UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

The effect of anxiety and defensiveness on testing expectation theories of decision making Wilson, William Taylor 1965

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-UBC_1966_A8 W54.pdf [ 4.97MB ]
Metadata
JSON: 831-1.0104725.json
JSON-LD: 831-1.0104725-ld.json
RDF/XML (Pretty): 831-1.0104725-rdf.xml
RDF/JSON: 831-1.0104725-rdf.json
Turtle: 831-1.0104725-turtle.txt
N-Triples: 831-1.0104725-rdf-ntriples.txt
Original Record: 831-1.0104725-source.json
Full Text
831-1.0104725-fulltext.txt
Citation
831-1.0104725.ris

Full Text

THE EFFECT OF ANXIETY AND DEFENSIVENESS ON TESTING EXPECTATION THEORIES OF DECISION MAKING by William Taylor Wilson B.Sc., University of B r i t i s h Columbia, 1963 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS i n the Department of Psychology We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA November 1965 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an advanced degree a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r a g r e e t h a t p e r -m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . , I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i -c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department o f Psychology The U n i v e r s i t y o f B r i t i s h C o l umbia Vancouver 8, Canada Date December 24, 1965 ABSTRACT The general purpose of this study was to examine one approach to the study of the relationship of personality variables to expectation theories of gambling. The Coombs and Bezembinder (1964) method of testing expectation theories of gambling be-havior was used to determine how many, among a group of 77 sub-jects, obeyed each of four expectation theories. These four expectation theories were: EV theory, assuming the maximization of the product of objective prize values and actual probabilities of winning; EU theory, assuming maximization of the product of subjective prize values and actual probabilities of winning; SEV theory, assuming the maximization of the product of objective prize values and subjective probabilities of winning; and SEU theory, assuming the maximization of the product of subjective value of the prize and the subjective probability of winning. The Coombs and Bezembinder method consists of comparing an estimate of an individual's consistency of choices independent of expectation theory assumptions with estimates of consistency under assumptions basic to each of the four expectation theories. A lower value of the consistency estimate under assumptions of a given expectation theory than the value calculated independently of any expectation theory assumptions leads to rejection of that i i p a r t i c u l a r theory as a model fo r the subject's behavior. The Coombs and Bezembinder technique f o r determining whether an i n d i -v i d u a l obeys the four expectation theories leads to the p r e d i c t i o n of an ordering of the expectation theories with respect to the number of subjects who do not s a t i s f y them. The procedure i n the present study involved the presen-t a t i o n of 96 pairs of one-outcome gambles to 77 subjects i n an introductory psychology c l a s s . A subject was required on each p a i r to choose between a gamble combining high r i s k with a large p r i z e and a gamble combining a low r i s k with a small p r i z e . I t was found that EV theory was rejected f o r 57 subjects, EU theory f o r 31 subjects, SEV theory f o r 26 subjects and SEU theory for 14 subjects. The hypothesis of monotonicity i n the number of rejections for the two sequences SEU-SEV-EV and SEU-EU-EV was accepted. A second hypothesis, that a higher proportion o f females w i l l obey the expectation theories than w i l l males, was rejected. The subjects were subdivided into high and low anxious and high and low defensive groups groups on the basis of scores obtained on the Alpert and Haber Test Anxiety Scale and the Crowne and Marlowe Defensiveness Scale. An examination of the data was suf-f i c i e n t to r e j e c t the hypothesis that more low anxious-low defensive and high anxious-high defensive subjects would obey the four ex-pectation theories than would subjects who were either low anxious-high defensive or high anxious-low defensive. i i i There were, however, some s t a t i s t i c a l l y s i g n i f i c a n t results obtained on the basis o f several ad hoc analyses. Fewer high defensive males than low defensive males appeared to obey SEV theory. Furthermore, fewer males who were e i t h e r high anxious-high defensive or low anxious-low defensive obeyed SEU and SEV theory than did males who were either low anxious-high defensive or high anxious-low defensive. On the basis of these r e s u l t s , i t was recommended that further research be conducted on the re-lationships of personality variables to expectation theories of gambling. I t was noted that the use of the Tversky method of t e s t i n g expectation theories would permit the simultaneous examination of two approaches to the r e l a t i o n s h i p of personality variables to d e c i s i o n making (personality variables versus propensity f o r r i s k and personality variables versus r a t i o n a l i t y of d e c i s i o n ). F i n a l l y , with respect to technique, i t was recommended that better ways of assessing personality variables be found and the subjects be f u l l y trained and run i n d i v i d u a l l y through the experiment. i v TABLE OF CONTENTS PAGE Abstract i i INTRODUCTION 1 PROCEDURE 20 Method 20 Analysis of Data 23 RESULTS 37 DISCUSSION 55 SUMMARY AND CONCLUSIONS 62 BIBLIOGRAPHY 65 APPENDIX I. Discussion of Five Theorems on Expectation Theory 69 APPENDIX I I . Sample Booklet 76 v i LIST OF TABLES (continued) TABLE PAGE 13. C l a s s i f i c a t i o n of Subjects S a t i s f y i n g Each o f the Four Expectation Theories by Personality Types . . 50 14. The Number of High and Low Anxious and High and Low Defensive Subjects F a l l i n g Into the Upper and Lower Quartiles on the Coombs and Bezembinder PI Consistency Estimate . 53 15. The Number of Male and Female Subjects S a t i s f y i n g Each of the Four Expectation Theories 54 INTRODUCTION Every action that an i n d i v i d u a l makes i n a multi-stimulus world implies the previous elimination of one or more al t e r n a t i v e s . What determines the course of action that an i n d i v i d u a l w i l l pursue? In attempting to answer this question one must assume that the i n d i v i d u a l i s motivated to choose the course of action that w i l l r e s u l t i n his receiving the greatest amount of benefit. The s p e c i a l i z e d aspect of t h i s problem dealt with by decision theory i s the problem of predicting an organism's behavior when i t i s faced with a series of a l t e r n a t i v e actions having f i n i t e p r o b a b i l i t i e s of leading to the desired consequence. Suppose the i n d i v i d u a l i n s t r i v i n g toward a desired goal i s faced with several a l t e r n a t i v e courses of action leading to unknown consequences. Suppose further that not a l l of the possible courses of action lead to the desired goal. That i s , there i s an element of r i s k involved. What factors govern the behavior of the individual? A simple paradigm for these conditions i s a gambling s i t u a t i o n . The i n d i v i d u a l i s required to predict one or more outcomes from a l i s t of possible outcomes r e s u l t i n g from a given action such as r o l l i n g dice. The action i s then performed and the 2 outcome noted. The individual receives or forfeits some benefit (usually a sum of money) depending upon whether the actual outcome is a member of the set of predicted outcomes that the individual chooses. The hedonistic answer, that an individual will make his choice in such a manner as to maxi-mize the resulting benefit to himself, introduces a number of problems. For instance, how can an outside observer deter-mine the benefit received by the individual? As early as the eighteenth century Bernouilli (1954) introduced the concept of "utility" to denote the lack of agreement between the paper value of money received as a prize in a gambling situation and its value as perceived by the recipient (subjective value). Furthermore, the list of bene-fits received by the individual may not be exhausted by the enumeration of goods and monies that change hands as a result of a given gamble. In the following paragraphs four models or theories which attempt to deal with the difficulties arising out of an attempt to predict an individual's behavior in a gambling situation will be discussed and evaluated. Edwards (1954) described four models which have been used as a basis for the experimental study of gambling deci-sions. These are defined by the equations used to calculate t h e i r values, and are: E V = (1) E U = (2) S E V= sjy1 (3) S E U= SjUi (4) where p^ i s the actual p r o b a b i l i t y that a given event i w i l l occur, where s^ ^ i s the subjective p r o b a b i l i t y that the player feels that the event w i l l occur and where i s the actual monetary gain accruing to the i n d i v i d u a l as a r e s u l t of the occurrence of the event i . i s the subjective value or u t i l i t y of the money accruing to the i n d i v i d u a l as a r e s u l t of the occurrence of event i . A l l four models of decision-making derive from the mathematical r e s u l t that i f an a l t e r -native associated with a prize of value v occurs with a pr o b a b i l i t y p, then, i n the long run, an i n d i v i d u a l would expect to end up with p x v d o l l a r s i f he i n v a r i a b l y selects —^ this a l t e r n a t i v e . Thus, i f the p r o b a b i l i t y of winning a $10.00 bet i s .5 then at the end of 100 bets, f o r example, one would expect to have won $500.00 (100 x .5 x $10 = $500). The other models incorporate other factors which may determine the choice of a given i n d i v i d u a l . EU theory allows f o r the introduction of Bernoulli!'s concept of subjective u t i l i t y f o r money. SEV theory permits the introduction of subjective 4 pr o b a b i l i t y preference, a concept popularized by Edwards (1953) as a r e s u l t of his discovery that some individuals prefer, o r tend to prefer, events that have .5 p r o b a b i l i t y to those which have .75 p r o b a b i l i t y of winning when the EV of both bets was maintained at a constant value. SEU theory allows the incorporation of both BernouiIll's concept of u t i l i t y and Edwards' concept of subjective p r o b a b i l i t y . The basic assumption underlying any of the expectation models of decision making i s that the i n d i v i d u a l w i l l choose an a l t e r -native that w i l l maximize the value of the p a r t i c u l a r expectation[representing the given model. Tests of the -theory become empirical measures of how many individuals behave i n a manner consistent with a given theory. An example may help to c l a r i f y the a p p l i c a t i o n of these theories. Con-si d e r the following four outcome gamble where i s the p r o b a b i l i t y of the value Vj.: Event P i V± EV (Mathematical Expectation) 1 .2 $9.00 $1.80 2 .3 $8.00 $2.40 3 .5 $4.00 $2.00 4 .5 $4.40 $2.20 Assume that the prize f o r the occurrence f o r event 2 i s a $2.40 non-transferable pass to a concert which, however, 5 occurs on a day on which i t i s impossible f o r the i n d i v i d u a l to attend. The prize for event number 3 i s a $4.00 t i c k e t f o r a play that the i n d i v i d u a l was, i n f a c t , going to attend, but f o r which the tickets were sold out with the r e s u l t that the i n d i v i d u a l was going to have to pay $5.00 from a t i c k e t scalper. Assume further that the cash p r i z e of $9.00 i s preferred a great deal more by the i n d i v i d u a l to the cash prize of $4.00. F i n a l l y , assume that the i n d i v i d u a l had a premonition that the outcome of event number 4 i s s l i g h t l y more l i k e l y than the outcome of event number 2, and that event number 1 i s only s l i g h t l y less l i k e l y than event number 3. I f EV theory was applied as a model f o r the individual's behavior we would predict that the i n d i v i d u a l would choose al t e r n a t i v e number 2, since this a l t e r n a t i v e provides the greatest mathematical speculation of return. I f EU theory i s used as a model of behavior, one would predict that the i n d i v i d u a l would choose a l t e r n a t i v e number 3 because the subjective u t i l i t y of the t i c k e t which i s the prize i s $5.00, r a i s i n g the expected value of that a l t e r n a t i v e to .5 x $5.00 or $2.50. I f SEV theory i s used as a model, one assumes l i n e a r u t i l i t y f o r money, but allows f o r idiosyncracies i n p r o b a b i l i t y pre-ference, with the r e s u l t that one would predict that the i n d i v i d u a l would choose event number 4. F i n a l l y , i f SEU theory i s used as a model which permits consideration of both 6 the subjective effects of probability and prize preference, one would predict that the individual would chose alternative number 2 because the probability preference i s only slightly less than for alternative number 4, but the prize is very much more preferred. Pruitt (1962) has made an excellent evaluation of these models and the following discussion relies heavily on that source. The evaluation of the three models was made by Pruitt on the basis of three c r i t e r i a : f i r s t , the range of alternatives within which each model has predictive value, secondly, the accuracy with which each model predicts decision within i t s range; and thirdly, the examination of whether a model containing a subjective parameter is any better than the comparable model containing an objective parameter within the ranges i n which both models have predic-tive power. The objective models EV, SEV, and EU are applicable only to situations where the objective parameters (i . e . , probability and value) are defined and known by the gambler. The EV model seems inadequate i n the range of gambles i n which there are small probabilities of winning large amounts of money, such as i n the purchase of lottery tickets ( A l l a i s , 1953; Bernouilli, 1954). EV seems to be a poor 7 predictor of choice behavior when the difference between two gambles i n EV as defined by equation (1) i s small, provided that the range of gambles involves only moderate probabili-ties and values of outcomes. The predictive power of EV theory increases, however, as the difference i n expected value increases. Thus, i n the case i n which the EV value of two bets i s identical, EV theory predicts that there w i l l be no preference, whereas Coombs and Kormorita (1958), Edwards (1953) have found consistent preferences. Mosteller and Nogee (1951) found an increase i n correct predictions from 42 percent to 67 percent when the difference i n EV between two bets was increased from less than 50 cents to $2.50. Edwards (1954a) suggests that the less complex the bets be-tween which a decision is made, the more accurate the EV model w i l l be as a predictor of decisions. Regarding the SEV model, Pruitt (1962) states that i f a person w i l l bet at a l l on a chance event, he w i l l put a l l his money on that event, a prediction which i s obviously not upheld at the race track and i n many other situations. This interpretation of SEV, however, seems to rest on a false interpretation of subjective probability. Just because an individual subjectively prefers one probability does not mean 8 that he automatically assigns a zero p r o b a b i l i t y to every other possible event. Thus, recognizing the p o s s i b i l i t y of other outcomes, there i s no reason f o r him to r i s k a l l h is c a p i t a l on a si n g l e a l t e r n a t i v e . The SEV model i s moderately accurate i n predicting choices between simple bets which d i f f e r i n p r o b a b i l i t y of winning o r losing but whose outcome involves the same l e v e l of money (Edwards, 1953, 1954a, 1954b, 1954c; Coombs and P r u i t t , 1960). Suppes and Walsh (1959) found that SEV pre-d i c t i o n s on two outcome bets with a subjective p r o b a b i l i t y of winning of .5 were correct only 57 percent of the time. This f i n d i n g indicates a second generalization. The SEV model i s only s l i g h t l y , i f any, better than the EV model i n the case i n which the p r o b a b i l i t i e s of winning and losing remain constant but the monetary outcome l e v e l v a r i e s . The major d i f f i c u l t y associated with EU and SEU models i s that u t i l i t y i s d i f f i c u l t to measure. Edwards (1961) developed a method of measuring u t i l i t y and testing EU theory but l a t e r discarded i t as i n v a l i d . An attempt at measuring u t i l i t y by Coombs and Kormorita (1958) on three subjects had 29 out of 30 correct predictions but a l l 30 would have been correct under the assumption that the subjects preferred a 9 higher p r o b a b i l i t y of winning to a lower p r o b a b i l i t y of winning. Mosteller and Nogee (1951) generated u t i l i t i e s f o r money and then used these u t i l i t i e s to t e s t EU theory. They found 66 percent correct predictions as opposed to 50 percent correct predictions with the EV model fo r a number of d i f f e r e n t levels of p r o b a b i l i t y and money. The predic-tions increased i n accuracy up to 93 percent i n pairs i n which the difference was over $2.50. Unfortunately, accuracy i n EV predictions w i l l also increase as the money value increases. This f a c t p a r t i a l l y v i t i a t e s the above findings. There seems, however, to be some ambiguity i n i n t e r p r e t a t i o n o f the measurement of u t i l i t y since the subjective difference i n u t i l i t y could be as well explained by a p r o b a b i l i t y pre-ference. Consider the gamble A which i s won with a p r o b a b i l i t y .5 and prize valued at $6 and i s preferred to gamble B which i s won with a p r o b a b i l i t y .48 and prize valued at $6.50. One might explain the choice as a prefer-ence for the p r o b a b i l i t y .5 or as a preference f o r the prize of value $6. The conclusion stated by P r u i t t (1962) i s that EU theory i s somewhat better than EV theory when differences between EU i n paired choices are small and improves markedly as differences increase, subject to ambiguities i n the measurement of u t i l i t y . 10 Most tests of the SEU model have used a chance event (p = .5) with a result that SEU theory remains essentially untested over a large range of probabilities. Suppes and Walsh (1959) used a die with three sides covered with the nonsense symbol ZOJ and three sides covered with the non-sense symbol ZEJ so as to eliminate preconceived probability preferences. The subject was forced to choose between two gambles with prize values ranging from 4-40 cents to -40 cents. The outcome of the gamble was determined by the r o l l of the die. They found that the SEU model predicted 58 percent correct choices as compared to 57 percent for the SEV model. Davidson, Suppes and Siegel (1957) attempted to measure u t i l i t y of phonograph records by a linear program-ming method. Using the u t i l i t i e s obtained, they tried to measure the subjective probabilities attached to various gambles and thus to test for SEU maximization. The results showed that SEU theory was able to predict correctly a minimum of 67 percent and a maximum of 71 percent of the choices. Pruitt (1962) has suggested two directions for re-search on gambling behavior: traditional models may be refined or new models may be postulated and tested. With 11 regard to the second alternative, he introduces a pattern and level of risk model (PLR) which i s applicable only to bets having at least one negative outcome. This model has had some success in predicting choices i n situations where the other four models f a i l but has the weakness that modifi-cations are required to explain choices i n situations i n which no negative alternatives occur. A second alternative i s one based on measuring the variance preference. Thus two bets could have the same ex-pectation value and the same probability of winning or losing but one bet could involve a large prize for winning and a large negative prize for losing while the other gamble involves small losses and gains. The former gamble has a large v a r i -ance while the latter gamble has a small variance. Studies i n this area have been made by Royden, Suppes and Walsh (1959); Davidson, Suppes and Siegel (1957); Coombs and Kormorita (1958); and Coombs and Pruitt (1960). These models have made some correct predictions i n cases i n which the u t i l i t y models have failed but a major shortcoming is that the methods of measuring variance preferences can be interpreted in terms of a u t i l i t y function for money. This means^if an individual has a variance preference then he prefers only bets of a certain dollar value. This i s the equivalent to having a u t i l i t y 12 function for money since a u t i l i t y function implies a predi-lection toward bets of a given dollar value. The preceding discussion suggests that one of the basic problems i n testing the SEU or SEV models i s in the d i f f i c u l t y of forming a "good" and independent estimate of both u t i l i t y and subjective probability. Coombs and Bezembinder (1964) have developed a method of testing these theories without requiring an estimate of subjective pro-bability and u t i l i t y . They have reported that for 36 subjects EV theory was rejected for 34 subjects, EU theory was rejected for 7 subjects, SEV theory was rejected for 7 subjects, and SEU theory was rejected for 2 subjects. Thus, i n this context, SEU had the widest general applicability for prediction of choice behavior. Coombs and Bezembinder*s method depends on five theorems which are described i n detail i n Appendix I. These theorems deal with the prediction of gambling choice between pairs of bets when the prize values and probabilities are varied sequentially. Pairs of bets can be presented to the subject i n such a manner that the mathematical expectation (EV) of the gambles with the higher probability of winning is 13 originally greater than the mathematical expectation value of gambles with the lower probability of winning. As the pro-bability of winning i s systematically decreased for both members of the pair, these five theorems predict a point after which the subject should choose the bet with the lower pro-bability of winning in order to maximize value and u t i l i t y of the prize. The f i r s t part of this introduction has discussed re-search attempting to develop more adequate methodological techniques for determining the validi t y of the four expectation theories of gambling behavior. Gambling behavior, however, is a subset of the class of behaviors known as risk taking. Several research studies have dealt with finding personality correlates of risk taking behavior. An examination of this research on personality correlates of risk taking might, therefore, help to isolate the personality types best obeying the four expectation theories. Slovic (1964) has reviewed much of the experimental work done i n an attempt to find personality correlates of behavior. Atkinson (1957) found that subjects with high motives to achieve preferred bets with intermediate probability of success whereas subjects with high motives to avoid failure 14 chose bets with high or low probabilities of success. A further study by Atkinson (1960) corroborated this finding. Scodel, Ratoosh and Minas (1959) found individuals with high need achievement, theoretical aesthetic values and fear of failure chose more conservative bets. Brody (1963) found that subjects rated high i n need achievement and low in test anxiety as measured by the Handler Sarason Test Anxiety questionnaire (Mandler and Sarason, 1952) tended to increase confidence rapidly i n the correctness of their choice up to the 50 percent level and then to increase more slowly as compared to low need-achievement high test anxiety persons. Further, a high need for achieve- ~ ment was found to be associated with a higher estimate of the subjective probability of success for a given bet. Rim (1964) found that individuals high on an extroversion scale took greater risks i n a gambling situation than those scoring low on the scale. In a group situation those individuals scoring moderate in an extroversion and a neurotocism scale shifted much more toward r i s k i e r behavior from their individual behavior than did either those scoring low or high on the scales. Stone (1964) found scholastic performance was 15 negatively related to u t i l i t y for risk whereas an agreeing response set was positively related. Intelligence and anxiety as measured by the Taylor Manifest Anxiety Scale i n this study was found to be unrelated to u t i l i t y for risk. Kogan and Wallach (1964) have measured the effect of test anxiety as defined by the Alpert and Haber (1960) scale and defensiveness as defined by the Crowne and Marlowe (1960) scale on risk taking behavior. The results from this study indicate that individuals classed as high anxious and high defensive and low anxious-low defensive tended to exhibit a more stable or consistent pattern of risk taking behavior than low anxious-high defensive or high anxious-low defensive individuals. The similarity i n behavior of opposite groups was explained as the result of two mechanisms. In the case of the high anxious-high defensive individuals, consistency of behavior was said to have resulted from the development of a common pattern of action derived to meet an emotionally arousing situation. In the low anxious-low defensive groups, the consistency of behavior over several situations was said to be due to a rational or cognitively perceived similarity among these situations. There were three types of situations 16 examined i n the experiment. F i r s t , s k i l l games, that i s , games i n which the individual was required to bet on the attaining of a goal that depended only on his own s k i l l . Second, actual gambles or situations in which the individual won or lost a sum of money dependent on the chance occurrence of an event. Thirdly, hypothetical gambles which consisted of asking an individual how much he would be willing to risk on the occurrence or non-occurrence of a certain chance event. It was found that females tended to behave more similarly across these three types of situations than did males. The above studies deal with three personality correlates cCrisk taking behavior: need for achievement, defensiveness and anxiety. The need for achievement has been further re-duced into a need to avoid failure and a need to succeed. Defensiveness would seem, a p r i o r i , to be closely related to the f i r s t of these components, the need to avoid failure. The apparent contradictory findings on the effect of anxiety on risk taking behavior reported by Stone (1964), Kogan and Wallach (1964) and Brody (1963) diminish upon examination of the types of anxiety being measured i n each case. The latter two studies use a form of test anxiety and report a positive relationship with risk taking behavior while Stone, using the 17 Taylor Manifest Anxiety Scale reports no relationship to risk taking behavior. One possible conclusion to be drawn from these findings i s that test anxiety and whatever is common to defensiveness and need achievement are related to risk taking behavior. It would seem appropriate to use these personality findings to attempt to form a more adequate test of the ex-pectation models of gambling behavior. A possible outcome of this type of investigation would be that some personality types obey one expectation theory while other personality types obey another. I t would seem desirable to apply the personality findings to the Coombs and Bezembinder method of testing ex-pectation theories for several reasons. It avoids the problem of measuring u t i l i t y and subjective probability independently and i t presents a series of gambles covering a range of pro-babilities rather than just a single probability. The sequence of presented gambles allows the presentation of a range of expected values of the various theories. The Coombs and Bezembinder method of testing the four expectation theories of gambling involved the presentation of a set of gambles three times and the evaluation of the consistency of behavior on identical gambles. If the individual behaved i n 18 a random manner, he was discarded from the analysis. It could thus be determined, for a l l Ss satisfying the Coombs and Bezembinder randomness criterion, whether personality variables affecting behavioral consistency had any effect on whether an individual obeyed the restrictions of the four expectation theories. Kogan and Wallach (1964) found that low anxious-low defensive and high anxious-high defensive individuals were more consistent i n their behavior than the other two groups of indi-viduals. It would thus be expected that these two groups would tend to obey the restrictions imposed by some of or a l l of the expectation models of gambling behavior to a greater extent than the other two groups. Furthermore, since Kogan and Wallach have already found that females were more consistent than males i n their risk taking behavior, i t might also be reasonable to expect that females would tend to obey the four expectation theories to a greater extent than males. The principal purpose of the present study was to test these predictions i n an experiment consisting of the presentation of the Coombs and Bezembinder gambles con-joined with the anxiety and defensiveness questionnaires to male and female subjects. Questionnaire scores s p l i t at the median for the entire group provide a basis for dividing the subjects into four personality groups: low anxious-low defensive, high anxious-19 high defensive, low anxious-high defensive, high anxious-low defensive. These groups can then be separately evaluated for conformance to the four expectation theories as can the male and female subgroups. Another prediction is suggested by the Coombs and Bezem-binder technique for testing the four expectation theories. This technique requires that an individual satisfy an increasingly stringent set of assumptions as the number of "subjective" parameters representing his behavior is reduced from two i n the case of SEU theory to zero in the case of EV theory. There are thus two possible sequences of expectation theories i n which the number of rejections should be monotonically increasing: SEU-SEV-EV and SEU-EU-EV. These predictions may be formalized with the following experimental hypotheses: 1) More low- anxious-low defensive and high anxious-high defensive individuals w i l l obey the given expectation theories than w i l l members of the other personality groups. 2) A higher proportion of females w i l l obey the expec-tation theories than w i l l males. 3) Monotonicity w i l l be observed i n the number of rejections of each theory for the two sequences: SEU-SEV-EV and SEU-EU-EV. 19a The experimental design also permits a preliminary, purely exploratory analysis of possible differences among indi-vidual personality types obeying the four expectation theories. PROCEDURE 20 Method Fif t y six male and twenty one female f i r s t year University of British Columbia psychology students were tested as a group during the last half hour of a regular class period. A l l Ss were provided with identical test booklets containing instructions, 96 pairs of gambles and two personality questionnaires. Instructions for the task read as follows: Please Do Not Open the Booklet Until Told To Do So At the top of the booklet indicate whether you are male or female. Do not place your name on the booklet. Pictured below is a pair of gambles. Each pattern has a spinner that rotates. If the spinner stops on the shaded area of a particular pattern, the prize is the amount of money stated above the pattern. If the spinner stops on the unshaded portion of the pattern, nothing is won or lost. Place a check inside the gamble which you would chose i f you were only allowed to play one of the gambles. On the following pages are a series of pairs of gambles. 21 Place a checkmark in the member of each pair that you would prefer to play. Make a choice on each pair. Following the series of patterns, there are two questionnaires. Please f i l l these out according to the instructions at the top of the questionnaire. At the end of the experiment, ten people w i l l be chosen at random from the class to play one of the* gambles on the questionnaire for the cash prize indicated on the booklet. Each of these persons must, however, play the gamble in the way i n which he indicated his choice on his booklet. The contents of this booklet have been reproduced in Appendix I. The 96 pairs of gambles were each presented on separate pages of the booklet i n order to reduce the effects of previous choices upon the gamble at hand. The gambles were presented i n a form similar to the pair that appears in the above instructions differing only i n the fact that a sum of money representing the prize was indicated above each gamble. The amount of money and the probability of winning the gamble were varied systematically although the choice for the subject was always between a high risk-high prize gamble and a low risk-low prize gamble. The f i n a l sections of the booklet consisted of the Alpert and Haber Test Anxiety Scale and the Crowne and Marlowe Defensiveness Scale. The Alpert and Haber Test Anxiety Scale i s based upon the principle that specific anxiety and general 22 anxiety are different concepts operationally measured on two separate scales. The test consists of two scales, a f a c i l i -tation scale of nine items of the prototype "Anxiety helps me to do better during examinations" and a debilitation scale stating the inverse sort of relationship. The items are randomly mixed from both scales into a single questionnaire. Each item i s answered on a five point scale designated by the adverbs: always, often, sometimes, seldom, and never. Alpert and Haber (1960) report a test - retest r e l i a b i l i t y coefficient over a ten week period of .83 for the f a c i l i t a t i n g scale and .87 for the debilitating scale. The Crowne and Marlowe Defensiveness Scale was developed in an attempt to measure social desirability independently of psychopathology. A set of items was drawn from the population of "behaviors" which are culturally sanctioned and approved but improbable, (e.g. "I never hesitate to go out of my way to help someone in trouble."). Crowne and Marlowe (1960) report an internal consistency for this test (Kuder-Richardson) of .88 and a test - retest r e l i a b i l i t y coefficient over a one month interval of .89. The instructions were read aloud by the experimenter who also solicited questions about the task before allowing the Ss to proceed. The Ss then indicated which gamble i n 23 each of 96 pairs they preferred. Space was provided at the top of the booklet for the subject to indicate his sex. In order to promote a more r e a l i s t i c approach to the gambling task, the subjects were told that at the conclusion of the experiment, ten individuals would be chosen at random to play one of the gambles for real money. The only res-t r i c t i o n imposed was that each of the selected Ss must choose the member of the selected pair of gambles that he had pre-viously indicated i n his test booklet. Since there was no possibility of losing money and a good chance of winning, i t was reasoned that this would provide sufficient motivation for completing the booklets as though real gambles were involved. The fact that a number of students expressed interest i n the experiment and participated i n a discussion after the testing was completed provided qualitative evidence for the correctness of this reasoning. Analysis of Data The analysis consisted of two major steps; f i r s t , the examination of each of the subject's gambling patterns to determine i f the assumptions of any of the four expectation theories were met. Second, i n order to provide a better test 24 of the expectation theories, the subjects were further divided into personality groups by sp l i t t i n g scores on the questionnaires at the median. The method for determining whether an individual obeys the assumption of a given expectation theory was done on a computer to avoid many tedious hand computations. The 96 pairs of gambles in the test booklet were com-posed of three identical sets of 32 pairs of gambles. The probability of winning and value of prize for each of these 32 pairs is given by Coombs and Bezembinder (1964) and are reproduced in Tables 1 to 4. Employing the Coombs and Bezembinder technique, i t was determined whether each subject obeyed any or a l l of the expectation theories. This technique i s a two step process. F i r s t , for each of the 32 different gambling patterns, i t was noted how many times the subject chose the l e f t alternative on each of the three presentations. From this, an estimate of the individuals consistency in behavior independent of any theoretical restrictions was ob-tained. The second step involves the estimate of the subject's consistency i n his choice behavior under the additional restriction imposed by the four expectation theories. If this new consistency estimate is no smaller than the original 25 Table 1 Expected Values of the gambles comprising Set I Set I: VL - .80, v r = 1.20 p l EV L p r EV r .9 .72 .8 .96 .8 .64 .7 .84 .7 .56 .6 .72 .6 .48 .5 .60 .5 .40 .4 .48 .4 .32 .3 .36 .3 .24 .2 .24 .2 .16 .1 .12 Table 3 Expected values of the gambles comprising Set III Set III: v i = 2.80, v r = 3.20 Pl EVx p r EV r .9 2.52 .8 2.56 .8 2.24 .7 2.24 .7 1.96 .6 1.92 .6 1.68 .5 1.60 .5 1.40 .4 1.28 .4 1.12 .3 .96 .3 .84 .2 .64 .2 .56 .1 .32 Table 2 Expected values of the gambles comprising set II Set II: v1 = 1.70, v = 2. r P l EV 1 * r EV r .9 1.53 .8 1.84 .8 1.36 .7 1.61 .7 1.19 .6 1.38 .6 1.02 .5 1.15 .5 .85 .4 .92 .4 .68 .3 .69 .3 .51 .2 .46 .2 .34 .1 .23 Table 4 Expected values of the gambles comprising Set IV Set IV: v L = 3.70, v r = 4.30 Pl EVX Pr EVr .9 3.33 .8 3.44 .8 2.96 .7 3.01 .7 2.59 .6 2.58 .6 2.22 .5 2.15 .5 1.85 .4 1.72 .4 1.48 .3 1.29 .3 1.11 .2 .86 .2 .74 .1 .43 26 estimate of consistency, i t i s assumed that one can not reject that particular expectation theory as a model for the subject's gambling behavior. The method used to estimate the required consistencies w i l l be explained i n more detail i n the follow-ing paragraphs. In order to estimate a subject's consistency without regard to theory (Pi), i t was determined for each of the 32 different pairs of gambles in Tables 1 to 4 whether the subject had chosen the l e f t gamble 0,1,2, or 3 times out of the three times the pair was presented to him. If the individual chose the l e f t gamble three or two times, a one i s placed opposite the representation of the particular gamble i n Tables 1 to 4. If he chooses the l e f t gamble, zero or one time, a zero i s placed i n the corresponding entry. An example of this method of recording the data appears i n Table 5. Because the scores of 3,2,1 and 0 are reduced to scores of 1 and 0, this method of recording the data w i l l be called the reduced matrix form. This reduction of the data makes i t much easier for the sub-sequent calculations to be made. The consistency of an individual i s related to the probability that he w i l l make his dominant choice on a given gamble. Thus, i f he is completely consistent he w i l l make 27 Table 5 Sample Scoring of Data i n Reduced Matrix Form (p L is the probability of winning i f the l e f t gamble i s chosen) P L P r Set I .9 ,8 0 .8 .7 0 .7 .6 0 .6 .5 0 .5 .4 0 .4 .3 0 .3 .2 1 .2 .1 1 Set II Set III Set IV 0 0 0 0 0 0 1 0 1 1 0 1 1 0 1 1 0 1 1 1 1 1 1 1 his dominant choice a l l the time ( i . e . , always choose the l e f t gamble or always choose the right gamble). If PI^ is the probability that the individual makes his dominant choice on any pair of gambles and i s the proportion of times that the individual i s observed to have made either three l e f t choices or three right choices, then i t follows that L i i s equivalent to the probability of making three l e f t choices plus the 28 probability of making three right choices. L t = ( P I t ) 3 + (1 - P I ± ) 3 (1) Similarly 1 - L i i s the probability of making two right choices and one l e f t choice plus the probability of making one right choice and two l e f t choices. Recall that PI i s the probability that a subject w i l l make his dominant choice on a given pair of gambles. A subject who had a PI value of .67- or larger (two standard deviations above the chance level) was accepted for further analysis. In order to evaluate an individual's consistency, i t is necessary to know the probability (P^) that the individual's dominant choice w i l l appear i n the reduced matrix form of the data. This i s the probability that he made his dominant choice three times or that he made his dominant choice twice i n any of three Bernouilli sequences. 1 - L i = 3 P I i 2 ( l - P l i ) + 3PIi(l - P l i ) 2 (2) These equations have the solution: (3) p i = P I i 3 + 3 P I 1 2 ( l - Pl i ) (4) Pi i s the required estimate of the individual's con-sistency of behavior - independent of any of the four expec-29 tation theories. The next step is the evaluation of the consistency of the individuals under restrictions imposed by each of the expectation theories. 1) EU theory. EU theory involves the assumption that an individual w i l l try to maximize the product of the actual probability of winning and the subjective value of the prize. Examination of Tables 1 to 4 reveals that the order of presen tation of gambles is such that the probability of winning decreases while the prize value remains constant. Thus, the subjective value should also remain the same. The EU value w i l l , therefore, decrease i n the same ratio as the EV value. As can be seen from row seven in Table 1, a point exists where a l l pairs below that point, EV's and EU's on the l e f t side are greater than those on the right. Since there are only two possible entries i n the table (0 and 1) any two scores can form only four different patterns: 0 0 1 1 • 0» 1» 1» 0 * Let a latent pattern be one that represents a person's real preferred set of two choices and l e t a manifest pattern be 30 the pattern occurring i n the reduced matrix. The only admis-sible latent patterns are: ®, ^, since the pattern referred to as a violating pattern, indicates that the individual has made one of his choices i n a way that does not maximize his EU value. The latter result follows from the fact that the EU value is continually decreasing for the l e f t side gambles less than for the right side gambles. Therefore, i f the right side originally has a higher EU value, the only permissible change i n preference would be from right to l e f t Let pi be the probability that a particular manifest ( i . e . , observed) pattern occurs i n the reduced data matrix when a given latent pattern is the person's real preferred choice pattern. Let q i be the probability that the manifest choice does not correspond to the latent choice. Table 6 i s a reproduction of a table i n Coombs and Bezembinder (1964) showing (in terms of p^ and , the conditional probability that a particular manifest pattern i s observed when a given latent pattern i s present. To understand this table, examine the f i r s t entry, the one corresponding to the J manifest pattern and the latent pattern. Both top and bottom entries are 0. This means that the right hand choice i s 31 Table 6 Conditional Probability That a Particular Manifest Pattern is Observed Given the Presence of a Particular Latent Pattern Manifest Pattern 0 1 0 1 0 1 1 0 Latent ® p ± 2 q ± 2 p ^ Pattern ^ ^ ^ - - ^ ^ - ^ ^ ^ ^ ^ ^ ^ ^ ^ - ^ ^ 1 2 2 1 <li Pi Pi<lt Pi<li 0 1 P i ^ i p i q i Pi qi preferred for both entries. The conditional probability that 2 0 1 the two pairs agree is Pi x p^ = Pi . Patterns Q and ^ are defined as compatible because these patterns indicate no change i n choice preference. Pattern ^ i s defined as a confirming pattern because this pattern indicates a choice change i n keeping with SEU assumptions, and J is defined as a violating pattern because this indicates a choice change that violates the assumptions of SEU theory. 32 Let be the proportion of latent compatible patterns for subject i . Since there are no violating latent patterns, 1-T^ i s the proportion of latent confirming patterns. The probability of obtaining a compatible pattern i n the reduced data matrix of individual i is the probability of obtaining the compatible pattern ^ or j given that the latent pattern i s a latent compatible pattern ( ( p i 2 + q^ 2)Ti) plus the pro-bability of getting a compatible pattern given that the latent pattern is not compatible (2p^q^(1-T^)). At = ( p i 2 + q i 2 ) T i + 2p iq i(l-T i) = r/N (5) In the above equation, r, s, t and N are equal res-pectively to the number of compatible, manifest confirming, manifest violating, and total number of patterns observed i n the individual's reduced data matrix. Similarly the probab-i l i t y (B^) of a confirming pattern being manifest is B i = EiSi T i + Pi 2< 1" Ti> = S / N <6> Equations (5) and (6) may be solved for p^ and T i . Remember-ing that p i is the probability that the observed pattern of two scores agrees with the individual's preferred choice, i t can be employed as an estimate of the individual's consistency of behavior under restrictions of EU theory. 33 2) SEV theory. SEV theory involves the subjective value of probability and real value of prize. If Tables 1 to 4 are read along rows of constant probability, analogous patterns to those examined for EU theory w i l l be observed. In this instance the pattern 1 0 is the violating pattern. Thus, the same procedure may be carried out for estimating p^ using row patterns instead of column patterns. The values of r, s, and t change, however, (r+s+t = 2 x 8 = 16). 3) SEU theory. The patterns involved i n testing SEU theory are a l l of the second order minors of the reduced data matrix. A second order minor i s any combination of four elements from a larger array such that the elements in each respective row of the minor are taken from the same row i n the matrix and the elements from each respective column are taken from the same column i n the original matrix. The following is a second order minor with elements A, B, C, and D; One of the Bezembinder and Coombs theorems (theorem 5 in Appendix I) states that i f the assumptions of SEU theory are accepted and i f the entries of the minor are numerical ratios of expectations, then the determinant of every second order minor w i l l equal zero (AD - BC = 0). If O's and l«s are the only entries i n the minors then there are 2^ or 16 A B C D 34 possible second order minor patterns. If the determinants must equal zero then two of these patterns are inadmissible or violating: and (since AD - BC # 0). The re-maining 14 of these patterns are of two types. A pattern is compatible i f given any three elements of the pattern there is no constraint on the value of the last element of the pattern to make the determinant equal to zero. A pattern is either confirming or violating i f there exists at least one particular subset^ of three elements of the pattern for which the fourth element i s predicted by theory. If the fourth element cor-responds to theoretical predictions, the pattern is called confirming, i f not, the pattern is called violating. A similar consideration of probabilities as that made for EU theory provides the set of equations: A i = (£i 4 + 4p iq i 2 + q ^ ) ^ + O p ^ + £ iq i 3)(l-T i) = r/N (7) B i = 4<£i 3Si + £iqi 3) Ti + P i 4 + S P i ^ 2 + q i 4 ) U - T i ) = s/N (8) These may be solved for p^ which provides an estimate of the individual's consistency i n behavior under SEU theory. 35 4) EV theory. In the reduced data matrix there are 32 pre-dictions of either a 0 or a 1. Two of these involve equal expectation values for both l e f t and right members of the pair and hence using only the assumption of EV theory, i t i s not possible to predict which of the two choices the individual w i l l prefer. The expectation and variance of the number of times these predictions w i l l f a i l , given p_i, are: E(vi) = 30qt (9) VAR(vi) = 30 piqi ( 1 Q ) In the case of EU, SEV, and SEU theories, the consistency value p^ according to a particular theory is compared with the value P^ of the consistency of the individual independent of any theory. An individual i s accepted as supporting a particular theory i f is greater than P^. That i s , the individual's consistency estimate under assumptions of a given expectation theory i s larger than the consistency estimate under assumptions that are independent of any expectation theory. In the case of EV theory, the number of violations of the requirements of the theory is compared with the expected value of violations ( v i o l ) . A Z ratio ( ( v i o l - 30 q L)/ sqrt (30 PiSO ) is formed and EV is accepted at the 05 per-36 cent level of significance. The second major step in the data analysis was testing the hypothesis that high anxious-high defensive and low anxious-low defensive subjects conformed more closely to the four expectation theories than low anxious-high defensive and high anxious-low defensive subjects. The two personality questionnaires were scored i n the manner prescribed by Alpert and Haber (I960) and Crowne and Marl owe (I960). A subject was designated high or low defensive and high or low anxious on the basis of whether his score was above or below the median score for the group on the questionnaires. In order to explore the possibility that various sex and person-a l i t y patterns would diffe r e n t i a l l y conform to the four theories, the whole group and then the male and female subjects were classified into: high and low anxious, high and low defensive, low anxious-low defensive, high anxious-high defensive, low anxious-high defensive, high anxious-low defensive. Chi square and binomial tests were then performed to determine whether there was any di f f e r e n t i a l conformity to the four theories. 37 RESULTS Table 7 shows the proportion of 3/0 splits (RL^), the average probability of an individual making his dominant choice on any pair of gambles (PIj_), and the individual's consistency estimate without regard to theory (P^). The latter (P^), is equivalent to the probability that the individual's dominant choice w i l l appear in the reduced data matrix. Three subjects S19, S46, and S70 were eliminated from further analyses because their PI^ was below the criterion for minimum consistency (PI less than .67). The proportion of latent compatible pat-terns (T) calculated separately under EU, SEV, and SEU theory for each subject appears i n Table 8. The estimate of the individual's consistency level (P) without theoretical assumptions c r i t i c a l to expectation theory, the consistency level assuming EU theory ( P E U ) J t n e consistency assuming SEV theory (P S E V^ » a n d t' i e consistency assuming SEU theory (P S E U ^ a r e presented in Table 9. Table 10 presents the results of the Coombs and Bezembinder method of testing whether a significant number of the subject's choices violated EV theory. On the basis of these data, i t was determined that EV theory was rejected for 57 subjects, EU theory was rejected for 31 subjects SEV theory was rejected for 26 subjects and SEU theory was re-jected for 14 subjects. Bartholomew tests of homogeneity for 38 Table 7. The Proportion of 3/0 Splits (RL), the Average Probability of an Individual Making his Dominant Choice on any Pair of Gambles (PI) and the Individual's Consistency Estimate Independent of Any Expectation Theory (P) for 77 Subjects Subiect RL PI P 1 .50 .79 .885 2 .66 .87 .952 3 .53 .81 .902 4 .72 .90 .969 5 .34 .68 .754 6 . .66 .87 .952 7 .50 .79 .885 8 .56 .82 .917 9 .81 .93 .987 10 .34 .68 .754 11 .72 .89 .969 12 1.00 1.00 1.000 13 1.00 1.00 1.000 14 .84 .94 .991 15 .63 .85 .942 16 .84 .94 .991 17 .78 .92 .982 118 1.00 1.00 1.000 19 .28 .60 .651 20 .75 .91 .976 21 .75 .91 .976 22 .91 .97 .997 23 .41 .73 .819 24 1.00 1.00 1.000 25 .78 .92 .982 26 .72 .90 .969 27 .78 .92 .982 28 .97 .99 .999 29 .91 .97 .997 30 .50 .79 .885 31 .50 .79 .885 32 .72 .90 .969 33 .91 .97 .997 34 .81 .93 .987 35 .97 .99 .999 36 .59 .84 .930 39 Table 7. (continued) Subject RL PI P 37 .72 .90 .969 38 .53 .81 .902 39 1.00 1.00 1.000 40 1.00 1.00 1.000 41 .56 .82 .917 42 .81 .93 .987 43 .84 .94 .991 44 .84 .94 .991 45 .44 .75 .844 46 .25 .51 .515 47 .91 .97 .997 48 .97 .99 .999 49 .84 .94 .991 50 .91 .97 .997 51 .47 .77 .866 52 .56 .82 .917 53 .72 .90 .969 54 .50 .79 .885 55 .66 .87 .952 56 .38 .70 .789 57 .81 .93 .987 58 .59 .84 .930 59 .75 .91 .976 60 .78 .92 .982 61 .91 .97 .997 62 .78 .92 .982 63 .53 .81 .902 64 .63 .85 .942 65 .88 .96 .995 66 .47 .77 .866 67 .53 .81 .902 68 .97 .99 .999 69 .94 .98 .999 70 .28 .60 .651 71 .78 .92 .982 72 .69 .88 .961 73 .63 .85 .942 74 .59 .84 .930 75 .81 .93 .987 76 .78 .92 .982 77 .78 .92 .982 J 40 Table 8. Proportion of Latent Compatible Patterns (T) Calculated Separately Under the Assumptions of EU, SEV, and SEU Expectation Theories. Subiect EU SEV SEU 1 .37 1.00 .86 2 .68 .85 .00 3 1.00 .94 .88 4 1.00 1.00 .88 5 .46 .77 .78 6 .54 .75 .64 7 .50 1.00 .00 8 .30 1.00 .76 9 .44 1.00 .88 10 1.00 .68 .43 11 .49 .94 .88 12 1.00 1.00 1.00 13 1.00 1.00 1.00 14 .71 1.00 .88 15 .64 1.00 1.00 16 .46 1.00 .88 17 1.00 1.00 1.00 18 1.00 1.00 1.00 19 20 .54 .56 .59 21 .71 .94 .88 22 1.00 .56 .88 23 .56 1.00 0.00 24 1.00 1.00 1.00 25 .54 1.00 .69 26 .52 .50 .55 27 .49 .94 .88 28 .43 1.00 1.00 29 .62 1.00 .88 30 .64 .85 .00 31 .25 .91 .00 32 1.00 1.00 .88 33 .35 .94 .88 34 .88 .88 .84 35 .76 1.00 1.00 36 .50 1.00 .72 37 .47 .94 .88 Table 8. (continued) Subiect EU SEV 38 .59 .84 39 1.00 1.00 40 1.00 1.00 41 .46 .81 42 .56 .88 43 .66 1.00 44 .39 .56 45 .63 .56 46 47 .49 .94 48 1.00 1.00 49 .75 1.00 50 .43 1.00 51 .51 .69 52 .92 .69 53 1.00 .49 54 .45 .75 55 .31 .94 56 .46 .81 57 .86 .93 58 .42 .81 59 .67 .44 60 .75 1.00 61 .45 1.00 62 .46 1.00 63 .72 .94 64 .46 .81 65 .48 .97 66 .58 .82 67 .56 .91 68 1.00 .93 69 .44 .93 70 71 .59 .88 72 .32 1.00 73 1.00 1.00 74 .63 1.00 75 .28 1.00 76 .52 .85 77 1.00 1.00 SEU .57 1.00 1.00 0.00 .84 .84 No Convergence .76 .88 .88 1.00 1.00 1.00 .73 .69 1.00 0.00 .64 .73 .73 .99 .59 1.00 .84 0.00 .73 .73 .64 .61 .68 .88 .88 .79 .00 .87 .64 .84 .64 42 ordered alternatives (Bartholomew, 1959) were made to test for the monotonicity of the number of rejections i n the SEU-SEV-EV sequence and the SEU-EU-EV sequence. These were both significant at beyond the .01 level of confidence. (Since no s t a t i s t i c a l test incorporating the non-independence of the data was available, this test was used i n spite of the fact that some of the individuals obeyed more than one theory.) Table 11 shows the number of high and low anxious and high and low defensive subjects (based on a median s p l i t of scores on the personality questionnaires) obeying each of the four expectation theories.*" Chi square tests conducted separ-ately for males, females, and the total group revealed that fewer high defensive males than low defensive males obeyed SEV theory (p less than .05). It was also noted that for a l l four expectation theories high anxious subjects were either equally or more frequently rejected than low anxious subjects, but none of these differences reached s t a t i s t i c a l significance. Eight individuals f a l l i n g right at the median on the anxiety scale were dropped from classification on the anxiety variable in this and subsequent portions of the analysis. Twelve individuals f a l l i n g right at the median on the defensiveness questionnaire and five additional individuals who failed to com-plete the defensiveness questionnaire properly were dropped from classi f i c a t i o n on the defensiveness variable in this and subsequent portions of the analysis. 43 Table 9. Estimate of the Subjects' Consistency Level Independent of Any Theoretical Assumptions C r i t i c a l to Expectation Theory (P), and Under the Assumptions of EU, SEV, and SEU Theories ( P E U , P S E V, P S E U ) . Subject P ! E J J PSEV PSEU 1 .885 .684 .875 .902 2 .952 .973 .922 .500 3 .902 .490 1.000 .999 4 .969 .484 .938 .999 5 .754 .849 .914 .864 6 .952 1.000 1.000 1.000 7 .885 .629 .865 .500 8 .917 .607 .750 .884 9 .987 1.000 .938 .999 10 .754 .500 .794 .857 11 .969 1.000 1.000 .999 12 1.000 1.000 1.000 1.000 13 1.000 1.000 1.000 1.000 14 .991 1.000 .938 .999 15 .942 .852 .933 .500 16 .991 .964 .938 .999 17 .982 1.000 1.000 1.000 18 1.000 1.000 1.000 1.000 19 20 .976 1.000 1.000 1.000 21 .976 .731 1.000 .999 22 .997 .938 1.000 .999 23 .819 .782 .750 .500 24 1.000 1.000 1.000 1.000 25 .982 .900 .813 .999 26 .969 .589 1.000 1.000 27 .982 1.000 1.000 .999 28 .999 1.000 1.000 1.000 29 .997 .986 .938 .999 30 .885 .714 .711 .500 31 .885 .500 .839 .500 32 .969 .500 .938 .999 33 .997 1.000 1.000 .999 34 .987 .990 1.000 .970 35 1 .999 1.000 1.000 1.000 44 Table 9. (continued) Subiect P P S E V pSEU 36 .930 .714 1.000 .999 37 .969 1.000 1.000 .999 38 .902 .795 .922 .929 39 1.000 1.000 1.000 1.000 40 1.000 1.000 1.000 1.000 41 .917 .774 .820 .500 42 .987 1.000 1.000 .999 43 .991 .853 .933 .999 44 .991 .980 1.000 .999 45 .844 .877 1.000 No Convergence 46 47 .997 1.000 1.000 .999 48 .999 1.00 1.000 1.000 49 .991 1.000 1.000 1.000 50 .997 1.000 1.000 1.000 51 .866 .742 .914 .889 52 .917 .810 1.000 .827 53 .969 1.000 1.000 1.000 54 .885 .500 .854 .500 55 .952 1.000 1.000 1.000 56 .789 .980 1.000 .999 57 .987 .939 1.000 .999 58 .930 .916 .928 .933 59 .976 1.000 1.000 1.000 60 .982 1.000 1.000 1.000 61 .997 1.000 .933 .999 62 .982 1.000 1.000 1.000 63 .902 1.000 1.000 .999 64 .942 1.000 1.000 .999 65 .995 1.000 1.000 1.000 66 .866 .734 .866 .945 67 .902 .965 1.000 .999 68 .999 1.000 .938 .999 69 .999 1.000 1.000 .999 70 71 .982 .772 1.000 .999 72 .961 .500 .797 .500 73 .942 1.000 .938 .999 74 .930 .746 .726 .871 75 .987 1.000 .933 .999 76 .982 1.000 .922 1.000 77 .982 1.000 1.000 1.000 45 Table 10. Results of the Coombs and Bezembinder Method of Testing Whether a Subject Should be Rejected for EV Theory (* indicates significance at the .01 level; ** indicates significance at the .001 level) Subject Number of Violations Z Value of EV Theory  1 11 4.32** 2 9 6.49** 3 17 8.63** 4 16 15.99** 5 5 -1.01 6 1 -0.37 7 13 5.46** 8 13 6.95** 9 7 10.72** 10 11 1.54 11 3 2.21* 12 15 0.00 13 15 0.00 14 4 7.31** 15 9 5.67** 16 5 9.27** 17 9 11.68** 18 15 0.00 19 20 0 -0.85 21 11 12.35** 22 10 32.75** 23 7 0.74 24 9 0.00 25 9 11.68** 26 13 12.81** 27 6 7.54** 28 5 50.15** 29 9 29.44** 30 13 5.46** 31 11 4.32** 32 10 9.63** 33 6 19.53** 34 7 10.72** 35 11 110.45** 36 10 5.66** 46 Table 10. (continued) Subject Number of Violations Z Value of EV Theory 37 6 5.39** 38 8 3.10** 39 9 0.00 40 9 0.00 41 6 2.32*> 42 6 9.10** 43 13 24.93** 44 3 5.35** 45 6 0.66 46 47 6 19.53** 48 9 90.35** 49 5 9.27** 50 5 16.22** 51 10 3.20** 52 7 2.98** 53 6 5.39** 54 16 7.18** 55 3 1.35 56 3 -1.49 57 14 22.06** 58 ' 7 3.51** 59 0 -0.85 60 5 6.16** 61 4 12.92** 62 6 7.54** 63 4 0.65 64 3 0.98 65 1 2.05* 66 9 2.66** 67 10 4.33** 68 14 140.60** 69 4 19.76** 70 71 13 17.20** 72 13 11.23** 73 4 1.76* 74 13 7.81** 75 5 7.48** 76 4 4.78** 77 15 19.96** Table,11. The Number of Anxious and the Number of Defensive Subjects (median s p l i t with persons scoring the median value dropped from classification) Satisfying Each of the Four Expectation Theories. M A L E Expec. Total High Low High Low Theory Satis. Anx. Anx. Def r Def. SEU 60 17 22 16 20 F E M A L E High Low High Low Anx. Anx., Def. Def. 8 T O T A L High Low High Low Anx. Anx. Def.. Def. 24 30 23 24 SEV 48 13 17 19 19 23 14 23 EU EV 43 12 14 10 14 17 2 6 5 4 6 7 6 4 4 4 5 2 18 21 16 18 6 10 10 6 4> 48 Table 12. Classification of Subjects Satisfying Each of the Four Expectation Theories by Extreme Personality Types (Upper and Lower Quartile Split) Expectation Total High Low High Low Theory Satisfying Anxious Anxious Defensive Defensive SEU 60 13 13 15 16 SEV 48 12 9 8 17 EU 43 11 7 8 13 EV 17 3 4 4 4 49 It seemed possible, however, that the masking effects of borderline anxious and borderline defensive individuals could have confused the results. In order to overcome this possible shortcoming, a subject was therefore redefined as high or low with respect to either of the two personality traits i f his score on the particular personality questionnaire f e l l within the upper or lower quartile of the total distribution of scores. The number of these newly defined personality types obeying each of the four expectation theories appears i n Table 12, It was determined that fewer high defensive subjects obeyed SEV theory than did low defensive subjects (p less than .01). This result was consistent with the finding reported for males under the original definition of high and low personality groupings. No new significant differences between personality groups satisfy-ing the expectation theories emerged i n this re-analysis, however. Table 13 presents the number of individuals in each of the four personality subgroups (low anxious - low defensive, high anxious - high defensive, low anxious - high defensive and high anxious - low defensive, based on a median s p l i t of the scores on the personality questionnaires) obeying each of the expectation theories. Since for a l l four expectation theories there were either fewer or an equal number of low anxious - low defensive males than high anxious - low defensive males and Table 13. Classification of Subjects Satisfying Each of the Four Expectation Theories by Personality Types (median s p l i t with persons scoring at the median value dropped from classi f i c a t i o n ) . M A L E F E M A L E T 0.1 A L High Def. Low Def. High Def. Low Def. High p e f . Low Def. Expec. Total High Low High Low High Low High Low High Low High *Low Theory Satis. Anx. Anx. Anx. Anx. Anx. Anx. Anx. Anx. Anx. Anx. Anx. Anx. SEU 60 3 12 12 6 4 3 1 3 7 -15 13 9 SEV 48 1 7 11 6 3 2 1 3 4 , 9 12 9 EU 43 2 7 8 5 4 2 1 3 6 * 9 9 8 EV 17 0 4 2 2 3 2 1 1 3 6 3 3 Cn O 51 fewer high anxious - high defensive males than low anxious -high defensive males, i t was decided to do an ad hoc s t a t i s t i c a l analysis of the relationships among these subgroups. Tests of the difference between the proportions of subjects obeying the four expectation theories revealed that fewer males who were either high anxious - high defensive or low anxious - low defen-sive obeyed SEU and SEV theory than did males who were either low anxious - high defensive or high anxious - low defensive (p less than .05). The numbers of high and low anxious and the numbers of high and low defensive subjects with scores i n the upper or lower quartile of the distribution of scores on the PI consist-ency criterion are shown i n Table 14. The difference i n the proportions of high anxious and low anxious subjects f a l l i n g into the high PI group and those proportions f a l l i n g into the low PI group was not s t a t i s t i c a l l y significant (chi square test). A chi square test also failed to reveal any significant d i f -ferences between the proportions of high defensive and low defensive subjects f a l l i n g into the high PI groups and those proportions f a l l i n g into the low PI groups. Table 15 presents the number of male and female subjects who do and do not satisfy each of the four expectation theories. Chi square tests failed to reveal any significant differences 52 between the number of males and females who did satisfy each of the four expectation theories and the number who did not. 53 Table 14. The Number of High and Low Anxious and High and Low Defensive Subjects Falling into the Upper and Lower Quartiles on the Coombs and Bezembinder PI Consistency Estimate LOW PI HIGH PI High Anxious 11 8 Low Anxious 7 8 High Defensive 6 11 Low Defensive 11 6 54 Table 15. The Number of Male and Female Subjects Satisfying Each of the Four Expectation Theories. M A L E F E M A L E Do Not Do Not Satisfy Satisfy Satisfy Satisfy SEU 45 8 15 6 SEV 36 17 12 9 EU 30 23 13 8 EV 9 44 8 13 55 DISCUSSION The proportion of subjects in the present study rejected for EV, EU, SEV and SEU theory respectively are .77, .42, .35 and .19. The hypothesis of monotonicity in the number of rejections for the two sequences, SEU-SEV-EV and SEU-EU-EV, was accepted on the basis of Bartholomew (1959) tests of homogeneity for ordered alternatives which were significant at beyond the nominal .01 level of confidence for both sequences. (Since the data unfortun-ately did not meet the assumptions of independence among components required of Bartholomew's tests, the "nominal" levels in the pre-sent instance only provide approximations of the "true" significance levels.) The proportion of subjects i n the present study rejected for EV, EU, SEV and SEU theory may be compared with the respective proportions .94, .24, .33 and .09 reported by Coombs and Bezembinder (1964). Thus, i n both studies a relatively large proportion of subjects was found to satisfy SEU theory, a small proportion was found to satisfy EV theory and intermediate proportions were found to satisfy SEV and EU theory. This pattern of findings i n the two studies was similar despite differences i n the type of subject used. The present study used university students only whereas university students comprised only part of the subject population for the Coombs and Bezembinder study. The remainder of Coombs and 56 Bezembinder*s subjects were adults from a low socio-economic area. These parallel findings for ostensibly diverse subjects suggest that these proportions may be generalizable to the population as a whole. The hypothesis that a higher proportion of females than males would obey the four expectation theories was rejected be-cause no significant differences were found between the proportion of males and females satisfying each of the four expectation theories. One of the major hypotheses of this study was that more low anxious-low defensive and high anxious-high defensive subjects would obey the four expectation theories than high anxious-low defensive and low anxious-high defensive subjects. Since the pro-portions obeying the expectation theories were the reverse of those predicted, this hypothesis was rejected. I t was noted that fewer high anxious than low anxious males (based on the median sp l i t ) conformed to each of the four expectation theories. Since none of the associated proportions reached s t a t i s t i c a l significance, i t is probable that the finding under the original definition of high and low anxious was the result of the smaller number of high anxious subjects (30) than low anxious subjects (36) i n the population tested. This dis-crepancy i n number arose from the elimination of subjects whose score tied at the median value for the questionnaire. When high 57 and low anxious subjects were redefined on the basis of upper and lower quartile divisions on the questionnaire scores, this result reverses direction, but not significantly so. If the finding that fewer high anxious subjects obeyed the expectation theories i s not entirely accounted for by the discrepancy between the number of high and low anxious subjects i n the population, the reversal when a quartile division of the questionnaire scores i s employed raises the possibility that anxiety may have a non linear effect on the individual's potentiality to satisfy the expectation theories. That i s , high and low anxiety might inhibit types of rational behavior<r required to satisfy the expectation theories: whereas intermediate anxiety might i n i t i a t e behavior required for the satisfaction of the expectation theories. There could be several possible explanations for the general inconclusiveness of these results. The manner and speed with which the subjects completed the task presents one possibility. The subjects completed the task i n a very short period of time (about 30 minutes). It was, at f i r s t , feared that this might have produced data of questionable r e l i a b i l i t y . These fears were par-t i a l l y allayed, however, by the degree of interest and personal involvement expressed by the subjects at the conclusion of the experiment. It was decided, therefore, to accept the data as a f a i r test of the hypotheses i f few subjects needed to be dropped from the analysis on the basis of the Coombs and Bezembinder 58 consistency criterion. Since only three out of seventy-seven subjects were eliminated on this basis, the data were deemed acceptable for further analysis. A second possible explanation for the non significant results may be that the two personality tests, the Alpert and Haber Test Anxiety Scale and the Crowne and Marlowe Defensiveness Scale, were not valid measures of defensiveness and anxiety. There have been relatively few studies in the literature dealing with either of these two scales and i t is not altogether clear whether the two scales measure what they purport to measure. These two personality tests were chosen, however, because their reported r e l i a b i l i t i e s were high and because Kogan and Wallach (1964) used them i n the study that led to the formulation of the f i r s t two hypotheses tested i n this experiment. These two hypotheses were based on the assumption that the type of consistency i n behavior noted by Kogan and Wallach was the same as the type of consistency in behavior required by the Coombs and Bezembinder method of testing expectation theories. The failure of this study to confirm the experimental hypotheses may be due, i n part, to the untenability of this assumption. Kogan and Wallach reported that certain personality types performed consistently (as far as risk taking behavior was concerned) across a series of qualitatively different risk taking situations (actual gamble, hypothetical gamble, and s k i l l game). Coombs and 59 Bezembinder, however, require a consistency of behavior across a series of risk taking situations that d i f f e r only quantitatively i n the amount won and the probability of winning. Consistency i n behavior across a series of different situations may not neces-sa r i l y imply consistency in behavior within a series of modifi-cations of the same situation. Thus, the consistency in behavior reported by Kogan and Wallach would not necessarily imply the type of consistency required by Coombs and Bezembinder for satis-faction of the expectation theories. This study did reveal two s t a t i s t i c a l l y significant re-sults with regard to the personality variables. SEV theory was obeyed by fewer high defensive males than low defensive males under the "median s p l i t " definition of defensiveness and fewer high defensive males and females than low defensive males and females under the "quartile s p l i t " definition of defensiveness. Further-more, fewer males who were either low anxious-low defensive or high anxious-high defensive obeyed SEU and SEV theory than males who were either low anxious-high defensive or high anxious-low defensive. These results provide no sound basis on which to state the exact effect of the personality variables on the satisfaction of expectation theories, but they suggest that these personality variables might indeed play a role i n governing this type of behavior. I t i s therefore the conclusion of this study that 60 further research directed toward discovery of personality cor-relates of gambling behavior would be f r u i t f u l . Two major recommendations can be made regarding future research i n this area. F i r s t , a new method of testing expectation theories of gambling recently developed by Tversky (1965) might prove to be a more valid way of testing the expectation models of behavior. Tversky's method appears to improve upon the Coombs and Bezembinder method for several reasons: i t is a more demanding test since there are no gambling patterns that are not tests of the expectation theory; additivity of the subjective probability and u t i l i t y components is tested directly rather than assumed which in turn permits the separate evaluation of subjective pro-bability and u t i l i t y ; and i t i s possible to evaluate a u t i l i t y - f o r -gambling-index. The power of Tversky's methodology and the inconclusive-ness of the results of the present study which may, i n part, be due to the fact that two distinct types of consistency (as men-tioned above) are involved i n the study of risk taking behavior, suggest that research on the role of personality variables i n de-cision making could take two distinct directions. Both of these directions could be tested by the Tversky technique. The direction taken by Kogan and Wallach which relates personality variables to propensity for risk could thus be examined by the Tversky technique i n which the utility-for-gambling-index could be directly related 61 to a host of personality variables. The direction taken i n this study was to relate personality variables to rationality of decision ( i . e . , satisfaction of some expectancy theory). The Tversky technique would make possible the study of the inter-relation of these two aspects of decision making to personality variables. The second recommendation is that this type of experiment might yield clearer results i f the subjects were trained for a period of time and run individually rather than as a group through a real gambling situation. If the group situation must be used, i t is recommended that a mechanical device such as a slide pro-jector for the presentation of gambles be employed in order to pace the subjects through the task. If these recommendations were followed, the present data suggest that meaningful relationships between personality vari-ables and expectation theories of gambling behavior might indeed be discovered. 62 SUMMARY AND CONCLUSIONS The general purpose of this study was to examine one approach to the study of the relationship of personality variables to expectation theories of gambling* The Coombs and Bezembinder (1964) method of testing, expectation theories of gambling be-havior was used to determine how many, among a group of 77 sub-jects, obeyed each of four expectation theories. These four expectation theories were: EV theory, assuming the maximization of the product of objective prize values and actual probabilities of winning; EU theory, assuming maximization of the product of subjective prize values and actual probabilities of winning; SEV theory, assuming the maximization of the product of objective prize values and subjective probabilities of winning; and SEU theory, assuming the maximization of the product of subjective value of the prize and the subjective probability of winning. The Coombs and Bezembinder method consists of comparing an estimate of an individual's consistency of choices independent of expectation theory assumptions with estimates of consistency under assumptions basic to each of the four expectation theories. A lower value of the consistency estimate under assumptions of a given expectation theory than the value calculated independently of any expectation theory assumptions leads to rejection of that 63 particular theory as a model for the subject's behavior. The Coombs and Bezembinder technique for determining whether an indi-vidual obeys the four expectation theories leads to the prediction of an ordering of the expectation theories with respect to the number of subjects who do not satisfy them. The procedure i n the present study involved the presen-tation of 96 pairs of one-outcome gambles to 77 subjects i n an introductory psychology class, A subject was required on each pair to choose between a gamble combining high risk with a large prize and a gamble combining a low risk with a small prize. It was found that EV theory was rejected for 57 subjects, EU theory for 31 subjects, SEV theory for 26 subjects and SEU theory for 14 subjects. The hypothesis of monotonicity i n the number of rejections for the two sequences SEU-SEV-EV and SEU-EU-EV was accepted. A second hypothesis, that a higher proportion of females w i l l obey the expectation theories than w i l l males, was rejected. The subjects were subdivided into high and low anxious and high and low defensive groups on the basis of scores obtained on the Alpert and Haber Test Anxiety Scale and the*Crowne and Marlowe Defensiveness Scale. An examination of the data was sufficient to reject the hypothesis that more low anxious-low defensive and high anxious-high defensive subjects would obey the four expectation theories than would subjects who were either low anxious-high defensive or high anxious-low defensive. There were, however, some s t a t i s t i c a l l y significant results 64 obtained on the basis of several ad hoc analyses. Fewer high defensive males than low defensive males appeared to obey SEV theory. Furthermore, fewer males who were either high anxious-high defensive or low anxious-low defensive obeyed SEU and SEV theory than did males who were either low anxious-high defensive or high anxious-low defensive. On the basis of these results, i t was recommended that further research be conducted on the re-lationships of personality variables to expectation theories of gambling. It was noted that the use of the Tversky method of testing expectation theories would permit the simultaneous examination of two approaches to the relationship of personality variable*to decision making (personality variables versus propensity for risk and personality variables versus rationality of decision). Finally, with respect to technique, i t was recommended that better ways of assessing personality variables be found and the subjects be f u l l y trained and run individually through the experiment. BIBLIOGRAPHY 65 A l l a i s , M., Le comportement de l'homme rationnel deuant le risque. Econometrika. 1953, 2J., 503-546. Alpert R. and Haber, R.N. Anxiety i n academic achievement situations. J. Abnorm. Soc. Psychol.. 1960, 61, 207-215. Atkinson, J. W. Motivational Determinants of Risk Taking Behavior. Psycho1. Rev.. 1957, 64, 359-372. Atkinson, J. W., Bastian, J. R., Earl, R.W., and Litwin, G. H. The Achievement Motive, Goal Setting and Probability Preferences. J. Abnorm. Soc. Psychol.. I960, 60, 27-36. Bartholomew, D. J. A Test of Homogeneity for Ordered Alternatives. Biometrika. 1959, 46, 36-48. Bernouilli, D. Exposition of a new theory on the measurement of risk. Econometrika. 1954, 22, 23-26. Brody, N. Need Achievement, Test Anxiety, and Subjective Probability of Success i n Risk Taking Behavior. J. Abnorm. Soc. Psychol.. 1963, 66, 413-418. Cohen, J. and Hansel, C.E. Experimental risk taking. Jahebt  Psychol, u. Psychotherapie. 1955, 3, 382-388. Coombs, C. H. A theory of psychological scaling. Bull. Engng. Res. Inst. Univer. Mich., 1952, No. 34. Coombs, C. H. and Beardsley, D. C. Decision making under uncertainty. In R. M. Thrall, C. H. Coombs, and R. L. Davis (eds.), Decision Processes, New York: 1954, Wiley. Coombs, C. H. and Bezembinder, T.G.G. Testing expectation theories without measuring u t i l i t y of subjective pro-bability. Michigan Mathematical Psychology Program. 1965, 2. Coombs, C. H. and Kormorita, S.S. Measuring u t i l i t y of money through decisions. Amer. J. Psychol.. 1958, 71, 383-389. 66 Coombs, C. H. and Pruitt, D.G. A study of decision making under risk. Rept. No. 2900-33-T.- Willow Run Laboratories, Univ. of Michigan, Ann Arbor, Mich., 1960. Crowne, D. P. and Marlowe, D. A new scale of social desirability, independent of psychopathology. J. Consult. Psycho1.. 1960, 24, 349-354. Davidson, D., Suppes, P., and Siege1, S. Decision Making; An  Experimental Approach. Stanford, California: Stanford Univer. Press, 1957. Edwards, W. Probability preferences among bets with differing expected values. Amer. J. Psychol.. 1954a, 67, 56-67. Edwards, W. The r e l i a b i l i t y of probability preferences. Amer. J. Psychol.. 1954b, 67, 68-95. Edwards, W. Variance preferences i n gambling. Amer. J. Psychol. 1954c, 67, 441-452. Edwards, W. Theory of decision making. Psychol. Bull.. 1954, 51, 380-417. Edwards, W. Behavioral decision Theory. Ann. Rev. Psychol.. 1961, 12, 473-495. Festinger. Studies in decision: II. an empirical test of a quantitative theory of decision. J. Exp. Psychol.. 1943, 32, 411-432. Fisher, I. The Nature of Capital and Income. New York: Macmillan, 1906. Kogan and Wallach. Risk Taking: a study i n cognition and personality. New York: Holt, Rinehart and Winston, 1964. Marks, R. W. The effect of probability, desirability and "privilege" on the stated expectations of children. J. Pers., 1951, 19, 332-351. Mandler, G. and Sarason, S. A study of Anxiety and learning. J. Abnorm. Soc. Psychol.. 1952, 47, 166-173. Mosteller, F. and Nogee, P. An experimental measurement of u t i l i t y . J. Polit. Econ.. 1951, 59, 371-404. 67 Pruitt, D. G. Pattern and level of risk i n gambling decisions. Psvcho1. Rev.. 1962, 69, 187-201. Rim, Y. Personality and group decision involving risk. Psychol. Rec. 1964, 14, 37-44. Royden, H. L., Suppes, P. and Walsh, R. A model for the experimental measurement of u t i l i t y of gambling. Behav. Sci., 1959, 4, 11-18. Scodel, A., Ratoosh, P. and Minas, J.S. Some personality cor-relates of decision making under conditions of risk. Behav. Sci.. 1959, 4, 19-28. Shuford, E. H. A comparison of subjective probability for elementary and compound events. Rept. No. 20. The Psychometric Laboratory, Univ. of North Carolina, Chapel H i l l , N.C., 1959. Slovic, P. Assessment of risk taking behavior. Psychol. Bull.. 1964, 61, 220-233. Stone, L. A. The influence of selected individual difference variables upon validi t y for risk. J. Gen. Psychol., 1964, 70., 29-32. Suppes, P. and Walsh, K. A non linear model for the experimental measurement of u t i l i t y . Behav. Sci.. 1959, 4, 204-211. Tversky, A. Additivity analysis of a test of u t i l i t y theory. Michigan Mathematical Psychology Program. 1965, 2. APPENDIX I 69 DISCUSSION OF FIVE THEOREMS ON EXPECTATION THEORY The following discussion relies heavily on the discussion by Coombs and Bezembinder (1964). Pairs of gambles are examined to see how the ratios of the expectation values of the members of the pair according to a given theory vary as increments are added or subtracted from the probability (P), and the value ( V ) components of the gambles. The particular gamble of the given pair which has a higher probability to win a smaller amount than the other, w i l l be denoted by the subscript 1* for l e f t , the other member w i l l be denoted by the subscript r for right. A prime w i l l always denote the addition of an increment to the corresponding unprimed quantity (P' is greater than P). Assume that the subjective probability ( W ) i s s t r i c t l y monotonic with objective probability and u t i l i t y ( U ) for money is s t r i c t l y monotonic with money. Thus, for any pair of gambles, is greater than W R and U ^ is less than U R . For each pair of gambles the ratio of their expectations may be formed ( i . e . E ^ ( W ^ U ^ ) / E R ( W R U R ) = A). There are five theorems dealing with the relationships of these ratios for the various pairs of gambles. Theorem I: For V R , V ^ , P r and the change i n P r greater than 0, E V theory requires that: i f P J V J ^ = A P R V R 70 and i f ?{Vl = BP£Vr then A is greater than B. Interpretation: If we have a pair of gambles as above, the effect of adding a fixed f i n i t e increment i n probability to both ? i and P r is to decrease the ratio of the expectation of the gamble on the l e f t to the expectation of the gamble on the right. Theorem 2: For VrV-^Pr and the change i n V r greater than 0, i f ?iVx = A P rV r and i f P ^ = C P rV£ then C is greater than A. Interpretation: The effect of adding a fixed f i n i t e increment to both and V r is to increase the ratio of the expectation of the gamble on the l e f t to the expectation of the gamble on the right. Theorem 3: For UjU r greater than 0 and U(V=0)=0 EU theory requires that: i f PjUi = A P rU r and i f ?{\Ji = B Pi-Ur then A is greater than B. Interpretation: The effect of adding a fixed f i n i t e increment to both ? i and P r i s to decrease the ratio of the 71 expectation of the gamble on the l e f t to the expectation of the gamble on the right. Theorem 4: For VjV r greater than 0 and W(P„ greater than 0) is greater than 0, SEV requires that: i f WjV^ = A WrVr and i f WLV[ = C WrV^ then C is greater than A . Interpretation: The effect of adding a fixed f i n i t e increment to both V^ and V r is to increase the ratio of the expectation of the gamble on the l e f t to the expectation of the gamble on the right. Theorem 5: For U (V=0) =0 and W (P greater than 0) is greater than 0, SEU theory requires that: i f W J U ' L = A WrUr and i f w j l ^ = B W^Ur and i f WjU[ = C W r u £ and i f w j u [ = D then BC = A D Interpretation: The effect of adding fixed f i n i t e increments to the U^ and U r components and the and Wr com-ponents is to make the product of the ratios of the expectations of the l e f t and right gambles before adding the increment and 72 after adding the increment equal to the product of the ratios of expectations of the l e f t and right gambles formed by adding the increment only to the W components and by adding the incre-ment only to the U components. These theorems suggest that with the correct choice of the P r, P^, V r, V^, and systematic increments to these values, a particular theorem w i l l predict that the individual w i l l change from consistently preferring the l e f t gamble to consis-tently preferring the right gamble. If the choice of a l e f t gamble over a right gamble i s denoted by the symbol 1 and the other choice by the symbol 0, EV theory w i l l predict that an individual's data matrix (see Table 5) w i l l contain no l's above a zero and that the cut-off point in the rows between the l's and the 0's w i l l occur at the point of equal expected value for the members of a pair. EU theory makes the same prediction except that the cut-off point is unknown. In an analogous way, EV and SEV theory predict that no zeros w i l l occur to the right of a 1 i n the data matrix with EV theory predicting a specific cut-off point. Finally, the second order minor of the data matrix can be examined in terms of Theorem 5 to see i f SEU theory is violated. 73 L i s t of Symbols Used i n the Coombs and Bezembinder Calculation Individual's consistency estimate independent of any expectation theory. Probability that the subject w i l l make his dominant choice on any pair of gambles. The proportion of times that the individual makes either 3 l e f t choices or three right choices out of three times that a gamble is presented. The probability that a particular manifest pattern occurs i n the reduced data matrix when a given latent pattern is the person's dominant choice pattern (this is equivalent to the consistency estimate under assump-tion of a given expectation theory). The probability that a manifest choice does not corr-espond to the latent choice. The proportion of latent compatible patterns for the subject. The probability of obtaining a compatible pattern i n the reduced data matrix. The probability of obtaining a confirming pattern in the reduced data matrix. Number of compatible patterns i n the reduced data matrix. 74 Number of manifest confirming patterns i n the reduced data matrix. the number of manifest violating patterns in the reduced data matrix. The total number of patterns in the reduced data matrix that test a given expectation theory. APPENDIX II 76 SAMPLE BOOKLET PLEASE DO NOT OPEN THE BOOKLET UNTIL TOLD TO DO SO. At the top of the booklet indicate whether you are male of female. Do not place your name on the booklet. Pictured below i s a pair of gambles. Each pattern has a spinner that rotates. If the spinner stops on the shaded area of a particular pattern, the prize i s the amount of money stated above the pattern. If the spinner stops on the unshaded portion of the pattern, nothing is won or lost. Place a check inside the gamble which you would choose i f you were only allowed to play one of the gambles. On the following pages are a series of pairs of gambles. Place a checkmark in the number of each pair that you would prefer to play. Make a choice on each pair. Following the series of patterns, there are two questionnaires. Please f i l l these out according to the i n -structions at the top, of the questionnaire. 77 At the end of the experiment, ten people w i l l be chosen at random from the class to play oner»of the gambles on the questionnaire for the cash prize indicated on the booklet. Each of these persons must, however, play the gamble i n the way in which he indicated his choice on his booklet. Listed below are a number of statements concerning personal attitudes towards taking exams. Please underline the word which best describes you personally. 1. Nervousness while taking an exam or test hinders me from doing well. Never Seldom Sometimes Often Always 2. I work most effectively under pressure, as when the task is very important. Never Seldom Sometimes Often Always 3. In a course where I have been doing poorly, my fear of a bad grade cut down my efficiency. Never Seldom Sometimes Often Always 4. when I am poorly prepared for an exam or test, I get upset, and do less well than even my restricted knowledge should allow. Never Seldom Sometimes Often Always 5. The more important the examination, the less well I seem to do. Never Seldom Sometimes Often Always 6. While I may (or may not) be nervous before taking an exam, once I start, I seem to forget to be nervous. Never Seldom Sometimes Often Always 79 7. During exams or tests, I block on questions to which I know the answers, even though I might remember them as soon as the exam i s over. Never Seldom Sometimes Often Always 8. Nervousness while taking a test helps me do better. Never Seldom Sometimes Often Always 9. When I start a test, nothing is able to distract me. Never Seldom Sometimes Often Always 10. In courses in which the total grade i s based mainly on one exam, I seem to do better than other people. Never Seldom Sometimes Often Always 11. I find that my mind goes blank at the beginning of an exam, and i t takes me a few minutes before I can function. Never Seldom Sometimes Often Always 12. I look forward to exams. Never Seldom Sometimes Often Always 13. I am so tired from worrying about an exam, that I find I almost don't care how well I do by the time I start the test. Never Seldom Sometimes Often Always 14. Time pressure on an exam causes me to do worse than the rest of the group under similar conditions. Never Seldom Sometimes Often Always 15. Although "cramming" under pre-examination tension is not effective for most people, I find that i f the need arises, I can learn material immediately before an exam, even under considerable pressure, and successfully retain i t to use on the exam. Never Seldom Sometimes Often Always 16. I enjoy taking a d i f f i c u l t exam more than an easy one. Never Seldom Sometimes Often Always 17. I find myself reading exam questions without understanding them, and I must go back over them so that they w i l l make sense. Never Seldom Sometimes Often Always 18. The more important the exam or test, the better I seem to do. Never Seldom Sometimes Often Always 19. When I don't do well on a d i f f i c u l t item at the beginning of an exam, i t tends to upset me so that I block on even easy questions later on. Never Seldom Sometimes Often Always 81 Listed below are a number of statements concerning personal attitudes and t r a i t s . Read: each item and decide whether the statement is true or false as i t pertains to you personally. If i t is true place a T beside the item, i f i t is false place an F. 1. Before voting I thoroughly investigate the qualifications of a l l the candidates. 2. I never hesitate to go out of my way to help someone i n trouble. 3. It is sometimes hard for me to go on with my work i f I am not encouraged. 4. I have never intensely disliked anyone. 5. On occasion I have had doubts about my a b i l i t y to succeed in l i f e . 6. I sometimes feel resentful when I don't get my way. 7. I am always careful about my manner of dress. 8. My table manners at home are as good as when I eat out in a restaurant. 9.. If I could get into a movie without paying and be sure I was not seen I would probably do i t . 10. On a few occasions, I have given up doing something because I thought too l i t t l e of my a b i l i t y . 11. I like to gossip at times. 12. There have been times when I f e l t like rebelling against people i n authority even though I knew they were right. 82 13. No matter who I'm talking to, I'm always a good listener. 14. I can remember "playing sick" to get out of something. 15. There have been occasions when I took advantage of someone. 16. I'm always willing to admit i t when I make a mistake. 84 $1.70 #2.30 $1.70 12.30 #1.70 $2.30 $1.70 $2.30 $1.70 ' *2.30 • $ 0 . 8 0 $ 0 . 8 0 #1.20 #1 .20 ,6 $1 . 2 0 

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0104725/manifest

Comment

Related Items