34 0.25 0.34 0.39 QRE (λ = 0.76) 0.58 0.18 0.28 0.28 0.29 0.39 Table 71: Defender Strategies for 6 Gate Settings in Cluster 2 Game Model Gate 1 Gate 2 Gate 3 Gate 4 Gate 5 Gate 6 PT 0.27 0.34 0.23 0.27 0.2 0.7 PT-Attract 0.33 0.39 0.27 0.34 0.37 0.3 COBRA (α = 0.15) 0.43 0.52 0.45 0.41 0.01 0.18 COBRA (α = 0.5) 0.44 0.52 0.47 0.4 0 0.17 DOBSS 0.35 0.44 0.38 0.32 0 0.51 QRE (λ = 0.45) 0.46 0.5 0.32 0.36 0.12 0.23 QRE (λ = 0.76) 0.45 0.49 0.36 0.36 0.1 0.26 Table 72: Defender Strategies for 6 Gate Settings in Cluster 3 67 Game Model Gate 1 Gate 2 Gate 3 Gate 4 Gate 5 Gate 6 PT 0.39 0.06 0.34 0.13 0.38 0.72 PT-Attract 0.48 0.06 0.36 0.15 0.4 0.55 COBRA (α = 0.15) 0.31 0.08 0.49 0.28 0.33 0.52 COBRA (α = 0.5) 0.36 0 0.5 0.25 0.36 0.53 DOBSS 0.15 0.05 0.47 0.24 0.47 0.62 QRE (λ = 0.45) 0.38 0 0.47 0.26 0.31 0.58 QRE (λ = 0.76) 0.35 0 0.47 0.28 0.34 0.57 Table 73: Defender Strategies for 6 Gate Settings in Cluster 4 68 Appendix H: Defender Strategies for 9 Gate Settings The defender strategies for 9 gate settings are given in the tables 74 to 77. Game Model Gate 1 Gate 2 Gate 3 Gate 4 Gate 5 Gate 6 Gate 7 Gate 8 Gate 9 PT 0.43 0.29 0.15 0.19 0.54 0.19 0.71 0.11 0.4 PT-Attract 0.19 0.31 0.16 0.22 0.57 0.2 0.73 0.19 0.42 COBRA (α = 0.15) 0.27 0.42 0.27 0.03 0.54 0.31 0.61 0 0.55 COBRA (α = 0.5) 0.25 0.42 0.17 0 0.6 0.24 0.73 0 0.6 DOBSS 0.49 0.38 0.26 0.06 0.49 0.29 0.55 0 0.49 QRE (λ = 0.45) 0.23 0.42 0.29 0.22 0.55 0.26 0.61 0 0.42 QRE (λ = 0.76) 0.25 0.43 0.29 0.18 0.53 0.27 0.59 0 0.46 Table 74: Defender Strategies for 9 Gate Settings in Cluster 1 Game Model Gate 1 Gate 2 Gate 3 Gate 4 Gate 5 Gate 6 Gate 7 Gate 8 Gate 9 PT 0.21 0.24 0.28 0.21 0.3 0.44 0.77 0.17 0.38 PT-Attract 0.29 0.14 0.33 0.24 0.34 0.42 0.58 0.2 0.46 COBRA (α = 0.15) 0.13 0.17 0.36 0.3 0.45 0.44 0.6 0.23 0.32 COBRA (α = 0.5) 0 0.23 0.37 0.35 0.52 0.48 0.63 0.14 0.28 DOBSS 0.09 0.32 0.32 0.33 0.41 0.37 0.69 0.21 0.26 QRE (λ = 0.45) 0.17 0.15 0.48 0.15 0.48 0.42 0.62 0.24 0.29 QRE (λ = 0.76) 0.16 0.16 0.45 0.2 0.47 0.39 0.61 0.24 0.32 Table 75: Defender Strategies for 9 Gate Settings in Cluster 2 Game Model Gate 1 Gate 2 Gate 3 Gate 4 Gate 5 Gate 6 Gate 7 Gate 8 Gate 9 PT 0.04 0.67 0.69 0.26 0.23 0.14 0.12 0.26 0.6 PT-Attract 0.07 0.76 0.31 0.32 0.28 0.21 0.25 0.31 0.49 COBRA (α = 0.15) 0 0.78 0.27 0.46 0.41 0.23 0.01 0.4 0.46 COBRA (α = 0.5) 0 0.92 0.3 0.44 0.42 0.02 0 0.42 0.48 DOBSS 0 0.63 0.52 0.37 0.37 0.16 0 0.39 0.56 QRE (λ = 0.45) 0.16 0.66 0.31 0.22 0.38 0.33 0.08 0.37 0.49 QRE (λ = 0.76) 0.12 0.68 0.32 0.3 0.38 0.31 0.05 0.37 0.48 Table 76: Defender Strategies for 9 Gate Settings in Cluster 3 69 Game Model Gate 1 Gate 2 Gate 3 Gate 4 Gate 5 Gate 6 Gate 7 Gate 8 Gate 9 PT 0.15 0.22 0.33 0.3 0.72 0.36 0.38 0.08 0.49 PT-Attract 0.18 0.29 0.29 0.33 0.8 0.34 0.43 0.11 0.25 COBRA (α = 0.15) 0.32 0 0 0.41 1 0.39 0.58 0 0.3 COBRA (α = 0.5) 0.24 0.2 0.06 0.44 0.77 0.43 0.51 0 0.35 DOBSS 0.23 0.16 0 0.44 0.67 0.5 0.45 0 0.54 QRE (λ = 0.45) 0.33 0.4 0.09 0.33 0.67 0.3 0.56 0.09 0.23 QRE (λ = 0.76) 0.31 0.35 0.05 0.36 0.69 0.34 0.54 0.07 0.28 Table 77: Defender Strategies for 9 Gate Settings in Cluster 4 70 Appendix I: Defender Strategies for 12 Gate Settings The defender strategies for 12 gate settings are given in the tables 78 to 81. Game Model Gate 1 Gate 2 Gate 3 Gate 4 Gate 5 Gate 6 PT 0.71 0.07 0.42 0.45 0.08 0.33 PT-Attract 0.28 0.1 0.47 0.51 0.1 0.4 COBRA (α = 0.15) 0.19 0.12 0.42 0.64 0.17 0.35 COBRA (α = 0.5) 0.21 0 0.43 0.8 0 0.37 DOBSS 0.57 0.07 0.52 0.52 0.13 0.38 QRE (λ = 0.45) 0.24 0.28 0.38 0.41 0.33 0.17 QRE (λ = 0.76) 0.26 0.23 0.4 0.49 0.28 0.24 Game Model Gate 7 Gate 8 Gate 9 Gate 10 Gate 11 Gate 12 PT 0.1 0.21 0.41 0.75 0.21 0.26 PT-Attract 0.15 0.24 0.29 0.84 0.34 0.3 COBRA (α = 0.15) 0.15 0.43 0.23 0.76 0.06 0.5 COBRA (α = 0.5) 0 0.44 0.24 0.98 0 0.54 DOBSS 0.08 0.36 0.42 0.59 0 0.41 QRE (λ = 0.45) 0.32 0.48 0.27 0.68 0 0.46 QRE (λ = 0.76) 0.26 0.45 0.27 0.67 0 0.45 Table 78: Defender Strategies for 12 Gate Settings in Cluster 1 Game Model Gate 1 Gate 2 Gate 3 Gate 4 Gate 5 Gate 6 PT 0.42 0.07 0.24 0.22 0.1 0.87 PT-Attract 0.55 0.1 0.29 0.32 0.14 0.58 COBRA (α = 0.15) 0.36 0 0.43 0.16 0.05 0.61 COBRA (α = 0.5) 0.34 0.13 0.42 0.29 0.2 0.54 DOBSS 0.29 0.11 0.36 0.21 0.17 0.74 QRE (λ = 0.45) 0 0.06 0.51 0.29 0.36 0.64 QRE (λ = 0.76) 0.09 0.09 0.48 0.27 0.32 0.62 71 Game Model Gate 7 Gate 8 Gate 9 Gate 10 Gate 11 Gate 12 PT 0.27 0.23 0.3 0.11 0.62 0.55 PT-Attract 0.33 0.3 0.35 0.16 0.66 0.22 COBRA (α = 0.15) 0.48 0.34 0.57 0.08 0.78 0.14 COBRA (α = 0.5) 0.45 0.38 0.5 0.22 0.51 0.02 DOBSS 0.39 0.31 0.44 0.19 0.65 0.14 QRE (λ = 0.45) 0.53 0.41 0.53 0.22 0.45 0 QRE (λ = 0.76) 0.51 0.39 0.52 0.21 0.5 0 Table 79: Defender Strategies for 12 Gate Settings in Cluster 2 Game Model Gate 1 Gate 2 Gate 3 Gate 4 Gate 5 Gate 6 PT 0.25 0.64 0.31 0.62 0.14 0.16 PT-Attract 0.31 0.34 0.39 0.7 0.23 0.24 COBRA (α = 0.15) 0.46 0.28 0.45 0.75 0.14 0.24 COBRA (α = 0.5) 0.51 0.34 0.5 0.8 0.01 0.15 DOBSS 0.39 0.57 0.36 0.63 0.08 0.17 QRE (λ = 0.45) 0.47 0.32 0.47 0.69 0.19 0.28 QRE (λ = 0.76) 0.46 0.33 0.46 0.66 0.17 0.27 Game Model Gate 7 Gate 8 Gate 9 Gate 10 Gate 11 Gate 12 PT 0.17 0.34 0.78 0.22 0.13 0.25 PT-Attract 0.22 0.39 0.43 0.28 0.18 0.3 COBRA (α = 0.15) 0.35 0.45 0.39 0.26 0.25 0 COBRA (α = 0.5) 0.32 0.49 0.43 0.31 0.16 0 DOBSS 0.29 0.48 0.51 0.33 0.2 0 QRE (λ = 0.45) 0.3 0.42 0.45 0.23 0.22 0 QRE (λ = 0.76) 0.32 0.43 0.43 0.24 0.24 0 Table 80: Defender Strategies for 12 Gate Settings in Cluster 3 72 Game Model Gate 1 Gate 2 Gate 3 Gate 4 Gate 5 Gate 6 PT 0.4 0.25 0.25 0.15 0.57 0.28 PT-Attract 0.44 0.27 0.27 0.17 0.51 0.41 COBRA (α = 0.15) 0.51 0.41 0.42 0.24 0.34 0 COBRA (α = 0.5) 0.62 0.45 0.46 0.16 0.28 0 DOBSS 0.47 0.38 0.4 0.24 0.28 0 QRE (λ = 0.45) 0.56 0.39 0.41 0.32 0.02 0.14 QRE (λ = 0.76) 0.56 0.4 0.41 0.31 0.11 0.01 Game Model Gate 7 Gate 8 Gate 9 Gate 10 Gate 11 Gate 12 PT 0.61 0.17 0.2 0.53 0.51 0.1 PT-Attract 0.28 0.19 0.21 0.56 0.56 0.12 COBRA (α = 0.15) 0.28 0.31 0.36 0.62 0.5 0 COBRA (α = 0.5) 0.22 0.28 0.35 0.6 0.59 0 DOBSS 0.57 0.3 0.34 0.56 0.45 0.02 QRE (λ = 0.45) 0.24 0.43 0.31 0.51 0.66 0 QRE (λ = 0.76) 0.26 0.4 0.32 0.53 0.62 0.06 Table 81: Defender Strategies for 12 Gate Settings in Cluster 4 73 Appendix J: Defender Strategies for 15 Gate Settings The defender strategies for 15 gate settings are given in the tables 82 to 85. Game Model Gate 1 Gate 2 Gate 3 Gate 4 Gate 5 Gate 6 Gate 7 Gate 8 PT 0.6 0.11 0.32 0.17 0.2 0.48 0.24 0.24 PT-Attract 0.47 0.17 0.36 0.12 0.23 0.51 0.29 0.27 COBRA (α = 0.15) 0.65 0 0.33 0.34 0.31 0.58 0.19 0.38 COBRA (α = 0.5) 0.66 0 0.36 0.3 0.29 0.64 0.14 0.4 DOBSS 0.59 0 0.32 0.31 0.3 0.53 0.21 0.36 QRE (λ = 0.45) 0.36 0.01 0.38 0.38 0.3 0.54 0.43 0.28 QRE (λ = 0.76) 0.45 0 0.38 0.37 0.3 0.53 0.36 0.35 Game Model Gate 9 Gate 10 Gate 11 Gate 12 Gate 13 Gate 14 Gate 15 PT 0.47 0.24 0.43 0.3 0.55 0.35 0.29 PT-Attract 0.5 0.12 0.53 0.34 0.34 0.4 0.28 COBRA (α = 0.15) 0.62 0.28 0 0.38 0.29 0.23 0.42 COBRA (α = 0.5) 0.63 0.21 0 0.43 0.37 0.23 0.36 DOBSS 0.56 0.41 0.09 0.36 0.32 0.25 0.39 QRE (λ = 0.45) 0.59 0.16 0.27 0.39 0.15 0.45 0.3 QRE (λ = 0.76) 0.58 0.19 0.21 0.38 0.18 0.4 0.32 Table 82: Defender Strategies for 15 Gate Settings in Cluster 1 Game Model Gate 1 Gate 2 Gate 3 Gate 4 Gate 5 Gate 6 Gate 7 Gate 8 PT 0.16 0.26 0.36 0.33 0.82 0.48 0.1 0.18 PT-Attract 0.28 0.29 0.42 0.39 0.44 0.35 0.17 0.22 COBRA (α = 0.15) 0 0.46 0.45 0.47 0.42 0.33 0 0.3 COBRA (α = 0.5) 0 0.54 0.48 0.53 0.44 0.34 0 0.24 DOBSS 0 0.43 0.38 0.4 0.7 0.44 0 0.26 QRE (λ = 0.45) 0 0.36 0.48 0.47 0.41 0.31 0.28 0.27 QRE (λ = 0.76) 0 0.4 0.47 0.46 0.43 0.31 0.19 0.28 74 Game Model Gate 9 Gate 10 Gate 11 Gate 12 Gate 13 Gate 14 Gate 15 PT 0.09 0.63 0.11 0.2 0.46 0.62 0.2 PT-Attract 0.12 0.7 0.14 0.28 0.41 0.56 0.23 COBRA (α = 0.15) 0.16 0.73 0.2 0.18 0.38 0.54 0.38 COBRA (α = 0.5) 0.01 0.94 0.11 0.01 0.4 0.56 0.4 DOBSS 0.15 0.62 0.2 0.12 0.55 0.4 0.35 QRE (λ = 0.45) 0.1 0.63 0.11 0.36 0.35 0.46 0.41 QRE (λ = 0.76) 0.16 0.65 0.13 0.32 0.36 0.44 0.4 Table 83: Defender Strategies for 15 Gate Settings in Cluster 2 Game Model Gate 1 Gate 2 Gate 3 Gate 4 Gate 5 Gate 6 Gate 7 Gate 8 PT 0.15 0.41 0.9 0.07 0.24 0.31 0.39 0.46 PT-Attract 0.19 0.48 0.53 0.1 0.28 0.15 0.43 0.44 COBRA (α = 0.15) 0.23 0.42 0.54 0 0.4 0.16 0.57 0.45 COBRA (α = 0.5) 0.02 0.18 0.06 0 0.33 0.02 0.54 0.3 DOBSS 0.23 0.4 0.54 0.02 0.36 0.4 0.51 0.46 QRE (λ = 0.45) 0.34 0.38 0.45 0.21 0.4 0.17 0.5 0.38 QRE (λ = 0.76) 0.32 0.38 0.44 0.17 0.4 0.17 0.52 0.39 Game Model Gate 9 Gate 10 Gate 11 Gate 12 Gate 13 Gate 14 Gate 15 PT 0.21 0.4 0.54 0.22 0.08 0.28 0.34 PT-Attract 0.3 0.45 0.59 0.29 0.11 0.32 0.38 COBRA (α = 0.15) 0 0.53 0.66 0.14 0 0.4 0.5 COBRA (α = 0.5) 0 0.43 0.47 0 0 0.22 0.45 DOBSS 0.03 0.47 0.61 0.16 0.02 0.37 0.45 QRE (λ = 0.45) 0.15 0.49 0.54 0.22 0.08 0.3 0.41 QRE (λ = 0.76) 0.12 0.48 0.56 0.21 0.08 0.31 0.44 Table 84: Defender Strategies for 15 Gate Settings in Cluster 3 75 Game Model Gate 1 Gate 2 Gate 3 Gate 4 Gate 5 Gate 6 Gate 7 Gate 8 PT 0.49 0.58 0.37 0.2 0.86 0.53 0.04 0.05 PT-Attract 0.56 0.36 0.49 0.18 0.56 0.59 0.08 0.07 COBRA (α = 0.15) 0.6 0.35 0.32 0.18 0.56 0.65 0 0.01 COBRA (α = 0.5) 0.71 0.27 0.15 0.26 0.61 0.76 0 0 DOBSS 0.52 0.29 0.23 0.34 0.74 0.56 0 0.02 QRE (λ = 0.45) 0.41 0 0.29 0.16 0.6 0.61 0.11 0.24 QRE (λ = 0.76) 0.46 0.09 0.32 0.18 0.6 0.61 0.06 0.19 Game Model Gate 9 Gate 10 Gate 11 Gate 12 Gate 13 Gate 14 Gate 15 PT 0.1 0.25 0.13 0.56 0.38 0.17 0.3 PT-Attract 0.13 0.28 0.27 0.45 0.43 0.2 0.36 COBRA (α = 0.15) 0.18 0.45 0 0.45 0.54 0.3 0.43 COBRA (α = 0.5) 0.01 0.49 0 0.51 0.6 0.23 0.43 DOBSS 0.17 0.41 0 0.6 0.47 0.28 0.38 QRE (λ = 0.45) 0.09 0.43 0.2 0.49 0.51 0.42 0.45 QRE (λ = 0.76) 0.13 0.44 0.1 0.49 0.51 0.4 0.43 Table 85: Defender Strategies for 15 Gate Settings in Cluster 4 76 Appendix K: Participants’ Choices for Cluster 1 The participants’ choices for cluster 1 are illustrated in the Tables 86 to 90 and Figures 23 to 27. Game Model Gate 1 Gate 2 Gate 3 PT 0.19 0.52 0.29 PT-Attract 0.19 0.62 0.19 COBRA (Alpha = 0.15) 0.1 0.71 0.19 COBRA (Alpha = 0.5) 0.14 0.62 0.24 DOBSS 0.1 0.71 0.19 QRE (Lambda = 0.45) 0.1 0.71 0.19 QRE (Lambda = 0.76) 0.14 0.48 0.38 Table 86: Participants’ Choices (in fraction) for 3 Gate Settings in Cluster 1 Figure 23: Participants’ Choices against Gate Number for 3 Gate Settings in Cluster 1 77 Game Model Gate 1 Gate 2 Gate 3 Gate 4 Gate 5 Gate 6 PT 0.52 0.14 0 0.19 0.14 0 PT-Attract 0.52 0 0.05 0.33 0.1 0 COBRA (Alpha = 0.15) 0.33 0 0.14 0.38 0.14 0 COBRA (Alpha = 0.5) 0.38 0 0.19 0.29 0.1 0.05 DOBSS 0.57 0.05 0.29 0.1 0 0 QRE (Lambda = 0.45) 0.62 0.05 0.05 0.14 0.05 0.1 QRE (Lambda = 0.76) 0.71 0.1 0.1 0.1 0 0 Table 87: Participants’ Choices (in fraction) for 6 Gate Settings in Cluster 1 Figure 24: Participants’ Choices against Gate Number for 6 Gate Settings in Cluster 1 78 Game Model Gate 1 Gate 2 Gate 3 Gate 4 Gate 5 Gate 6 Gate 7 Gate 8 Gate 9 PT 0 0.05 0.05 0.05 0.14 0 0.67 0.05 0 PT-Attract 0 0.1 0 0 0.05 0.14 0.71 0 0 COBRA (Alpha = 0.15) 0.1 0.14 0 0.05 0 0.1 0.62 0 0 COBRA (Alpha = 0.5) 0.19 0 0 0.1 0.05 0.1 0.52 0 0.05 DOBSS 0.1 0.05 0.29 0 0.14 0.14 0.24 0 0.05 QRE (Lambda = 0.45) 0.1 0.05 0.1 0.1 0.38 0 0.29 0 0 QRE (Lambda = 0.76) 0 0.29 0.05 0.05 0.29 0.1 0.19 0.05 0 Table 88: Participants’ Choices (in fraction) for 9 Gate Settings in Cluster 1 Figure 25: Participants’ Choices against Gate Number for 9 Gate Settings in Cluster 1 79 Game Model Gate 1 Gate 2 Gate 3 Gate 4 Gate 5 Gate 6 PT 0 0.05 0.1 0 0.14 0.33 PT-Attract 0 0 0 0 0.05 0.76 COBRA (Alpha = 0.15) 0 0.05 0.14 0 0.24 0.33 COBRA (Alpha = 0.5) 0 0 0.05 0 0.05 0.1 DOBSS 0 0 0.05 0 0 0.05 QRE (Lambda = 0.45) 0 0 0.05 0 0 0 QRE (Lambda = 0.76) 0 0 0.05 0 0.38 0.1 Game Model Gate 7 Gate 8 Gate 9 Gate 10 Gate 11 Gate 12 PT 0 0 0 0 0.19 0.19 PT-Attract 0 0 0 0 0.05 0.14 COBRA (Alpha = 0.15) 0 0 0 0 0.05 0.19 COBRA (Alpha = 0.5) 0 0 0 0 0.76 0.05 DOBSS 0 0 0 0.05 0.86 0 QRE (Lambda = 0.45) 0 0 0 0 0.95 0 QRE (Lambda = 0.76) 0 0 0 0 0.48 0 Table 89: Participants’ Choices (in fraction) for 12 Gate Settings in Cluster 1 Figure 26: Participants’ Choices against Gate Number for 12 Gate Settings in Cluster 1 80 Game Model Gate 1 Gate 2 Gate 3 Gate 4 Gate 5 Gate 6 Gate 7 Gate 8 PT 0 0 0 0 0.43 0 0 0 PT-Attract 0 0 0.24 0 0.38 0 0 0 COBRA (Alpha = 0.15) 0 0 0.05 0 0.81 0 0 0 COBRA (Alpha = 0.5) 0 0.05 0.05 0 0.62 0 0 0 DOBSS 0 0 0.19 0 0.48 0 0.05 0 QRE (Lambda = 0.45) 0 0.1 0.1 0 0.43 0.05 0 0 QRE (Lambda = 0.76) 0 0.05 0.1 0 0.52 0.05 0 0 Game Model Gate 9 Gate 10 Gate 11 Gate 12 Gate 13 Gate 14 Gate 15 PT 0 0 0.05 0 0.52 0 0 PT-Attract 0 0 0.1 0 0.29 0 0 COBRA (Alpha = 0.15) 0 0 0.05 0 0.1 0 0 COBRA (Alpha = 0.5) 0 0.05 0.05 0 0.19 0 0 DOBSS 0 0.1 0 0 0.19 0 0 QRE (Lambda = 0.45) 0 0 0.14 0 0.19 0 0 QRE (Lambda = 0.76) 0 0.1 0.05 0 0.14 0 0 Table 90: Participants’ Choices (in fraction) for 15 Gate Settings in Cluster 1 Figure 27: Participants’ Choices against Gate Number for 15 Gate Settings in Cluster 1 81 Appendix L: Participants’ Choices for Cluster 2 The participants’ choices for cluster 1 are illustrated in the Tables 91 to 95 and Figures 28 to 32. Game Model Gate 1 Gate 2 Gate 3 PT 0.48 0.44 0.08 PT-Attract 0.32 0.56 0.12 COBRA (Alpha = 0.15) 0.28 0.68 0.04 COBRA (Alpha = 0.5) 0.24 0.72 0.04 DOBSS 0.32 0.6 0.08 QRE (Lambda = 0.45) 0.24 0.76 0 QRE (Lambda = 0.76) 0.16 0.84 0 Table 91: Participants’ Choices (in fraction) for 3 Gate Settings in Cluster 2 Figure 28: Participants’ Choices against Gate Number for 3 Gate Settings in Cluster 2 82 Game Model Gate 1 Gate 2 Gate 3 Gate 4 Gate 5 Gate 6 PT 0.36 0.16 0.12 0 0.16 0.2 PT-Attract 0.16 0 0.12 0.72 0 0 COBRA (Alpha = 0.15) 0.08 0 0.52 0.16 0.2 0.04 COBRA (Alpha = 0.5) 0.12 0 0.6 0.2 0.08 0 DOBSS 0.32 0.12 0.48 0.04 0 0.04 QRE (Lambda = 0.45) 0.76 0.08 0 0.12 0 0.04 QRE (Lambda = 0.76) 0.72 0.04 0 0.2 0 0.04 Table 92: Participants’ Choices (in fraction) for 6 Gate Settings in Cluster 2 Figure 29: Participants’ Choices against Gate Number for 6 Gate Settings in Cluster 2 83 Game Model Gate 1 Gate 2 Gate 3 Gate 4 Gate 5 Gate 6 Gate 7 Gate 8 Gate 9 PT 0.04 0.08 0 0.04 0.04 0 0.76 0.04 0 PT-Attract 0 0.04 0.04 0.04 0 0.12 0.72 0 0.04 COBRA (Alpha = 0.15) 0.08 0.08 0 0.04 0.04 0.04 0.72 0 0 COBRA (Alpha = 0.5) 0 0.04 0.04 0 0 0.08 0.72 0.08 0.04 DOBSS 0 0.04 0.2 0 0.36 0 0.32 0.08 0 QRE (Lambda = 0.45) 0 0.08 0 0.04 0.44 0.04 0.32 0.04 0.04 QRE (Lambda = 0.76) 0.08 0.12 0.16 0 0.24 0 0.32 0 0.08 Table 93: Participants’ Choices (in fraction) for 9 Gate Settings in Cluster 2 Figure 30: Participants’ Choices against Gate Number for 9 Gate Settings in Cluster 2 84 Game Model Gate 1 Gate 2 Gate 3 Gate 4 Gate 5 Gate 6 PT 0 0 0 0 0.04 0.48 PT-Attract 0 0 0.04 0 0.08 0.6 COBRA (Alpha = 0.15) 0 0.04 0.04 0 0.16 0.4 COBRA (Alpha = 0.5) 0 0.04 0.04 0 0.08 0.28 DOBSS 0 0 0 0 0 0.2 QRE (Lambda = 0.45) 0.08 0 0.08 0 0 0.12 QRE (Lambda = 0.76) 0 0 0.04 0 0.16 0.16 Game Model Gate 7 Gate 8 Gate 9 Gate 10 Gate 11 Gate 12 PT 0 0.04 0.08 0.04 0.12 0.2 PT-Attract 0 0 0 0 0.08 0.2 COBRA (Alpha = 0.15) 0 0 0 0 0.08 0.28 COBRA (Alpha = 0.5) 0 0 0 0 0.56 0 DOBSS 0 0 0 0 0.8 0 QRE (Lambda = 0.45) 0 0 0 0 0.68 0.04 QRE (Lambda = 0.76) 0 0 0 0 0.6 0.04 Table 94: Participants’ Choices (in fraction) for 12 Gate Settings in Cluster 2 Figure 31: Participants’ Choices against Gate Number for 12 Gate Settings in Cluster 2 85 Game Model Gate 1 Gate 2 Gate 3 Gate 4 Gate 5 Gate 6 Gate 7 Gate 8 PT 0.08 0 0 0 0.16 0.04 0 0 PT-Attract 0 0.04 0.16 0 0.52 0 0.04 0 COBRA (Alpha = 0.15) 0 0 0.12 0 0.48 0 0 0 COBRA (Alpha = 0.5) 0.04 0 0.12 0 0.44 0 0 0 DOBSS 0.04 0 0.28 0 0.4 0 0 0 QRE (Lambda = 0.45) 0.04 0.04 0.24 0 0.48 0 0 0 QRE (Lambda = 0.76) 0 0 0.12 0 0.44 0.04 0 0.04 Game Model Gate 9 Gate 10 Gate 11 Gate 12 Gate 13 Gate 14 Gate 15 PT 0 0.04 0 0 0.6 0.04 0.04 PT-Attract 0 0 0.04 0 0.2 0 0 COBRA (Alpha = 0.15) 0 0 0.04 0 0.36 0 0 COBRA (Alpha = 0.5) 0.04 0.04 0 0 0.32 0 0 DOBSS 0 0 0.08 0 0.2 0 0 QRE (Lambda = 0.45) 0 0 0.04 0 0.16 0 0 QRE (Lambda = 0.76) 0 0 0 0 0.36 0 0 Table 95: Participants’ Choices (in fraction) for 15 Gate Settings in Cluster 2 Figure 32: Participants’ Choices against Gate Number for 15 Gate Settings in Cluster 2 86 Appendix M: Participants’ Choices for Cluster 3 The participants’ choices for cluster 1 are illustrated in the Tables 96 to 100 and Figures 33 to 37. Game Model Gate 1 Gate 2 Gate 3 PT 0.22 0.43 0.35 PT-Attract 0.35 0.48 0.17 COBRA (Alpha = 0.15) 0.17 0.74 0.09 COBRA (Alpha = 0.5) 0.22 0.65 0.13 DOBSS 0.22 0.57 0.22 QRE (Lambda = 0.45) 0.13 0.83 0.04 QRE (Lambda = 0.76) 0.17 0.74 0.09 Table 96: Participants’ Choices (in fraction) for 3 Gate Settings in Cluster 3 Figure 33: Participants’ Choices against Gate Number for 3 Gate Settings in Cluster 3 87 Game Model Gate 1 Gate 2 Gate 3 Gate 4 Gate 5 Gate 6 PT 0.26 0.26 0.09 0.13 0.22 0.04 PT-Attract 0.04 0 0.17 0.57 0.17 0.04 COBRA (Alpha = 0.15) 0.17 0 0.43 0.13 0.26 0 COBRA (Alpha = 0.5) 0.17 0.09 0.39 0.3 0.04 0 DOBSS 0.09 0.04 0.52 0.04 0.17 0.13 QRE (Lambda = 0.45) 0.7 0.09 0 0.17 0.04 0 QRE (Lambda = 0.76) 0.57 0.13 0.09 0.09 0.04 0.09 Table 97: Participants’ Choices (in fraction) for 6 Gate Settings in Cluster 3 Figure 34: Participants’ Choices against Gate Number for 6 Gate Settings in Cluster 3 88 Game Model Gate 1 Gate 2 Gate 3 Gate 4 Gate 5 Gate 6 Gate 7 Gate 8 Gate 9 PT 0.04 0.04 0 0.09 0.13 0.04 0.61 0.04 0 PT-Attract 0.04 0.13 0 0.13 0 0.17 0.52 0 0 COBRA (Alpha = 0.15) 0.04 0.13 0.04 0 0.04 0.09 0.61 0.04 0 COBRA (Alpha = 0.5) 0.22 0.04 0.04 0.04 0.13 0 0.48 0.04 0 DOBSS 0.09 0 0.26 0.04 0.17 0.13 0.17 0.13 0 QRE (Lambda = 0.45) 0.04 0.17 0 0.13 0.26 0.04 0.35 0 0 QRE (Lambda = 0.76) 0.13 0.22 0.09 0.04 0.13 0.04 0.35 0 0 Table 98: Participants’ Choices (in fraction) for 9 Gate Settings in Cluster 3 Figure 35: Participants’ Choices against Gate Number for 9 Gate Settings in Cluster 3 89 Game Model Gate 1 Gate 2 Gate 3 Gate 4 Gate 5 Gate 6 PT 0 0.04 0.17 0 0.09 0.26 PT-Attract 0 0.04 0.17 0.04 0 0.43 COBRA (Alpha = 0.15) 0 0.04 0 0 0.13 0.35 COBRA (Alpha = 0.5) 0 0 0.04 0 0 0.13 DOBSS 0 0 0.04 0.04 0 0.09 QRE (Lambda = 0.45) 0.17 0 0.04 0 0 0.04 QRE (Lambda = 0.76) 0.04 0 0 0 0.13 0.17 Game Model Gate 7 Gate 8 Gate 9 Gate 10 Gate 11 Gate 12 PT 0 0 0 0.17 0.04 0.22 PT-Attract 0.09 0 0 0.04 0.09 0.09 COBRA (Alpha = 0.15) 0 0 0 0.09 0.04 0.35 COBRA (Alpha = 0.5) 0 0.04 0.04 0 0.61 0.13 DOBSS 0.04 0.04 0 0.13 0.52 0.09 QRE (Lambda = 0.45) 0 0.04 0 0 0.65 0.04 QRE (Lambda = 0.76) 0 0 0.04 0.04 0.43 0.13 Table 99: Participants’ Choices (in fraction) for 12 Gate Settings in Cluster 3 Figure 36: Participants’ Choices against Gate Number for 12 Gate Settings in Cluster 3 90 Game Model Gate 1 Gate 2 Gate 3 Gate 4 Gate 5 Gate 6 Gate 7 Gate 8 PT 0.04 0.04 0 0 0.26 0 0.09 0 PT-Attract 0 0.17 0.3 0 0.17 0.04 0 0.04 COBRA (Alpha = 0.15) 0.09 0.04 0.3 0.04 0.3 0 0 0 COBRA (Alpha = 0.5) 0.09 0.09 0.17 0 0.39 0 0 0 DOBSS 0.13 0.04 0.26 0.09 0.35 0 0 0 QRE (Lambda = 0.45) 0.04 0.04 0.17 0.04 0.48 0 0 0 QRE (Lambda = 0.76) 0.09 0.04 0.22 0.04 0.26 0.04 0 0 Game Model Gate 9 Gate 10 Gate 11 Gate 12 Gate 13 Gate 14 Gate 15 PT 0.04 0.04 0 0.04 0.39 0 0.04 PT-Attract 0.04 0 0.04 0 0.17 0 0 COBRA (Alpha = 0.15) 0 0 0.04 0.04 0.13 0 0 COBRA (Alpha = 0.5) 0 0 0 0 0.26 0 0 DOBSS 0 0 0.04 0 0.04 0 0.04 QRE (Lambda = 0.45) 0 0.04 0.09 0 0.04 0 0.04 QRE (Lambda = 0.76) 0 0 0.09 0 0.17 0.04 0 Table 100: Participants’ Choices (in fraction) for 15 Gate Settings in Cluster 3 Figure 37: Participants’ Choices against Gate Number for 15 Gate Settings in Cluster 3 91 Appendix N: Participants’ Choices for Cluster 4 The participants’ choices for cluster 1 are illustrated in the Tables 101 to 105 and Figures 38 to 42. Game Model Gate 1 Gate 2 Gate 3 PT 0.36 0.41 0.23 PT-Attract 0.23 0.32 0.45 COBRA (Alpha = 0.15) 0.18 0.64 0.18 COBRA (Alpha = 0.5) 0.18 0.55 0.27 DOBSS 0.18 0.55 0.27 QRE (Lambda = 0.45) 0.18 0.59 0.23 QRE (Lambda = 0.76) 0.23 0.45 0.32 Table 101: Participants’ Choices (in fraction) for 3 Gate Settings in Cluster 4 Figure 38: Participants’ Choices against Gate Number for 3 Gate Settings in Cluster 4 92 Game Model Gate 1 Gate 2 Gate 3 Gate 4 Gate 5 Gate 6 PT 0.5 0.14 0 0.14 0.14 0.09 PT-Attract 0.32 0.05 0.05 0.5 0.09 0 COBRA (Alpha = 0.15) 0.23 0.09 0.23 0.14 0.27 0.05 COBRA (Alpha = 0.5) 0.27 0 0.32 0.32 0.09 0 DOBSS 0.32 0.09 0.45 0.14 0 0 QRE (Lambda = 0.45) 0.64 0.09 0 0.23 0 0.05 QRE (Lambda = 0.76) 0.68 0.23 0 0.05 0 0.05 Table 102: Participants’ Choices (in fraction) for 6 Gate Settings in Cluster 4 Figure 39: Participants’ Choices against Gate Number for 6 Gate Settings in Cluster 4 93 Game Model Gate 1 Gate 2 Gate 3 Gate 4 Gate 5 Gate 6 Gate 7 Gate 8 Gate 9 PT 0.14 0.05 0.05 0 0 0 0.64 0.05 0.09 PT-Attract 0.09 0.09 0.05 0 0.05 0.14 0.59 0 0 COBRA (Alpha = 0.15) 0.14 0.05 0.05 0 0.05 0.18 0.5 0 0.05 COBRA (Alpha = 0.5) 0.18 0 0.05 0 0 0.09 0.64 0 0.05 DOBSS 0.09 0 0.18 0 0.32 0 0.36 0.05 0 QRE (Lambda = 0.45) 0.09 0.09 0.14 0.05 0.45 0 0.14 0 0.05 QRE (Lambda = 0.76) 0.05 0.14 0.14 0 0.45 0 0.18 0.05 0 Table 103: Participants’ Choices (in fraction) for 9 Gate Settings in Cluster 4 Figure 40: Participants’ Choices against Gate Number for 9 Gate Settings in Cluster 4 94 Game Model Gate 1 Gate 2 Gate 3 Gate 4 Gate 5 Gate 6 PT 0 0.05 0.14 0.09 0.05 0.32 PT-Attract 0 0.05 0 0 0.09 0.36 COBRA (Alpha = 0.15) 0 0.14 0.09 0 0.14 0.27 COBRA (Alpha = 0.5) 0 0 0.05 0 0 0.27 DOBSS 0.09 0.05 0.18 0 0 0.09 QRE (Lambda = 0.45) 0.09 0.05 0 0 0 0.09 QRE (Lambda = 0.76) 0.05 0 0 0 0.27 0.23 Game Model Gate 7 Gate 8 Gate 9 Gate 10 Gate 11 Gate 12 PT 0 0 0 0 0.05 0.32 PT-Attract 0 0 0.05 0 0.14 0.32 COBRA (Alpha = 0.15) 0 0 0 0 0.09 0.27 COBRA (Alpha = 0.5) 0 0.05 0 0 0.55 0.09 DOBSS 0 0 0 0 0.55 0.05 QRE (Lambda = 0.45) 0 0 0 0 0.73 0.05 QRE (Lambda = 0.76) 0 0 0 0 0.36 0.09 Table 104: Participants’ Choices (in fraction) for 12 Gate Settings in Cluster 4 Figure 41: Participants’ Choices against Gate Number for 12 Gate Settings in Cluster 4 95 Game Model Gate 1 Gate 2 Gate 3 Gate 4 Gate 5 Gate 6 Gate 7 Gate 8 PT 0.05 0.05 0.05 0 0.14 0 0 0 PT-Attract 0 0.09 0.18 0 0.32 0 0 0 COBRA (Alpha = 0.15) 0.05 0 0.05 0 0.55 0 0.05 0 COBRA (Alpha = 0.5) 0.09 0 0.05 0 0.45 0.05 0.09 0 DOBSS 0.14 0.05 0.14 0 0.27 0 0 0 QRE (Lambda = 0.45) 0.09 0 0.14 0 0.5 0 0 0 QRE (Lambda = 0.76) 0.18 0 0.05 0 0.36 0.05 0.05 0 Game Model Gate 9 Gate 10 Gate 11 Gate 12 Gate 13 Gate 14 Gate 15 PT 0.05 0.09 0 0 0.55 0.05 0 PT-Attract 0 0.05 0.14 0.05 0.18 0 0 COBRA (Alpha = 0.15) 0 0.05 0.05 0.05 0.18 0 0 COBRA (Alpha = 0.5) 0 0.05 0.05 0 0.18 0 0 DOBSS 0 0.09 0.09 0 0.18 0 0.05 QRE (Lambda = 0.45) 0 0 0.09 0 0.14 0.05 0 QRE (Lambda = 0.76) 0 0 0.05 0 0.27 0 0 Table 105: Participants’ Choices (in fraction) for 15 Gate Settings in Cluster 4 Figure 42: Participants’ Choices against Gate Number for 15 Gate Settings in Cluster 4 96 Appendix O: Average Response Times (Uncapped) The average response times of the participants for various game models are illustrated in Tables 106 to 109 and Figures 43 to 46. Game Model 3 Gates 6 Gates 9 Gates 12 Gates 15 Gates PT 11.476 11.905 12.381 19.333 9.619 PT-Attract 9.238 16.952 15.429 15.571 17 COBRA (Alpha = 0.15) 8.952 13 16.048 16.429 19.476 COBRA (Alpha = 0.5) 7.619 16.381 14.905 21.905 13.905 DOBSS 11.857 11.286 19.714 10.19 16.619 QRE (Lambda = 0.45) 7.524 11.81 16.667 11.905 15.619 QRE (Lambda = 0.76) 9.571 10 31 17.81 14.476 Table 106: Average Response Times (Uncapped) for Cluster 1 Figure 43: Average Response Times (Uncapped) against Number of Gates for Cluster 1 97 Game Model 3 Gates 6 Gates 9 Gates 12 Gates 15 Gates PT 15.08 19.08 15.2 22.08 17.12 PT-Attract 10.32 16.76 14.96 13 17.08 COBRA (Alpha = 0.15) 10.08 18.44 16.04 18.56 21.88 COBRA (Alpha = 0.5) 10.6 15.8 16.48 13.72 19.96 DOBSS 11.52 17.08 22.52 15.16 19.68 QRE (Lambda = 0.45) 8.16 12.2 20.64 12.84 17.12 QRE (Lambda = 0.76) 11.48 11.76 24.96 14.2 17.76 Table 107: Average Response Times (Uncapped) for Cluster 2 Figure 44: Average Response Times (Uncapped) against Number of Gates for Cluster 2 98 Game Model 3 Gates 6 Gates 9 Gates 12 Gates 15 Gates PT 12.783 18.261 24.13 21.522 21.304 PT-Attract 12.261 20.087 20.783 22.783 26.217 COBRA (Alpha = 0.15) 11.652 20.696 24.609 24.087 20.913 COBRA (Alpha = 0.5) 11.957 22.652 13.304 23.087 19.652 DOBSS 10.435 18.391 28.261 17.043 23.478 QRE (Lambda = 0.45) 10.783 12.304 24.696 14.826 36.435 QRE (Lambda = 0.76) 11 14.043 25.609 21.435 19.609 Table 108: Average Response Times (Uncapped) for Cluster 3 Figure 45: Average Response Times (Uncapped) against Number of Gates for Cluster 3 99 Game Model 3 Gates 6 Gates 9 Gates 12 Gates 15 Gates PT 11.409 21.5 19.227 19.955 14.545 PT-Attract 10 17 14.545 12.636 17.955 COBRA (Alpha = 0.15) 10.455 19 63.227 15.773 16.727 COBRA (Alpha = 0.5) 10.591 14.591 18.409 18.5 20.136 DOBSS 8.545 11.364 22.091 16.227 12.682 QRE (Lambda = 0.45) 9.591 12.273 23.182 12.682 13.682 QRE (Lambda = 0.76) 11.727 18.955 20.182 14.273 17.5 Table 109: Average Response Times (Uncapped) for Cluster 4 Figure 46: Average Response Times (Uncapped) against Number of Gates for Cluster 4 100 Appendix P: Average Response Times (Capped) The average response times of the participants for various game models with a constraint of 120 seconds are illustrated in Tables 110 to 113 and Figures 47 to 50. Game Model 3 Gates 6 Gates 9 Gates 12 Gates 15 Gates PT 11.476 11.905 12.381 19.333 9.619 PT-Attract 9.238 16.952 15.429 15.571 17 COBRA (Alpha = 0.15) 8.952 13 16.048 16.429 19.476 COBRA (Alpha = 0.5) 7.619 16.381 14.905 21.905 13.905 DOBSS 11.857 11.286 19.714 10.19 16.619 QRE (Lambda = 0.45) 7.524 11.81 16.667 11.905 15.619 QRE (Lambda = 0.76) 9.571 10 31 17.81 14.476 Table 110: Average Response Times (Capped) for Cluster 1 Figure 47: Average Response Times (Capped) against Number of Gates for Cluster 1 101 Game Model 3 Gates 6 Gates 9 Gates 12 Gates 15 Gates PT 15.08 19.08 15.2 22.08 17.12 PT-Attract 10.32 16.76 14.96 13 17.08 COBRA (Alpha = 0.15) 10.08 18.44 16.04 18.56 21.88 COBRA (Alpha = 0.5) 10.6 15.8 16.48 13.72 19.96 DOBSS 11.52 17.08 22.52 15.16 19.68 QRE (Lambda = 0.45) 8.16 12.2 20.64 12.84 17.12 QRE (Lambda = 0.76) 11.48 11.76 24.96 14.2 17.76 Table 111: Average Response Times (Capped) for Cluster 2 Figure 48: Average Response Times (Capped) against Number of Gates for Cluster 2 102 Game Model 3 Gates 6 Gates 9 Gates 12 Gates 15 Gates PT 12.783 18.261 24.13 21.522 21.304 PT-Attract 12.261 20.087 20.783 22.783 26.217 COBRA (Alpha = 0.15) 11.652 20.696 22.565 24.087 20.913 COBRA (Alpha = 0.5) 11.957 22.652 13.304 20.783 19.652 DOBSS 10.435 18.391 28.261 17.043 20.174 QRE (Lambda = 0.45) 10.783 12.304 24.696 14.826 28.696 QRE (Lambda = 0.76) 11 14.043 25.609 21.435 19.609 Table 112: Average Response Times (Capped) for Cluster 3 Figure 49: Average Response Times (Capped) against Number of Gates for Cluster 3 103 Game Model 3 Gates 6 Gates 9 Gates 12 Gates 15 Gates PT 11.409 18.227 19.227 19.955 14.545 PT-Attract 10 17 14.545 12.636 17.955 COBRA (Alpha = 0.15) 10.455 19 27.864 15.773 16.727 COBRA (Alpha = 0.5) 10.591 14.591 18.409 18.5 20.136 DOBSS 8.545 11.364 22.091 16.227 12.682 QRE (Lambda = 0.45) 9.591 12.273 23.182 12.682 13.682 QRE (Lambda = 0.76) 11.727 18.955 20.182 14.273 17.5 Table 113: Average Response Times (Capped) for Cluster 4 Figure 50: Average Response Times (Capped) against Number of Gates for Cluster 4
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Asset Metadata
Creator
Goenka, Mohit
(author)
Core Title
Computational model of human behavior in security games with varying number of targets
School
Viterbi School of Engineering
Degree
Master of Science
Degree Program
Computer Science
Publication Date
04/19/2011
Defense Date
03/30/2011
Publisher
University of Southern California
(original),
University of Southern California. Libraries
(digital)
Tag
artificial intelligence,behavioral sciences,COBRA,DOBSS,game theory,Human behavior,OAI-PMH Harvest,PT,PT-Attract,QRE,Stackelberg
Language
English
Contributor
Electronically uploaded by the author
(provenance)
Advisor
Tambe, Milind (
committee chair
), John, Richard S. (
committee member
), Maheswaran, Rajiv T. (
committee member
)
Creator Email
mgoenka@usc.edu,mohitgoenka@gmail.com
Permanent Link (DOI)
https://doi.org/10.25549/usctheses-m3757
Unique identifier
UC1336112
Identifier
etd-Goenka-4204 (filename),usctheses-m40 (legacy collection record id),usctheses-c127-469124 (legacy record id),usctheses-m3757 (legacy record id)
Legacy Identifier
etd-Goenka-4204.pdf
Dmrecord
469124
Document Type
Thesis
Rights
Goenka, Mohit
Type
texts
Source
University of Southern California
(contributing entity),
University of Southern California Dissertations and Theses
(collection)
Repository Name
Libraries, University of Southern California
Repository Location
Los Angeles, California
Repository Email
cisadmin@lib.usc.edu
Tags
artificial intelligence
behavioral sciences
COBRA
DOBSS
game theory
PT
PT-Attract
QRE
Stackelberg