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A Multidimensional Similarities Analysis Of Twelve Choice Probability Learning With Payoffs

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A Multidimensional Similarities Analysis Of Twelve Choice Probability Learning With Payoffs

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Content This dissertation has been microfilmed exactly as received 67— 1 7 ,7 0 7 YO UNG , F o r r e s t W e sle y , 1 9 4 0 - A M ULTIDIM ENSIONAL. SIM ILAR ITIES A N A LY SIS O F TW E LV E CHOICE PR O B A B IL IT Y LEARNING WITH P A Y O F F S . U n iv e r s ity o f S ou th ern C a lifo r n ia , P h .D ., 1967 P s y c h o lo g y , e x p e r im e n ta l U niversity Microfilms, Inc.. A nn Arbor, M ichigan A MULTIDIMENSIONAL SIMILARITIES ANALYSIS OF TWELVE CHOICE PROBABILITY LEARNING WITH PAYOFFS by Forrest. Wesley Young A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (Psychology) August 1967 UNIVERSITY O F S O U T H E R N CA LIFO RN IA T H E G RADUA TESCH OOL UNIVERSITY PARK LOS ANOELES. C A LIFO RN IA 9 0 0 0 7 This dissertation, written by ® ® r ^ - . ® 5 T . . S. under the direction of h.X.P...Dissertation Com m ittee, and approved by a ll its mem bers, has been presented to and accepted by the Graduate School, in partial fulfillm ent of requirements fo r the degree of D O C T O R OF P H I L O S O P H Y Dean DISSERTATION C O M M IT T E E ..... I Chairman ACKNOWLEDGEMENTS The author is indebted to the members of his com mittee , Professors Daniel Davis and Richard Wolf, and ;particularly to his committee chairman. Professor Norman Cliff, for the valuable guidance and assistance received j ;throughout the preparation of this dissertation and during i his graduate studies at the University of Southern Califor nia. Thanks are also due to Stanley Terebinski, System Development Corporation, who initiated my interest in probability learning. The generosity of the following organizations in providing free computer time is also acknowledged: Honey well Corporation; University of Southern California Com puter Sciences Laboratory; System Development Corporation; Western Data Processing Center; and Health Sciences Com puting Facility, University of California at Los Angeles. Finally, the author is particularly indebted to his wife, Suzanne, for her encouragement and patience. TABLE OF CONTENTS Page ACKNOWLEDGEMENTS....................................... ii LIST OF TABLES............................................ LIST OF FIGURES..........................................vii Chapter I. INTRODUCTION .................................. 1 II. REVIEW OF PREVIOUS RESULTS ................. 4 III. STIMULUS SAMPLING THEORY ..................... 8 Philosophical Foundations General Axiomatization of Stimulus Sampling Theory Specific Models IV. PILOT EXPERIMENT .......................... 20 Method Results and Discussion V. EXPERIMENTAL DESIGN..............................28 Subjects Design Apparatus Procedure Analysis VI. RESULTS: ORDINAL PREDICTIONS, ANALYSIS OF VARIANCE, AND DISCRIMINANT ANALYSIS. . . 43 Chapter Page Ordinal Predictions Analysis of Variance Discriminant Analysis VII. RESULTS: LEARNING MODELS . . . . . . 56 VIII. RESULTS: MULTIDIMENSIONAL SCALING . . . 62 Scaling Group Hypothetical Learning Subjects Utility Learning Group Probability Learning Group Value Learning Group IX. DISCUSSION........................... 84 Choice Frequencies and Scaling Spaces Stress and Distances Learning Models X. SUMMARY .................................. 96 REFERE N C E S . 99 APPENDIX A. Instructions for Learning Groups and Scaling Groups............... 106 APPENDIX B. Stimulus Pair Orders ................. 114 LIST OF TABLES Table Page 1. Event. Characteristics— Pilot Experiment . . 23 2. Event Characteristics and Spatial Orders for Main Experiment.........................31 3. Analysis of Variance of Ranked Choice Frequencies...................................48 4. Discriminant Analysis— Between Group Significances .............................. 52 5. Discriminant Analysis— Number of Subjects Correctly Classified ....................... 54 6. Results of Model Analysis Obtained and Predicted Frequencies. .................... 58 7. Results of ZWS Model Analysis— Minimum Chi Values of the Parameters and Chi Squares...................................59 8. Results of AWS Model Analysis— Minimum Chi Values of the Parameters and Chi Squares ...............61 9. Coordinates for Hypothetical Subjects . . 63 10. Scaling Coordinates Based on Group Point of View from Group U S ......................65 11. Scaling Coordinates Based on First Point of View from Group U S ......................70 12. Scaling Coordinates Based on Group Point of View from Group U L ......................73 v Table Page 13. Scaling Coordinates Based on Group Point of View from Group P L ...................... 79 14. Scaling Coordinates Based on Group Point of View from Group V L ...................... 82 vi LIST OF FIGURES Figure Page 1. Transition Chain for Stimulus Sampling M o d e l ........................................ 15 2. Choice Frequency versus Expected Value Pilot Experiment..............................25 3. Goodman & Kruskall's Gamma on Blocks of 60 T r i a l s .....................................46 4. Scale Value versus Expected Value . . . . 67 5. Expected Value versus Two Dimensional Solution for Group U L ....................... 68 6. Two Dimensional Scaling Solution for Group UL ................................. 74 7. Points of View for Group U L .................... 76 vii CHAPTER I INTRODUCTION "In the natural environment of the living being . . . there is . . . no perfect certainty that this or that will, or will not, lead to a certain end, but only a higher or lesser degree of probability." This statement of Brunswick's (1939), along with his and Humphreys' (1939) experiments, are the beginning of the study of probability learning. Goodnow (1958), in a review of the literature, states that probability learning experiments "have come to be regarded as model situations for studying decision making, learning, and problem-solving. The classical paradigm for the probability learn ing (PL) experiment has become quite familiar. On each of a series of trials the subject (S) makes a choice from an experimenter-defined set of alternative responses, then receives from the experimenter a signal indicating whether or not the choice was correct. In the original variant of the PL experiment (Humphreys, 19 39; Brunswick, 1939; Grant, Hake and Hornseth, 1951) , S predicted which of two alterna- tive events would occur, and at the end of each trial, exactly one of these events occurred. The event's occur rence had fixed probability on all trials, and was inde pendent of the Ss response. This variant is called the 2-choice, non-contingent fixed probability paradigm. More recently, some modifications have been intro duced into the classical paradigm. Neimark (1956) and others (Gardner, 1957; 1958; Messick and Rapoport, 1965a; 1965b; 1966) have studied multi-choice probability learning by having the £ choose one of 3, 4, or even 10 events on each trial. Estes (1954) and others (Detambel, 1955; Koehler, 1961; Woods, 1959a; 1959b) have studied probabil ity learning where the reinforcement probabilities are contingent on the response the S gives. Estes (1957) and others (Young, 1967a) have observed the S's behavior when the event probabilities vary as a function of the trial number. In addition, Myers (1961) and others (Katz, 1964; Young and Peeke, 1966; Young, 1967b) have studied the effect on probability learning of varying the amount of reinforcement. The experiment reported here combines two of the modifications into a single experiment. Specifically, the experiment involved four groups. The Ss in three of the groups were required to predict one of 12 events on each of 300 trials. Each of the 12 events had a certain prob ability of occurrence on each trial, and a certain payoff. The probabilities and payoffs were constant for all trials The payoffs and probabilities varied between the three groups. In addition, each of these groups was asked to rate the similarity of these 12 events, after the probabil ity learning experiment was completed. The fourth group only rated the events on their similarity. The experi mental hypothesis was that the Ss would predict each event with a frequency which is an increasing monotonic function of the expected value of the event, and a linear function of the scale numbers of the events. CHAPTER II REVIEW OF PREVIOUS RESULTS In the classical 2-choice, noncontingent, fixed probability experiment, the early results (Neimark and Shuford, 1959) indicated, with a degree of replicability quite unusual in human learning, that the Ss predicted an event about as often as the event occurred. This finding, called probability matching (PM) , runs counter to the "rational man" concept in game theory and classical deci sion theory (von Neuman and Morgenstern, 1944), since these theories predict that if a rational man is confronted with a random series of events he will axways choose the more frequent one. This is the optimum strategy since it achieves the maximum number of correct predictions. This method of responding, the "rational" way, is very rare indeed. This finding, that curves of response proportions match event probabilities, holds up for several hundred trials. In some experiments falling in this category a slight "overshooting" begins to appear after about 250 trials. For example, groups run by Gardner (1957) with reinforcement probabilities of .60 and .70 yielded mean response proportions of .62 and .72 respectively, over the final portion of a 450 trial series. In order to determine whether this behavior is asymptotic, several studies have been run for 1,000 trials, and one Russian study was run a total of 2,000 trials (Cole, 1965). In general, these studies indicate that PM is asymptotic behavior. As more and more studies have accumulated, some tentative generalizations can be made about conditions under which PM occurs. PM tends to occur in the classical paradigm when the experimental task and instructions induce the to indicate his expectation on each trial (Estes and Straughan 1954; Friedman, Burke, Cole, Estes, Kelly and Millward, 1963) or when they emphasize being correct on every trial (Estes and Johns, 1958; Goodnow, 1955; Goodnow and Post man, 1955) . Overshooting tends to occur In the classical paradigm when instructions, or the experimental procedures convince S he is confronted with a sequence of truly random events (Edwards, 1961; Rubenstein, 1959; Peterson and Ulehla, 1965) or when they indicate the S should maxi mize his successes over blocks of trials (Das, 1961). In the multi-choice paradigm, it is almost always found that overmatching occurs. This effect becomes stronger after 400 or 500 trials. The extent of the over matching depends on the number of events. Estes (1964) reports only slight overshooting in a 3 event task, whereas Gardner (1957) found distinct overshooting. Gardner (1958) varied the number of events and held the probability of the most frequent event constant. He found that for the .70 alternative, the prediction frequencies were .73, .75, .81, and .83 for 2, 3, 4, and 7 events respectively. In a parallel manner he found, for a .60 alternative the fre quencies were .61, .61, .69, and .87 for 2, 3, 5, and 8. events. Beach (1965) , in a 1,500-trial, 10-alternative design also obtained overshooting of the most likely event. These studies, along with others (e.g.. Shipper, 1966), indicate that PM breaks down with increasing sever ity as the number of events is increased. An interesting exception to this finding is a recent experiment by Bower (1966) which was a 4 event, 384 trial design. In place of the usual prediction of an event on each trial, Bower had his Ss predict a pair of events on each trial. Similarly on each trial a pair of events was correct. The six pairwise event probabilities were very closely matched by the six pairwise prediction frequencies. In the variant of the classical paradigm which includes payoffs (monetary reinforcements) it is generally found that overmatching occurs. In one of the first o studies of this type (Brackbill, Kappy, and Starr, 1962a; 1962b), it was found that, in a 2-choice situation with children, maximizing behavior increased as payoff increased,, even though event probabilities were held constant. Simi lar results have been obtained by other investigators (Siegel and Andrews, 1962). In a more extensive study, Myers and his associates (Myers, Fort, Katz, and Suydam, 1963) obtained similar results in college sophomores. In a series of studies combining both multi-choice and payoff Messick and Rapoport (1965a; 1965b; 1966) have also found similar results. CHAPTER III STIMULUS SAMPLING THEORY This chapter is organized into three sections. The first section presents a brief discussion of the philo sophical underpinnings of stimulus sampling theory; the second presents a general axiomatization of stimulus sam pling theory; and the third presents two specific stimulus sampling models derived from the axiomatization. One pur pose of the research reported here is to test the effi ciency of each model as a descriptor of the results of the research. Philosophical Foundations In 1950/ Estes published a paper which became the foundation of stimulus sampling theory. The ideas pre sented in this paper (Estes, 1950) were expanded by Estes and Burke (1953; 1955) and by Estes (1954). Later Estes stated (Estes, 195 7) that he intended his theory to be applied to situations where the S is confronted with the same stimulating situation at the start of each trial. 9 The S responds with one of a finite set of responses, after which he is presented with one of a finite set of reinforcing events, exactly one reinforcing event corre sponding to each possible response. For example, in the PL paradigm, the responses are the S's predictions as to which event will occur on each trial, and the reinforcing events are "right" or "wrong." Estes' original work has been expanded into what is now known as stimulus sampling theory. Stimulus sam pling theory is a way of conceptualizing the experimental and psychological processes, and it is a way of presenting these conceptualizations in mathematical terms. The general modus operandi of the theorist in this school is to present a set of axioms concerning the sampling of stimuli, the conditioning of stimuli, and the responding to stimuli. From these axioms are derived predictions of behavior. Stimulus sampling theory views behavior as being elicited by antecedent stimulus events that are in some fashion associated to responses. The undefined terms in the theory are "stimulus," "response," "association,1 1 and "reinforcement." These terms, not including "response," derive their meaning from the way they are interrelated In the axioms of the theory. "Association" refers to the way that stimulus and response are functionally related. We refer to association when we say that response is i conditioned to stimulus Sj• Note that stimulus sampling i ~ |theory postulates nothing about neural pathways, or the I |like. The phrase "functionally related" refers to the ! ; assertion that the probability of response A. is related i ; to the presence or absence of stimulus Sj. The term "reinforcement" refers to a theoretical event that produces changes in associations of stimuli present during the time the reinforcing event occurs. i Stimulus sampling theory classifies reinforcing events according to the way stimulus-response associations are affected by their presence. If responses A^, A2, ..., Ar have been identified, then there will be reinforcing events E^, e2, ..., Er. If stimulus Sj is associated with response A2, for example, then reinforcing event E2 will leave the association unchanged, whereas any other re inforcing event may, with probability p, change the asso ciation in some manner. The change in association from one conditioning state to another is assumed to happen in an all-or-none fashion. That is, at any point in time, each stimulus is associated with exactly one response. The term "stimulus" is not precisely defined by stimulus sampling theory, although its role is precisely represented. A given stimulus situation is represented in terms of a set (finite) of stimulus elements, the size of which is denoted by N. These elements are completely distinct from each other, which means that the response associated with is unaffected if the response asso ciated with Sj is changed. On any particular trial the N stimuli can be divided into two mutually exclusive and exhaustive subsets, one of which is the "sampled" subset, and the other which is the "unsampled" subset. The sam pling procedure is usually assumed to be a random one. The response rules are usually very simple. Only the sampled stimulus elements are assumed to affect the response. As stated earlier, each stimulus element is associated with one response. Finally, it is assumed that each stimulus element has equal weight in determining the response. Stated differently, the probability that the S makes response A^ is given by the number of elements in the sample that are conditioned to A^ divided by the num ber of sampled stimuli. 12 General Axiomatization of Stimulus Sampling Theory The original axiomatic systems presented by Estes (1957) considered only one characteristic of the reinforc ing events. That characteristic was the probability of reinforcement. Atkinson (1962), however, has presented an expanded set of axioms which not only recognize the probability of reinforcement, but also the amount of reinforcement. This model was specifically designed to cope with results obtained in PL paradigms which in corporated payoff as a characteristic of the reinforcing events. The following axioms are presented by Atkinson (1962) as the basis for his stimulus sampling model of the multi-choice PL experiment with payoffs. Representation Axioms R1 There are N stimulus elements, labelled S^, S2, . . . , S jij. R2 There are r responses, labelled A^, A2, ..., Af. R3 There are r+1 reinforcing events, labelled Eq , E1' E2' --- Er’ One of these reinforcing events occurs on each trial. 13 Stimulus Sampling Axiom SI Exactly one of the N elements is sampled on each trial. Conditioning-State Axiom CS1 An element is either strongly or weakly conditioned to a single response. Conditioning Axioms Cl At the start of each trial, each element is con ditioned to exactly one response. C2 Elements that are not sampled on a trial do not change their conditioning state. C3 If event Ei occurs (i^O), then (a) if the sam pled element is strongly conditioned to the A^ response it remains so, and (b) if the sampled element is weakly conditioned to the A^ response there is probability that it becomes strongly conditioned. C4 If event Ej occurs (ij^j^O) , then (a) if the sampled element is strongly conditioned to the A^ response there is probability v^ that it becomes weakly conditioned to Aj_, and (b) if the sampled element is weakly conditioned to the Aj_ response there is probability that it becomes weakly conditioned to the Aj response. C5 If event EQ occurs on a trial, there is no change in conditioning of the sampled element. C6 The probabilities u^, v^, and are independ ent of the trial number and of events on preceding trials. Response Axiom R1 If the sampled element is conditioned to the A^ response (either strongly or weakly) then that response will occur with probability 1. Figure 1 illustrates the transitions that are possible under Axioms C3 and C4 for the two-response case. Note that an element can transit only to a directly ad joining state. For the matrix 0 of outcomes (or payoffs), where element o^j represents the amount of reinforcement received (for example the amount of money) when the £ 5 predicts event i and event j occurs, the following restrictions on the parameter space apply: E,:1-w Ei :v Ei :v Ei :w 2 :v2 E2:1"w1 Figure 1. Transition Chain for Stimulus Sampling Model k, 1, such that i j * j, and k f t 1. 3. O^u. , v. , w.< 1. -» i' ir Restriction 3 is the usual probability conservation restriction. Restrictions 1 and 2 insure that the transi tion probabilities are strictly monotonic with the payoffs (amount of reinforcement or outcome) of the events. This is the device used by Atkinson to introduce utility (amount of reinforcement) into the stimulus sampling framework. Notice that there are 3r parameters in this model. Since there are such a large number of parameters to be estimated, the general model has been solved only for specific subcases. Each subcase involves arbitrary assumptions concerning the value of some of the parameters, thereby reducing the number to be estimated. The following section presents two such subcases. Specific Models There are two major assumptions which can be made I 17 about the values of Vj_, the transition probabilities from | the strong to the weak state. The two assumptions lead to two different specific models, each of which will be ! fitted to the results of the research to be reported. One possible assumption which cam be made about I i the values of v., is that v. = w. (for all i) which makes 1 1 1 ! the transition chain more symmetric (see Figure 1) since i ! the transition probabilities under this assumption are the same from strong to weak and from weak to strong when i the S_makes an incorrect prediction. Atkinson (Myers and Atkinson, 1964) made precisely this assumption. For this subcase (which is referred to as Atkinson's weak-strong model, AWS) of the general model Atkinson has derived a set of theorems for the general two response case, and in j the noncontingent symmetric payoff three response case. Atkinson has shown that AWS predicts PM in the classical paradigm when the are all zero, and when the w^ are all equal. In this case AWS is a one-stage, N-element model which has a two-state transition matrix (the weak states) , and which has the single conditioning parameter w. At kinson has also shown that this special case of AWS is precisely the same as the model described by Estes (1959) and by Atkinson and Estes (1963) which they called the 18 "pattern" model. Myers and Atkinson also show that when the w^ are all equal to the , without the restriction that the u^ all be zero, then p^ =*TT^/( J\ + (l-7T^)r which is a result predicted by the "scanning" model developed by Estes (1962), and which (according to Myers and Atkinson) has been shown to give a fairly good account of several sets of data obtained in the two-choice non contingent situation involving payoffs. The second major assumption which can be made con cerning the value of v. is that v. = 0 (for all i). This i x assumption reduces the general model to a two-stage, N- element model which has a 2r state transition matrix with 2r conditioning parameters u^ and w^. (This subcase is referred to as the zero weak-strong model, ZWS.) This particular assumption has not appeared in the literature. It is chosen for investigation because the transition matrix is absorbing. If we assume further that the u^ are all zero, and that the w^ are all equal, then in the class ical paradigm ZWS predicts PM since these assumptions reduce ZWS to the same one-stage, N-element model with the same single conditioning parameter as the "pattern" case of AWS. Notice, however, that when the w^ are all equal 19 to the without the restriction that the all be zero, that the transition matrix does not reduce to the same one as the "scanning" case of AWS. AWS and ZWS, therefore, do not make the same predictions in the two-choice, non contingent situation involving payoffs. As a part of this research both versions of Atkinson's model will be fitted to the data of the experi ment. It will become clear when the experimental design is presented, that there are five free parameters for both models. The model fitting procedure is computer oriented, and is an algorithm for minimizing chi-square. Since both models have the same number of parameters, a statistical test of the difference in goodness-of-fit of the two models is possible. This test involves comparing the minimum chi-square for the two variables. It should be pointed out that this test is only approximate because the chi-square values which are being compared are not inde pendent . CHAPTER IV PILOT EXPERIMENT j | The brief review of the literature given in the i first section of this proposal shows that PM is a common i and reliable finding in the two-choice no-payoff PL experi- : ment. It also showed that as the number of alternatives is increased or as the amount of payoff is increased that j human Ss tend to pick the most frequent alternative more often than it occurs. The experiment reported here was intended to investigate the relationship between choice frequency and both the probability and payoff of the events in a within Ss design. No research has been done which varied both probability and payoff of the events within Ss in order to determine the relative importance of both variables. The research reported here also intended to deter mine the relationship between the final choice frequency of the events, and the subjective expected utility of the events. Several theorists (Siegel, 1964; Messick and 20 Rapoport, 1965a; 1965b; 1966) have investigated the rela tionship of choice frequency and expected value and expected utility, but no research has related choice fre quency and subjective expected utility (SEU). The research reported here also intended to deter mine the adequacy of the two models of PL presented earlier. Predictions based on these models are generated for the current study. In the experiment presented here, it was predicted that the asymptotic frequency with which human Ss choose one of a set of events is a monotonic function of the expected values of the events. More specifically, it was hypothesized that human Ss asymptotic choice frequency should be a linear function of the subjective expected utility of the events, where the SEU function is derived by multidimensional scaling methods (Shepard, 1962a; 1962b Kruskal, 1964a; 1964b) in conjunction with several multi variate techniques. In order to test the most basic of these hypoth eses, i.e., that the asymptotic frequency with which human Ss choose one of a set of_events is a monotonic function of the expected values of the events, a pilot experiment was performed. 22 Method Subjects. There were 18 Ss obtained from intro ductory psychology classes. Results for only 15 Ss are reported because of equipment breakdowns. Design. There was one group of Ss. Each £ was run 200 trials. Each trial began with a ready signal. The £ was instructed to predict one of 12 events when he saw the ready signal. After making his prediction the £ was informed as to which event had occurred on that trial, how much the event paid off, and his total earnings to that trial. The matrix of payoffs and event probabilities is presented in Table 1, along with the expected values of the events. Apparatus. The instructions and the series of events were automatically presented by a teletype terminal of the System Development Corporation time sharing system. The responses were automatically recorded by the system. The ready signal was an asterisk typed by the teletype. The £s responded by typing one of the numbers 1-12. No other responses were allowed. The Ss could take as much time as they desired to respond, although the teletype 23 TABLE 1 EVENT CHARACTERISTICS— PILOT EXPERIMENT Event. I 2 3 4 5 6 7Ti .0375 .0375 .0375 .0375 .0725 .0725 vi 1 2 3 4 1 2 EV± .0375 .0750 .1125 .1500 .0725 .1450 Event 7 8 9 10 11 12 771 .0725 .0725 .1400 .1400 .1400 .1400 vi 3 4 1 2 3 4 EV± .2175 .2900 .1400 .2800 .4200 .5600 24 - urged them to respond faster if ten seconds elapsed. Results and Discussion The mean relative choice frequencies for the Ss were obtained over the last 20 trials. These frequencies were converted to proportions. To test the expected value hypothesis, Kendall's tau was calculated between the rank ordered choice fre quencies and the rank ordered expected values. It was found to be highly significant (Z = 3.76, p<C..0001) . The Spearman rank order correlation coefficient was .95. To test the probability-matching hypothesis, Kendall's tau was calculated between the ranked choice frequencies and the ranked event probabilities. The tau (corrected for ties) was not significant. The relationship between choice frequency and expected value is indicated in Figure 2. It is clear from the figure, and on the basis of the statistical anal yses, that probability matching was not found. These results fall closely in line with those of Messick and Rapoport (1965a). On the basis of this experiment, then, it appears likely that the expected value hypothesis was supported; £ iG 50 f r e e n c * 40 30 20 10 • • ,10 .20 .30 .40 Expected Value .50 .60 Figure 2. Choice Frequency versus Expected Value, Pilot Experiment ro U) 26 however, there were some major criticisms of the pilot experiment. The main criticism was: From the single group it was not possible to determine whether the same results would have been obtained if only one of the two independent variables of probability and payoff had been varied. That is, it would have been possible to obtain the results above even though one of the variables, say probability, was not having any effect. For this reason, two new groups were added to the final design. One group was an ordinary probability learning group, and the other was a value learning group. The probability learning group was added to indicate whether the effect for payoff in the main group was due to the payoff variable, and the value learning group was added to yield parallel information concerning probability. In addition, a second phase was added to the de sign. This phase, which was added to all three groups, was a scaling phase. It allowed the determination of the form of the SEU function for each group. This phase consisted of a similarities experiment in which the Ss were asked to rate all pairs of events on their similarity. A final modification in the pilot design was intro- 27 duced to compare the learned expected value function with that communicated to the Ss via instructions. This group simply received the standard scaling experiment: i.e., they were instructed about the payoff and probability of each event, and then were asked to rate the events on their similarity. CHAPTER V EXPERIMENTAL DESIGN The purposes for the experimental design intro duced in this chapter are as follows: 1. To provide a valid test of the two stimulus sampling models developed earlier in the paper. 2. To determine the effect of event probability and event payoff on the frequency with which a chooses the event. 3. To determine whether the results of a multi dimensional scaling analysis of the judgments of the similarity of the several events can be related to the choice frequencies. 4. To provide a valid test of the subjective expected utility hypothesis as a descriptor of both the choice frequencies and the results of the scaling analysis. Subjects Fifty-nine Ss were obtained from the introductory 28 29 psychology classes for a "gambling" experiment, which they j were told involved monetary reward. They reported individ ually to an air-conditioned room containing the apparatus. i j Design j There were four groups of Ss. The groups are ; designated: (a) the utility learning group (UL); (b) the j probability learning group (PL); (c) the value learning group (VL); and (d) the utility scaling group (US). The first three groups are sometimes referred to as the "learn ing" groups, and the last group is sometimes called the "scaling only" group. There were 15 Ss in the learning groups, and 14 in group US. For groups UL, PL, and VL, the experimental pro- i i cedure was divided into two phases: The first phase was called the learning phase, during which the Ss were pre sented with a series of 300 trials; the second phase was called the scaling phase, during which the Ss were asked to rate the similarity (with respect to how much one would win over a long time) of all pairs of the 12 events. The second phase consisted of 66 pair-comparison judgments. Group US was informed of the value and probability of each of the 12 events, and only completed the scaling phase of the experiment. Groups UL, PL, and VL differed from each other in the following ways. For group UL the 12 events varied in both their probability of occurrence and in their payoff. The payoffs and probabilities are presented in Table 2. Four different spatial orders of the events were used, the Ss being randomly assigned to a particular order. For group PL the events had the same probabilities listed in Table 2, but the payoffs for all events were one-half cent. Group VL is the opposite of group PL; the events had the same payoffs listed in Table 2, but were all equiprobable (p=l/12). Groups PL and VL, then/served as comparison groups for the first phase of group UL, and group US served as a comparison for the second phase of group UL. The dependent variables in these groups is the frequency with which each S predicted each event, particularly during the last 50 trials, and the similarity judgments. Apparatus A 8-1/2" x 43" sheet of 1/8" black plastic was mounted (on its long edge) on a piece of wood. On the back of the plastic were mounted 12 Bemak Plastics IBM card holders. On the front of the plastic could be TABLE 2 EVENT CHARACTERISTICS AND SPATIAL ORDERS FOR MAIN EXPERIMENT Event P Z V Q A D G H M U J S Probability .05 .05 .05 .05 .07 .07 .07 .07 .13 .13 .13 .13 Payoff .5 1.0 1.5 2.0 .5 1.0 1.5 2.0 .5 1.0 1.5 2.0 i EV .025 .050 .075 .100 .035 .070 .105 .140 .065 .130 .195 .260 Orders 1 2 3 4 5 6 7 8 9 10 11 12 1 H J G A D Z M V P S Q U 2 V H Q D G P A S J Z U M 3 Z G A H P U S M V J Q D 4 J H Q A Z P M V G U D S u> H 32 mounted a second piece of 1/8" black plastic 4M x 43". I Glued to the front of the large plastic sheet were 12 white transfer type letters (capitals, 3/4"). The letters I were centered in front of the card holders. For groups UL and PL, each card holder contained 100 IBM cards (96 for group VL). The apparatus was placed on a table. On the for- i ] : ward edge of the table, a Bemak Plastics card tray was placed. The tray could hold up to 300 cards. Procedure The Ss reported to the experimental room individ ually. They were told by E to sit in front of the appara tus, and to fill out the front page of the instruction booklet and read the instructions. The instructions are given in Appendix A. If the S asked any questions, the relevant part of the instructions was paraphrased. Ss in the learning groups proceeded to draw an IBM card from any one of the 12 IBM card holders. On the top i edge (the colored edge) of the card were printed two letters and a number. The left letter was the same as the ; letter on the front of the apparatus which identified the card holder. The number indicated the payoff of the card. The right, letter indicated the correct event: i.e., if the two letters were the same, then the S won one-half the amount indicated by the number, and if they were not the same he won nothing. If the S won (if the letters were the same) the £ called out the number on the card and the E added this number to the current total on a cal culating machine. The then drew a second card. This process was repeated until the S had filled the card holder to overflowing. The Ss proceeded at their own rate, usually requiring about twenty minutes to complete the task. Following the learning task, the Ss performed the . scaling task. The scaling task was introduced by a second set of instructions (Appendix A) which informed the Ss to judge the similarity (with respect to the overall win nings) of each pair of events. The Ss indicated the simi larity by circling a number on a scale from 1 to 9. The number 1 indicated a high degree of similarity and the number 9 a high degree of' dissimilarity. This phase of the experiment took the Ss approximately ten minutes. At the end of the scaling task the S was given his winnings from the first phase, and was told that that was the end of the experiment. 34 The Ss in the scaling only group were instructed to judge the similarity of the gambles whose characteris tics were indicated on the board in front of them. They were told that the characteristics represented the results of the S that was run before them, and that they were to judge the similarity with respect to overall winnings of each event. Ss were assigned to groups randomly according to a random number table. Letters were assigned to the events in a random order, and then four different spatial orders were generated (utilizing a random number table) to present the events. The orders are given in Table 2. A single order of the 66 pairs was used for all Ss in the scaling phase. This order is presented in Appendix B. Before each S entered the experimental room, the IBM cards were thoroughly shuffled, to insure that each S received a unique event series. Analysis This section is organized into two parts. The first lists the various analyses which were performed and their objectives, and the second presents the algorithm for the model-fitting analyses. 35 An analysis of variance was performed on the rank ordered individual choice frequencies. The analysis was performed on each group separately to determine the effect of probability and payoff on the choice frequencies. The analysis was performed on the frequencies summed over all 300 trials. It was performed on the rank ordered frequen cies under the assumption that the rank order was a trans formation of the original frequencies which produced more nearly equal variances across events and compensated for the high degree of skew in the raw data. A multiple discriminant analysis was performed on the raw similarities judgments. This analysis was per formed as an indirect test of the effectiveness of the learning phase of the experiment. It is an indirect test because it is not performed on the learning data itself, but on the scaling data. The analysis rests on the assump tion, then, that the results of the second phase depend directly on the learning which occurred during the first phase. That is, any between groups differences will be related to differences in event characteristics during the learning phase even though the analysis is based on the scaling phase. Stated differently, group differences are assumed to be due to differences in prior experience. 36 The analysis is in fact a test of the efficacy of the learning phase because the only differences in the stimuli between the groups is in their prior historyr i.e., the learning process for the three learning groups, and the instructions for the scaling group. If the discriminant analysis discloses significant group differences, they will be interpreted as follows: between any of the three learning groups, the differences will be attributed to the difference in variables used to describe the events; i.e., between groups UL and VL, for example, a significant difference will be interpreted as due to the presence of the probability varied)le in group UL. Between group US and the learning groups significant differences will be interpreted as being due to the instructional variable. A multidimensional scaling analysis was performed on the similarities data. The euialysis was performed by program TORSCA (Young and Torgerson, 196 7), a semi-metric algorithm. The program is essentially the same as those of Shepard (1962a; 1962b) and Kruskal (1964a; 1964b) with the exception that a rational, metric configuration is used to begin the scaling process, whereas the Shepard and Kruskal algorithms use an arbitrary configuration to 37 start: the process. A points of view analysis (Tucker and Messick, 1963) was performed to obtain the distances for input to the scaling procedure. This analysis was performed on the raw similarities judgments, and it served as a means of investigating individual differences. Finally, a minimum chi-square analysis was used to fit the stimulus sampling models to the learning data. For both models the parameter estimation was based on the choice frequencies over the last 50 trials, which were assumed to be asymptotic. The details of this algorithm will be now described. The algorithm to fit the models is based on a search procedure which can be viewed as looking through a microscope at the parameter space, finding the area con taining the minimum chi-square between the frequencies and the predictions, then looking at this part of the space under a higher magnification. By repeating this procedure many times, a chi-square is found which is arbitrarily close to the minimum. Minimizing chi-square analytically is very diffi cult for the models presented here, because of the com plexity of the transition matrix. Thus, a computer-based 38 search algorithm was developed and used. This algorithm involves starting with arbitrary initial values for the parameters, generating expected asymptotic frequencies based on these parameter values, and calculating chi- square (between these expectations and the choice fre quencies over the last 50 trials). Then one of the parameters is incremented by an arbitrary amount, new expectations are computed, and a new chi-square is cal culated. If the new chi-square is smaller than the pre vious (smallest)one, it is stored, and the parameter values are stored. This process is repeated until the parameter reaches an arbitrary maximum value, at which point it is reset to its initial value, and a second parameter is incremented. When all parameters have reached their maximum value, new initial values of the parameters are computed in such a way that the area of the parameter space immediately surrounding the minimum chi-square is searched again, but with a much smaller increment. The search is stopped when an arbitrarily small increment is reached. The method of generating frequencies is different for each of the two models. For the ZWS model, a method is used which can only be used with absorbing transition 39 matrices. The method utilizes what Kemeny and Snell refer to as the fundamental matrix (Kemeny and Snell, 1960) . Any absorbing transition matrix can be rearranged, by per muting rows and columns, into the form I O R S known as the canonical form, where I is an identity matrix, O is a zero matrix, R is a matrix giving the transition probabilities from the transient to the ab sorbing states, and S is a matrix giving the transition probabilities between the transient states. For the AWS model, R is a diagonal matrix, and S is a symmetric matrix. Based on the canonical form of the transition matrix, the fundamental matrix is defined as N = (I - S)"1 This matrix has several interesting properties, one of which is that if it is premultiplied by R B = RN then the elements b^j give the probability of being ab- 40 aorbed in state j when the process started in state i. If V is the vector whose elements give the probability of starting the process in state i, then F = VB = VR(I - S)_1 is the vector whose elements fj give the probability of absorbing in state j. When F is multiplied by the appropriate scalar, it gives the expected asymptotic response frequencies for the set of parameter estimates used. Because the AWS model has a transition matrix which is classified by Kemeny and Snell (1960, p. 37) as a regular transition matrix, it is not possible to use the same method for generating asymptotic predictions as was used for the ZWS model which has an absorbing transi tion matrix. For a regular transition matrix P, it can be proved that the powers Pn approach a probability matrix A, and that each row of A is the same vector OC^ whose elements give the probability for being in state i. Since each row is the same, these probabilities are inde pendent of the initial state, and of the initial probabil ity vector V. When the row OC^ is multiplied by the appropriate scalar, we have the desired asymptotic predic 41 tions. The method used to obtain these predictions, then, is simply to multiply the transition matrix by itself until it converges on A, i.e., until the difference D = Pn - pn+1 less than some arbitrary number. It was decided to assume that the process had converged on A when max(dij) ^.001. The minimum chi-square method has been used here and elsewhere (Atkinson, Bower, and Crothers, 1965) because it allows a direct statistical test of the goodness of fit of the model. To perform this test, it is only necessary to determine chi-square and its degrees of free dom. It is known (Atkinson, Bower, and Crothers, 1965; McNemar, 1962) that the degrees of freedom is equal to the number of frequency categories (transition states) minus the number of parameters. In our case this is equal to 12-5=7 degrees of freedom. If the value for chi-square is significant then it indicates that the frequencies deviate significantly from the models' predictions. On the other hand a very small chi-square indicates that chance sampling of the frequency population would lead to a better fit less than one time in one hundred, indicating, perhaps, non- random sampling. The minimum chi-square method also provides a 42 method for directly comparing the fi-ts of several models. We can form the ratio xf o fi x|/ af2 where df is the degrees of freedom of the model and the subscript indexes the model. The value of this ratio indicates whether one of the models fits better than the other. If the ratio is less than 1, then model 1 fits better, and if it is greater than 1 then model 2 fits better. When the chi-squares are independent, then this ratio has an F distribution with df^ and df2 degrees of freedom, and an F test can be made to determine whether one of the models fits significantly better than the other (Atkinson, Bower, and Crothers, 1965). Unfortunately, in our case the two chi-squares are not independent since they utilize the same frequencies. It is possible, however, to interpret the ratio as an approximate F test, keeping in mind possible sources of error, and limitations in interpretation. CHAPTER VI RESULTS: ORDINAL PREDICTIONS, ANALYSIS OF VARIANCE, AND DISCRIMINANT ANALYSIS Ordinal Predictions The basic hypothesis states that the asymptotic choice frequencies (referred to hereafter as frequencies) will be monotonically related to the expected values of the events. This hypothesis was tested for each of the three learning groups. The obtained frequencies over the last 50 trials for group UL are presented in the first column of Table 7. (See Chapter VII.) The Spearman rank order correlation between frequency and expected value was .76, which was significant at the .003 level (z=2.76 based on Kendall's Tau of .606). It is seen that the basic hypothesis was strongly supported. Inspection of the data indicated that the choice frequencies might be even more highly related to the payoffs of the events, so a Spearman correlation was calculated between these two variables. The correla- 43 tion was .91, which was significant at the .001 level j (z=3.66, Tau=.806). The inspection of the data was borne out; even though the choice frequencies are strongly related to the expected values of the events, they are more strongly related to the event payoffs. To complete the comparisons, the correlation between the choice fre quencies and the event probabilities was calculated. The Spearman rank order correlation was .12. This is not a significant value. The results for group PL indicate a strong rela tionship between the frequencies over the last 50 trials and the event probabilities. (The frequencies are pre sented in Column 4 of Table 7.) The Spearman correlation was .66, which was significant at the .01 level (Tau-.501, z=2.28). The highly probable events were clearly differ entiated from the other events. The results of the correlational analysis for group VL indicate an even stronger relationship between the frequencies over the last 50 trials and the payoffs. (The frequencies are presented in Column 7 of Table 7.) The Spearman correlation was quite high, at .88. This is a highly significant value (Tau=.776, z=3.53, p=.0002). As in the PL group, the source of this effect is apparently 45 the fact that the highest payoff events were chosen a great deal more often than the remaining events. Further correlational analyses were performed to investigate the learning process. The choice frequencies were obtained (for each event, summed over Ss) for each block of 60 trials. A rank order correlation was obtained between the choice frequencies and the expected values for each block of trials and for each group. It is pre dicted that if learning is taking place the correlation for each block of trials should be larger than that for the preceding block of trials. The Goodman-Kruskal index of association was used in these analyses because of the large number of ties, and because of its ease of use. The results of this analysis are presented in Figure 3. As can be seen, the prediction of the order of the values is perfectly supported for group PL (p=l/120), and significantly supported for the remaining two groups (p=5/120). These analyses indicate, then, that the Ss in each group were learning the expected values of the events. It is interesting to note that the indices are smallest for group PL, and still appear to be growing. This may indicate that not enough trials were given for 00 VL 75 UL 50 PL 25 .00 1 2 3 4 5 Trial Blocks o \ Figure 3. Goodman & Kruskall's Gamma on Blocks of 60 Trials 47 tills group to learn the probabilities fully. It may also indicate that the probabilities were a more difficult, criterion to learn. This interpretation is also supported by noticing that the correlations for group VL are some what higher than those for group UL, indicating again that the probability of an event is more difficult to learn than the value. Analysis of Variance Three analyses of variance were performed, one for each learning group. The input to these analyses was the rank ordered choice frequencies for each S in the group. The frequencies were taken over the entire 300 trials. The results are presented in Table 3. The analysis for group UL was a 3 x 4 factorial with replications. For groups VL and PL the analysis was a one-way design with replications. The replications were broken down into a within Ss term and a between Ss term. For group UL, the error term for the first order effects is the interaction term, and for groups PL and VL it is the within Ss term. As can be seen from Table 3, both probability and payoff significantly affected the choice frequencies for group UL, and payoff contributed a significant effect for group TABLE 3 ANALYSIS OF VARIANCE OF RANKED CHOICE FREQUENCIES Group Source SS df MS F P UL Probability 161.2 2 80.6 31.35 .001 Value 415.4 3 138.5 53.87 .001 P x V 15.4 6 2.6 .28 — Between Ss 1544.1 168 9.2 — PL Probability 33.8 2 16.9 .86 — Within Ss 232.9 9 25.9 2.33 — Between Ss 1727.4 156 11.1 — VL Value 394.1 3 131.4 20.30 .005 Within Ss 95.7 8 11.9 1.22 — Between Ss 1640.2 168 9.8 — o o 49 VL. Both these groups are in line with predictions. The results for group PL are somewhat different, however, in that no significant effect could be attributed to probability. Although the analysis of variance for group PL appears to be in conflict with the correlation analysis presented in the previous section, it should be kept in mind that the correlation was performed on data from the last 50 trials, whereas the analysis of variance was per formed on data over all 300 trials. The correlation analysis was performed on the last 50 trials because it asked the question: Are the asymptotic frequencies re lated to the expected values? The analysis of variance, on the other hand, was performed to determine whether probability and payoff had a significant effect on the choice frequencies. Since this analysis had to be per formed with Ss as replications, all 300 trials were used to reduce the error in the ranking of the individual choice frequencies. Together the analysis of variance and the correla tion analyses indicate the following: For group UL the asymptotic choice frequencies are significantly correlated with expected value (as well as with payoff), and both 50 payoff and probability significantly affect the overall choice frequencies. For group VL the asymptotic choice frequencies are significantly correlated with the value of the events, and the value variable significantly affected the overall choice frequencies. For group PL the asymptotic choice frequencies are significantly cor related with the probabilities of the event, but the over all choice frequencies are not affected by the probability variable. The interpretation of these findings is straight forward for groups UL and VLs the variables which com bine to make expected value are being used by the Ss in determining which event to choose. The interpretation for group PL is the same although not quite so straightforward. The analyses are interpreted as indicating that the Ss were using probability as a basis for deciding which event to choose, but only at the end of the experiment. This interpretation, which is also supported by the correlational analysis of the blocked frequencies, can be easily related to the experimental apparatus. At the beginning of the experiment the Ss knew nothing of the event probabilities or payoffs. But after only a few trials the S^ became aware of the payoffs (in groups UL and 51 VL) because they were printed on the cards they were drawing. The probabilities, on the other hand, were not printed on the cards, and the Ss could only become aware of them after a large number of trials. It is seen, therefore, that the weaker probability effect can be easily related to the apparatus and design. Discriminant Analysis A step-wise discriminant analysis was performed on the reordered similarities data. This analysis was performed by BMD08M, a program in the BIMED series (Dixon, 1965). The information afforded by this analysis is of two types: (a) between groups significances on the basis of a subset of the variables which maximally discriminates the several groups; and (b) classification of the subjects into the groups they are most likely to have been a member of, on the basis of the same subset of variables. For the analysis of this data there were 66 vari ables, 4 groups, and 58 Ss (one S from group PL did not complete the similarities experiment). The results of the between-groups discrimination are presented in Table 4. On the basis of only one of the 66 variables (the similarity of stimuli 5 and 11) a TABLE 4 DISCRIMINANT ANALYSIS BETWEEN GROUP SIGNIFICANCES Number of Variables Overall UL-VL UL-Pl PL-VL US-UL US-VL US-PL 1 .001 - - .05 .001 .001 .001 2 .001 .001 - .001 .001 .001 .001 12 .001 .001 .05 .001 .001 .001 .001 14 .001 .001 .01 .001 .001 .001 .001 m N> 53 highly significant, overall between groups F was obtained (F=15.5, df=3,54, p<.001). On the basis of the same judgment, the pairwise F's between group US and all the other groups was highly significant. (For US and UL: F=25.1j US, VL: F=17.5? US, PL: F=46.8, all df's-1,54, and all p's <.001). On the basis of one more of the 66 variables groups PL and VL were highly different (F=14.8, df=2,53, p<.001) as were groups UL and VL (F=8.7, df=2,53, p<.001) . On the basis of only two of the 66 similarities judgments, then, each of the four groups can be signifi cantly discriminated from every other group, except groups UL and PL. It was not possible to significantly discrimi nate between groups UL and PL until 12 of the 66 variables were included in the discriminant function (F=2.25, df—12,43, p<.05) and 14 variables were required to increase the significance to less than .01 (Fas2.56, df=14,41) . The results of the classification of the subjects is presented in Table 5. On the basis of the first vari able alone, twelve of the fourteen Ss in group US were classified correctly, and eleven of the fifteen Ss in group PL were correctly classified. On the basis of the first seven variables entered into the discriminate func- 54 TABLE 5 DISCRIMINANT ANALYSIS— NUMBER OF SUBJECTS CORRECTLY CLASSIFIED Number of Variables UL VL PL US 1 3 3 11 12 2 4 10 8 12 3 7 9 9 11 4 8 10 10 12 5 10 11 10 12 6 8 12 10 13 7 10 14 10 14 8 11 14 13 14 9 11 14 11 14 10 11 14 11 14 11 11 14 13 14 12 13 14 12 14 13 13 14 13 14 14 13 14 13 14 55 tion all Ss in groups VL and US were correctly classified, and ten of the fifteen Ss in both groups UL and PL were correctly classified. On the basis of 14 variables, all except four Ss were correctly classified. CHAPTER VII 1 RESULTS : LEARNING MODELS Both models discussed earlier were fitted to the raw choice frequencies for the learning groups. In the process of fitting these models to the data a complication developed. This complication centered around the fact that five Ss in group UL and one S in group PL drew all 100 cards for one of the events. Specifically, for group UL four Ss drew all the cards for the event with the highest expected value, and one S drew all the cards for the event with payoff of 1.5 cents and probability of .13. For group PL the S drew all the cards for one of the most probable events. It was felt that using the responses made after the 100th card was drawn was misleading since this action tended to deflate the choice frequency of the preferred event. Two different courses of action were taken to compensate for the deflated choice frequencies. The first action was to simply assume that the Ss would have conti nued making the same response had they been able to. The 56 57 second course of action involved simply not analyzing the trials following the one on which the Ss ran out of cards. Although the two courses of action led to two sets of analyses for each group, the results of both were very similar. For this reason only one course of action is reported. The first course of action seemed to be more appropriate in terms of the models and the Ss1 behavior. By way of explanation, the Ss were observed, informally, to be very likely to repeat a response once it had been made. That is, it appeared that the Ss were suffering from posi tive recency, a finding which is rather unusual. It is mainly for this reason that the assumption that the Ss would have continued making the same response had they been able to appearsto be the more reasonable course of action. The results of the ZWS model fitting are presented in Tables 6 and 7. Table 6 presents the obtained and pre dicted frequencies for each group, and Table 7 presents the parameter values upon which the predicted frequencies are based. Table 7 also presents the value of chi-square between the obtained and predicted frequencies. As can be seen from Table 7, the model did not fit the data for any of the three groups. The chi-square values are all highly 1 2 3 4 5 6 7 8 9 10 11 12 TABLE 6 RESULTS OF MODEL ANALYSIS OBSERVED AND PREDICTED FREQUENCIES UL PL VL Obs Pred Obs Pred Obs Pred ZWS AHS ZWS AWS ZWS AWS 6 11.8 27.9 69 43.8 43.6 21 17.2 56.1 37 15.2 27.9 35 43.8 43.6 41 31.8 56.1 25 19.7 27.9 64 43.8 43.6 73 55.8 56.1 97 88.8 62.8 34 43.8 43.6 183 143.2 81.5 9 16.5 38.2 29 54.9 54.7 17 17.2 56.1 26 22.3 38.2 35 54.9 54.7 38 31.8 56.1 47 30.0 38.2 43 54.9 54.7 60 55.8 56.1 110 131.4 92.4 48 54.9 54.7 119 143.2 81.5 23 30.6 68.8 114 88.8 89.2 13 17.2 56.1 15 47.3 68.8 82 88.8 89.2 13 31.8 56.1 48 68.7 68.8 80 88.8 89.2 33 55.8 56.1 307 267.7 190.1 106 88.8 89.2 133 143.2 81.5 Ul oo r TABLE 7 RESULTS OF ZWS MODEL ANALYSIS— MINIMUM CHI VALUES OF THE PARAMETERS AND CHI SQUARES Group U1 U2 i °3 ; U4 W X2 UL 0.0000 0.0175 0.0425 0.7000 0.0412 89.5393 PL 0.0752 0.0754 62.5613 VL 0.0500 0.1500 0.3200 1.0000 0.6900 47.6498 i i n u> 60 significant (p<.001, df=7 for groups UL and VL, df«10 for PL), indicating that the predicted frequencies are sig nificantly different from the obtained frequencies. The results of the AWS model fitting are presented in Tables 6 and 8. As can be seen from Table 8, this model fits the data even worse than the previous one. Whereas the chi-square values ranged from 47 to 89 for the previous model, they range from 62 to 316 for AWS. Formal comparison of the several analyses con firmed the impressions already gained. The ZWS model fit significantly better than the AWS model for group VL (F=7.79, df=7,7, p<.01), but not for group UL (F=2.48) or group PL (F=1.01). Also, the AWS model fit the data of group PL better than group UL {F=5.09, df=7,10, p<.05) and than group VL (F=7.28, df=7,10, p<.01). It should be kept in mind that the between model comparisons are not strict tests because the chi-square values are not independent. TABLE 8 RESULTS OF AWS MODEL ANALYSIS— MINIMUM CHI VALUES OF THE PARAMETERS AND CHI SQUARE Parameter Group U. U0 U. U. W X2 1 2 3 4 UL 0.0000 0.0000 0.0000 1.0000 0.0337 221.5123 PL 0.3430 0.0500 62.0827 VL 0.0000 0.0000 0.0000 1.0000 0.0500 316.6080 CHAPTER VIII RESULTS: MULTIDIMENSIONAL SCALING The second phase of the experiment, the similar ities judgment phase, was analyzed according to the non metric procedure developed by Shepard (1962a, 1962b). The program TORSCA was used (Young and Torgerson, 1967). The scaling analyses were preceded by a Tucker and Messick points of views analysis (Tucker and Messick, 1963) as performed by VIEWS (Young and Pennell, 1967). The data from each group were submitted, sepa rately, to the points of view analysis. For each learning group, several unique points of view were submitted to the scaling procedure. The coordinates (representing the hypothetical Ss) used to compute the points of view are presented in Table 9. As can be seen, the first point of view analyzed for each learning group was simply the group point of view. The reasons which led to the remaining sets of coordinates will be explained presently. The coordinates used to compute the unique points 62 63 TABLE 9 COORDINATES FOR HYPOTHETICAL SUBJECTS Group Hypothetical Subject No. 1 Dimension 2 3 UL 1 1.000 .000 .000 2 .220 -.200 .140 3 .300 -.080 -.270 4 .310 .450 .000 5 .200 .500 -.500 PL 1 1.000 .000 .000 2 .220 .060 -.300 3 .230 -.050 -.290 4 .233 -.030 .320 5 .315 -.140 .040 VL 1 1.000 .000 .000 2 .361 -.080 -.200 3 .260 -.340 .220 US 1 1.000 .000 .000 2 .287 -.100 -.070 64 of view for group US are also presented in Table 9. Only two points of view were used for this group, the first one being the group point of view. The rationale for the co ordinates will be discussed presently. The results of the scaling only group will be pre sented first, followed by those for the learning groups. Scaling Group The results of the group point of view scaling are presented in Table 10. The stress of the one, two, and » three dimensional solutions was 14%, 7%, and 4%, respec tively. Even though the stress value for the one dimen sional solution was somewhat high, it was decided, after looking at all three solutions, that the configuration was really one dimensional. In three dimensional space the points lie on a curve roughly described as a spiral. As one travels along the spiral, he is travelling in an increasing (or decreasing) expected value path. In two- space the spiral is extended into a parabola, as is deter mined from the coordinates presented in Table 10. The one dimensional solution, also presented in Table 10, is clearly identified as an expected value solu tion. The relationship between scale value and expected 65 TABLE 10 SCALING COORDINATES BASED ON GROUP POINT OF VIEW FROM GROUP US One Dimension Two Dimensions 1 1 2 1 .781 .567 .563 2 .392 .505 .013 3 .155 .241 -.196 4 -.064 -.117 -.395 5 .675 .600 .396 6 .171 .242 -.167 7 -.040 -.060 -.421 8 -.311 -.461 -.368 9 .206 .260 -.312 10 -.258 -.412 -.125 11 -.638 -.665 .271 12 -1.071 -.701 .742 66 value is presented in Figure 4, and it is clear that the scale values are linearly and inversely related to the expected values. In Figure 5 the relationship is plotted between expected value and the two dimensional solution. It is striking that the relationship between expected value and the first dimension is linear, while the relation of expected value with the second dimension is quadratic. Similarly, for the three dimensional solution, the first dimension is a linear function of expected value; the second is a quadratic function; and the third is a cubic function of the expected value. It is interesting to note, that in Shepard and Carroll's terms (1966), the configuration is basically one dimensional, and these data appear to be a good candidate for Carroll's parametric-mapping theory (Carroll, 1966). There was some question as to whether these results might be due to individual differences, so it was decided to look closely at the subject space revealed by the points of view analysis. On the basis of this analysis, it appeared that three factors were sufficient to describe the judgment and subject spaces. It was found that five of the Ss occupied the same region of the subject 3-space, E x P e c t e d V a 1 u e .30 .25 .20 .15 .10 .05 —----- - 1.00 ■ - J - 1 -.75 i . ■ -.50 Figure 4. Scale Value versus Expected Value * • * . . . i . . * -.25 .00 .25 .50 Scale Value 75 E x P e c t e d V a 1 u e .30 .25 .20 .15 .10 .05 .1 • A « - 1.00 In— . . . I,. -.75 -.50 » - -.25 Figure 5. Expected Value versus Two Dimensional Solution for Group UL 69 and that the remaining Ss were scattered more or less evenly throughout the remainder of the space. Because these five Ss appeared to represent a consistent method of response, it was decided to use the mean of their co ordinates for a hypothetical S. These are the coordinates presented in Table 9. (Note that the means were derived visually from the plot of the S space, not arithmetically. This is true for all sets of hypothetical S coordinates.) The results of the analysis of the hypothetical S's judgments are presented in Table 11. The stress in one, two, and three dimensions was 11%, 6%, and 3%, re spectively. As can be seen by comparing Table 11 with Table 10, the results of this analysis are very similar to those for the preceding analysis, with the exception that each solution now has less stress. The subjects represented by the point of view serving as the basis for this analysis are, apparently, more consistent in their judgments than the group as a whole. This difference in consistency appears to be their only difference from the others. Hypothetical Learning Subjects For all three learning groups the hypothetical Ss 1 2 3 4 5 6 7 8 9 10 11 12 70 TABLE 11 SCALING COORDINATES BASED ON FIRST POINT OF VIEW FROM GROUP US One Dimension Two Dimensions 1 1 2 .799 .588 .543 .437 .462 .097 .205 .247 -.151 -.081 -.084 -.344 .764 .623 .441 .217 .270 -.167 -.025 -.000 -.423 -.333 -.392 -.490 .257 .281 -.211 -.292 -.405 -.244 -.734 -.743 .179 1.215 -.846 .770 were generated on the basis of the learning phase. It is important to note that the learning data were used in con junction with the scaling data in developing the hypothet ical Ss. The reasoning on this point is that any individ ual differences which do exist in the scaling data should be related to similar individual differences in the learn ing data. Furthermore, it was hypothesized that if a group of Ss could be broken down into subgroups on the basis of individual differences, then the Ss in these sub groups should appear in the same region of the subject- space. If, in fact, regions are found in the subject-space which reflect individual differences in the learning data, then a hypothetical £[ is generated for each region. The question remains as to exactly where in the learning data we should look for the individual differ ences . Since the dependent variable for the learning data is choice frequency, it was felt that this is where the individual differences should be searched for. It was decided that choice frequencies totaled over all 300 trials would be used to help increase the reliability of the individual choice frequencies. The final question is: How many dimensions of the subject-space should be used? It was decided, somewhat arbitrarily, to use the first three dimensions, because in some cases three dimensions were needed to differentiate the subgroups. Utility Learning Group The results of the group point of view scaling are presented in Table 12 and Figure 6. The stress of the one, two and three dimensional solutions was 18%, 11% and 8%, respectively. The two-dimensional solution appeared to be the overall best solution. Its stress value was high, but the solution was easily interpretable (if un expected) , and appeared highly similar to the first two dimensions of the 3-space solution. Finally, the third dimension was not easily interpreted. The most salient characteristic of the solution was its dichotomous nature. Stimuli 4, 8, and 12 were in one part of the space, and the remaining stimuli were in a different part. Stimuli 4, 8, and 12 are the three gam bles with the highest payofft 2 cents. It is apparent, then, that the solution space is divided into a high payoff and a low payoff area. Looking more closely at the distribution of the points in the low payoff sector of the space, and at their 1 2 3 4 5 6 7 8 9 10 11 12 73 TABLE 12 SCALING COORDINATES BASED ON GROUP POINT OF VIEW FROM GROUP UL One Dimension Two Dimensions 1 1 2 162 -.183 -.025 158 -.235 .081 093 -.113 .184 180 .250 -.348 061 -.039 -.058 134 -.108 .075 176 -.207 -.040 407 .474 .598 102 -.135 -.172 236 -.298 -.114 208 -.342 .075 743 .936 -.256 74 Figure 6. 12 Two Dimensional Scaling Solution for Group UL High payoff region Medium and low payoff region 75 relation to the three points in the high payoff sector, it was discovered that the space could more accurately be described as three lines instead of two regions. Each line has four elements, and each element has the same prob ability. The lines, then, each represent one of the event probabilities: the shortest is for events with probability of .05, and the longest for events with probability of .13. Turning now to the hypothetical points of view for group UL, the Ss were divided into three subgroups on the basis of the choice frequencies. Figure 7 shows that the hypothesized relation between the subgroups and the sub- ject-space was found: the subject-space can be divided into sectors based on the subgroups. The coordinates presented in Table 9 are the mean of the coordinates of the Ss in each region. For group UL, the first hypothetical point of view is based on those Ss who apparently were unaffected by the event characteristics (probability and payoff). That is, their choice frequencies were approximately equal for all twelve events. This lack, of effect is apparent in the scaling solution. The stress values were very large: 41%, 19%, and 10%, for one, two, and three dimensions, respec tively. (Experience has shown that random input for 12 Third point of view a Secon£ point ^ of view A First point of view A Fourth point of view Figure 7. Points of View for Group UL points produces stress values similar to these.) The points were scattered about in a haphazard manner in the space, and no interpretation of the dimensions was possi ble. The second hypothetical point of view for group UL was based on Ss whose choice frequencies were approximately in the order predicted by the expected values. The stress for the solutions was compared} le to that for the group point of view: 16%, 13%, and 9%. The solution was highly comparable to that for the group point of view. The third hypothetical point of view was not based on any Ss, but was, instead, representative of an octant in the subject-space which was unoccupied. This point of view was used to determine what sort of judgments would have been given by Ss in that area. The results of this analysis are extreme. The stress values are 0%, 3%, and 4%, for one, two, and three dimensions. The program was able to find a perfect fit in 1-space, by collapsing all the points into two points. These two points represent the two regions uncovered by the analysis of the group point of view. That is, if there had been any Ss in this area of the subject space, they would have judged events 4, 8, and 12 as being the same, and the remaining events as being the same. In addition, they would have judged 78 the two sets as very different. This, then, is simply the extension of the results found for the group point of view. The fourth hypothetical point of view was based on Ss whose frequencies were unlike the other Ss. The scaling fit quite well, with stress values of 14%, 7%, and 3%. The solutions were quite similar to those for the main point of view, but with a slight difference. The distances between points 4, 8, and 12 were as large as the distances between them and the remaining 9 points. Again, the 9 remaining points we^e all grouped together. It is felt that possibly the Ss representing this point of view had experience with only the three high payoff events, and judged them to be different from each other on the basis of their differing probabilities. Probability Learning Group The configurations derived by scaling the group point of view for group PL are presented in Table 13. The stress in one, two, and three dimensions was 28%, 16%, and 7%, respectively. These values, being quite large, suggest that the data representing the main (group) point of view are not conducive to the ordinal assumptions for the 1 2 3 4 5 6 7 8 9 10 11 12 79 TABLE 13 SCALING COORDINATES BASED ON GROUP POINT OF VIEW FROM GROUP PL One Dimension Two Dimensions 1 2 150 -.178 -.077 018 -.013 -.122 229 -.278 .032 108 -.059 .105 026 .003 .060 139 -.146 .037 271 -.157 .238 271 .263 .163 411 .358 -.156 109 -.079 -.076 040 -.101 -.273 356 .388 .069 80 scaling analysis. This suggestion was further supported by looking at the results of the analysis, since it was impossible to develop any reasonable interpretation of the scaling space. In order to see if any interpretable scaling solu tion could be obtained, several hypothetical points of view were developed. These points of view were obtained in the same manner as were those for group UL: the choice frequencies were looked at to identify individual differ ences, and then hypothetical Ss were developed to reflect these differences. The results of the search (based on individual differences) for improved scaling solutions were negative. Each solution for each point of view listed in Table 9 had a higher stress value than the correspond ing solution for the group point of view. And, like the results of the group point of view analysis, the difficulty in interpreting the scaling space derived in each analysis was immense. The overall conclusion from the series of analyses is that the judgments were not conducive to multidimensional scaling. Value Learning Group The scaling configurations for group VL's main 81 point, of view are presented in Table 14. The stress in one, two, and three dimensions was 23%, 13%, and 8%, re spectively. These stress values compare in magnitude with those for group UL, and are somewhat better than those for group PL. The comparison with group UL goes even further. As can be seen in Table 14, the two dimensional solution is strongly dichotomous, as was the main solution for group UL. The parallel can be continued. Each of the two areas in the solution for group VL contains the same subset of points as each of the two areas in group UL's solution. That is, one of the two areas contains events 4, 8, and 12, the high value events (2 cents), while the other area contains the remaining events, which are all of lower value. In one important detail, however, the solution for group VL is different from that for group UL. The space for group VL contains no identifiable vectors as did the space for group UL. This is a fortunate finding, since the vectors were related to event probability which is not a salient event characteristic for group VL. We turn now to the hypothetical points of view. The first point of view listed in Table 9 is simply the 1 2 3 4 5 6 7 8 9 10 11 12 82 TABLE 14 SCALING COORDINATES BASED ON GROUP POINT OF VIEW FROM GROUP VL One Dimension Two Dimensions 1 1 2 112 -.228 .031 049 -.174 .193 016 -.037 .285 274 .511 -.029 092 -.111 -.206 033 -.025 -.021 095 -.166 .087 190 .351 .339 134 -.228 -.104 141 -.256 -.025 179 -.156 -.274 355 .521 -.276 83 coordinates for the subject that appeared to give the "best" set of choice data. That is, his choice frequencies appeared to be highly related to the event values. The scaling space for this S appeared to be very similar to the space for the entire group; that is, the space was highly dichotomous. The stress values were somewhat better than those for the entire group, being 18%, 10%, and 6% for one, two, and three dimensions, indicating that this S was consistent in his mode of response. The second, and last hypothetical point of view listed in Table 9 is simply the coordinates for the S whose choice frequencies appeared least affected by the events' payoffs. The stress values for the three solu tions were 37%, 22%, and 15%, which are all very high, suggesting, perhaps, that there is no consistency to the judgments representing this point of view. The scaling space for this is completely uninterpretable, also suggesting more or less random judgments. CHAPTER IX DISCUSSION Three topics will be taken up in this discussion. The first one to be considered is the question of why the choice frequencies and scaling spaces were not entirely as predicted. The second topic to be discussed concerns why the stress values were large, and whether the distances were non-Euclidian. The third and final topic of discus sion centers around the question of why the learning models did not fit the data accurately. Choice Frequencies and Scaling Spaces The obtained choice frequencies are slightly dis appointing even though the hypothesized relation between them and the expected values was highly significant. This disappointment lies in the finding that the payoffs alone were even more highly related to the choice frequencies than the expected values. It is apparent that the two sets of data are con sistent with each other, and this is an interesting fact 84 85 in itself; but the question remains: Why were the choice frequencies for the learning data not as predicted? And why were the scaling results not as predicted? Answering the second question is somewhat easier than answering the first. In fact, it is clear why the scaling results were as they were. For group UL, the main effect observed in the scaling solutions was a payoff effect with the high payoff events separated from the low payoff events. The payoff effect was also the strongest effect in the learning data as was shown by the analysis of variance. A secondary effect observed in the scaling data was a small probability effect— there were curves of equi-probable events in the scaling space. Similarly, a small (but significant) probability effect was noted in the learning data. Finally, an expected value effect was noticed in the scaling data as an interaction of the prob ability and payoff effects. The vector lengths varied: the shortest one was for the less probable events; the longest for the most probable. As hypothesized, this effect relates to a similar one in the learning data. The apparent reason that the scaling data was not as predicted was that the learning data (upon which it depended) was not as predicted. That is, the scaling 86 spaces were not simple one dimensional spaces because the choice frequencies were not simple and "one dimensional." The strong dependency of the scaling results on the learning process can also be used to explain the scaling spaces for groups PL and VL. But, in reality, the explanation does not explain. It simply moves the ques tion back one more step to the question: Why did the learning not take place as predicted? "Because utility theory is wrong," is an answer to this question which can be tentatively ruled out. If we were to accept this answer we would find ourselves in the embarrassing position of accepting the theory sometimes and not at other times. We found that the theory predicted the results of the pilot experiment very nicely, and we accepted the theory. We can not now reject it altogether. It may be that there are differences in the design of the pilot experiment and in the current experiment which les sened the appropriateness of the theory. And it may be that these differences can be attributed to the radically different equipment used for the two experiments. It is apparent that the differences between the results for the pilot group and group UL lies in the im portance of event probability as a determiner of a S's 87 behavior. For group UL, -there was a smaller probability effect than in the pilot group. It is obvious that if there is small effect for probability, then the expected value hypothesis will not be supported as strongly as the payoff hypothesis (assuming an effect for payoff). This is, in fact, what happened for group UL. Why, we ask, was there a probability effect for the pilot group, and only a very weak one for UL? We can begin to answer this question by asking another, more fundamental one: What are the differences in equipment and procedure between the pilot group and group UL? Be sides the obvious equipment differences, there were some slight procedural differences. One procedural difference between the pilot group and the learning groups was that Ss in the learning groups could run out of cards for a specific event, eliminating the event from those which could be chosen. For the pilot group, however, the number of events could never be re duced. In fact, for the learning groups the probability of a given event paying off varied slightly each time it was chosen. This was not so for the pilot group. That is, one of the major differences between the pilot group and the remaining groups was that for the pilot group the 88 -trials were independent (the events were sampled with replacement) whereas for the remaining groups the trials were not completely independent (the events were sampled without replacement). It is not felt that this difference in design led to the difference in results, because the relative order of event probabilities should have remained unchanged (at least through the major portion of the trials), even though the level of probability was fluctu ating somewhat. Another difference between the pilot experiment and group UL centers around the area of contingent-noncon- tingent events. That is, it may have appeared to the Ss in groups UL, VL, and PL, that the probability of their drawing a card with an "A" on it was contingent on where they drew the card from. The Ss may have thought, in other words, that the probability of a given event occur ring depended on what choice they made. This was not, in fact, the case (except to a slight extent caused by the slight variations in event probability discussed in the previous paragraph), although it may have appeared to be the case. For the pilot experiment, on the other hand, this appearance of contingency did not occur. It is not clear exactly how this apparent difference affected the 89 results, nor what contribution it made to the differences in the outcomes of the pilot experiment and group UL. A third difference between the pilot experiment and the learning groups centers around the "history" of the event process. The history of an event process refers to how many of the previous events the Ss can see. For the pilot experiment, the Ss could look back at as many previous events as desired. But the Ss in the three learning groups were not allowed to look back at any pre vious events. This procedural variable, then, took on diametrically opposed values for the two designs. It is felt that the difference in history may be the critical difference between the pilot experiment and the three learning groups. It appears reasonable to assume that the Ss in the pilot experiment would utilize probability more effectively in their decisions than the Ss in the learning groups because these pilot Ss were able to look back at previous events and determine how often each event paid off. It also appears reasonable to assume that the Ss in the three learning groups, not being able to look back at previous events, would not be as heavily influenced by event probability. This line of argument can be easily tested by designing an experiment whose 90 critical variable is the event history. Stress and Distances We now turn to the multidimensional scaling anal ysis. As stated earlier, the scaling spaces show consist ent agreement with the choice frequencies. There is, however, one other aspect of the analyses which should be discussed. The values obtained for the stress index were uniformly large. The large values indicate that the Euclidian dis tance model does not fit the similarities as well as desired. Even though there are several possible sources for a lack of good fit, it is felt that the discriminant analysis revealed the major source: the similarities data are non-Euclidian, at least for some groups. The conclusion that some of the groups may be generating non-Euclidian judgments is reached on the basis of the following argument: for groups UL and US the stimuli have highly similar characteristics. That is, a given stimulus has the same probability and payoff for both groups UL and US, the main difference is that these charac teristics are imparted by learning for group UL and in structions for group US. Since the stimuli have similar 91 characteristics the judgment of the similarity of a given pair should be comparable between the two groups, i£ the same metric is being used. But if the two groups are using different metrics then there should be a consistent pattern of differences between their judgments. In looking at the pattern of similarity judgments it is particularly important to observe the cross-modal stimulus pairs, since: (1) the magnitude of the distance between cross-modal stimuli depends on the value of the Minkowski metric; and (2) the magnitude of the distance between two inter-modal stimuli is independent of the value of the Minkowski metric. Specifically, the distance between two cross-modal stimuli increases as the Minkowski constant decreases, while the distance between two inter- modal stimuli does not change. If we observe (1) that the cross-modal judgments are consistently larger for one of the groups than for the other; and (2) that the inter-modal judgments are the same, then we can conclude that there is a difference in Minkowski metric between the groups. This pattern of relationships was found in this study via the discriminant analysis. That is, of the first 14 stimuli pairs which were entered into the dis criminant function, 12 were cross-modal. Of these 12, the 92 judgment, for group US was larger than that for group UL in most cases. This may indicate, then, that group US has a metric more like a "city block" metric than group UL has. We can tentatively conclude, then, that the judgments given by the Ss in group US can be described by a space having a smaller Minkowski constant than the judgments given by Ss in group UL; i.e., the judgment space for the scaling only group is more "city-block like." This conclusion is particularly interesting in light of the results obtained by various investigators (Attneave, 1962; Shepard, 1964; Torgerson, 1958) which suggest that Euclidian judgments are obtained when the dimensions are less distinct and the objects behave as "unitary wholes,” and that city-block judgments are ob tained when the dimensions are obvious or "perceptually distinct." The conclusion reached in the previous para graph, that the judgment space for the scaling only group is more "city-block-like" is in line with these findings. More research is needed in this area. Learning Models The learning models did not fit the data from this experiment. Why this was the case is not clear, although 93 several tentative suggestions can be made. These sugges tions fall into the general category of possible violations of assumptions underlying the models. As mentioned earlier, the event probabilities fluctuated somewhat from trial to trial due to the fact that the number of possible responses for a given event was finite. That is to say, the model tacitly assumes that the events are sampled with replacement, whereas they were actually sampled without replacement. This violation may be of some importance, especially when one considers that the extent of the violation became extreme when the Ss ran out of cards. Another possible reason the models did not fit the data centers around the issue of whether the events were contingent or noncontingent. Although it may have seemed to the Ss that the experiment was a contingent design, it actually was not. This is an interesting question which needs investigation: Does a model which assumes noncon tingent reinforcement probabilities fit data from designs which appear to be contingent (but actually are noncontin gent) less well than data from truly noncontingent designs? This question reduces to a more fundamental one: Is the data gathered from an apparently contingent design different 94 from that gathered from either an actually contingent or an actually noncontingent design? Another reason that the models did not fit the data may be that the "independence of path" assumption (at the very heart of any Markov model) is inappropriate. Some informal observations indicated that the Ss tended to choose the same event again, once he had chosen it; i.e., there seemed to be a positive recency effect. Why this should be the case is not clear, since this is not a very common finding. More formal research should be done on this question. As a final note on the efficacy of the learning models, the models fit the data from the probability learn ing groups better than for either of the other groups. This is a particularly telling finding, since the specific raison d'etre of both models discussed in this paper was to cope with the payoff variable. This finding is telling because the probability group, for which the models fit best, is the only group without the payoff variable. Com pounding this finding is the fact that the AWS model fit exceedingly poorly for the value learning group, the group whose only variable is the one the model was specifically designed to handle. These results may be due to the find 95 ing that: the value variable had a stronger influence on the choice frequencies than was anticipated. CHAPTER X SUMMARY An experiment was designed with four purposes: The first was to provide a test of two stimulus sampling models of choice behavior in people. The second was to determine the effect of event probability and event payoff on the frequency with which a subject chooses the event. The third was to determine whether judgments of the simi larity of pairs of the events can be easily related to the choice frequencies. The final purpose was to provide a test of the subjective expected utility hypothesis of human choice behavior. The experimental design involved four groups of subjects. In three of the groups each subject was required to choose one of twelve gambles on each of 300 trials. The gambles each had a specific probability of paying a specific amount of money. For one group, three probability levels (.05, .07, and .13) were combined factorially with four payoff levels (.5 cents, 1.0 cents, 1.5 cents, and 96 97 2.0 cents) to produce 12 events with unique expected values. For the second group the same probability levels were used, but all 12 events paid .5 cents. For the third group the four payoff levels were used, but all events were equiprobable. After completing the gambling phase of the experiment each subject was asked to rate all 66 pairs of the twelve gambles on their overall similarity. The fourth group only performed the similarities phase of the experiment. 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Psychometrika, 1962, 27, 219-246. Shepard, R. N., and Carroll, J. D. Parametric representa tion of nonlinear data structures. In P. R. Krishnaiah (ed.). Proceedings of the International Symposium on Multi variate Analysis. New Yorks Academic Press, 1966. Shipper, L. Context effects in probability learning and decision making. Psychological Reports, 1966, 18, 131-138. Siegel, S. Theoretical models of choice and strategy behaviors stable state behavior in the two-choice uncer tain outcome situation. Psychometrika, 1959, 24^, 303-316. 105 Siegel, s., and Andrews, J. M. Magnitude of reinforcement and choice behavior in children. Journal of Experimental Psychology, 1962, 63, 337-341. Siegel, S. Choice, strategy, and utility. New York: McGraw-Hill, 1964. Torgerson, Warren S., Multidimensional scaling of similar ity. Psychometrika, 1965, 30, 379-394. Tucker, L., and Messick, S. An individual differences model for multidimensional scaling. Psychometrika, 1963, 28, 333-367. Von Neuman, J ., and Morgenstern, O. Theory of games and economic behavior. Princeton: Princeton University Press, 1944. Woods, P. J. The effects of motivation and probability of reward on two-choice learning. Journal of Experimental Psychology, 1959, 57, 380-386. Woods, P. J. The relationship between probability differ ences and learning rate in a contingent partial reinforce ment situation. Journal of Experimental Psychology, 1959, 58, 27-30. (b) Young, Forrest W. Twelve choice probability learning with payoffs. Psychonomic Science, 1967, 1_, 353-354. (a) Young, Forrest W. Periodic probability learning. Psycho nomic Science. 1967, £, 71-72. (b) Young, Forrest W., and Peeke, H. V. S. Probability learn ing in the goldfish I: aversive reinforcement. Psycho nomic Science. 1966, £, 373-374. Young, Forrest W., and Pennell, R. J. VIEWS, an IBM/360 program for points of view analysis. Behavioral Science, 1967, 12, 166. Young, Forrest W., and Torgerson, W. S. TORSCA, an IBM 7094 program for Shepard-Kruskal similarities analysis. Behavioral Science, 1967, in press. APPENDIX A INSTRUCTIONS FOR LEARNING GROUPS AND SCALING GROUP INSTRUCTIONS For Learning Groups This is a gambling experiment. You will be playing one of twelve gambles on each of a series of trials. These gambles are represented by the black plastic board in front of you, and IBM cards in containers on the back of the board. Behind each letter on the board is a deck of IBM cards. Please draw a card from one of the containers. DRAW THE CARD NOW. Looking at the card you have just drawn, you will notice that there are two letters and a number printed on the colored edge. Look at the left-hand letter on the card. The letter in front of the container you drew the card from is the same as the left-hand letter on the card. If the right-hand letter on the card is the same as the left-hand one, then you win a certain number of pennies. The number of pennies you win is one-half the number printed on the card. If the letters are not the same, you neither win nor lose. When you win please tell the experi menter the number on the card. He will add your winnings on the calculator. After you have drawn the card please 107 108 place it in the tray in front of you. Place it FACE DOWN with the colored edge TOWARD you. After placing the card in the tray, you are to draw a second card. Continue doing this until the tray is full. This will take about half an hour. It is important that you do as well as you can, and that you earn as much money as you can. Remember that when you win a gamble, you are to call the experimenter, who will total your winnings on the calculator. After you have filled the tray up so that it won't hold any more cards, you will go on to the next page of this booklet, but not until you fill the tray completely. If you have any questions, please ask the experimenter. Otherwise you may start. DO NOT TURN THE PAGE UNTIL YOU HAVE FILLED THE TRAY COM PLETELY . 109 In the remainder of this booklet, you will be asked to compare the gambles to each other. You will be given pairs of gambles, and will be asked to judge their difference. You should ask yourself how different the first gamble Is from the second one In terms of how much they paid during the course of the previous experiment. You are to report your opinion of the difference by marking the appropriate number to the right of the pair. Use a 1 for almost no difference, and a 9 for great difference. Intermediate spaces are used for intermediate degrees of difference. Let us look at an example. One pair is little great difference difference M - Q 1 2 3 4 5 6 7 8 9 Look at the gambles M and Q in front of you, to deter mine how much they paid over the entire course of the previous experiment. If you feel they paid the same amount, then circle the 1. If you feel they were very different in terms of payoff, then circle the 9. The task is fairly simple; you look at the two gambles, decide how different they are, and record your judgment by circling the appropriate number. You should 110 work quickly, recording your first impression. The task is not at all a test of your ability. We are using it as a means of studying how one interprets the differences between gambles. Each person's interpre tation is different from everyone else's, so we expect people to differ somewhat in the way they answer. Please answer each item. One's impressions are usually quite accurate even if he is unsure, and omitting items will cause more difficulty in this research them answering when you are unsure of how you feel. If you have any questions, please ask the experi menter. PLEASE TURN THE PAGE WHEN YOU ARE READY TO DO SO. INSTRUCTIONS For Scaling Group This is a gambling experiment. You will not actually be gambling, instead you will be ashed to judge the similarity of some gambles which the subject before you made. The gambles are represented by the letters and numbers on the board in front of you. The row of letters simply serves to identify each gamble. There are twelve gambles. The top row of numbers indicates the amount of money, in pennies, that the gamble pays. The second row of numbers indicates the number of times the previous subject won by playing that gamble. In the remainder of this booklet, you will be asked to compare the gambles to each other. You will be given pairs of gambles, and will be asked to judge their difference. You should ask yourself how different the first gamble is from the second one in terms of how much they paid during the course of the previous experiment. You are to report your opinion of the difference by marking the appropriate number to the right of the pair. Use a 1 111 1X2 for almost no difference, and a 9 for great difference. Intermediate spaces are used for intermediate degrees of difference. Let us look at an example. One pair is little great difference difference M - Q 1 2 3 4 5 6 7 8 9 Look at gambles M and Q in front of you, to determine how much they paid over the entire course of the previous experiment. If you feel they paid the same amount, then circle the 1. If you feel they were very different in terms of payoff, then circle the 9. The task is fairly simple; you look at the two gambles, decide how different they are, and record your judgment by circling the appropriate number. You should work quickly, recording your first impression. The task is not at all a test of your ability. We are using it as a means of studying how one interprets the differences between gambles. Each person's interpretation is different from everyone else's, so we expect people to differ somewhat in the way they answer. Please answer each item. One's impressions are 113 usually quite accurate even if he is unsure, and omitting items will cause more difficulty in this research them answering when you are unsure of how you feel. If you have any questions, please ask the experi menter . PLEASE TURN THE PAGE WHEN YOU ARE READY TO DO SO. APPENDIX B STIMULUS PAIR ORDERS 1. G “iJ 2 . Z - U 3. A - J 4. D - M . 5 . A — U 6. P — Z 7. A - H 8. A - Q 9. D — G 10. V - Z 11. G — S 12. P - H 13. V - U 14. A - M 1 5. G — Q 1 6. Q - U 1 7. Q - H 18. J — Z 19. V - H 20. M . — H 2 1. M - Z 2 2. S _ Z 23. J — H • CM A - P 25. S - H 26. V - Q « CM S - U • GO N A — D t o V O . P - U 30. J - P 31. D - U 32. D - Z 33. A - Z 34. S - V 35. M - Q 36. G — M 37. A - V 38. G — H 39. M — S 40. A - G 41. D - V CM D - Q 43. P - S 44. A 115 — H 45. J — V 46. D - H 47. J — U • 00 J - M 49. A - S 50. M - P 51. P - Q 52. M - U 53. G - V 54. Z - Q 55. J - Q 56. G - P 57. S — Q 58. G — u 59. H — u 60. G — z 61. J - S 62. M - V 63. D - J 64. P - V 65 . D - s 66 . D p

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Asset Metadata

Creator Young, Forrest Wesley (author)

Core Title A Multidimensional Similarities Analysis Of Twelve Choice Probability Learning With Payoffs

Degree Doctor of Philosophy

Degree Program Psychology

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

Tag OAI-PMH Harvest,psychology, experimental

Language English

Contributor Digitized by ProQuest (provenance)

Advisor Cliff, Norman (committee chair), Davis, Daniel J. (committee member), Wolf, Richard M. (committee member)

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c18-580987

Unique identifier UC11360858

Identifier 6717707.pdf (filename),usctheses-c18-580987 (legacy record id)

Legacy Identifier 6717707.pdf

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Document Type Dissertation

Rights Young, Forrest Wesley

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the au...

Repository Name University of Southern California Digital Library

Repository Location USC Digital Library, University of Southern California, University Park Campus, Los Angeles, California 90089, USA