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A computerized valuation model for minig companies Wright, James Kirkland 1973

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SOME BEHAVIORAL ASPECTS OP ELICITING UTILITY (USING THE MACCRIMMON-TODA METHOD FOR ORDINAL UTILITY AND THE STANDARD GAMBLE METHOD FOR CARDINAL UTILITY) by EUGENE WONG BSc.,University of B r i t i s h Columbia, 1971 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE IN BUSINESS ADMINISTRATION i n the Management Science department of the Faculty of Commerce We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA October 1973 A.D. In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make i t freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department The University of British Columbia Vancouver 8, Canada Abstract This study investigates some behavioral aspects and properties of e l i c i t i n g u t i l i t y . Previous investigations devoted to empirical u t i l i t y measurement have stemmed from the work of experimentalists who have applied various u t i l i t y models in an e f f o r t to measure u t i l i t y . However, empirical studies devoted to investigation into behavioral factors which may bias the measurement are lacking and i t i s t h i s gap i n the u t i l i t y l i t e r a t u r e that prompted our empirical study. We chose to examine the standard gamble method for deriving von Heumann-Morgenstern cardinal u t i l i t y and the MacCrimmon-Toda method for deriving indifference curves. The domain of choice involved hospital days i n bed with r i s k of additional days. The analysis consisted of id e n t i f y i n g relationships between behavioral factors and properties of choice predictions obtained by the methods. Furthermore, the study also provided a means for comparing properties of the two methods for e l i c i t i n g u t i l i t y . Among other findings, the res u l t s show that not a l l subjects expressed agreement with the appropriateness of s p e c i f i c axioms of behavior which underly some methods f o r e l i c i t i n g u t i l i t y and that not a l l people express constant s e n s i t i v i t y over a l l stimuli l e v e l s . The two res u l t s in themselves suggest that a p r i o r i assumptions regarding " r a t i o n a l i t y " and i n f i n i t e s e n s i t i v i t y may have to be reexamined. The preferences e l i c i t e d by both methods seem to suggest that the subjects follow a l i n e a r rule to trade-off i i i sure outcome and r i s k . Although correspondence between test-retest preferences predicted by the standard gamble was generally closer than that for the MacCrimmon-Toda method, the MacCrimmon-'Toda method had generally better predictive a b i l i t y . Our re s u l t s also indicate that certain behavioral factors seem to aff e c t preferences predicted by the methods as we hypothesized. This observation has implications for p r a c t i c a l measurement of u t i l i t y since "successful" application of methods for e l i c i t i n g preferences depends upon our awareness of which behavioral factors may bias the measurement. Table_of_Contents Chapter 1 - An Introductory Discussion ..................... 1 Chapter 2 - Methods Of Estimating U t i l i t i e s ....18 Section I - Introductory Discussion ., ......18 Section II - C l a s s i f i c a t i o n Of Methods 19 Section III - Discussion Of Methods .....................26 Banking Method ., 26 Magnitude Estimation Method .29 Preference Ratio Method 31 Ordered Metric I Method 34 Ordered Metric II Method .............................. 36 Successive Comparisons Method (Churchman-Ackoff Method) ......38 Single Trade-Off ........39 Double Trade-Off ,.40 Single Transformation 44 Double Transformation 45 Cancellation Method 47 Two-Stage Rating Model 50 Unweighted Rating Model 52 Linear Model 53 M u l t i p l i c a t i v e Model ..54 Hybrid Model .56 Multidimensional Scaling Method 57 Chapter 3 - Cardinal U t i l i t y Curves 6 Indifference Curves .60 Section I - Von Neumann-Morgenstern Cardinal U t i l i t y ....60 In troduct ion .................60 Discussion Of Previous Experiments ....................72 Discussion Of Our Experiment .....76 Section II - Ordinal U t i l i t y Represented By Indifference Curves ...........78 Introduction 78 Discussion of Previous Experiments ....80 Discussion Of Our Experiment .83 Chapter 4 - Results, Analysis, And Conclusions ............89 Section I - Introduction ..89 Section II - Discussion Of Experimental Design ...,..,.,.97 Test-Retest Correspondence Of Preferences ........,,,..101 Intermethod Correspondence Of Preferences ...,...,.,,.,101 Section III - Results And Analysis .,. 103 Dependent Variables (Inconsistency Measures) - P r o f i l e Of Results ............. ,...104 The Independent Variables ,......,..,105 Independent Variables - P r o f i l e Of Results .......... 109 Linear Associations Among Independent Variables .....110 Linear Associations Between Independent And Dependent Variables .....112 The Questionnaire Items ............................... 114 Questionnaire Responses - P r o f i l e Of Results ........116 Linear Associations Among The Questionnaire Responses , 118 Linear Associations - Questionnaire Responses S Dependent Variables ,........,..,119 Comparison Of Scores For Inconsistency Measures .......121 MacCrimmon-Toda Vs Standard Gamble Test-Retest ......123 Test-Retest (for Each Method) Vs Averaged Intermethod v i . . . .125 Intermethod (for Each Session) Vs Averaged Intermethod , , .126 Goodness Of Prediction . . . . . . . . . . 1 2 8 Bibliography Appendix A . Appendix B . 129 141 142 v i i L i st_of_Figures 1 Decision Tree For C l a s s i f i c a t i o n Of Methods ............ 19 2 Figure For Double Trade-Off Method ..................... 40 3 Double Valued Indifference Curves 42 4 Figure For Double Transformation Method ................45 5 Double Valued Transformation Curves ....................47 6 Figure For Cancellation Method 47 7 Identifying Accept And Reject Regions ..................85 8 Identifying Accept And Reject Regions ..................85 9 Identifying Accept And Reject Regions 86 10 Identifying Accept And Reject Regions .................86 11 Test-Retest Model .......101 12 Intermethod Model 101 13 inconsistency Measures - P r o f i l e Of Results ...104 14 Independent Variables - P r o f i l e Of Results ......110 15 Spearman Correlation Matrix For Independent Variables .112 16 Spearman Correlations Between Independent And Dependent Variables , 114 17 Questionnaire Responses - P r o f i l e Of Results ..........116 18 Spearman Correlation Matrix For Questionnaire Responses ......119 19 Spearman Correlations - Questionnaire Responses 8 Dependent Variables ............121 20 MacCrimmon-Toda vs Standard Gamble Test-Retest ...,,,..124 21 Test-Retest (for Each Method) vs Averaged Intermethod .125 22 Intermethod (for Each Session) vs Averaged Intermethod 127 23 Numerical Estimates Of Probab i l i t y Expressions .,,.....14 v i i i Acknowledgements I would l i k e to especially thank Dr. Ilan Vertinsky, the chairman of my thesis committee, for his valuable assistance and guidance during the undertaking of t h i s thesis project. Also, I would l i k e to thank him for providing me with a research assistantship i n the Health Systems Group of the Resource Science Centre during ay graduate years and allowing me generous use of the resources available to t h i s research group. I would l i k e to thank the committee members Dr. Dean Oyeno and Dr. J e f f Sidney. Last but c e r t a i n l y not least I would l i k e to extend my appreciation to the 23 subjects who volunteered their time and e f f o r t toward th i s study. IK Dedicated to my parents 1 £hap.ter_ 1,~^ftn mlB^£9§^S^9fy„Pig9M§glgS-2l-§ o mg.. philosophical Our very destinies are determined by the decisions that we make i n our dai l y l i v e s . When confronted with choices, we may experience d i f f i c u l t y i n reaching a decision due to our ignorance of the decision environment, i . e . our ignorance of the p r o b a b i l i t i e s of outcomes, the scope of actions av a i l a b l e , the decision framework, etc. However, more importantly, we may not be aware of what we actually "want" or at least are unable to present our preferences or desires in a communicable (eg. easily understandable, unambiguous, meaningful) manner although one often assumes that the nature of human needs and desires are obviously apparent. In order to make meaningful recommendations concerning decisions to a decision-maker one must be able to e l i c i t his subjective values. An important consideration i s whether there i s some basic "sensation' 1 that determines for him the value that he attaches to objects, actions, etc. Can feelings of pleasure and pain be used as a basis for value judgements? Bentham (1907) f e l t that pleasure and pain alone guide us to what we ought to do, as well as to what we s h a l l do. However, the immediacy of pleasure sensations do not give us an adequate assessment of values since sensations may be f e l t but are d i f f i c u l t to communicate. Even i f communication i s possible, v e r i f i c a t i o n or v a l i d a t i o n of sensation i s extremely d i f f i c u l t i f not impossible. Therefore, the methodological problem i s to 2 translate pleasure and pain response into some measure which can be communicated and manipulated. Philosophers who had o r i g i n a l l y advocated the p r i n c i p l e of pleasure as a basis for value had not gone to the point of finding a reasonable way of measuring pleasure, a n t i t h e t i c a l l y , economists had not been so concerned with the philosophical issues of value as the measurability or quantification of pleasure, or what the economists termed u t i l i t y * . What do we mean by measurement? Measurement involves the assignment of numbers to objects, observations, outcomes, etc. whereupon "allowable" operations performed on these numbers w i l l reveal new information about the e n t i t i e s measured. Alchian (1953) i d e n t i f i e s three main aspects i n the process of measurement: (1) the purpose of the measurement, (2) the method of measurement, and (3) the ar b i t r a r i n e s s of the assigned numbers. Underlying the three main aspects i s the assertion that measurement i s always invented and never discovered since measurement i s not a property which i s inherent i n e n t i t i e s . The method of assignment of numbers to e n t i t i e s i s determined merely by i t s convenience for the purpose. Thus, the a r b i t r a r i n e s s of aspect 2 i s limited by aspect 1, but for no other reason than convenience or manageability. The 1 The term i s inherited from Bentham and his u t i l i t a r i a n philosophy; i n the context of u t i l i t a r i a n i s m , which i s the doctrine that the greatest happiness of the greatest number s h a l l be the end and aim of a l l s o c i a l and p o l i t i c a l i n s t i t u t i o n s , u t i l i t y i s the standard of morality - actions are r i g h t i f they promote happiness. a r b i t r a r i n e s s of the assigned numbers (aspect 3) i s inherent i n the r u l e used f o r ass i g n i n g numbers (aspect 2 ) . In order to perform c e r t a i n operations with these numbers i n such a way as not to a l t e r the information t h a t these numbers were assigned to convey, the s t r u c t u r e of the measurement must be isomorphic to some numerical s t r u c t u r e which i n c l u d e s these operations. The f o l l o w i n g four b a s i c measurement s c a l e s are of s p e c i a l i n t e r e s t here: (1)nominal, ( 2 ) o r d i n a l , (3) i n t e r v a l , and (4) r a t i o . The nominal or c l a s s i f i c a t o r y s c a l e i s the weakest of the four measurement s c a l e s . The s c a l i n g r u l e c o n s i s t s of p a r t i t i o n i n g a given c l a s s i n t o a set of subclasses that are mutually e x c l u s i v e and equivalent i n the property being s c a l e d . Numbers are then assigned to objects that belong to the same subclass. A one-to-one transformation w i l l preserve equivalence. In other words, a l l s c a l e s derived by a one-to-one transformation have the same equivalence property. In an o r d i n a l or ranking s c a l e , numbers are assigned to ob j e c t s such that group membership and the ordering r e l a t i o n s h i p (>) may be i d e n t i f i e d . Any monotonic transformation w i l l preserve the equivalence and ordering r e l a t i o n s h i p i n an o r d i n a l s c a l e . When the sc a l e has the c h a r a c t e r i s t i c s of an o r d i n a l s c a l e and furthermore when the distances between any two numbers on the s c a l e are of known " s i z e " , i . e . when the same numerical d i f f e r e n c e between any two numbers on the s c a l e r e f l e c t the same d i f f e r e n c e i n the property measured. 4 an i n t e r v a l scale measurement has been achieved. The differences between numbers on the i n t e r v a l scale are isomorphic to the structure of arithmetic. It i s t h i s property that makes the i n t e r v a l scale a quantitative scale. An i n t e r v a l scale i s unique up to a li n e a r transformation. A r a t i o scale has a l l the c h a r a c t e r i s t i c s of an i n t e r v a l scale and in addition has a true "zero point" as i t s o r i g i n , i . e . the number zero i s assigned to an object which possesses the true zero i n the property being scaled. M u l t i p l i c a t i o n by a positive constant i s a linear transformation and thus w i l l preserve the i n t e r v a l properties; i n addition the true property "zero" w i l l s t i l l be assigned the number zero (zero multiplied by a positive constant i s s t i l l zero). Thus the r a t i o scale i s unique up to m u l t i p l i c a t i o n by a positive constant. Furthermore, the scale i s isomorphic to the structure of arithmetic and i s thus a high order l e v e l of measurement. In regards to the purpose of measurement, the normative purpose of u t i l i t y measurement i s to provide a basis for decision and policy. The decision-maker has to decide which type of measurement i s most suitable for his purpose and his choice of measurement involves a tradeoff between economy of measurement and need for d e t a i l . H i s t o r i c a l l y , economists had been interested i n u t i l i t y as a descriptive theory; they were concerned with the measurability of u t i l i t y i n an attempt to explain consumer behavior. Economists had designated the term u t i l i t y (out of 5 t r a d i t i o n as indicated i n i n the previous footnote) to the subjective value of commodities. Experimental psychologists, interested i n psychophysics, had been measuring human response as a function of measurable physical stimulus whereupon the measurements are used to construct a subjective scale d i r e c t l y from the subject»s own quantitative estimates of the scale values of a series of s t i m u l i . From the functional r e l a t i o n s h i p , measures of responses were derived. Fechner's law was an early attempt to express a functional r e l a t i o n s h i p between the physical magnitude and the psychological magnitude of the sensation i t arouses. Brightness may be measured i n b r i l s ; loudness may be measured i n sones 2. Can u t i l i t y be measured i n u t i l s ? It was i n the l i g h t of t h i s question that economists were searching for a measure of u t i l i t y . Experimental economists were not so interested i n the i n t e r n a l pleasure-f e e l i n g aspect as in the external choice behavior of the subject. The behavioristic i n t e r e s t i s i n the observed behavior which arises from simple sensations. Thus we have an observer interpreting observed behavior versus the subject introspecting his own fee l i n g s . For the behaviorist, the mind of the person i s the observed behavior. For the in t r o s p e c t i o n i s t , there i s no such equivalence. Obviously, d i r e c t empirical refutation of introspection i s impossible. The closest we can come to validation of introspection i s to 2 B r i l s and sones are both subjective scales that are discussed by Stevens (1959). 6 examine i t s a b i l i t y to derive theorems of behavior which i t claims to enunciate. Introspectionism regards simple sensations such as pleasure as primitive notions which are not definable i n any other terms. Therefore, in order to avoid interpreting or valid a t i n g introspection i t seems natural that values be operationally defined in terms of choice. However, even defining values by choice does not guarantee that behavioristic expressions w i l l not be e r r a t i c , whimsical, careless, etc. Thus, although our conception of values spring from inner impulsive emotive f e e l i n g s , our b e h a v i o r i s t i c expressions of our values must be guided by judgement of one sort or another in order to be useful information. Can standards of judgement be provided that w i l l form not only ground rules for measurement of u t i l i t i e s but also norms by which one "should" abide i n making judgements? In order to provide a framework for measurement of u t i l i t y , judgement of values (eg. the exhibition of preferences) i s r e s t r i c t e d to that observed under so c a l l e d r a t i o n a l behavior. A set of i n t u i t i v e l y plausible assumptions (axioms) of r a t i o n a l i t y provide the formal conditions for r a t i o n a l patterns of preference. In t h i s context, "inconsistent" behavior may be interpreted as behavior which contradicts one of the axioms of r a t i o n a l i t y . However, i t i s conceivable that a person's behavior contradicts one of the axioms yet he s t i l l considers himself to be consistent (or r a t i o n a l ) . Here, a person questions the axioms as c r i t e r i a for r a t i o n a l i t y . Savage (1954) suggests that a person's acceptance of or agreement with an axiom may 7 be i n d i c a t e d by the degree to which he t r i e s to act i n accord with i t or by h i s w i l l i n g n e s s to r e v i s e h i s behavior when made aware of h i s v i o l a t i o n with the axiom. The obvious i n t e r p r e t a t i o n of the axioms i s that they p r o v i d e the d e f i n i t i o n s of r a t i o n a l i t y . Secondly, i n terms of p r o v i d i n g ground r u l e s f o r measurement of u t i l i t y , the axioms of r a t i o n a l i t y may be i n t e r p r e t e d as r u l e s which must be obeyed before meaningful measurement i s p o s s i b l e . L o o s e l y speaking, t h i s means t h a t e r r a t i c or c a r e l e s s behavior cannot be i n t e r p r e t e d i n terms of values. In a d d i t i o n , these axioms may be t r a n s l a t e d i n t o p r e s c r i p t i v e statements r e g a r d i n g behavior, i . e . one's behavior should abide by these axioms; i n t e r p r e t a t i o n i s a l s o p o s s i b l e i n terms of p r e d i c t e d r a t i o n a l behavior, i . e . one's behavior w i l l be i n accord with these axioms i f c h o i c e s are f r e e and r a t i o n a l . E x i s t e n c e of c e r t a i n proposed u t i l i t y f u n c t i o n s i s dependent on the axiomatic system upon which the p r o p o s i t i o n i s based. In t h i s sense, the axioms take on the same r o l e as axioms i n mathematics from which theorems are d e r i v e d . Thus, the e x i s t e n c e of a c e r t a i n proposed u t i l i t y f u n c t i o n (or i n other words, the p o s s i b i l i t y of f i n d i n g a s c a l e of numbers t h a t express the u t i l i t i e s i n a convenient way) i s l o g i c a l l y e q u i v a l e n t (<=>) to the hypothesis that behavior i s c o n s i s t e n t with the u n d e r l y i n g axioms. E m p i r i c a l t e s t of the p r o p o s i t i o n t h a t a given u t i l i t y f u n c t i o n e x i s t s i s thus d i s p l a c e d to the axioms t h a t imply i t . I t i s a much e a s i e r task to t e s t the axiomatic system statement by statement (in 8 i s o l a t i o n from the theorized u t i l i t y function) than i t i s to question the v a l i d i t y of the u t i l i t y function (in i s o l a t i o n from the axiomatic system which theorized i t ) . I t i s a general observation that the stronger the measurement (ratio i s stronger than i n t e r v a l , than ordinal, than nominal) of the proposed u t i l i t y scale, the stronger the axioms needed to derive i t . The ordinal scale of u t i l i t y e s s e n t i a l l y requires only the basic axiom of t r a n s i t i v i t y of preferences for i t s existence. It i s therefore not unusual that the p o s s i b i l i t y of u t i l i t y measurement in ordinal terms i s f a i r l y well agreed upon among value theory philosophers (see Davidson et a l , 1955). In p a r t i c u l a r . Perry wrote: ... The important feature of preference i s that i t arranges the objects of any given in t e r e s t in a order, r e l a t i v e l y to one another, and i n a manner that cannot be reduced either to the in t e n s i t y or to the inclusiveness of the i n t e r e s t . This order of preference has i t s own c h a r a c t e r i s t i c magnitudes, which determine comparative values ... Preference generates a similar (his reference to the color spectrum) t r a n s i t i v e , asymmetrical r e l a t i o n among i t s terms. It i s t r a n s i t i v e because i f b i s preferred to a and c to b, then c i s preferred to a ... The three terms i n order of preference may be said to constitute a "stretch" which i s greater than any of i t s included stretches. Thus the stretch a-c i s greater than the stretch a-b, or we may say that c l i e s beyond b. (1950, pages 635-636) On the other hand he and other value theory philosophers, who had accepted o r d i n a l i t y , expressed scepticism about the p o s s i b i l i t y of measurability on a higher scale. One suspects that early ( h i s t o r i c a l l y ) 9 r e j e c t i o n of the c a r d i n a l i t y (interval scale) of u t i l i t y l i e i n the f a i l u r e of expressing the cardinal proposition in terms of an acceptable axiomatic system. As a r e s u l t , the argument for rejec t i o n revolved around the quantitative aspect of c a r d i n a l i t y . Thus, the early c a r d i n a l i t y proposition was rejected for mistaken reasons. As a t y p i c a l example concerning t h i s point, Hhellwright, who rejected a higher-than-ordinal measure, wrote: We may on a particular occasion prefer reading a book to taking a walk: the former, then, we say, would give us (on t h i s occasion) the greater pleasure. But i s t h e r e any conceivable sense i n which_wg_could_say that the i n t e n s i t y o f t h e pleasure to be got from reading i s twice rather than three times_or 0Bg_ 3 N J - L§^h?l! ^ i l e s t . the i n t e n s i t y of the pleasure to_be_gptfrgm_walking? (the emphasis i s mine) Would we not, by trying to make our comparison of i n t e n s i t i e s mathematically exact, reduce i t to meaninglessness? (1949, page 87) Along the same l i n e s , Lewis wrote: ...numerical measure cannot be assigned to an i n t e n s i t y of pleasure, or of pain, unless a r b i t r a r i l y . I n t e n s i t i e s have degree, byt_they are not extensive,or measurable .magnitudes JgJli£fe-£aS-feg .added_ or subtracted. (the emphasis i s mine) That i s , we can- presumably-determine a s e r i a l order of more or less intense pleasure, more and less intense pains, but we cannot assign a measure to the i n t e r v a l between such. (1946, page 490) However, fo r economists the question of measurability took on more pragmatic and less philosophical tones. The early objectors to any search for a cardinal u t i l i t y claimed that ordinal u t i l i t y (re marginal rates of substitution) could explain the aspects of economic behavior (eg. market behavior under certainty) which the proposed cardinal 10 u t i l i t y (re marginal diminishing u t i l i t y ) could. However, c a r d i n a l i t y revived i t s e l f i n the face of welfare economics and r i s k y 3 choices, A group of economists, of which Jevons, Benger, and Marshall are t y p i c a l , asserted that the ordinal measurability of r e l a t i v e preferences at the l e v e l of introspection implies the p o s s i b i l i t y of an i n t e r v a l scale for u t i l i t y 4 . In a Jevonsians experiment, the person i s asked to rank differences in u t i l i t y of e n t i t i e s . Any numerical indices that preserve the ordinal relationship between u t i l i t y differences w i l l be related by a linear transformation. Thus the u t i l i t y index i s said to be c a r d i n a l . On the basis of a Jevonsian u t i l i t y index, the rule of maximization of expected u t i l i t y was used as a prescription for choices between wagers even though the 3 In an attempt to develop an a n a l y t i c framework for dealing with the degree of knowledge we have of our decision environment, various p a r t i t i o n s between complete knowledge and complete ignorance have been attempted. The most accepted i s a p a r t i t i o n into unmeasurable uncertainty, measurable uncertainty, and certainty. For measurable uncertainty, which i s sometimes c a l l e d r i s k , the p r o b a b i l i t y d i s t r i b u t i o n of the uncertain event i s known, while for unmeasurable uncertainty, a d i s t r i b u t i o n i s not given. * The construction of a scale based upon ordered differences was f i r s t discussed by Coombs (1950). He then c a l l e d such a scale an ordered metric and placed i t between the ordinal and the i n t e r v a l scale. The p o s s i b i l i t y of an ordinal scale on i n t e r v a l s implies the p o s s i b i l i t y of a nominal scale on i n t e r v a l s , i . e . i n p r i n c i p l e the operations s u f f i c i e n t for an ordered metric scale ought to be s u f f i c i e n t to determine an i n t e r v a l scale. Stevens (1959) concludes that the ordered metric scale appears i n practice to be a type of unfinished i n t e r v a l scale. s The term was suggested and used by Heldon (1950) and had since been adopted by Ellsberg (1951). * Formally, a wager i s a set of alternative mutually exclusive outcomes, each of which occurs with stated p r o b a b i l i t i e s . The terms "prospect" and "gamble" are often used i n place of "wager". 11 index was not derived from any r i s k behavior 6. This p r i n c i p l e of maximization governing r i s k behavior i s not implied by Jevon's concept of u t i l i t y or by the methods of measuring i t . In t h i s regard, the emphasis upon mathematical expectation i s arbitrary and any other normative c r i t e r i a for choice i s equally meaningful. I t was not u n t i l the appearance of the von Neumann-Morgenstern u t i l i t y theory that i t was shown formally that the existence of an i n t e r v a l scale of u t i l i t y , which i s derived from exhibited choice behavior under r i s k , implies the norm of maximization of expected u t i l i t y . Previously, there had been suggestions that such a norm was i m p l i c i t i n r a t i o n a l behavior. Ramsey (1926) had e a r l i e r hypothesized that a person acts upon his b e l i e f s i n such a way as to maximize the o v e r a l l good. Bernoulli (1738) defined the moral expectation of a decision, which was to be maximized, as the mathematical expected value in accord with the expected u t i l i t y model. The hypothesis of the von Neumann-Morgenstern model i s that the maximization of expected u t i l i t y i s s u f f i c i e n t for predicting the free choice behavior of a r a t i o n a l person. From the viewpoint of o b j e c t i v e 7 p r o b a b i l i t y , the expected u t i l i t y concept seems plausible because the weighted sum of the outcome u t i l i t i e s i s the expected long-run u t i l i t y of the action. Using t h i s u t i l i t y model, the method of e l i c i t i n g u t i l i t y consists of observing an individual's 7 The concept of objective pr o b a b i l i t y w i l l be discussed l a t e r i n t h i s chapter. 12 behavior i n the very simplest of risk y s i t u a t i o n s : a choice between a constructed wager and a certain outcome (Swalm, 1966). From the observations of a person's choices i n these simple constructed r i s k s i t u ations, von Neumann and Hdrgenstern theorized that i t was possible to predict his choice i n more complicated r i s k s i t u a t i o n s . From a person's introspection into his preferences for certain outcomes, the Jevons school assumed that the same predictive a b i l i t y claimed by the von Neumann-Morgenstern model was possible. The obvious operational difference between the two cardinal measures i s however masked by the s i m i l a r i t y i n their algebraic formulations; the two schools summarize their r e s u l t s by algebraic expressions (namely statements expressing maximization of expected u t i l i t y ) that are mathematically equivalent 8. By employing wagers to i n d i r e c t l y e l i c i t u t i l i t i e s , the von Neumann-Horgenstern theory avoided many of the objectionable methodological features of the Jevonsian and other dir e c t methods 9 of measuring u t i l i t y . In a psychophysics experiment, a series of v i s u a l i n t e n s i t i e s 8 The confusion i s also due to the use of the same term "cardinal u t i l i t y " to denote the result of two di f f e r e n t operations. On the operational approach, Bridgeman states: " I f we have more than one set of operations we have more than one concept, and s t r i c t l y there should be a separate name to correspond to each d i f f e r e n t set of operations." (1927, page 10) 9 Some of these dir e c t methods w i l l be discussed in Chapter 2. 13 whose brightness i s to be judged can be presented without attaching numerical indices to the v i s u a l i n t e n s i t i e s , but i n a u t i l i t y experiment magnitudes of wealth whose u t i l i t y i s to be judged cannot be presented without attaching numerical indices to the magnitudes of wealth. As a r e s u l t , one of the two a p r i o r i arguments against the method of d i r e c t estimation of u t i l i t y i s that the subject may judge the physical magnitude of the stimulus and not the subjective e f f e c t thereby committing what the psychophysicists c a l l the stimulus error. Also by l i m i t i n g concern to choice behavior among wagers, the von Neumann-Horgenstern u t i l i t y index i s more firmly t i e d to a be h a v i o r i s t i c basis than to an introspective basis and thus f a c i l i t a t e s easier interpretations. Direct quantitative estimates may prove d i f f i c u l t to make because of t h e i r introspective nature (Stevens, 1959). While for the d i r e c t methods of measuring u t i l i t y in which r i s k l e s s choices (or certain outcomes) are presented fo r which the u t i l i t y alone i s maximized, the introduction of r i s k y choices into the measurement of u t i l i t y presents an additional cue, namely that of p r o b a b i l i t y . Formally, objective probability measures the agreement between outcomes of repeatable physical events on the one hand and outcomes of hypothetical mathematically defined events on the other. This t r a d i t i o n a l interpretation of the pr o b a b i l i t y concept i s based on the r e l a t i v e frequency with which the event occurs i n a long series of observations. 14 Alberoni (1962) described an experiment in which the subjects were shown a seguence of binary events and were to l d to predict future seguences. The subjects i n v a r i a b l y associated the observed r e l a t i v e frequencies with p r o b a b i l i t i e s and were increasingly confident that the events were due to "chance" as more event occurrences conformed to their expectations were observed. However, t h i s interpretation of probability as frequency f a i l s when one considers probability of events that cannot be repeated i n order to y i e l d long-run r e l a t i v e frequencies. Such i s the case i n p r a c t i c a l decision problems. As an example, consider an event in which a certain company X w i l l make more than one m i l l i o n d o l l a r s annual p r o f i t as a r e s u l t of introducing a new product Y into the market. The management of company X w i l l surely not consider stopping the introduction of product Y into the market because there i s not enough data to construct long-run r e l a t i v e frequencies of outcomes. It i s i n situations such as t h i s that the decision-maker must subjectively assess the l i k e l i h o o d of such an event occurring without recourse to the long-run frequency concept. It i s this c r i t i c i s m of the objective concept of probability that led Savage (1954) to conclude that "the grounds for adopting an o b j e c t i o n i s t i c view are not overwhelmingly strong". Is there empirical evidence to suggest that people do not always behave i n accord with the objective view of probability? In experiments of probability r e v i s i o n , subjects tended to revise t h e i r p r o b a b i l i t i e s i n the same d i r e c t i o n as 15 dictated by Baye's theorem but i n a much more conservative manner (Edwards, Lindman, P h i l l i p s , 1965). The experiments of Edwards and P h i l l i p s (1966) revealed that when subjects make datum-by-datum revisions throughout a sequence of data, the f i n a l subjective probability i s far more conservative than the s t a t i s t i c a l l y calculated combination of series of single estimates made by subjects for each datum in the sequence. Alberoni (1962) had subjects estimate various binomial sampling d i s t r i b u t i o n s . The sums of the estimated p r o b a b i l i t i e s for the d i f f e r e n t outcomes consistently t o t a l l e d about 0.85, considerably less than the 1.0 required by p r o b a b i l i t y theory. These experimental re s u l t s lead one to adopt an additional concept of probability interpretation, that known as subjective p r o b a b i l i t y . The subjective view interprets probability as a measure of an i n d i v i d u a l ' s confidence in the truth of a p a r t i c u l a r proposition. S t a t i s t i c i a n s have devoted much e f f o r t i n the development of formal procedures for dealing with f a l l i b l e information in making inferences about prevailing and future states of the decision environment. Mathematical d e f i n i t i o n s of probability are meant to provide an analytic procedure for c a l c u l a t i n g probability. However, the i n t e l l e c t u a l process i s much more subjective and in situations where the objective measurement of probability i s unnecessary, uneconomical, unreasonable, or unattainable, the probability of the occurrence of an event may be described by the measure of the degree to which the b e l i e f of the occurrence of the event i s substantiated by a group of people. A b e l i e f 16 i s well substantiated i f i t i s accepted by most people (breadth) and i s intense (depth) (Churchman, 1961). Therefore, while the von Neumann-Horgenstern theory has made the cardinal u t i l i t y notion with i t s implied norm of maximization of expected u t i l i t y a more meaningful concept than the Jevonsian theory, the empirical derivation of an u t i l i t y index has introduced an additional stimulus that i s subjectively interpreted by the subject, namely that of pro b a b i l i t y . Thus, value researchers who want to measure subjective estimates of outcome values ( u t i l i t i e s ) using models of risky choice must r e a l i z e the inevitable confounding of subjective estimates of values with that of subjective interpretation of p r o b a b i l i t i e s f or which the unfortunate consequence i s that the subject's estimates of proba b i l i t y and u t i l i t y may be mutually biased. The d i f f i c u l t y of i s o l a t i n g p r o b a b i l i t y from u t i l i t y in experiments has been one of the methodological problems inherent i n the construction of the von Neumann-Horgenstern u t i l i t y index. Thus, experimenters (eg. Hosteller and Nogee, 1951) who have obtained u t i l i t y indices have had to make assumptions regarding subjective estimates of probability, while experimenters (eg. Preston and Baratta, 1948) who have obtained subjective probability measures have had to make assumptions regarding the u t i l i t y function. In t h i s chapter, some of the concepts of u t i l i t y measurement which have been highlights i n the history of u t i l i t y theory have been discussed. In Chapter 2 we w i l l 17 examine some of the numerous a l t e r n a t i v e ways of e l i c i t i n g and measuring u t i l i t y , each with i t s own assumptions r e g a r d i n g concepts of u t i l i t y and c a p a b i l i t i e s of human judgement. In Chapter 3, we w i l l r e t u r n to the von Neumann-Horgenstern model and focus our a t t e n t i o n upon t h i s u t i l i t y index and the i n d i f f e r e n c e map concept, both of which have s p e c i a l t h e o r e t i c a l and b e h a v i o r a l a s p e c t s that are of i n t e r e s t to us. 18 S g c t i o n . I _ - IS££2^SS^9£Y mPl§£ qssion I f at the o u t s e t of the decision-making p r o c e s s , the decision-maker does not possess c l e a r and a r t i c u l a t e n o t i o n s of values t h a t he p l a c e s upon action-outcomes, one must c r e a t e such n o t i o n s by i n d u c i n g him to c o n s c i o u s l y express a c l e a r concept of worth f o r a l t e r n a t i v e s r e l e v a n t to a p a r t i c u l a r d e c i s i o n . There are a v a i l a b l e a number of d i f f e r e n t methods that w i l l p r ovide some measure of u t i l i t y from e l i c i t e d value judgements. For each u t i l i t y e s t i m a t i o n method, the u n d e r l y i n g model d e f i n e s the c o n d i t i o n s under which e s t i m a t i o n i s v a l i d . The u n d e r l y i n g premise of each model i s t h a t from a c a r e f u l o b s e r v a t i o n and a n a l y s i s of an i n d i v i d u a l ' s s t a t e d p r e f e r e n c e s i n comparisons between h y p o t h e t i c a l l y c o n s t r u c t e d c h o i c e s under c o n t r o l l e d l a b o r a t o r y c o n d i t i o n s , one may a r r i v e at a p r e s c r i p t i v e as we l l as p r e d i c t i v e f o r m u l a t i o n of h i s choice behavior i n r e a l d e c i s i o n s from the deduced u t i l i t y index. Each method has i t s own assumptions r e g a r d i n g concepts of u t i l i t y , type of d e c i s i o n a l t e r n a t i v e s , and c a p a b i l i t i e s of human judgement. T h e r e f o r e , a c l a s s i f i c a t i o n of methods i s u s e f u l to the decision-maker i n h e l p i n g him to decide which method i s most s u i t a b l e f o r h i s purposes. The c l a s s i f i c a t i o n scheme t h a t was devised resembles t h a t of a d e c i s i o n t r e e . At each node i n the t r e e the 19 decision-maker's choice of a set of method properties (indicated at the branch) directs him along one branch of the tree. By successively encountering a node, choosing a set of method properties, and branching, the decision-maker a r r i v e s at a method with the chosen properties. The methods appear at the very ends of the branches. An i l l u s t r a t i o n of the numbering scheme i s given here. Suppose that a branch i s assigned the code X (for example, X i s 1-1). Furthermore, suppose that the branch enters a node which branches N ways, Then each of these N branches i s assigned a unique number X-I where 1=1,...,H. Thus, a number code assigned to a branch w i l l indicate which branches of the tree were i t s predecessors. The decision tree i s summarized by Figure 1 . A description of the tree i s given i n Section I I . Section_II - _ C l a s s i f i c a t i o n . o f methods Each of the c l a s s i f y i n g properties of the decision tree w i l l be l i s t e d here. {1) Unidimensipnal U t i l i t y Unidimensional u t i l i t i e s take into account only one fa c t o r . They are represented a l g e b r a i c a l l y by U(X) and geometrically by a l i n e i n 2-dimensional space. (2) Multidimensional U t i l i t y Multidimensional u t i l i t y take into account the o o o rt 11 tr cs o o rti ft* a H i o CT o ri M i-3 CD CU H (D c H fD O ro <v H* CA H" O o rt M (0 ro o n o M p> If) cn H* H» H 1 o o 3 o r*> multidimensional alternatives \-2 Ju1p;o wonts 1.1-1 _^ T . ^ n - i t u J * e o t i t r - a t l o n r r f l t h o - l unidlmannicnal u t i l i t y 1 multidimensional u t i l i t y u n l d i m e n s i o n a l elternitlvee 2-1 preference r a t i o -ethod a u c c i G s l v o comparison (Churcfcrcan-AcKoff) method tnon-linear :node 1 ' 12-1-2 multiplicative model m u l t i d i m e n s i o n a l A l t e r n a t i v e s 2-2  linear model 2-1-1 geometric representation 2-2-1 algebraIc representation ranking method ordered metric I method use of p r o b a b i l i t i e s 1-2-1 standard cambie method (von Vorfienotern H t y ) ' no use of probabilities 1-2-2 additivity not nt-edod 2,-2-1-1 coefficients determined by subjective weighting '**' Jyd|»e*.ente jl-2-1 -2 ordernd tretric TI wffthod two-stage rating modal coefficients set equal to 1 2-1-1-2 unweighted rating model additivity needed 2-2-1-2 linear r-o<l+\ hybrid r s i e l I'acCrlr.non-Toda nothod {indifference curves) •uitidin.ensional sealing  cancellation method u t i l i t y indices of botr- factors x Involved are estimated 1-2-2-1 double trade-off double tronaforr-ation 1 u t i l i t y Index of one of the two factors involved is estimated 1-2-2-2 single trade-off elngle t r a n s f o r a t i o n ro ti ro co ro a rr ro ro r r tr o Ql Ui to o 21 al g e b r a i c a l l y by U(Xl,...,Xn) and geometrically by a hypersurface in n+1-dimensional space. In general, multidimensional u t i l i t i e s are more r e a l i s t i c but less manageable for analysis. It w i l l be shown l a t e r that certain assumptions can be made which w i l l separate multidimensional u t i l i t i e s i n t o several unidimensional u t i l i t i e s , U i ( X i ) . £ } M 2 - X J u d «g e m en t s_0 f _ U n i d i m e n s i o n a 1_ A1 tern at i ve s The subject i s required to evaluate unidimensional alternatives which can be represented geometrically as points on a l i n e . 11x2-21_Judgements_Of_Multi^ The subject i s required to evaluate multidimensional alte r n a t i v e s which can be represented geometrically as points i n n-dimensional space. Evaluation consists of taking into account the contributions of several factors. Examples of multidimensional alternatives are company p r o f i l e s ( (p r o f i t , prestige, number of personnel, number of branch offices) for example) and automobile performance ( (maximum speed, acceleration, s t a b i l i t y on corners, engine size) for example). On the other hand, judgements between engine sizes of d i f f e r e n t automobiles or p r o f i t s of d i f f e r e n t companies would involve only one factor; t h i s i s the case with unidimensional a l t e r n a t i v e s . One of the d i f f i c u l t i e s expected to be encountered by presenting multidimensional alternatives to the subject i s his i n a b i l i t y to take into account simultaneously a l l the component factors. 22 K i l l e r (1956) showed that there are li m i t a t i o n s on the number of conceptual units that can be handled at any one time. Shepard et_a1 (1961) has shown that learning became slower and more d i f f i c u l t as the number of attr i b u t e s to be simultaneously judged increased. The re s u l t s of DeSoto (1961) and Osgood e t _ a l (1956) revealed that subjects are unable to adequately grasp the notion of multidimensionality in making judgements. These pychological r e s u l t s seem to suggest that s i m i l a r cognitive burden w i l l a r i s e in u t i l i t y assessment of multidimensional a l t e r n a t i v e s . J J - J - J ) ^ _ T h e _ H e tho d_Is_ Base d_U Judqemgnts_On_Utilities The subject i s asked to assign numbers to factor le v e l s according to his evaluation of their r e l a t i v e u t i l i t i e s . 1 I r J r ? ) ^The^Meth^cl_ Is_Based Upon.. Preference Judgements Or "Direct".Inequality.Judqements_On_Utilities In the case of ranking, the subject i s required to rank factor l e v e l s by preference while i n the ordered metric method, the subject i s asked to rank u t i l i t y differences. llzirll_2he_Method_Is_Based_U Inequality IJJudgements_On_U t i l i t i e s IlrirIL-Thg,. Method_yses_Probabilit The subject i s asked to compare between constructed a l t e r n a t i v e s which involve probability. The problems of confounding u t i l i t y with probability were discussed in 23 Chapter 1. This confounding may not only confuse the subject but may also confuse the experimenter i n his analysis. Methods using p r o b a b i l i t i e s may present d i f f i c u l t i e s to subjects who do not adequately understand the pr o b a b i l i t y concept. For these subjects, Gustafson e t _ a l (1972) suggest the use of two t r a i n i n g devices, the probability wheel and the p r o b a b i l i t y bar, which are both routinely used at the Stanford Research I n s t i t u t e . Ilz2r21_The_Method_Does_Not_D^ (1—2—lyl) The Methpd_Is Based_Upon Indifference_Judgements JJ~2~lz2}.„The Method_Is Based Dpon P£g|grence_Judgements In general, preference judgements are less demanding than indifference judgements. Preference judgements do not generally require a precise perception of subjective magnitude while indifference judgements do. Indifference judgements assume the existence of i n f i n i t e s e n s i t i v i t y of the subject, i . e . that the discrimination band (between preference of A over B and of B over A) i s i n f i n i t e s i m a l l y narrow. However t h i s i s not the case and psychophysicists were quite aware of t h i s gradation of discrimination appearing i n t h e i r empirical r e s u l t s (see Torgerson, 1958 for example). Mosteller and Nogee (1951) suggest that the width of the discrimination band in their u t i l i t y experiment i s a c h a r a c t e r i s t i c that distinguishes groups of people. (1-2-2-1) The Utility.Curves Of_B Factors Involved Are Estimated iJ-2-2-21_The_Utili^_Curv lnvolved_Is_Estimated 24 For the single trade-off and single transformation methods, the u t i l i t y curve of one factor must be known beforehand. U t i l i t y Is Based Opon A Lingar_Mgdel A type of additive u t i l i t y i s assumed. j2-172) The Algebraic Expression Thg_Multidimensional D t i l i t y Is Based Dpojn A Non-Linear, Model The second order mixed p a r t i a l derivatives do not vanish. The model allows for inclusion of in t e r a c t i o n terms. ^2-1^2rll_lil§_Coef f i c i e n t A rg^Pgtg£mined_By Subjectiye Weighting The c o e f f i c i e n t s are estimated by the subject according to his evaluation of the r e l a t i v e importance of each factor i n determining o v e r a l l u t i l i t y . (2-1-1-2} The C o e f f i c i e n t s Are Set_Egual_Tp_1 The factors are given equal weight; i . e . i t i s assumed a p r i o r i that the factors are of equal importance i n determining o v e r a l l u t i l i t y . 12-2-1), The Multidimensional U t i l i t y _ I s Represented Geometrically The u t i l i t y i s represented i n terms of a surface or in terms of r e l a t i v e "distance" between points in a multidimensional space. Even though n-dimensional space for n>2 i s d i f f i c u l t to represent on paper, the interpretation 25 lends i t s e l f more readi l y i n terms of geometric rather than algebraic analogies. .12-2-2) _The_Multidimensional 0 t i l i t y m I S , Represented The o v e r a l l u t i l i t y i s described by an algebraic eguation involving component u t i l i t i e s . Certain assumptions concerning aggregation of component u t i l i t i e s are needed. j2z2~ j\zl]._lhS-^^Liy. ^^%9l^-9 ^ -h-U^ili^Y. Index Does Not_Assume M d i t i v i t y I t i s often u n r e a l i s t i c to assume a d d i t i v i t y . J2-2-1-2j_The Derivation Of A D t i l i t y Index Assumes £§§lil-¥4£r.9!-.c9mPPnent U t i l i t i e s The a d d i t i v i t y assumption asserts that the multidimensional u t i l i t y equals the sum of i t s component u t i l i t i e s ( U <X1,. . . ,Xn) =U 1 (X1) +.. . +Un (Xn) ). This se p a r a b i l i t y implies that the various components of the multidimensional a l t e r n a t i v e contribute independently to i t s o v e r a l l u t i l i t y . The additive u t i l i t y model dominates the l i t e r a t u r e on r i s k l e s s choice because i t i s a simple model. Usinq an a d d i t i v i t y assumption, complex choices can be judqed by judging simpler ones and analyzed by analyzing simpler ones. See Fishburn (1965a, 1965b, 1966) for further information regarding a d d i t i v i t y . 26 Sec t ion_I II_-_D iscussion_jgf _me thods 1 0 Each of the methods which appear i n the decision tree w i l l be discussed here. Ranking.Method The actual application of the method requires the subject to either d i r e c t l y rank his preference among a l i s t of factors or to indicate his preference between paired comparisons. Eckenrode (1965) used various modifications of the ranking method as part of a study for comparison of u t i l i t y assessment methods. In addition to the ranking method, the p a r t i a l paired comparisons I, the p a r t i a l paired comparisons I I , and the complete paired comparisons method (the terminology i s that used by Eckenrode) were used. In the p a r t i a l paired comparisons I method, the factors are represented on the rows and columns of an upper triangular matrix. The subject i s asked to indicate i n each matrix entry, the more preferred of the pair of factors which are the coordinates of the matrix entry. Buel (1960) used t h i s format for paired comparisons. In t h i s format, each factor i s paired only once with every other factor. In the p a r t i a l paired comparisons II method, the subject i s presented with *° The standard gamble method for derivation of von Neumann-Morgenstern u t i l i t y and the MacCrimmon-Toda method for derivation of indifference curves w i l l be discussed i n Chapter 3. 27 a l i s t of factor pairs. He i s then asked to c i r c l e the more preferred factor of each pair. Again each factor i s paired only once with every other factor. The complete paired comparisons method has the same format as the p a r t i a l paired comparisons II method. However, each factor pair appears twice, once i n the order A-B and once i n the order B-A. In presenting a l i s t of paired comparisons, i t seems common sense to arrange the pairs i n such a fashion so that pairs involving a p a r t i c u l a r factor should be as far apart as possible and that no p a r t i c u l a r factor should appear preponderantly in one position. P h i l l i p s (1964) has generated tables of orders for stimulus pairs based upon n s t i m u l i (n-3,...,15). His table of paired comparisons i s based upon the works of Ross (1934) and Wherry (1938) and yi e l d s (1) the maximum possible spacing between pairs involving the same stimulus, and which ensure (2) that every stimulus appears an equal number of times i n the f i r s t and second positions but there i s to be the minimum possible number of pairs which have the same stimulus occurring either f i r s t i n a pair or second i n a pair, twice running (Wherry, 1938 described t h i s undesirable feature as "space e r r o r " ) . A problem which may be encountered in the paired comparison method i s the v i o l a t i o n of the t r a n s i t i v i t y axiom. If the subject has firm b e l i e f in th i s axiom, he w i l l consider the inconsistency as a s l i p i n judgement and w i l l accordingly r e c t i f y the rank. However, i f the subject has 28 doubts about the v a l i d i t y of the axiom and refuses to r e c t i f y the inconsistency, the evaluated preferences may have limited value. In a study cited by Goode and Hatt (1952), 150 adults were asked to pairwise compare a l i s t of f i v e occupations. Out of t h i s test sample, less than 9 percent expressed i n t r a n s i t i v e comparisons. Suzuki (1957) obtained preference rankings of naval equipment models from naval o f f i c e r s . Beach (1972) obtained preference rankings of college courses from subjects. Charact e r i s t i c s The ranking method i s very simple to apply and demands very l i t t l e from the subject. The method of paired comparisons may be advantageous over d i r e c t ranking a l i s t i n reducing the number of factors per judgement. With dire c t ranking of a l i s t of a l l factors, the large number of factors per judgement may be a s t r a i n upon the cognitive a b i l i t i e s of the subject. With a large number of factors, the subject may be unable to perceive adequately a l l the factors involved. However, for the method of paired comparisons, the process of evaluation may be too laborious and time-consuming for the impatient subject (using P h i l l i p s 1 table, 15 paired comparisons are needed for 6 fa c t o r s , 21 for 7 factors, 28 f o r 8 factors, and so on). The r e s u l t i n g scale i s ordinal. Despite the weakness of the r e s u l t i n g scale, the ranking method i s useful when 29 augmented with other methods which y i e l d a measurement scale that i s of higher order than ordinal- In f a c t , the ranking procedure i s inherent, i f not e x p l i c i t l y apparent, in other u t i l i t y measurement methods. Magnitude,Estimation Method The subject indicates his assessment of the r e l a t i v e worth of a factor by assigning i t s u t i l i t y to a position on a l i n e a r subjective scale. The subjective scale i s normally anchored at two points. The usual procedure for anchoring i s to choose the l e a s t and the most preferred factors to be at the extreme ends of the scale. The unit and the o r i g i n for the subjective scale are a r b i t r a r y . Torgerson (1958) employs a two-way c l a s s i f i c a t i o n of d i f f e r e n t empirical procedures used i n obtaining measurements by the d i r e c t magnitude estimation technigue. Four procedures r e s u l t from pairwise combinations of single stimulus or multiple s t i m u l i with limited categories or li m i t e d categories. In a t y p i c a l single stimulus presentation, several factors are presented to the subject one at a time i n random order. The subject estimates the u t i l i t y of a factor after the presentation of each factor. In the multiple s t i m u l i presentation, a l l the factors are available and each factor i s allowed to influence d i r e c t l y the judgement of each other factor. In the case of single stimulus, i t i s l i k e l y that the subject w i l l change his reference points (which he bases 30 his judgements upon) as he progresses through the series of f a c t o r s . However, in the case of multiple stimuli the subject i s instructed to rearrange the u t i l i t i e s so that there i s less p o s s i b i l i t y that d i f f e r e n t reference points w i l l be used for d i f f e r e n t factors. However, there are the usual advantages of the single stimulus presentation over the multiple s t i m u l i presentation as was discussed in regards to the ranking method. Limited-category scales provide subjective scales of discrete categories while unlimited-category scales provide continuous scales. Limited-category scales with odd number of categories allow the p o s s i b i l i t y of indicating neutral preference while even number of categories do not. At selected points on either scale, annotations may be used to help the subject to quantify his subjective feelings (eg. very desirable, desirable, e t c . ) . However this leaves open the danger of misinterpretation of the annotations. In psychophysics experiments, Stevens (1959) has shown that subjects apparently do not judge equal i n t e r v a l categories of limited category scales to be equal; the r e s u l t s indicated that categories at one end of the scale tended to include a greater i n t e r v a l of subjective magnitude than at the other end. Steven's explanation i s that the subject's s e n s i t i v i t y i s not uniform over the subjective scale. A given difference may seem less impressive at the other end. If the asymmetry in the subject's s e n s i t i v i t y i s too s i g n i f i c a n t , i t i s incorrect to assume an underlying l i n e a r subjective scale. This psychophysical phenomenon suggests 31 that similar nonlinearity may e x i s t i n u t i l i t y scales. Using the magnitude estimation method, the u t i l i t i e s of the following e n t i t i e s have been empirically assessed: job a t t r i b u t e s (Vroom, 1971), hospital ward conditions (Huber,Sahney, and Ford,1969), college courses (Beach, 1972) . C h a r a c t e r i s t i c s The method demands greater s e n s i t i v i t y from the subject than the ranking method. In order to help the subject to quantitatively rate the factors, the subject may i n i t i a l l y be asked to rank them. The assumed l i n e a r i t y of the subjective scale may not be v a l i d . In t h i s method, i t i s assumed that the subject i s capable of d i r e c t l y perceiving the r a t i o between two subjective magnitudes. Having assigned the u t i l i t y of one factor l e v e l a r b i t r a r i l y to specify the unit, the u t i l i t i e s of the remaining factor l e v e l s may be determined by the empirically derived r a t i o s . Hetfessel (1947) proposed the constant-sum method which was an alternative form of expressing f r a c t i o n r a t i o s . The subject i s instructed to divide, say 100 units, between two selected factors according to t h e i r r e l a t i v e u t i l i t i e s . For example, the 32 subject may decide that one factor should be assigned 30 out of 100 units more than the other. In th i s case, he would d i s t r i b u t e 35-65 and the corresponding r a t i o would be 35/65. If the subject i s not capable of reporting r a t i o s i n general, he may s t i l l be capable of making v a l i d bisection ( i . e . half or double r a t i o s or Metfessel's 50-50) judgements. In the case of bisection judgements, the requirements placed upon the subject are considerably less than i n the case of general r a t i o judgements. The eguisection method i s a method employing the repeated application of bisection judgements. The task of the subject i s to select n-1 of the remaining factors (after the least and most preferred factors have been fixed to the endpoints) so that the n-1 associated u t i l i t i e s divide the u t i l i t y i n t e r v a l between the two endpoints into n equal i n t e r v a l s . For the preference r a t i o method, Torgerson (1958) further c l a s s i f i e s the method into direct-estimate and prescribed-ratio methods. The es s e n t i a l difference i s that in the direct-estimate method, the subject i s presented with two factor l e v e l s for which he i s to provide a subjective estimate of the r a t i o of the i r u t i l i t i e s while i n the prescribed-ratio method, the subject i s asked to report the factor l e v e l f o r which the r a t i o of i t s u t i l i t y to that of the u t i l i t y of the standard i s equal to a r a t i o prescribed by the experimenter. 33 In the prescribed-ratio method, the subject must be able to choose any factor l e v e l between the two endpoints; therefore the factor levels must be continuous. U t i l i t y values may be obtained only for cer t a i n factor l e v e l s . For the prescribed r a t i o of 1/m, only those factors whose u t i l i t i e s equal k«[ m to the ath power] for a = 1,2,3,... (k i s an a r b i t r a r i l y assigned unit of u t i l i t y ) may be assessed. On the other hand, the factor l e v e l s need not be continuous for evaluation by the direct-estimate method. In addition, a l l factor l e v e l s of inte r e s t to the assessor may be evaluated. With n factor l e v e l s , n-1 r a t i o s ( a l l compared to the standard) are obtained. However, the method may also be extended to pairwise comparison; Comrey (1950) proposed that a l l the factor l e v e l s (to be assessed) serve in turn as standards, thus giving n(n-1)/2 r a t i o s from which a u t i l i t y scale can be derived (see Comrey, 1950 and Torgerson, 1958 for c a l c u l a t i o n formulas). Comrey's procedure consists of reporting comparative judgements i n the same manner as advocated by Metfessel (1947). Klahr (1969) used Comrey»s procedure in obtaining u t i l i t y estimates of college admission a t t r i b u t e s . Galanter (1962) used preference r a t i o s to obtain a u t i l i t y scale for money. Beach (1972) had subjects make preference r a t i o judgements i n order to derive u t i l i t y scales for college courses. Dudek and Baker (1956) obtained u t i l i t y scales for neckties through preference r a t i o judgements of subjects. 34 C h a r a c t e r i s t i c s This method demands greater s e n s i t i v i t y from the subject than the magnitude estimation method. Some subjects may f e e l uncomfortable about conceptualizing in terms of r a t i o s . Naive subjects may have to be trained in the concept of r a t i o s . Reported judgements may be severely limited to cert a i n e a s i l y interpretable or commonly used r a t i o s . The procedure for deriving a u t i l i t y index from r a t i o judgements i s very s e n s i t i v e to errors in judgement. Errors w i l l be multiplied by estimated r a t i o s . Ordered Metric I^Method The term "ordered metric" i s attributed to Coombs (1950). In deriving Coomb's d i r e c t ordered metric, the factor l e v e l s are f i r s t ranked and then the increments between adjacent u t i l i t i e s are ranked. For example, i f the subject can not only state that his preferences are A to B and B to C but also state that his preferences of A to B i s greater than his preference of B to C, then his u t i l i t y index i s represented by a t r i p l e t of numbers which s a t i s f y : U (A) >0 (B) >0 (C) and U (A)-U (B) >0 (B)-U (C) . Suppes and Winet (1955) suggest that from an alte r n a t i v e construction of choices and an assumption of a d d i t i v i t y , an ordered metric along one dimension may also be obtained. For example, suppose A>B>C. The subject i s then hypothetically given A and B. Next he i s reguired to choose 35 between trading B for C, or A for D. If he trades B for C, U (A and C) >D (D and B) U1 (A) +01 (C) >U1 (D) + U1 (B) by a d d i t i v i t y U1 (A)-U1 (B) >U1 (D) -01 (C) Siegel (1956) extends Coomb's derivation of a di r e c t ordered metric by devising a procedure for deriving a higher ordered metric scale i n which a l l possible combinations of contiguous increments between adjacent u t i l i t i e s are ordered. Siegel proposes his maximin rule which provides an e f f i c i e n t means of e l i c i t i n g judgements. His maximin rule i s a rule which maximizes the amount of information (necessary to construct the scale) from a minimum number of reported judgements. The rule dictates the order in which the judgements should be examined i n the derivation of the scale (see Hurst and Siegel, 1956). Becker and Siegel (1962) obtained a u t i l i t y scale of college grades from a higher ordered metric scale. C h a r a c t e r i s t i c s The subject may f i n d i t too d i f f i c u l t to compare between differences i n preferences. These choices are not commonly made and may seem a r t i f i c i a l to the subject. The ordered metric places bounds on the location of each factor on the u t i l i t y scale. The bounds become more r e s t r i c t i v e as more ordered metric relationships are obtained. 36 Ol^®£e3_Metric_II_Kethod Siegel (1956) has reported a method of obtaining a higher ordered metric from interrogation of a subject about his preference ordering of gambles. The subject i s f i r s t required to rank the factors. The r e s u l t i n g ordinal scale gives us information about the preferences between factors but nothing about the r e l a t i v e magnitudes of factors. However, by further requiring the subject to rank his preferences between constructed 50-50 gambles, one may derive an ordered metric (or higher ordered metric) by use of the von Neumann-Morgenstern theorem on maximization of expected u t i l i t y . For example**, (A,D;1/2)> (B,C;1/2) => 1/2 (U (A) *U (D) ) >1/2 (U (B)+U (C) ) => 0 (A) -U (B) >U (C) -0 (D) . The i n i t i a l ranking of certain outcomes automatically gives us information about the ordering of some gambles (called the orderable pairs) but i t i s the ordinal relations between non-orderable pairs of gambles which contain the information necessary to change an ordinal scale to a higher-ordered scale. However, not a l l the non-orderable r e l a t i o n s must be found in order to obtain the information necessary for higher-ordered metric scaling (maximin r u l e ) . Farrer (1964) found the predictive a b i l i t y of higher ordered metric scales (constructed for cigarettes) to be 1 1 (A,B;p) represents a wager whose outcome i s A with probability p or B with probability 1-p. 37 quite stable over a one month period. Using the von Neumann-Morgenstern theorem, ordered metric scales of u t i l i t y have been obtained for money (Coombs and Komorita, 1958) and various appliances (Coombs and Beardslee, 1954) while higher ordered metric scales have been obtained for books (Siegel, 1956) and cigarettes (Hurst and Siegel, 1956; Farrer, 1964). Characte r i s t i c s The use of gambles instead of certain outcomes in constructing a higher ordered metric presents an additional cue to be considered in the subject's judgement, namely that of p r o b a b i l i t y . However, most subjects should have no problem i n conceptualizing p r o b a b i l i t i e s of 1/2 (the analogy of heads and t a i l s i s usually used). The ordered metric places bounds on the location of each factor on the u t i l i t y scale. The bounds become more r e s t r i c t i v e as more ordinal relationships between gambles are found (i.e. the more ordered metric relationships that are obtained) . 38 §H c!ggs§iyg,.. c 95!E§£l§Q n§-Bgthod (Churchman-Ackoff Method^ In t h i s method, i t i s assumed that the factors involved are u t i l i t y independent and unidimensional. The method consists of successively comparing a factor (or level) with a group of factors (or factor levels) . At each comparison, the subject i s required to assign tentative numbers to u t i l i t i e s of factors such that they are compatible with the currently expressed preferences of the subject. For each of the remaining comparisons, the previously assigned u t i l i t i e s are refined according to the currently expressed preference. Hopefully, the assigned u t i l i t i e s w i l l converge during the l a t t e r comparisons. Churchman and Ackoff (1953) give one possible way of using the successive comparisons method. B r i e f l y stated, (1) Rank the factors. Let X1>X2>X3>...>Xn. (2) Assign U(X1) = 1.00 and numbers to U (X2),. . . ,0 (Xn) according to th e i r r e l a t i v e preferences. (3) Compare X1 with X2+X3+...Xn. (3i) If X1 >,= ,< X2+X2+. . .+Xn, adjust U (X2) , 0 (X 3) , . . . , 0 (Xn) so that D(X1)=1.00 >, = ,< U (X2) +0 (X3) +. . . +U (Xn) . (4) Compare X1 with X2+X3 + . . .+Xn-1. Adjust U (X2),...,U (Xn-1) according to expressed preference. (5) Compare X1 with X2+X3 + ...+Xi where i=n-1(-1)3. Adjust U (X2) ,U (X3) ,... ,U (Xi) accordingly. (6) Continue u n t i l Xn-2 i s compared with Xn-1+Xn. The f i n a l refined values of U (X1) , 0 (X2), . . . ,U (Xn) form the 1 2 The "+" i s a l o g i c a l "and" and not an algebraic "plus". 39 u t i l i t y index for the f a c t o r s . Churchman and Ackoff (1953) c i t e various actual applications of t h i s method in quality control and corporate planning. C h a r a c t e r i s t i c s The method consists of successive refinement of estimates. The assigned u t i l i t i e s may not converge i n which case the assignment of u t i l i t i e s i s too e r r a t i c to be u s e f u l . Furthermore, the assumption of u t i l i t y independence among the factor l e v e l s involved may be too a r t i f i c i a l to s a t i s f y any r e a l s i t u a t i o n . As an a l t e r n a t i v e (to the one previously cited) v a r i a t i o n of the method, the comparisons for judgement may be randomly chosen as suggested by Churchman and Ackoff (1953). Single Trade-off An indifference curve i s estimated for two factors by f i n d i n g points X and Y such that (X, Y ) ^ (Xr,Yr) where Xr and Yr are the reference points. From a knowledge of one of the two u t i l i t y curves, one can estimate the other u t i l i t y curve. Suppose that we have an estimate of U1 ( X ) . Then, from the a d d i t i v i t y assumption, U (X,Y)=U1 (X)+U2 (Y)=U1 (Xr)+U2 (Yr)=c say, for (X,Y) on the 40 in d i f f e r e n c e curve. Therefore U2 (Y) =c-U1 (X) where c i s an arbit r a r y constant, c may be assigned a r b i t r a r i l y while preserving i n t e r v a l properties. C h a r a c t e r i s t i c s A u t i l i t y curve of one of the two factors oust be estimated previously by another method. From t h i s given u t i l i t y curve, the other factor's u t i l i t y curve may be estimated from the trade-off or indifference curve. The a d d i t i v i t y assumption i s needed. Jtouble_Trade-off Two in d i f f e r e n c e curves, r e l a t i v e l y close to each other, are estimated by finding points X and Y such that (X,Y)^ (Xri, Yr1) and (X,Y)~-(Xr2, Yr2) for the reference points Xrl,Yr1,Xr2,Yr2. Figure 2 i l l u s t r a t e s the two curves. A " f l i g h t of s t a i r s " i s then drawn between the two curves by a connected s e r i e s of horizontal and v e r t i c a l l i n e segments Indifference curve 1: (1) U (Xl,Y1)=d say (2) 0(X2,Y2)=d (3) 0(X3,Y3)=d Indifference curve 2: (4) 0 (X2, Y1)=e say (5) 0(X3,Y2)=e (6) U (X4,Y3) =e 41 Figure 2 - Figure f o r Double Trade-off Method Subtracting (4) from (1) : 0 (X1,Y1)-U (X2,Yl)=d-e=c say Then 01 (X1)-U1 (X2)=c by a d d i t i v i t y S i m i l a r l y : 01 (X2)-U1 (X3) =c and 01 (X3)-U1 (X4)=c Subtracting (5) from (1): 0 (X1,Y1)-D (X3,Y2) =d-e=c Then 02 (Y1)-02 (Y2) =c-(01 (X1)-U1 (X3)) by a d d i t i v i t y =c- (2c) =-c Si m i l a r l y : .02 (Y2)-U2 (Y3)=-c Thus, the successive points on the indifference curves touched by the s t a i r s define egual increments of u t i l i t y for 42 each factor, By assigning an a r b i t r a r y number to the u t i l i t y of a certain l e v e l of one factor and also assigning an arb i t r a r y number to the increment of u t i l i t y (c) for that factor (these a r b i t r a r y assignments w i l l preserve i n t e r v a l properties) the u t i l i t y curve for that factor may be estimated. The same procedure i s followed for obtaining the other factor's u t i l i t y curve. Ch a r a c t e r i s t i c s From two indifference curves, certain l e v e l s of both factors which define constant adjacent increments of u t i l i t y may be estimated. From these i d e n t i f i e d factor l e v e l s , piecewise l i n e a r representation of each factor's u t i l i t y curve may be used as an approximation to the u t i l i t y curve. The closer together the two indifference curves are, the more the number of points on each u t i l i t y curve that may be i d e n t i f i e d . However, the closer together the two indifference curves which the subject t r i e s to estimate, the greater the chance of inconsistency ( i . e . crossing of two indifference curves) occurring. For the case of indifference curves that are double valued (this may occur for u t i l i t y functions that are not monotonic with respect to the factor level) as in Figure 3 , the procedure of sketching a f l i g h t of s t a i r s may be t r i c k y or inappropriate. x -> Figure 3 - Double Valued Indifference Curves 44 Single Transformation A transformation curve i s constructed by estimating points (X,Y) for which (X,Yr)~(Xr,Y) where Xr and Yr are reference points. One possible procedure i s to estimate Y for successive a r b i t r a r i l y fixed X's, and then to estimate,X for successive a r b i t r a r i l y fixed l ' s . From a given u t i l i t y curve for one of the two factors, one can estimate the other factor's u t i l i t y curve. Suppose that we have an estimate of 01 (X). Then,.from the a d d i t i v i t y assumption, U (X,Yr)=U1 (X)+02 (Yr> 0 (X,Yr)=U (Xr,Y)=U1 (Xr) +U2 (Y) for points (X,Y) on the transformation curve. Then, U2 (Y) =U1 (X) + (U2 (Yr)-01 (Xr) )=01 (X)-c where c i s an a r b i t r a r y constant, c may be assigned a r b i t r a r i l y while preserving i n t e r v a l properties. C h a r a c t e r i s t i c s A u t i l i t y curve of one of the two factors must be estimated previously by another method. From t h i s given u t i l i t y curve, the other factor's u t i l i t y curve may be estimated from the transformation curve. The a d d i t i v i t y assumption i s needed. The transformation curve i s not as r e a d i l y interpretable as the indifference curve. 45 Double Transformation Two transformation curves, r e l a t i v e l y close to each other, are estimated by finding X and Y such that ( X , Y r 1 ) ^ (Xr1 ,Y) and (X,Yr 1) /s>(Xr2,Y) for the reference points XrT, Yr1, Xr2. Figure U i l l u s t r a t e s the two curves. A XI X2 X3 . Figure 4 - Figure for Double Transformation Method " f l i g h t of s t a i r s " i s then drawn between the two curves by a connected s e r i e s of horizontal and v e r t i c a l l i n e segments. For the (X,Yr1)*^(Xr1,Y) transformation curve: (1) D (X1,Yr1)=0 (Xr1,Y1) 46 (2) 0 (X2,Yr1) =0 (Xr1,Y2) (3) 0 (X3,Yr1)=0 (Xr1,Y3) For the (X,Yr 1) /-v/ (Xr2,Y) transformation curve: (4) D (X2,Yr1)=0 (Xr2,Y1) (5) 0 (X3,Yr 1) =U (Xr2,Y2) (6) U (X4,Yr1)=U (Xr2,Y3) By a d d i t i v i t y , (1) 0 (Xl,Yr1)=U1 (X1) +02 (Yr1)=U (Xr1 ,Yl)=0l (Xr1) +U2 (Y1) or U1 <X1) +U2 (Yr1)=01 (Xr1) +02 (Y1) By a d d i t i v i t y , (4) 0 (X2,Yr1)=0l (X2)+U2 (Yr1)=0 (Xr2,Yl)=Ol (Xr2)+02 ( Y1) or U1(X2)+02(Yr1)=U1 (Xr2)+ U2 (Y1) Subtracting (4) from (1) : 01 (X1) -01 (X2) =01 (Xr1)-01 (Xr2)=c say S i m i l a r l y : 01 (X2)-01 (X3) =c and U1 (X3)-U1 (X4) =c By a d d i t i v i t y , (5) 01 (X3)+02 (Yr 1) =U 1 (Xr2)+02 (Y2) Subtracting (5) from (1) : U1 (X1) -01 (X3) =U1 (Xr1)-01 (Xr2) +02 (Y1) -02 (Y2) 2c=c+02 (Y1)-02 (Y2) 02 (Y1)-D2 (Y2)=c S i m i l a r l y : 02 (Y2)-02 (Y3) =c Thus, the successive points on the transformation curves touched by the s t a i r s define constant adjacent increments of u t i l i t y f o r each factor. By assigning an ar b i t r a r y number to the u t i l i t y of a certain l e v e l of one facto r and also assigning an arb i t r a r y number to the increment of u t i l i t y for that factor (these a r b i t r a r y assignments w i l l preserve i n t e r v a l properties), the u t i l i t y 47 curve for that factor may be estimated. The same procedure i s followed to obtain the other factor's u t i l i t y curve. Charact e r i s t i c s From two transformation curves, certain l e v e l s of both factors which define constant adjacent increments of u t i l i t y may be estimated. From these i d e n t i f i e d factor l e v e l s , a piecewise l i n e a r representation of each factor's u t i l i t y curve may be used to approximate the u t i l i t y curve. For the case of transformation curves that are double valued as in Figure 5 , the procedure of sketching a f l i g h t of s t a i r s may be t r i c k y or inappropriate. The a d d i t i v i t y assumption i s needed. The transformation curve i s not as readi l y interpretable as the indifference curve. £§ncellation_Method The method consists of deriving two indifference curves from which a t h i r d one may be estimated. The method assumes a d d i t i v i t y of component u t i l i t i e s and i s i l l u s t r a t e d in Figure 6 . Two " f l i g h t of s t a i r s " are sketched between two derived indifference curves. From indifference curve 1: U (X1,Y2)=0 (X2,Y3) U1 (X1) +U2 (Y2)=U1 (X2) +U2 <Y3) by a d d i t i v i t y (1) <49 Y X -»-Figure 5 - Double Valued Transformation Curves From indifference curve 2: U (X2f Y1)=U (X3,Y2) 01 (X2)+02 (Y1)=01 (X3)+02 (Y2) by a d d i t i v i t y (2) Adding together (1) and (2): 01 (X1) +U2 (Y2) +U1 (X2) +02 (Y1) =U1 (X2) +U2 (Y3) +01 (X3) +U2 (Y2) The deletion of U1 (X2) and U2(Y2) from both sides of the equality sign (this i s the double cancellation property) r e s u l t s i n : U1 (X1)+02(Y1)=U1(X3)+02 (Y3) U (X1,Y1)=U (X3,Y3) (X1,Y1W (X3,Y3) 49 Figure 6 - Figure for Cancellation Method I t was shown from the above arguments that i f (X1,Y2) and (X2rY3) l i e on the same indifference curve and (X2,Y1) and (X3,Y2) l i e on the same indifference curve, then (X1,Y1) and (X3,Y3J l i e on the same indifference curve. This r e l a t i o n s h i p i s formally given as the Thomsen condition (see Krantz e t _ a l , 1971, page 250). By p a r a l l e l arguments, i t can be shown that a t r i p l e c a n c e l l a t i o n property e x i s t s (Reidmeister condition) for which three pairs of given related points on three in d i f f e r e n c e curves generate additional indifference points. 50 Higher order cancellation properties are also possible but involve complicated formulas for derivation of additional points. Chara c t e r i s t i c s The closer together the two derived indifference curves are, the more number of points on the t h i r d curve that may be estimated. The number of generated indifference points may be too few for adequately c u r v e - f i t t i n g an additional indifference curve. Two-stagg ..Rating, Model A linear model for o v e r a l l u t i l i t y i s developed from a two-stage process of c o l l e c t i n g data. The l i n e a r model i s : 0 (X1,... ,Xn)=W1«u"1 (X1) +...+Wi»Ui (Xi) + ...wn«un (Xn) (1) The subject i s required to estimate U i ( X i ) . (2) The subject i s required to assiqn numbers to Wi accordinq to his evaluation of each factor»s r e l a t i v e importance i n determining the o v e r a l l u t i l i t y . The procedure usually used for assigning numbers to Hi i s the magnitude estimation method. Studies (Sarbin, 1942; Smedslund, 1955) have shown that there are noticeable differences between subjective weights Hi and those determined "optimally" by f i t t i n g the proposed 51 l i n e a r regression model to the data. Shepard (1962) offers a plausible explanation by suggesting that subjective weighting involves comparing factors which occupy d i f f e r e n t dimensions. His analogy concerning the ease of comparing two s t i l l - l i f e paintings r e l a t i v e to the d i f f i c u l t y of comparing a painting with an abstract sculpture i s most appropriate. In Hoepfl and Huberts study (1970), evaluation of teaching a b i l i t y of hypothetical professors each described by six factors ( i . e . n=6) were obtained. Characteristics A multidimensional u t i l i t y i s decomposed into i t s component u t i l i t i e s . A benefit of t h i s decomposition i s that deriving u t i l i t i e s along one dimension demands l e s s from the subject than along several dimensions simultaneously. Validation of the model may be accomplished by c o l l e c t i n g subjective estimates of the o v e r a l l u t i l i t y and comparing these estimates with those predicted by the two-stage rating model (degree of agreement i s indicated by the c o r r e l a t i o n c o e f f i c i e n t ) . This a a se l f - e x p l i c a t e d model (a term used by Hoepfl and Huber, 1970) because the parameters Hi and Ui(Xi) are e x p l i c i t l y estimated by the decision-maker. If n i s large, the number of evaluations needed from the subject may be too large for p r a c t i c a l application of the model. 52 UBweighted_Rating_Model A l i n e a r model f o r o v e r a l l u t i l i t y i s used. The model eguation i s i d e n t i c a l to the two-stage ra t i n g model equation except that the subjective weiqhts Wi are set equal to 1. The model equation i s : U(X1,...Xn)=U1 (X1) + ...+Un(Xn) The subject i s required to estimate Ui (Xi). The method aviods the d i f f i c u l t y of assigning weights to the factors. C h a r a c t e r i s t i c s A multidimensional u t i l i t y i s decomposed into i t s component u t i l i t i e s . The weights for the factors are assumed a p r i o r i to be equal i n magnitude, i . e . i t i s asssumed that the component factors have egual importance in determining o v e r a l l u t i l i t y . Validation of the model may be accomplished by c o l l e c t i n g subjective estimates of the o v e r a l l u t i l i t y and comparing these estimates with those predicted by the two-stage rating model (degree of agreement may be indicated by the co r r e l a t i o n c o e f f i c i e n t ) . If n i s large, the number of evaluations needed from the subject may be too large for p r a c t i c a l application of the method. 53 Linear^ Model The method consists of c o l l e c t i n g multidimensional u t i l i t y judgements, deriving model parameters from these judgements, and then predicting future multidimensional u t i l i t y judgements from the model. The model used i s a lin e a r regression model with dummy variables (see Suits, 1957 regarding dummy variables). U (X1,.,.,Xn)=Uo+U11»X11+...+Uij«Xij+...+Unl«Xnl where Xij=0 i f the multidimensional attibute i n question does not possess the i t h factor at the j t h l e v e l , and Xij=1 otherwise. U(X1,...,Xn) are estimated by the subject for various l e v e l s of the component factors. The usual procedure i s to require the subject to rate multidimensional alternatives on a magnitude estimation scale. A l i n e a r regression i s applied to the model equation to obtain "optimal" (in a least squares sense) estimates of Oo and U i j . Future predictions may be made from t h i s model with the derived parameters. Characteristics The subject i s required to make u t i l i t y estimates on multidimensional alternatives. The linear model proposes a decomposition of multidimensional u t i l i t y into component parameters which are estimated by regression techniques. The subject may be reguired to make many 5U multidimensional judgements before the regression r e s u l t s can be considered s t a t i s t i c a l l y s i g n i f i c a n t . The model d i f f e r s from the two-stage rating model and the unweighted r a t i n g model in that the application of regression reveals which components i n the model do not make a s i g n i f i c a n t contribution to the ov e r a l l u t i l i t y . In the case of multiple regression, the F s t a t i s t i c associated with every estimated parameter gives an indication as to whether one should r e j e c t the n u l l hypothesis that Uij=0 at the s p e c i f i e d s i g n i f i c a n c e l e v e l . In the case of stepwise regression, the f i n a l equation of the stepwise i t e r a t i o n s w i l l contain only the s t a t i s t i c a l l y s i g n i f i c a n t component parameters. V a l i d a t i o n of the model i s indicated by B 2, the multiple c o r r e l a t i o n c o e f f i c i e n t . M u l t i p l i c a t i v e Model A non—linear configural model f o r ov e r a l l u t i l i t y i s used. The model used i s : v u o f r W""1 i=l ' The model equation may be transformed to: In(0(X1»...,Xn) =ln0o+A1-lnU1(X1)•...•&n»lnDn(Xn) Huber, Sahney, and Ford (1969) have suggested that such a model might more nearly represent the form of the subject's » 3 lnx=natural logarithm of x 55 actual u t i l i t y model than does the form of the previously discussed additive models especially i n those cases in which some factors e s s e n t i a l l y act as screening factors. The subject i s required to estimate Ui (Xi) and U(X1,...;xn) for selected component l e v e l s . The usual procedure i s to require the subject to rate multidimensional alternatives on a magnitude estimation scale. The parameters Uo and Ai are estimated by the application of regression techniques to the transformed model equation. C h a r a c t e r i s t i c s A multidimensional u t i l i t y i s decomposed non-linearly into i t s component u t i l i t i e s . The subject i s required to make multidimensional as well as unidimensional u t i l i t y judgements. The parameters are estimated by regression techniques. The subject may be required to make many multidimensional judgements before the regression r e s u l t s can be considered to be of s t a t i s t i c a l s i g n i f i c a n c e . This model d i f f e r s from the two-stage rating model and the unweighted rating model i n that the application of regression can reveal which components in the model do not make a s i g n i f i c a n t contribution to the ove r a l l u t i l i t y . In the case of multiple l i n e a r regression, the F s t a t i s t i c associated with every estimated parameter gives an ind i c a t i o n as to whether one should reject the n u l l 56 hypothesis that Aij=0 at the sp e c i f i e d significance l e v e l . , In the case of stepwise regression, the f i n a l equation of the stepwise i t e r a t i o n s w i l l contain only the s t a t i s t i c a l l y s i g n i f i c a n t component parameters. Hybrid Model The method consists of c o l l e c t i n g multidimensional and unidimensional judgements, deriving model parameters from these judgements, and then predicting future multidimensional u t i l i t y judgements from the model. The model used i s a lin e a r regression model: D(X 1, . • ,,Xn)=Uo+W1»01 (X1)•...+Wn«0n(Xn) The subject i s required to estimate Ui(Xi) as well as U(X1,...,Xn) for various levels of the component factors. A l i n e a r regression i s applied to the model equation to obtain "optimal" (in a l e a s t squares sense) estimates of Oo and Wi. Future predictions may be made from th i s model with the desired parameters. In Hoepfl and Huber»s study (1970), the hybrid model was used to describe the evaluation of teaching a b i l i t y of hypothetical professors, each described by s ix factors ( i . e . n=6). C h a r a c t e r i s t i c s The subject may be required to make many multidimensional judgements before the regression r e s u l t s 57 can be considered s t a t i s t i c a l l y s i g n i f i c a n t . The model equation d i f f e r s from that of the two-stage rating model equation i n that the co e f i c i e n t s for the hybrid model are determined objectively, i . e . by regression techniques. In the case of multiple regression, the F s t a t i s t i c associated with every estimated c o e f f i c i e n t gives an ind i c a t i o n as to whether one should reject the n u l l hypothesis that Wi=0 at the sp e c i f i e d significance l e v e l . In the case of stepwise regression, the f i n a l equation of the stepwise i t e r a t i o n s w i l l contain only the s t a t i s t i c a l l y s i g n i f i c a n t component parameters. Validation of the model i s indicated by R2, the multiple c o r r e l a t i o n c o e f f i c i e n t . Multidimensional Scaling_Method The subject's preference ordering i s represented in terms of "distance" in a multidimensional space. The subject i s required to make judgements about the s i m i l a r i t i e s of pairs of multidimensional a l t e r n a t i v e s (for n d i s t i n c t a l t e r n a t i v e s , n(n-1)/2 s i m i l a r i t y judgements can be obtained). Messick (1956) suggests an empirical procedure for obtaining s i m i l a r i t y judgements. In addition to s i m i l a r i t y judgements, the subject i s reguired to designate his most preferred alternative (ideal point). The method postulates that from a set of s i m i l a r i t y judgements, a 58 s p a t i a l configuration can be constructed in which the alternatives are arranged such that the inverse rank order of interpoint Euclidean distances i n the space corresponds to the rank order of s i m i l a r i t i e s given i n the input data ( i . e . pairs of more s i m i l a r alternatives are "closer" together). From t h i s postulation, the preference ordering of the a l t e r n a t i v e s i s d i r e c t l y related to the ordering of Euclidean distances in the space from the i d e a l point to each a l t e r n a t i v e , i . e . A i s preferred to B i f the Euclidean distance between A and the i d e a l point i s less than that between B and the i d e a l point. The search for a s p a t i a l representation involves a tradeoff between: (1) maximizing the inverse c o r r e l a t i o n of interpoint distances rank and s i m i l a r i t y measures rank by increasing the s p a t i a l dimension, and (2) achieving a more parismonious representation of the data by decreasing the s p a t i a l dimension. Klahr (1969) has shown that a s p a t i a l configuration constructed from s i m i l a r i t y judgements obtained from college admission o f f i c e r s was accurate in predicting the o f f i c e r s * preferences among college a p l l i c a n t s . C h a r a c t e r i s t i c s If many multidimensional a l t e r n a t i v e s are to be used i n constructing a s p a t i a l configuration, the re s u l t i n g number of s i m i l a r i t y judgements may be too large for p r a c t i c a l a p p l i c a t i o n . 59 A preference ordering i s e a s i l y i n t e r p r e t a b l e i n geometric terms. The l o c a t i o n of the i d e a l point i s c r u c i a l i n a c c u r a t e l y determining the preference o r d e r i n g . The search f o r a s p a t i a l r e p r e s e n t a t i o n r e q u i r e s very long and complicated c a l c u l a t i o n s . The experimenter must have access to an appropriate computer program f o r performing the c a l c u l a t i o n s or be able to w r i t e such a program him s e l f . 60 Chajjter_3_-_Car dinal_U Section I - yon Neumann-Mprggnstern Cardinal U t i l i t y iStroduction The method for deriving a von Neumann-Horgenstern cardinal u t i l i t y index (the method i s sometimes c a l l e d the standard gamble method) consists of presenting hypothetically constructed gambles to the subject from which a cardinal u t i l i t y may be derived from his expressed preferences. The method i s based upon the maximization of expected u t i l i t y which i s the c r u c i a l t h e o r e t i c a l r e s u l t a r i s i n g from the von Neumann-Morgenstern assumptions. However, previous to von Neumann and Morgenstern•s work, t r a d i t i o n a l mathematical treatment of risky situations had proposed the notion of maximization of expected value**. Such a notion does not seem plausible i n l i g h t of the fact that people buy insurance and l o t t e r y t i c k e t s . Premiums are more than the expected monetary gains from an insurance policy; t i c k e t prices are more than the expected monetary wins from l o t t e r y draws. Furthermore,a f a u l t y fundamental assumption inherent in t h i s notion was recognized by Daniel Bernoulli (1738). ** The expected value of a money gamble i s the sum of the products of monetary value of outcomes with t h e i r associated p r o b a b i l i t i e s . 61 One of the implications of expected value maximization i s that two persons w i l l have i d e n t i c a l preference towards the same risky s i t u a t i o n since the expected value of the risk y s i t u a t i o n i s i d e n t i c a l f o r both persons. This implication i s contradictory to our everyday experience. In order to circumvent t h i s impasse and to resolve the insurance paradox, Bernoulli proposed that people act so as to maximize expected u t i l i t y rather than expected value. Although Bernoulli i s credited with being the f i r s t to advocate the notion of expected u t i l i t y maximization, i t was not u n t i l 1944 when von Neumann and Mprgenstern•s book, Theory of Games and Economic Behavior , was published that a formal basis was provided for such a concept. In f a c t , they asserted that " i t can be shown that under the conditions on which the indifference curve analysis i s based very l i t t l e extra e f f o r t i s needed to reach a numerical u t i l i t y " (1944, page 17). While the a b i l i t y of a r a t i o n a l person to order preferences for certain outcomes i s needed for the ordinal theory, the extra e f f o r t required for the cardinal theory i s that he be able to also order probability combinations of outcomes. For example, suppose that a person expresses indifference between the certain outcome of $8.00 (which may be thought of as a gamble which w i l l r e s u l t i n the outcome of $8.00 with pr o b a b i l i t y 1 or any other outcome with probability 0) and a gamble i n which he i s offered a 40-60 chance of winning $10.00 or $0.00. From his expressed indiffe r e n c e , the von Neumann-Morgenstern theory implies that the u t i l i t y of $8.00 and the expected u t i l i t y of the 62 40-60 gamble are i d e n t i c a l , i . e. 0 ($8.00)=0.40»u ($10.00)+0.60-U ($0.00) Assigning u($10.00) = 1 and u($0.00)=0 (two a r b i t r a r y assignments w i l l preserve the properties of the cardinal u t i l i t y scale since i t i s measurable up to a linear transformation; for d e t a i l s see Chapter 1), u ($8. 00) = 0.40 (1) +0.60 (0)=0.40 By varying the p r o b a b i l i t i e s i n the gamble, the u t i l i t y for various monetary levels between $0.00 and $10.00 may be determined. Such a concept as expected u t i l i t y maximization i s c e r t a i n l y appealing since i t i s not only i n t u i t i v e l y plausible but also simple to comprehend as was demonstrated by the previous example. The e s s e n t i a l difference between the von Neumann-Morgenstern presentation of the proposition and the previous suggestions of such a maximization concept i s that their presentation demonstrated that the notion of expected u t i l i t y maximization was l o g i c a l l y equivalent to the acceptance of c e r t a i n basic axioms, which in themselves seemed to be reasonable assumptions about human behavior. The importance of developing the theory axiomatically i s that i f the axioms have empirical v a l i d i t y , the empirical meaning of the theory i s much more s i g n i f i c a n t than i f the theory's r e s u l t s are stated without j u s t i f i c a t i o n . Therefore, the axiomatic system upon which the von Keumann-Morgenstern cardinal u t i l i t y theory rests i s c r u c i a l l y important i n construct v a l i d a t i o n of the propositions which the theory claims to enunciate. In an attempt to gain 63 insight into the basic foundations of the von Neumann-Morgenstern cardinal u t i l i t y theory, the following discussion w i l l give a c r i t i c a l appraisal of each axiom underlying the theory and the re s u l t s which are derived from the axioms. The axioms which w i l l be discussed are: (1) the t r a n s i t i v i t y axiom, (2) the continuity of preferences axiom, (3) the sure-thing axiom, (4) the independence of ordering axiom, and (5) the compound-gamble axiom. The t r a n s i t i v i t y , axiom states that»s : &>B and B>C => A>C. At the l e v e l of introspection, t h i s axiom seems to be a reasonable assumption about human behavior; some experiments have supported t h i s view while the res u l t s of others have claimed otherwise. Marschak (1964) performed a casual experiment on his own graduate students to test for t r a n s i t i v i t y . Pairs of objects were presented i n the following order: (A1,B1) ; (A2,B2) ;... ; (Am,Bm) ; (B1,C1) ; (B2,C2) ;... ; (Bm,Cm) ; (C1,A1); (C2,A2);...; (Cm,Am). The objects considered were those relevant to graduate students: jobs, t r i p s , apartments, medical care, etc. The subject i s considered to be not consistent i f he prefers Ai to B i , and Bi to Ci yet prefers C i to Ai for any i . The r e s u l t s indicated that students s a t i s f i e d the t r a n s i t i v i t y axiom when m was small; eg. m=5. 1 5 A>B i s defined to mean that A i s preferred to B. 64 Michalos (1967) has argued that t r a n s i t i v i t y i s an inaccurate empirical generalization and unacceptable normative p r i n c i p l e although his arguments do not seem convincing. In May's (1954) experiment, the results showed that subjects made i n t r a n s i t i v e preferences. May's explanation was that the subjects' preference orderings , i n f a c t , are i n t r a n s i t i v e . However, Rose (1957) claimed that the i n t r a n s i t i v e e f f e c t i s an a r t i f a c t a r i s i n g i n the course of the experiment. For one thing, at the moment of evaluation, a person's preferences may be t r a n s i t i v e but during the course of the experiment, his preferences may remain t r a n s i t i v e but change. Thus, the apparent i n t r a n s i t i v i t y , r e f l e c t s the change i n preferences rather than " i r r a t i o n a l i t y 0 . For example,a person whose preference ordering i s A>B>C w i l l state that he prefers A to B and B to C but 30 minutes l a t e r he may change his preferences to C>A>B i n which case he w i l l say that he prefers C to A, which would give r i s e to an apparent i n t r a n s i t i v i t y . In experiments, the inevitable unintentional s l i p s i n judgement, hastiness in responding, and carelessness may contribute to misleading conclusions regarding t r a n s i t i v i t y . Rose (1957) found that the number of apparent i n t r a n s i t i v i t i e s i n the judgements of each of his subjects was inversely correlated with the time they took to complete the experiment, a result which suggests that hastiness in responding may have contributed to apparent i n t r a n s i t i v i t y . 65 In contrast to May's r e s u l t s , Papandreou (1953) offered experimental evidence to support the hypothesis that i n d i v i d u a l preference systems s a t i s f y the axiom of t r a n s i t i v i t y . Two dif f e r e n t experiments were conducted involving a t o t a l of 24 subjects and 17,604 responses (pair-wise choices). Both experiments gave re s u l t s which strongly support the t r a n s i t i v i t y hypothesis. Tullock (1964) has argued that the assumption of t r a n s i t i v i t y i s not p a r t i c u l a r l y dubious. Furthermore, he postulated that any apparent i n t r a n s i t i v i t y may be tested by presenting the subject with a choice among a l l elements of the i n t r a n s i t i v e loop simultaneously. If the subject's preference ordering i s actually i n t r a n s i t i v e , he w i l l be unable to choose among the elements because for any element of the i n t r a n s i t i v e loop there i s always a more preferred element. The a b i l i t y of May's subjects to rank apparently i n t r a n s i t i v e elements i s contradictory to the hypothesis that they had i n t r a n s i t i v e preference orderings. The axiom regarding continuity of preferences states that: A>B>C => there exists a pr o b a b i l i t y p (1>p>0) for which (A,C;p)>B and also there e x i s t s a pr o b a b i l i t y g (1>g>0) for which B> (A,C;g).»• The axiom asserts that there exists an intermediate value of p between 0 and 1 for which the probability 1 6 (X,Y;p) represents the gamble whose outcome i s X with p r o b a b i l i t y p or Y with probability 1-p. 66 "mixture" (i.e. (A,C;p)) of the l e s s preferred outcome C and the more preferred outcome A w i l l be preferred to the intermediately preferred outcome B. 1 7 This axiom implies that the probability p plays a c r i t i c a l r o le i n determining r e l a t i v e u t i l i t y magnitudes of outcomes. The p r o b a b i l i t y p for which (A,C;p)>B r e f l e c t s the r e l a t i v e positions of A, B, and C on the u t i l i t y scale. The lower l i m i t for p (for which (A,C;p)>B) w i l l increase as the "distance" on the u t i l i t y scale between A and B increases. Introspection would indicate that such an axiom be i n v a l i d i n cases where A and B are highly disparate in u t i l i t y value from C. For example, i f a l t e r n a t i v e A i s receiving 2 pennies, alternative B i s receiving 1 penny, and al t e r n a t i v e C i s being tortured to death, i t seems unlikely that a person w i l l prefer (22,being tortured to death;p) to receiving 12 no matter how much (1-p) i s lowered (for 1>p>0). One may dismiss the contradictory nature of t h i s example by claiming that such d i s p a r i t y in values never occur i n p r a c t i c a l s ituations or by asserting that a s u f f i c i e n t l y small probability (1-p) does not e x i s t since people cannot relate to very small p r o b a b i l i t i e s (eg. what i s the psychological significance of a p r o b a b i l i t y of one-m i l l i o n t h or o n e - t r i l l i o n t h ? ) . Most scholars have accepted 1 7 The discussion of the axiom w i l l be r e s t r i c t e d to the case for which (A,C;p)>B. From p a r a l l e l considerations, the discussion can be e a s i l y extended to the case for which B> (A,C;g). 67 t h i s axiom as a good enough approximation to actual behavior to be useful while others (such as Hausner and Wendel, 1952;Hausner, 1954;Thrall, 1954) have modified the cardinal theory of u t i l i t y in which t h i s axiom i s not assumed to hold. The sure-thin^ axiom states that: A>B => A>(A,B;p). Conversely, B>A => (A,B;p)>A. The preferences (>) are changed to indifferences (~) for p=1. 1 8 The gamble (A,B;p) w i l l r e s u l t i n one of either two outcomes. I f outcome A occurs, the person w i l l be i n d i f f e r e n t between t h i s outcome and the sure-thing A; i f outcome B occurs, the person w i l l prefer the sure-thing A to t h i s outcome. Thus, the gamble (A,B;p) has two possible outcomes A and B for which the sure-thing A i s at lea s t as preferred as either and i s d e f i n i t e l y preferred to one of them (namely the outcome B). Discussed along these l i n e s , the axiom c e r t a i n l y seems l i k e a reasonable description of human behavior as well as a norm by which human behavior should abide. Despite the i n t u i t i v e p l a u s i b i l i t y of t h i s axiom, Harschak (1950) has given mountain climbing and Russian Roulette as two examples for which the axiom apparently does not hold. Marschak claims that in mountain climbing, ( l i v i n g s afely,serious injury;p)>living safely even though l i v i n g 1 8 The following discussion w i l l apply to the former case of A>(A,B;p) from which a p a r a l l e l discussion i s i m p l i c i t for the l a t t e r case of (A,B;p)>A. 68 safely>serious injury and s i m i l a r l y for Russian Roulette, (staying alive,being killed;5/6)>staying a l i v e even though staying alive>being k i l l e d ; both instances seem to i l l u s t r a t e behavior contradictory to the axiom. However, Adams (1960) suggested that mountain climbing and Russian Roulette cannot be formulated i n such simple terms. He pointed out that Marschak had assumed that " l i v i n g s a f e l y " at home (i.e. the alternative to not mountain climb) i s i d e n t i c a l to " l i v i n g s afely" i n the alternative to mountain climb, an assumption which he f e l t was incorrect. S i m i l a r l y , Adams f e l t that in the case of Russian Roulette, the term "staying a l i v e " involved i n not par t i c i p a t i n g and in pa r t i c i p a t i n g does not assume the same meaning in both cases. I f the person par t i c i p a t e s , he has prestige to gain and i f he does not par t i c i p a t e , he has his courage to lose. Even i f Adam's explanation i s s u f f i c i e n t , i t would s t i l l seem that mountain climbing and Russian Roulette would contradict the sure-thing axiom because the undertaking of each a c t i v i t y biases the u t i l i t i e s of the outcomes involved in the gamble (the act of mountain climbing and playing Russian Roulette enhances the u t i l i t y of staying a l i v e ) . Thus, i f under ce r t a i n t y , A>B and the "love of gambling" i s s i g n i f i c a n t enough so that the r i s k involved i n (A,B;p) i s s u f f i c i e n t to enhance the u t i l i t y of B, the gamble (A,B;p) may be preferred to A, which i s behavior that i s incompatible with the axiom. Thus, love of r i s k or gambling 69 i s contradictory to the sure-thing axiom**. The axiom may be rescued by assuming that risk-taking i n s i t u a t i o n s of p r a c t i c a l i n t e r e s t w i l l influence the u t i l i t i e s of the outcomes to a n e g l i g i b l e extent. The axiom, regarding independence of,i.outcome_ordering , states that: (A,B;p) (B,ft;1-p). The psychological premise underlying t h i s axiom i s that the order of presentation of the gamble outcomes should not a f f e c t a person's subjective value of that gamble. For example, suppose that a gamble i s described by two cards, each written with an outcome and i t s associated p r o b a b i l i t y . The axiom asserts that i n presenting the gamble to the subject for judgement, the order i n which the cards are presented to the subject i s i r r e l e v a n t . The i n t u i t i v e appeal of t h i s axiom i s obvious. The compound-gamble axiom states that a person i s i n d i f f e r e n t between ( 1 ) a gamble which o f f e r s a further gamble as outcome and ( 2 ) a gamble which i s a reduced form of the f i r s t gamble, i . e . the p r o b a b i l i t i e s of outcomes are the s t a t i s t i c a l equivalent to the f i r s t gamble. In notational form, ( (A,B ; p 1 ) , B ; p 2 W ( A , B ; p 1 a p 2 ) The axiom i s confronted by three objections of 1 9 The love of r i s k used in t h i s context i s not to be confused with a person's preference for unfair gambles, a preference behavior indicated by a concave-upward u t i l i t y curve (Mosteller and Hogee, 1 9 5 4 dubbed the term •extravagant behavior* to describe this l a t t e r behavior.). Extravagant behavior i s compatible with the axiom while love of r i s k i s not. 70 psychological s i g n i f i c a n c e . (1 ) The axiom asserts that subjective estimates of p r o b a b i l i t i e s coincide with the s t a t i s t i c a l l y s p e c i f i e d p r o b a b i l i t i e s . Chapter 1 points out that such coincidence i s not apparent i n experimental r e s u l t s (eg. Edwards and P h i l l i p s , 1966). (2) The "love of gambling" i s incompatible with the dictates of the axiom. It may be argued that those people who react negatively to r i s k taking ( i . e . detest gambling) w i l l prefer (A,B;p1»p2) to ((A,B;p1),£;p2) since the l a t t e r gamble involves the possible p a r t i c i p a t i o n in two gambles while people who react p o s i t i v e l y to taking of r i s k s ( i . e . love gambling) w i l l prefer ( (A,B;p1),B;p2) to (A,B;p1«p2) since the former gamble involves the possible p a r t i c i p a t i o n i n two gambles. This l a t t e r observation of human behavior i s c a p i t a l i z e d upon by casino owners. Slot machines have three revolving wheels (representing three r i s k y situations) instead of one wheel (representing one ris k y situation) with equivalent p r o b a b i l i t i e s of winning. (3) The axiom asserts that people do not have prob a b i l i t y preferences. Although the reduced form of the more complex gamble ((A,B;p1),B;p2) i s s t a t i s t i c a l l y equivalent to the simpler gamble (A,B;p1»p2), a person may prefer ((A,B;p1),B;p2) to (A,B;p2) because he prefers the p r o b a b i l i t i e s p1 and p2 to p1*p2 (or vice versa). For example, suppose a person i s required to state his preference or indifference between ( (A,B;1/2),B;1/2) and 71 (A,B;1/4). Although the axiom claims that a person w i l l be i n d i f f e r e n t between the two gambles, i t i s quite conceivable that he w i l l prefer ((A,B; 1/2) ,B; 1/2) to (A,B;1/4) because he can more re a d i l y r e l a t e to a probability of 1/2 (head or t a i l , g i r l or boy newborn, etc.) than to p r o b a b i l i t i e s of 1/4 and 3/4. In a series of experiments, Edwards (1953, 1954a, 1954b, 1954c) provided experimental evidence to show that people have pro b a b i l i t y preferences that cannot be accounted by u t i l i t y considerations alone. This axiom i s perhaps the most controversial of the axioms not only because of the three previously mentioned objections but also because of i t s i m p l i c i t assumption that humans are i n f i n i t e s i m a l l y sensitive creatures. This assumption i s i m p l i c i t i n the other axioms but i s of c r u c i a l importance here because of the indifference r e l a t i o n (as contrasted to a preference relation) stated i n the compound-gamble axiom, If subject X s t r i c t l y prefers ( (A,B;0.58),B;0.37) to (A,B;0.2146), can one conclude that subject X violated the axiom since (0.58)• (0.37)=0.2146? From these previously stated axioms, von Neumann and Morgenstern (1947) were able to deduce that: (1) u t i l i t y of z may be represented by numbers, u(z). (2) x>y => u(x)>u(y) (3) u(x,y;p) = p«u (x) + (1-p) *u (y) (4) The u t i l i t y u (z) i s unigue up to a l i n e a r transformation; i . e . the u t i l i t y scale i s c a r d i n a l . The c a r d i n a l i t y r e s u l t implies that unit and o r i g i n for t h i s 72 u t i l i t y scale are arbit r a r y . The following discusses previous experimental attempts to measure cardinal u t i l i t y using the von Neumann-Morgenstern model. Other experiments employing elaborate modifications of the basic model (eg. stochastic model of choice, subjective expected u t i l i t y model) are not discussed here. Hosteller and Nogee*s (1951) experiment represented the f i r s t attempt to measure cardinal u t i l i t y using the von Neumann-Horgenstern model. National Guardsmen and Harvard undergraduates served as subjects and a game c a l l e d poker dice served as the betting task. The subject was repeatedly reguired to accept or refuse bets stated in terms of r o l l s at poker dice. Subjects played with $1.00, which they were given at the beginning of each experimental session. Subjects were provided with a table which informed them whether a given bet was f a i r , or better or worse than f a i r . The subjects were presented with bets of the following form: "You can bet or not bet f i v e cents against ten doll a r s that you can r o l l f i v e dice once and get a better poker hand than 44441," or "You have the opportunity of betting or not betting f i v e cents against t h i s double 73 o f f e r : i f you beat 22263 you w i l l receive 20 cents; i f you do not beat 22263 but do beat 66431, you w i l l receive three cents. If you do not beat either, you w i l l lose the f i v e cents you must r i s k to play. You w i l l r o l l the dice only once." The indifference o f f e r was operationally defined as the amount of money for which the subject would accept the bet 1/2 of the t i m e 2 0 . The experiment required approximately 30 hours from each subject. The r e s u l t s indicated that for Harvard undergraduates u t i l i t y for money was approximately proportional to money up to a point, a f t e r which marginal u t i l i t y decreased. However, for the Guardsmen, t h e i r u t i l i t y curves showed increasing marginal u t i l i t y for money, which indicated that they were w i l l i n g to play unfair gambles. Realizing that interpretations of p r o b a b i l i t i e s may be biased, Davidson, Suppes and Siegel (1957) set out to i d e n t i f y an event for which the subjective probability of occurrence i s equal to i t s subjective probability of non-occurrence so that t h i s event, rather than a stated p r o b a b i l i t y , could be used in a gamble. A coin toss (head or t a i l ) , a die r o l l (odd number or even number), a two coin 2o By providing t h i s operational d e f i n i t i o n of indifference, Mosteller and Nogee have t a c i t l y given a stochastic interpretation to the preference r e l a t i o n s . 74 toss (match or non-match), and some other simple games were t r i e d , but in each case most subjects showed a d e f i n i t e preference for one of the "equal-chance" outcomes. After much t r i a l and error, a s a t i s f a c t o r y event was found. The event was the toss of a s p e c i a l l y designed die. On three faces of the die, •ZOJ* was engraved while on the other three faces, 'ZEJ* was engraved. Two more die were s i m i l a r l y engraved with »HNH« and «XEQ», »QUG» and *QUJ*. Glaze ( 1 9 2 8 ) had shown that these nonsense s y l l a b l e s had p r a c t i c a l l y no association value. With a l l three die, their preliminary experimentation showed that the subjective pr o b a b i l i t y of occurrence i s equal to i t s subjective probability of non-occurrence; i . e . , the subject associated a pr o b a b i l i t y of 1/2 with the event. The subject was required to make pairwise comparisons between gambles. The instructions took on the following form: "I*11 present you with two alternatives. You are to choose one of them. For example, i f you want to bet on ZEJ against the f i e l d , you w i l l win 1 8 0 i f ZEJ comes up when you toss the die, but you w i l l lose 40 i f the f i e l d comes up (that i s , i f any side but the ZEJ comes up). On the other hand, i f you want to bet on the f i e l d , you w i l l win 6 0 i f the f i e l d comes up, but you w i l l lose 120 i f ZEJ comes up." From expressed preferences between these type of choices. 75 the experimenters were able to determine upper and lower bounds on th e i r subjects* u t i l i t y functions. The bounds were generally close together. The r e s u l t s showed that 15 out of the 19 subjects made choices as i f they were attempting to maximize expected u t i l i t y . In Swalm's experiment (1966), the u t i l i t y curves for money were derived from business executives. Realizing the problem of confounding subjective value of outcome with subjective p r o b a b i l i t y , Swalm attempted to minimize t h i s e f f e c t by using only p r o b a b i l i t i e s of 1/2. Although Swalm expected that the probability 1/2 could be e a s i l y related (eg. to the concept of a coin f l i p ) , many subjects f e l t that the hypothetically constructed situations involving 50-50 odds were u n r e a l i s t i c because most of the situations encountered by them i n t h e i r i n d i v i d u a l businesses were not 50-50 gambles. Subjects were repeatedly required to state the certainty equivalents of 50-50 gambles whose two outcomes are given by the experimenter and whose u t i l i t i e s were previously derived. The questions took on the following form: •'Suppose that you planned to purchase a general-purpose machine but a colleague proposed, instead, to buy a more e f f i c i e n t special-purpose machine. Both cost the same; the d i f f i c u l t y i s that the contract for which the special-purpose machine would be required has only a 50-50 pro b a b i l i t y of being 7 6 received. If i t i s received, the s p e c i a l -purpose machine w i l l y i e l d a p r o f i t of $250,000. If not, your net income w i l l be zero. On the other hand, the general-purpose machine w i l l produce a ce r t a i n savings of, say, $100,000. Which would you recommend?" The money amounts used i n the gambles were chosen to be meaningful to the subject. Losses as well as gains were involved. The r e s u l t s of the experiment indicated that: (1) the subjects did not maximize expected monetary value, (2) the u t i l i t y curves provide a basis for i d e n t i f y i n g risk-aversion (concave downward u t i l i t y curve) and extravagant behavior (concave upward u t i l i t y curve), (3) the u t i l i t y curves revealed d i f f e r e n t attitudes toward r i s k decisions among executives of the same company, and (4) most of the u t i l i t y curves indicated risk-aversion. ^§£fiss^on_of_our_Ex£eriment The standard gamble method was used to determine von Seuman-Borgenstern cardinal u t i l i t y for days in bed. The application of the method was repeated i n the second experimental session. The application of the method involved finding certainty equivalents to 50-50 gambles stated i n the context of a scenario. The essence of the scenario i s qiven here. 77 "You are suffering from a case of the f l u . Your family doctor informs you that in your p a r t i c u l a r case there are two possible treatments available. The f i r s t treatment w i l l r e s u l t in either a days of rest in bed or b days of rest i n bed with equal chances. The second possible treatment w i l l r e s u l t in a fixed number of days i n bed f o r ce r t a i n . State the number of days in bed for certain for which you w i l l be i n d i f f e r e n t between this treatment and the f i r s t one." I n i t i a l l y , a and b were chosen to be 0 and 15 days since our domain of i n t e r e s t was from 0 to 15 days. Then the subject was repeatedly asked to state c e r t a i n t y equivalents x to 50-50 gambles with outcomes a and b (whose u t i l i t i e s are known) from which the u t i l i t y of x was calculated according to the following formula: u (x)=0.50»u (a)+0.50»u (b) The u t i l i t i e s of 0 and 15 days were a r b i t r a r i l y set to 1 and 0 respectively. The gambles were presented in the form of the previously mentioned scenario, approximately 10 points were i d e n t i f i e d on each subject's u t i l i t y curve. The u t i l i t y curve was approximated by piecewise linear i n t e r p o l a t i o n . The procedure for determination of the u t i l i t y curve was repeated in the next session. 78 Section_II_-_Ordinal_Uti1 Curves Introduction C l a s s i c a l u t i l i t y theory was used by theorists (eg. Jevons, Menger, and Marshall) to establish consumer demand for commodities. When re l a t i o n s between u t i l i t i e s of d i f f e r e n t commodities were considered, Jevons, Menger, and Marshall assumed the a d d i t i v i t y of u t i l i t i e s . However, such an a d d i t i v i t y notion i s inconceivable for commodities which are not independent. This objection was resolved through recourse to Edgeworth fs (1881) notion of indifference curves for which the r e s t r i c t i v e a d d i t i v i t y assumption i s not needed and from which s u b s t i t u t a b i l i t y and complementary relationships between commodities may be interpreted. Furthermore, Pareto (1906) seriously doubted whether preferences could be measured on a numerical scale as implied by the c l a s s i c a l c a r d i nal theory. Rejecting c l a s s i c a l cardinal u t i l i t y i n favor of ordinal u t i l i t y , he asserted that the same conclusions about consumer demand that had been drawn from c l a s s i c a l cardinal measures could be drawn from analysis of indifference curves. In f a c t . Hicks and Allen (1934) and Samuelson (1938) derived a l l the usual conclusions about consumer behavior from analysis of indifference curves. Samuelson asserted that the structure of the theory of consumer choice could be derived from observation of choices among alternatives available to a consumer (the concept of revealed preference). The essence 79 of t h i s approach i s that each choice defines a point and a slope i n the choice space from which a family of slopes constitute an indifference hyperplane. O r d i n a l i s t s argued for abandoning cardinal u t i l i t y because analysis of indifference curves could deduce the same res u l t s in the area of r i s k l e s s choice as cardinal u t i l i t y could with i t s stronger assumptions. In general, the only required assumption for the derivation of indifference curves i s that concerninq weak ordering of commodity bundles, i . e . that the subject i s able to express preference or indifference between commodity bundle pairs and that his preference and indifference relations are t r a n s i t i v e 2 1 . In terms of the indifference map, the intersection of two indifference curves implies the v i o l a t i o n of t r a n s i t i v i t y of the indifference r e l a t i o n . This t r a n s i t i v i t y assumption i s also required i n the cardinal u t i l i t y theory but the other axioms required for cardinal u t i l i t y are not e s s e n t i a l to the ordinal theory. Thus a stronger u t i l i t y scale requires a stronger axiomatic system (this point i s discussed in Chapter 1). 2 1 T r a n s i t i v i t y was discussed i n Section I of th i s chapter. 80 Discussion _of Previous^Experiments The s c a r c i t y of studies devoted to experimental derivation of indifference curves i s a pa r t i c u l a r s t r i k i n g feature of the l i t e r a t u r e . There were two notable early experimental attempts to derive indifference curves. In 1931, Thurstone performed a simple experiment to e l i c i t preferences from a subject whereupon indifference curves were derived. Thurstone's research assistant served as the subject. Preference judgements between hat and overcoat combinations were required from the subjects. From these preference judgements, Thurstone was able to locate an indifference curve between two regions of the choice space, namely the region i n which the reference combination i s preferred to the combinations in the region and the region in which the hat and overcoat combinations i n the region are preferred to the reference combination. The same procedure was repeated for hats and shoes, and for shoes and overcoats. From f i v e psychophysical laws, Thurstone chose Fechner*s logarithmic law to f i t an indifference curve between the two regions. Four indifference curves were f i t t e d i n the hat X shoe space and 4 indifference curves were f i t t e d i n the hat X overcoat space. From these indifference maps, indifference curves in the shoe X overcoat space were predicted using the a d d i t i v i t y assumption. The predictions were surprisingly accurate but Thurstone (1953) hypothesized that the 81 consistencies "were the r e s u l t of careful instructions to assume a uniform motivational a t t i t u d e . " The second experimental determination of indifference curves was attempted by Housseas and Hart (1951). The experiment required 67 students to rank sets of three bacon-egg combinations. Vectors in the choice space were constructed based upon directions of preference between choices. Those vectors which were consistent with the experimenters* assumed saturation levels and convexity properties were used for constructing the indifference curves. By making a dubious assumption regarding homogeneity of preferences among the students, a composite indifference map was derived by curve f i t t i n g a set of vectors. A major difference between t h i s experiment and Thurstone*s i s that Rousseas and Hart provided a motivation for c a r e f u l consideration by forcing the subject to consume the top ranked egg-bacon combination. From the economist's point of view, the previously mentioned experiment (and generally any experimental derivations of indifference curves or any other forms of u t i l i t y for that matter) face objections of which the essence i s that hypothetically constructed choices made i n a controlled laboratory s i t u a t i o n do not r e f l e c t "actual" preferences. Characterizing the economist's standpoint, Wallis and Friedman (1942) expressed doubts about the a p p l i c a b i l i t y of deriving indifference curves on laboratory data for which MacCrimmon and Toda (1969) offered rebuttals. 82 In 1969, MacCrimmon and Toda presented an e f f i c i e n t method for determining indifference curves based upon dominance concepts. B r i e f l y , i f both commodities are monotonic i n preference ( i . e . more i s preferred to less or vice versa) expressed preference between a chosen point i n the choice space and the reference point leads to the i d e n t i f i c a t i o n of a rectangular subset of the acceptance region and a rectangular subset of the rejection r e g i o n 2 2 . The acceptance region consists of a l l points which are accepted i n l i e u of the reference point while the rejec t i o n region consists of a l l points which are rejected i n l i e u of the reference point. Expressed choice between other points i n the choice space and the reference point leads to i d e n t i f i c a t i o n of a greater portion of each region u n t i l the band between the i d e n t i f i e d portions of the acceptance and the r e j e c t i o n region i s s u f f i c i e n t l y narrow to locate an indifference curve. For the case of only one monotonic valued commodity, li n e s instead of rectangular subsets in each region are i d e n t i f i e d . In a t r a i n i n g session l a s t i n g about 2 hours, the 7 subjects were taught to derive their own indifference curves using the previously mentioned method. In order to provide an incentive for revealing true preferences, payoffs were made according to the derived indifference curves. The domain of choice included b a l l point pens, money, and French 2 2 The procedure w i l l be more f u l l y discussed l a t e r i n t h i s chapter. 83 pastries. In the f i r s t session, 7 indifference curves were derived i n the money X pen space for which i t was assummed that more i s preferred to less for both commodities (rectangular method). In the second session, 4 indifference curves were derived in the money X pastry space for which i t was assumed that more i s preferred to less for money but not necessarily for a l l le v e l s of pastries (line method). F i n a l l y subjects were given a serie s of pairwise comparison bundles on either side of the indifference curves drawn. Of 21 of these consistency checks given to each subject, an average of l e s s than one choice per subject was actually inconsistent with the appropriate indifference curve. Discussion of our Experiment The subject was required stated i n the context of scenario i s presented here. to choose among alternatives a scenario. The essence of the "You are in a bed recovering from a case of the f l u . The treatment prescribed to you by your physician i s complete rest in bed. However, your physician informs you that after you leave the bed, the f l u has some chance of immediately recurring and hence there w i l l be a p r o b a b i l i t y that you w i l l have to return to bed for an additional f i v e days. Choose between resting for : (i) a days in bed with £ pro b a b i l i t y of an 84 additional 5 days i n bed or ( i i ) b days i n bed with g probability of an additional 5 days i n bed." Indifference as well as s t r i c t preference judgements were allowed. The days presented were whole numbers and the p r o b a b i l i t i e s were given to the f i r s t decimal place (i . e . 0.0(0.1)1.0). The method we used i s adopted from the MacCriramon-Toda method for derivation of indifference curves. Because of the sp e c i a l nature of our alternatives ( i.e. outcome X r i s k instead of the usual outcome X outcome choice space), we have added certain assumptions i n addition to the monotonicity and t r a n s i t i v i t y assumptions which MacCrimmon and Toda used (1969). These added assumptions w i l l be e f f e c t i v e i n i d e n t i f y i n g additional portions of the reject and accept region. Assumptions we used are presented here. (1) monotonicity of preferences: An alternative i s preferred to a l l other alternatives which have more number of days in bed and more probability of an additional 5 days i n bed. An alt e r n a t i v e i s rejected in favor of a l l other alternatives which have less number of days in bed and less probability of an additional 5 days i n bed. (2) t r a n s i t i v i t y : S t r i c t preference and indifference r e l a t i o n s are t r a n s i t i v e . (3) Let (a,b) represent an alternative where the f i r s t coordinate represents number of days in bed and the second coordinate represents the associated probability of an 85 add i t i o n a l 5 days in bed. (a) (d1,p1) i s preferred to a l l other alternatives (d2,p2) where d2> <d1 + 5). <b) <d1,1) (d1+5,0) To i l l u s t r a t e the method, suppose that we wish to determine the indifference curve that passes through the reference point Ro=J10.tQ«,5QL . Suppose that our domain of inter e s t i s 0<days in bed<20 and 0<probability of an additional 5 days<1.0. The method i s equivalent to determining subsets of (1) an accept region A and (2) a r e j e c t region R, whose common boundary i s the indifference curve. The accept region consists of a l l points which are preferred to Ro, while the reject region consists of a l l the points which are rejected i n favor of Ro. Of course, the experimental application of the method cannot ever hope to i d e n t i f y the t o t a l accept and reject regions since t h i s would reguire an i n f i n i t e number of judgements from the subject. However the application of the method endeavors to i d e n t i f y as much of the accept and r e j e c t regions as p r a c t i c a l l y possible from which an indifference curve may be approximately located. (i) Osing assumption (1), subsets of the A and R regions are shaded as shown i n Figure 7 . ( i i ) Using assumption 3(a), the unshaded region i s further reduced as shown in Figure 8 . ( i i i ) The subject i s then presented with a point i n the 86 Figure 7 - Identifying Accept and Reject Regions fi ta fi o u P-Figure 8 - Identifying Accept and Reject Regions unshaded area and asked to compare th i s point with Ro. Suppose the subject prefers (12,0.20) to Ro. Using assumptions (1) and (2), a subset of the A region i s shaded as shown i n Figure 9 . (iv) ' Furthermore, using assumptions 3(a) and (2), a subset of the A region i s shaded as shown in Figure 10 . 87 day —» Figure 9 - Identifying Accept and Reject Regions 5 7 10 12 15 day-» Figure 10 - Identifying Accept and Reject Regions Thus, we see that for each choice between presented a l t e r n a t i v e s , the unshaded area i s further reduced. The choices were sequentially presented to the subject u n t i l the unshaded area was reduced as much as possible (using whole number of days and f i r s t place decimal p r o b a b i l i t i e s ) . A piecewise li n e a r interpolation was used to approximate the i n d i f f e r e n c e curve within the region unshaded area. 89 Chapter 4 7 Results, Analysis, and Conclusions Section_I_-_lntroduction The study focuses upon the application of two methods for e l i c i t i n g preferences from subjects, which have received s i g n i f i c a n t attention i n the l i t e r a t u r e . The methods are: (1) the HacCrimmon-Toda method for constructing indifference curves, and (2) the standard gamble certainty equivalence method for constructing von Heumann-Horgenstern cardinal u t i l i t y functions. While i n d i v i d u a l discussion of each method was completed i n Chapter 3, the aim of t h i s chapter i s to report on an experiment which was conducted for comparing the two methods i n terms of the following c r i t e r i a : (1) Test-retest correspondence of preference judgements (2) The existence of "personal attitudes" which a f f e c t the t e s t - r e t e s t correspondence (3) L i n e a r i t y of derived indifference points i n the day X probability space. (4) Goodness of prediction In addition, the study attempts to i d e n t i f y i n t e r -method correspondence of predictions of preferences. F i n a l l y , the study inguires into some possible relationships between te s t - r e t e s t and inter-method correspondences and attitudes (eg. concerning r a t i o n a l i t y . 90 in t e r p r e t a t i o n of p r o b a b i l i t i e s , and discrimination bands). The sample consisted of 23 commerce students, of which 14 were undergraduates and 9 were graduates. A l l have had some previous exposure to the concept of u t i l i t y , but none had ever participated i n experiments for e l i c i t i n g preferences. The domain of choice concerned decisions to stay i n hospital bed and take a r i s k of readmission for an additional period. A l l subjects have stayed in hospital at least once, but only 4 have stayed i n hospital for 3 or more days i n the year preceding the experiment. The study consisted of experimental derivations of indifference curves and u t i l i t y functions i n repeat tests with the same sample of subjects, as well as the administration of a guestionnaire i n the l a s t experimental session. The experimental sessions were conducted separately with each subject. In each session the subject was presented with two scenarios r e l a t i n g to choices involving hospital days i n bed and p r o b a b i l i t y of additional f i v e hospital days in bed. The f i r s t scenario presented the subject with three reference points consisting of days i n bed for sure and pro b a b i l i t y of additional fixed number of hospital days. The reference points in the day X probability space were respectively (4,0.50), (7,0.50), and (10,0.50). A method developed by MacCrimmon and Toda (1969) was employed to e l i c i t preferences from the subject i n r e l a t i o n to a given reference point and consequently deriving an indifference curve, i . e . a curve r e f l e c t i n g trade-offs which maintain 91 the welfare of the subject at an equal l e v e l . Following the application of this method, a von Neumann-Morgenstern u t i l i t y function for hospital days i n bed was derived for the subject using certainty equivalences for gambles involving chances of days i n bed. The order of application of each on the two methods of e l i c i t i n g preferences was reversed for the second session. This procedure was repeated twice for each subject with a minimum of one. week delay and a maximum of two weeks delay between sessions for each subject. In the t h i r d session the subject was asked to f i l l a questionnaire consisting of guestions related to the following themes: (1) Attitudes concerning the acceptance of pa r t i c u l a r fundamental r a t i o n a l i t y axioms. (2) Propensities for cer t a i n judgemental modes of evaluation. (3) Evaluation of various components of the experiment (eg. the scenario introducing the choice space for each method), (4) I d e n t i f i c a t i o n of the discrimination band in the choice space ( p r o b a b i l i t i e s and days). (5) The subjective i n t e r p r e t a t i o n of certain c o l l o g u i a l p robability expressions. The f i r s t theme of the questionnaire attempted to provide an i n d i c a t i o n of the subject's agreement with the appropriateness of some of the fundamental axioms underlying the methods used i n the experiment. We have used Savage's 92 defence of r a t i o n a l i t y axioms (1954) in designing the form of the questionnaire items. Savage argued that r a t i o n a l i t y provides the rules for reasonable behavior and that when a subject i s aware of v i o l a t i n g these rules he w i l l tend to revise his decisions. Our questionnaire items presented the subject with examples i l l u s t r a t i n g v i o l a t i o n s of some axioms. Then he was asked to rate his agreement with the need to revise the decision, on a L i c k e r t scale (ranging from 1=strongly disagree through 4=neutral to 7=strongly agree). The following are examples from the questionnaire of cases where the t r a n s i t i v i t y axiom and the compound-gamble axiom are v i o l a t e d : "George prefers driving a Ford Pinto to a Toyota MK II. Furthermore, he prefers driving a Toyota MK II to a Datsun 1600. Yet, from a rent-a-car which of f e r s a Datsun 1600 or a Ford Pinto at the same rent a l rate, George rents a Datsun 1600 instead of a Ford Pinto. Realizing t h i s "inconsistency", George should change his choice to Ford Pinto." 7=strongly agree-6-5-4-3-2-1=strongly disagree "A sweepstake t i c k e t e n t i t l e s the holder to either a prize of $1.00 or a chance in the grand f i n a l draw. The grand f i n a l draw prize w i l l be either $100.00 or $1.00. 93 another sweepstake t i c k e t e n t i t l e s the holder to a prize of $1.00 or $100.00. Both sweepstake t i c k e t s s e l l for the same price . Taking the chances of winning into account, Dan calculates that the p r o b a b i l i t i e s of winning each prize are the same for both sweepstakes. In spite of t h i s information, Dan i n s i s t s upon buying the second sweepstake ti c k e t and i s even w i l l i n g to pay s l i g h t l y more for t h i s t i c k e t . Dan should stop favoring the second sweepstake." 7=strongly agree-6-5-4-3-2-1=strongly disagree Questions r e l a t i n g to the second theme tapped attitudes concerning the general mode f e l t appropriate for this domain of decision making, eg. to what extent one prefers careful l o g i c a l judgement to spontaneous response to the problem s i t u a t i o n in health matters, again the subject was asked to rate his agreement on a Lickert scale. The following are examples of these questions: "In health matters, people ought to c a r e f u l l y evaluate t h e i r preferences among alternatives without being influenced by their mood or emotion at the moment of evaluation." 7=strongly agree-6-5-4-3-2-1=strongly disagree "In matters concerning i l l n e s s , people ought 94 to evaluate t h e i r preferences among alternatives before the i l l n e s s a c t ually occurs because under pain and discomfort thay may not be c l e a r l y aware of their preferences." 7=strongly agree-6-5-4-3-2-1=strongly disagree The t h i r d theme of the questionnaire focused on evaluation of the experiment and the methods used. The subject was asked to: ( 1 ) rate the realism of the presentation of the scenarios introducing the domain of choice for each method, and (2) compare the d i f f i c u l t y of judgements required by each method. F i n a l l y questions were directed to i d e n t i f y the confidence the subject has in the methods. To t h i s end two questions about each method were presented to the subject; one question was concerned with his willingness to have the e l i c i t e d preferences used in l i e u of his personal judgements when a s i t u a t i o n of choice ari s e s , while the other was concerned with the question of which preferences should dominate i n a decision making s i t u a t i o n - those which were obtained prior to the health s i t u a t i o n or judgements spontaneously made in the face of the s i t u a t i o n . Examples of the guestions are presented below: "Suppose that with reference to a particular health matter, a trained health personnel derives your indifference curves. If a situ a t i o n resembling the scenario arises in 95 „ r e a l l i f e , would you l e t a physician determine the decision for you from a c a r e f u l consideration of your indifference curves?" 7=without any doubt-6-5-4-3-2-1=with complete doubt "suppose that you were to ac t u a l l y encounter a si t u a t i o n where you had to compare two alternatives each involving days of rest in bed and associated probability of additional days in bed (as in our experiment). Furthermore suppose that the decision you actually make does not conform with your responses using method ... in t h i s experiment. In l i g h t of t h i s information, how important do you f e e l that i t i s for you to change your decision?" 7=extremely important-6-5-U-3-2-1=extremely unimportant The fourth theme i n the guestionnaire focuses upon measurement of "discrimination "bands", i . e . to what extent changes i n s t i m u l i such as "day" or " p r o b a b i l i t i e s " are perceived s i g n i f i c a n t by the subject and s i g n i f i c a n t l y a f f e c t his judgements. For example the subject was asked to rate the sig n i f i c a n c e of changes i n p r o b a b i l i t i e s from 0.5 to 0.55 as opposed to 0.6 to 0.8. We have selected values which r e f l e c t both the l e v e l s of p r o b a b i l i t i e s and the degree of change i n p r o b a b i l i t i e s . Similar comparisons were obtained for changes i n the number of days in hos p i t a l . 96 The fourth theme i n the questionnaire consisted of one question aiming at providing insight into the possible biasing e f f e c t s of using p r o b a b i l i t i e s in e l i c i t i n g preferences. We have attempted to associate c o l l o q u i a l expressions of r i s k with the objective scale of probability. The form of the question presented below i s based on a method which was used by Lichtenstein and Newman (1967) for a similar purpose. "What p r o b a b i l i t i e s do you associate with the following words (or phrases): (a) certain (b) unlikely (c) highly probable (e) uncertain (e) probable (f) impossible (g) extremely l i k e l y " " The f i n a l procedure used i n our experiments presented subjects with re-evaluation of "gambles" for which diammetrically opposed choices were indicated f o r the subject using the alternative methods. The subject was requested to make an additional judgement as to his preferences among these gambles. 23 The responses are presented i n appendix a. 97 Section II - Discussion of Experimental_Design The central indices chosen for evaluating and comparing the methods were: (a) te s t - r e t e s t squared deviations for each method (intramethod inconsistency), and (b) intermethod squared deviations of corresponding indifference predictions obtained by each method (intermethod inconsistency). In order to provide a base by which meaningful comparisons between the two methods of e l i c i t i n g preferences could be made, the u t i l i t y curve for hospital days derived by the standard gamble method was transformed into the day X probability space using the following procedure: (1) The u t i l i t y of the f i r s t reference point (4,0.50) was calculated from the u t i l i t y curve. The point (4,0.50) represents a "gamble" for which the outcome w i l l be 4 days with p r o b a b i l i t y 0.50 or 4+5=9 days with probability 0.50. Therefore, the u t i l i t y of the reference point i s 0ref=0.50»u (4) +0.50«u (9) which may be calculated since u(4) and u(9) can be found from the subject's empirically derived u t i l i t y curve. (2) From t h i s reference u t i l i t y , the (day,probability) trade-off equivalents to the reference point was calculated for p r o b a b i l i t i e s from 0 to 1 at increments of 0.05. The problem becomes that of finding X for a given p such that: u (X,p)=0ref or (1-p) »u (X)+p»u (X + 5)=0ref The equation takes on the same form as f (X)=0 for which the bisection search method was used to f i n d X for a given p 98 (for p=0 (0.05) 1) . Thus, from following these two procedures, we have obtained from the von Neumann-Morgenstern u t i l i t y curve, an indiffe r e n c e curve corresponding to the one we have derived using the MacCrimmon-Toda method. The MacCrimmon-Toda derived trade-off curve and the von Neumann-Morgenstern derived trade-off curve intersect at the reference point since the points on each trade-off curve are trade-off equivalents to the same reference point. The two procedures were s i m i l a r l y applied to the same u t i l i t y curve i n order to derive a second trade-off curve with points which are trade-off equivalents to the second reference po^nt of (7,0.50). The e n t i r e procedure applied to the two reference points was repeated for the u t i l i t y curve derived i n the second experimental session. Once the two trade-off curves derived by each method for each session were i d e n t i f i e d , the two measures of inconsistency, t e s t - r e t e s t and intermethod squared deviations could be calculated. An i n d i c a t i o n of the test-retest inconsistency for each method was provided by the squared deviations between the two trade-off curves derived for botL sessions for a given method. For the MacCrimmon-Toda method, the squared deviation between the two derived trade-off curves was denoted by (MT1-MT2)2 and defined as: (MT1-MT2)2 = H H (MT^ i J-MT^ i ) ) 2 i = l j=l 3 J 9 9 Where MT^(i)= the value of the f i r s t argument of the (day,probability) point on the MacCrimmon-Toda derived trade-off curve for the p r o b a b i l i t y of (21-i)/20, the f i r s t experimental session, and the j t h reference point, and MT^(i)=the value of the f i r s t argument of the (day,probability) point on the HacCrimmon-Toda derived trade-off curve for the pr o b a b i l i t y of (21-i)/20, the second experimental session, and the . jth reference point. S i m i l a r l y , the t e s t - r e t e s t squared deviation for the standard gamble method was denoted by (VH1-VM2)2 and defined as: 21 2 (VM1-VM2)2 = 2Z ZZ (VM ' * ( i)-VM? ( i ) ) 2 i = l j = l J 3 ; Where VM*(i) = the value of the f i r s t argument of the (day,probability) point on the standard gamble derived trade-off curve for the pr o b a b i l i t y of (21-i)/20, the f i r s t experimental session, and the j t h reference point, p and VM^(i) = the value of the f i r s t argument of the (day,probability) point on the standard gamble derived trade-off curve for the pr o b a b i l i t y of (21-i)/20, the second 100 experimental session, and the j t h reference point. An i n d i c a t i o n of the intermethod inconsistency was provided by the squared deviation between the trade-off curves derived from both methods for a given experimental session. For the f i r s t experimental session, the intermethod deviation was denoted by (MT1-VM1)2 and defined as: 2 1 2 (MT1-VM1)2 = X L E L (MT^(i)-vtri( i) . ) 2 i=l j=l J J For the second experimental session, the intermethod deviation was denoted by (MT2-VM2)2 and defined as: 21 2 (MT2-VM2)2 = E E - (MT?(i )-VM?(i ) ) 2 i=l j=l J • J _ F i n a l l y , an indication of the inconsistency between the trade-off curves for each method averaged over the two sessions was denoted by (av (MT) -av (VM)) 2 and defined as: (av(MT)-av(VM)) 2 = II E L (( ^ ( i ) ^ 2 ( i ) ) . ( ^ i l l ^ i l l 1=1 j=l • d 1 • The conceptual models underlying our experiment are based upon a number of propositions as to possible variables which a f f e c t t e s t - r e t e s t , intermethod, and averaged-method deviations. 1 0 1 £g§£-£g£g§t_Corres|3ondence_of _g,gef erenceg The f i r s t model which i s presented in Figure 11 hypothesizes that attitudes toward assumptions which underly each method, confidence i n the method (as a measure of motivation), propensities for certain judgemental modes of evaluation, realism of method scenario, the width of the discrimination band for probability and day s t i m u l i , bias in interpreting probability s t i m u l i , and the use of simple (linear) rules for judging preferences, would a l l tend to a f f e c t correspondence between responses in repeat tests and o r i g i n a l responses. As to the discrimination band, we have hypothesized that there i s an optimal l e v e l of s e n s i t i v i t y to s t i m u l i magnitudes,!.e. there exists a threshold s e n s i t i v i t y l e v e l , deviation from which w i l l lead to more pronounced differences i n t e s t - r e t e s t responses. Intermethod_Correspondence of Preferences The second model (presented i n Figure 1 2 ) d i f f e r s from the f i r s t only in the d e f i n i t i o n of the dependent variables: ( H T 1 - V M 1 ) 2 for the f i r s t session and ( M T 2 - V M 2 ) * for the second session. I t would seem that most of the independent variables which are hypothesized to a f f e c t test-retest consistency w i l l also a f f e c t intermethod consistency. However, the degree of association may d i f f e r between the dependent and the independent variables in the two cases. ACCEPTANCE OF "RATIONALITY" AXIOMS PROPENSITIES FOR JUDGEMENTAL KODES REALISM OF SCENARIO CONFIDENCE IN METHOD DISCRIMINATION FOR PROBABILITY AND DAY STIMULI BIAS IN . INTERPRETATION OF PROBABILITY LINEARITY TEST-RETEST CORRESPONDENCE (IVT1-1VT2)2 (VM1-VM2)2 Figure 11 - Tes t - re te s t Hodel 103 ACCEPTANCE: OF "RATIONALITY" AXIOMS PROPENSITIES FOR JUDGEf^SNTAL tfODSS REALISM OF SCENARIO CONFIDENCE IN KETHOD .DISCRIMINATION FOP PROBABILITY AND DAY STIMULI BIAS IN INTERPRETATION OF PROBABILITY LINEARITY INTERMETHOD CORRESPONDENCE (KTl-VMl) 2 . (WT2-VM2)2 Figure 12 - Intermethod Model 104 Dependent Variables (Inconsistency,, Measures)._.-_Prgfile_gf Results Figure 13 displays the p r o f i l e of inconsistency measures for each of three groups: undergraduates, graduates, and the t o t a l sample. The tabled responses bring to attention two s t r i k i n g patterns: (1) the median for each of the inconsistency measures for the graduate group i s lower than that for the undergraduate group, and (2) some inconsistency measures are higher i n value than others across a l l groups. One possible explanation for the f i r s t observation i s that the graduates, because of a longer s o c i a l i z a t i o n process and a greater s k i l l in mental computing, tended to make a much more conscientious e f f o r t in responding consistently and make fewer errors i n computing their responses. The second observation i s that the median for (MT1-MT2)2 i s higher than for (VM1-VH2)2 in both sample groups. The observation that the median for (MT2-VM2)2 i s lower than that for (MT1-VM1)2 for both sample groups seems to suggest that intermethod consistency may improve with experience (learning). The median for (av(MT)-av (VM)) 2 i s lower than that for (MT1-MT2)2 and (VM1-VM2)2 for a l l three samples; thus, averaging indifference curves derived by each method seems to improve intermethod consistency. 105 Undergraduates N=l4 Graduates N=9 Total M=23 e CO 01 P.'edian 6 c cd •H t3 CD & c cd CU Median 6 Range • c cd Median 6 Range (MT1-MT2)2 37.5 30.7 27.8 4.5-95.2 •8,5 6.6 8.2 0.0-22.2 26.1 20.8 26.2 0.0-95.2 (VM1-VM2)2 10.2 4.7 12.0 0.7-41.5 4.1 3.3 2.7 0.0- 9,3 7.8 3.3 9.8 0.0-41.5 (MT1-VM1)2 38.7 42.1 21.4 5.1-69.2 < 19.0 14.8 16.1 0.0-42.3 31.0 31.6 21.4 0.0-69.2 (MT2-VM2)2 28.1 25.5 9.9 10.5-52.0 16.9 12.2 15.2 0.0-49.1 23.7 23.6 13.1 0.0-52.0 ' iv (MTj-av(VM)) 2 22.7 23.6 8.0 9.4-34.8 15.0 8.4 14.0 0.0-41.5 19.7 20,9 l l . l 0.0-41,5 A l l entries are in units of (day) Figure 13 - Inconsistency Measures - P r o f i l e of Results The Independent_Variableg The independent variables, which were proposed in our te s t - r e t e s t and intermethod models, are operationally defined i n terms of our indices based upon questionnaire items. Responses to those questionnaire items which are relevant to a p a r t i c u l a r concept are aggregated to provide an index for that concept. The following discussion indicates the concepts and their respective d e f i n i t i o n s . 106 Refer to Appendix B f o r a l i s t of questionnaire items, (a) acceptance iof " r a t i o n a l i t y " axioms The independent variable, characterizing acceptance of " r a t i o n a l i t y " axioms, i s defined i n terms of an index which i s a simple l i n e a r combination of the Lic k e r t scales for the three questionnaire items that measure attitudes concerning acceptance of the t r a n s i t i v i t y axiom, the sure-thing axiom, and the compound-gamble axiom. The index i s normalized i n such a way that a value of 1 indicates strong disagreement with a l l three axioms, while a value of 7 indicates strong agreement with a l l three axioms. __b} p r o p g n s i t i e g f o r ^ judgemental, modes of evaluation The index f o r t h i s independent variable i s a linear combination of the Lickert scales for the three questionnaire items that measure attitudes concerning three modes of evaluation: (1) ca r e f u l unemotional evaluation, (2) p r i o r - t o - s i t u a t i o n evaluation versus i n - s i t u a t i o n evaluation, and (3) l o g i c a l systematic evaluation. The f i r s t two modes refer to health matters s p e c i f i c a l l y . The index i s normalized so that a value of 1 indicates an extreme negative attitude towards a l l three modes, while a value of 7 indicates an extreme positive attitude towards a l l three judgemental modes. jc) bias i n int e r p r e t a t i o n gf colloguial_probabij.it j statements The index for t h i s independent variable i s a linear combination of the responses for two par t i c u l a r 107 questionnaire items. One item requires the subject to assign a probability to the c o l l o q u i a l term "certain" while the other requires the subject to assign a probability to the c o l l o q u i a l term "impossible". The index measures the deviation of assigned p r o b a b i l i t i e s from the p r o b a b i l i t i e s of 100 percent and zero percent which are conventionally associated with the terms " c e r t a i n " and "impossible". The index i s normalized so that a value of 0 indicates no deviation while a value of 100 indicates extreme deviation, jd)^discrimination band_for_.probability and day stimuli Two indices were developed to measure s e n s i t i v i t y towards day s t i m u l i and probability s t i m u l i . The index for day s t i m u l i i s a li n e a r combination of Li c k e r t scales for the questionnaire items that measure s e n s i t i v i t y towards various day s t i m u l i . The index for prob a b i l i t y s t i m u l i i s defined s i m i l a r l y . The two indices are normalized so that a value of 1 indicates extreme i n s e n s i t i v i t y (wide discrimination band) while a value of 7 indicates extreme s e n s i t i v i t y (narrow discrimination band). Je) confidence in method The index for t h i s independent variable i s a linear combination of the Lic k e r t scales for the two questionnaire items that provide some measure of confidence i n methods: (1) willingness to use e l i c i t e d preferences i n l i e u of personal judgements when a choice s i t u a t i o n a r i s e s , and (2) willingness to revise spontaneous judgements which are contradictory to e l i c i t e d references. One index representing 108 confidence i n each method was developed. The indices are normalized so that a value of 1 indicates complete non-confidence while a value of 7 indicates complete confidence. | f [ realism of method scenario The value of the index for t h i s independent variable i s i d e n t i c a l to the response of the questionnaire item which requires the subject to rate the realism on a Lickert scale (7=extremely r e a l i s t i c to 1=extremely u n r e a l i s t i c ) . ftn index representinq realism of method scenario was developed for each of the two methods. In addition to the questionnaire items, four additional independent variables were defined to represent l i n e a r i t y of trade-off curves 2*: (1) RMT1= (R2 for MT curve of f i r s t session and f i r s t reference point+R 2 for MT curve of f i r s t session and second reference point)/2. (2) RVM1= (R2 for VM curve of f i r s t session and f i r s t reference point+R 2 for VM curve of f i r s t session and second reference point)/2. (3) RMT2= (R2 for MT curve of second session and f i r s t reference point+R 2 for MT curve of second session and second reference point)/2. (4) RVM2= (R2 for VM curve of second session and f i r s t reference point+R 2 for VM curve of second session and second 2 * These abbreviations w i l l be used in the following d e f i n i t i o n s and in any l a t e r discussions: MT=MacCrimmon-Toda, and VM=von Neumann-Morgenstern. 109 reference point)/2. Independent Variables,- of Results Figure 14 summarizes the responses for each of the independent variables for a l l three groups. The independent variable, characterizing acceptance of " r a t i o n a l i t y " axioms, indicates a central tendency towards acceptance (median i s greater than 4). The medians f o r the independent variable representing propensities for certa i n judgemental modes of evaluation, show that at least 501 of the subjects do not display a strong p o s i t i v e attitude. Furthermore, the medians for the independent variables representing confidence in method, show that at least 50% express a neutral or non-confidence in both methods of e l i c i t i n g preferences; the medians of the independent variables representing realism of scenario for each method, show that at least 50% display neutral b e l i e f . The mean R 2 values for the derived indifference curves are close to 1.00 for both undergraduate and graduate groups. I t i s quite conceivable that: (1) the subjects conscientiously follow a lin e a r r u l e , or (2) the subjects do not make a conscientious e f f o r t to follow a l i n e a r rule but the methods of e l i c i t i n g preferences induce them to provide judgements which r e l e c t l i n e a r i t y . The mean R 2 for the von Neumann-Horgenstern derived curves are higher than for the corresponding MacCrimmon-Toda derived curves. I f the subjects do follow a simple l i n e a r rule f o r response 110 judgements (suggested by a high R 2), the difference in R 2 values could be due to the fact that i t i s computationally more demanding to apply t h i s rule to a choice between pairs of gambles (MacCrimmon-Toda method) than i t i s to apply t h i s rule to choosing a sure-thing which i s judged to be i n d i f f e r e n t to a gamble (standard gamble method). Linear Associatign§_amonq Independent Variables Figure 15 displays the Spearman cor r e l a t i o n matrix of the independent variables. The c o r r e l a t i o n (0.50) between acceptance of " r a t i o n a l i t y " axioms and propensities for judgemental modes of evaluation i s s i g n i f i c a n t l y high. In other words, acceptance of p a r t i c u l a r " r a t i o n a l i t y " axioms i n simple preference situations correlates p o s i t i v e l y with agreement that evaluation of preferences should follow a l o g i c a l , consistent, and unemotional path of reasoning. The correlations between confidence in each method of e l i c i t i n g preferences and the propensities for judgemental modes of evaluation seem to suggest that subjects who agree with the appropriateness of the general " r a t i o n a l " modes of evaluation, also agree with the appropriateness of two pa r t i c u l a r modes of evaluation, namely the standard gamble method and the HacCrimmon-Toda method. The hypothesis that subjects who are confident in one p a r t i c u l a r method for e l i c i t i n g preferences w i l l also be confident in the ether method i s supported by the s i g n i f i c a n t l y positive c o r r e l a t i o n between confidence in the standard gamble method 111 Undergraduates N-14 Graduates »~9 Total N-23 C a « £ c • •a • £ 6 « c s. Mean c 09 -H £ • 6 Range . c 0) • ii a c a —H TJ V r- d c a ce ACCPPTArCS 0 ? "RATIONALITY" AXIOMS' 5.2 5 . 6 1 . 4 3 .0 -7 .0 5.5 5 . 5 1 . 0 4.0 -7 .0 5.4 5 . 6 1.2 3 . 0 - 7 . 0 PROPENSITIES FOR JUDGSI.3NTAL NODES 4.8 4 . 5 0 . 9 3 . 6 - 6 . 6 4 .5 4.4 0.6 3 . 6 - 5 . 6 .4.7 4.4 0.8 3 . 6 - 6 . 6 ' BIAS IN PROBABILITY INTERPRETATION M 0.4 7 . 4 0 . 0-22 . 5 . 2.3 1 . 0 3.4 0.0-10.0 3 . 6 0.4 6.1 0.0-22*5 DISCRIMINATION FOR PROBABILITY 4.2 4 . 5 1 . 0 1 . 8 . - 6 . 0 3.9 3 . 5 0.9 2 . 6 - 5 . 8 4.1 4.2 1.0 1.8-6.0 DISCRIMINATION FOR DAY 2.8 2.9 0 . 8 1 . 6 - 4 . 6 3 . 3 3.2 l i l 2 .0 -5 .0 3.0 2.9 0.9 1 .6 -5.0 •CONFIDENCE'IN EACCRIKiVON-TODA VZTHOD 3 - 3 3 . 2 1 . 1 2 . 0 - 5 . 5 3 . 5 3 . 3 1 . 5 • 1.0-6.0 3.4 3 . 3 1.2 1 . 0 - 6 . 0 CONFIDENCE IN STANDARD GAMBLE METHOD 3 . 5 3 . 5 1 . 0 2 . 0 - 6 . 0 3 . 6 3 . 7 1 . 5 1.0-6.0 3 . 5 3 . 6 1.2 1.0-6.0 REALISM OP f.'ACCRIKMON-TODA METHOD 4.8 5 . 3 1 . 5 1 . 0 - 6 . 0 4 . 5 4 . 6 1 . 5 2 . 0 - 7 . 0 4.7 5.0 1 .5 1.0-7.0 REALISE OF STANDARD CA.VBI.E METHOD 3.0 3.0 1.2 1.0-5.0 4 .6 4 . 5 1.4 3.0-7 .0 3 . 6 3.4 1.4 1.0-7.0 RMT1- LINEARITY OF 1ST SESSION KT CURVES 0 .91 0.92 0 . 0 3 0 . 8 6 - 0 . 9 8 0 . 9 4 0 . 9 6 0.07 O.76-I.OO O .92 0 . 9 3 O.05 0 .76-1.00 . RVI.U- LINEARITY OF 1ST SESSION V CURVES 0 . 9 6 0.99 0.04 0.88-0.99 0 . 9 9 0.99 0.00 0.97 -1 .00 0.97 0 . 9 9 0 . 0 3 0.88-1,0C ROT2- LINEARITY OF 2ND SESSION CT CURVES 0 . 9 1 0 .92 0 . 0 6 0 . 7 0-0 . 9 7 0 . 9 6 0 . 9 6 0 .03 0 . 9 1 -1.00 0 . 9 3 0 . 9 3 0 . 0 5 0 .70-1.OC RVM2- LINEARITY OF 2ND'SESSION TO CURVES 0 . 9 8 0 . 99 0,02 0 . 9 1-0 . 9 9 O..98 0.98 0 .01 0 . 9 6 - 1 . 0 0 0 . 9 8 0 . 9 9 0.02 0 . 9 1 - 1 . 0 0 Figure 14 - Independent Variables - p r o f i l e of Results and confidence i n the HacCrimraon-Toda method. 112 There i s a s i g n i f i c a n t l i n e a r c o r r e l a t i o n of acceptance of p a r t i c u l a r " r a t i o n a l i t y " axioms with confidence i n the MacCrimmon-Toda method but not with confidence in the standard gamble method. It i s quite conceivable that those who accept the appropriateness of r a t i o n a l behavior i n nonpersonal s i t u a t i o n s , express a similar attitude in personal situations only to a certain extent, i . e . are confident i n the predictions made by the MacCrimmon-Toda method (based upon weak assumptions of behavior) but are not confident i n the predictions made on the basis of the standard gamble method (based upon stronger assumptions of behavior). Linear Associations between Independent and Dependent Variables Figure 16 displays the Spearman correlations between the independent and dependent variables. The s i g n i f i c a n t positive c o r r e l a t i o n between (MT2-VM2)2 and propensities for r a t i o n a l modes of evaluation i s incompatible with our hypothesis since i t implies that for those who accept " r a t i o n a l " modes of evaluation, predictions based on alte r n a t i v e methods tend to show less correspondence. The s i g n i f i c a n t c o r r e l a t i o n between the discrimination for probability s t i m u l i with (VM1-VH2)2 but not with (MT1-MT2) 2 i s surprising since the MacCrimmon-Toda method 1 1 3 N=23 ACCEPTANCE OP "RATIONALITY" AXIOMS o K H O C fcc £ ro H i E M Z. k. K S ! C - O O O 1=3 BIAS IN INTERPRETATION OF PROBABILITY DISCRIMINATION FOR PROBABILITY cc o z o • I-l E-•< ' Z t-t Si t-i ze. O l O CO >->~ •< ca • z o sz £ CC o u «< 2K Z M ' o fcj O O X Z 6-fcl W C 2. »-t &.«: z o o o O f - CONFIDENCE IN STANDARD GAMBLE METHOD o I z o t-H cc o o O I-l < =c E . < * . S o o CO E CO o l-l o X •"•JE* • K W CC l i REALISM OF STANDARD GAMBLE METHOD SCENARIO ACCEPTANCE 0? "RATI0NALITY"AXI0KS •.50 +.30 +.38 + .52 PROPENSITIES FOR JUDGEMENTAL MODES +.51 +.69 + .50 BIAS IN INTERPRETATION OF PROBABILITY . DISCRIMINATION FOR PROBABILITY -DISCRIMINATION FOR DAYS +.33 + .29 CONFIDENCE IN -iMACCRIKMOK-TODA METHOD + .83 +.31 CONFIDENCE IN STANDARD GAMBLE METHOD REALISM OF MACCRIMMON-TODA METHOD SCENARIO REALISM OF STANDARD GAMBLE METHOD SCENARIO Only correlations significantly nonzero at 0.10 confidence level are shown. Figure 15 - Spearman Correlation Hatrix for Independent Variables requires subjects to make judgements upon choices involving 114 various l e v e l s of probability s t i m u l i , while the standard gamble method requires subjects to make judgements upon choices involving only the probability 1/2. The s i g n i f i c a n t correlations of the inconsistency measures with the l i n e a r i t y measures seem to suggest that l i n e a r i t y i s the dominant feature a f f e c t i n g consistency. The adoption of a simple rule for combining att r i b u t e s of outcomes reduces computational errors in evaluating preferences. The.Questionnaire. .Items Through the use of independent variables, questionnaire items of conceptual s i m i l a r i t y , were aggregated to form one composite score or index to represent the common concept. However, the use of an aggregated score to define an independent variable may disguise: (1) responses to in d i v i d u a l guestionnaire items which comprise that independent index (or variable), (2) associations between responses of questionnaire items comprising that index, (3) associations between responses of in d i v i d u a l questionnaire items comprising that index with responses of ind i v i d u a l questionnaire items comprising other indices, and/or (4) associations between i n d i v i d u a l questionnaire items and inconsistency measures (dependent variables). To circumvent these objections, the following discussions w i l l present (1) a p r o f i l e of questionnaire item responses, (2) a corr e l a t i o n analysis performed between questionnaire item responses, and 115 N=23 N E-« SC 1 w £-£ CJ CM > 1 > CM £ > 1 • »-» £ CJ CM f CM g CM > > at 1 1 a • ACCEPTANCE OF "RATIONALITY" AXIOMS PROPENSITIES FOR JUDGEMENTAL MODES +.35 BIAS IN INTERPRETATION OF PROBABILITY -.33 -.31 DISCRIMINATION FOR PROBABILITY -.33 DISCRIMINATION FOR DAYS -.38 -.4? CONFIDENCE IN MACCRIMMON-TODA METHOD CONFIDENCE IN STANDARD GAMBLE ^ THOD +.28 +.27 REALISM OF MACCR1KMON-TODA METHOD SCENARIO REALISM OF STANDARD ' GAMBLE METHOD SCENARIO -M RMT1- LINEARITY OF IST SESSION rcr CURVES -.50 -.83 -.66 RVM1- LINEARITY OF 1ST SESSION VM CURVES -M -M -.80 -.31 -.66 'RMT2- LINEARITY OF 2ND SESSION MT CURVES -.5* -M -.?e -.41 RVM2-LIKEARITY OF 2ND SESSION V?' CURVES • -.83 I I " 1 Only co r r e l a t i o n s s i g n i f i c a n t l y nonzero at the 0.10 confidence l e v e l are shown. Figure 16 - Spearman Correlations between Independent and Dependent Variables (3) a co r r e l a t i o n analysis performed between the 116 questionnaire item responses and the inconsistency measures (dependent v a r i a b l e s ) . Qggstionnaire R e s p o n s e s - P r o f i l e of Results Figure 17 displays the p r o f i l e of questionnaire responses for each of three groups. The most s t r i k i n g feature i s the large range of responses for most questionnaire items. Thus, although the students of the undergraduate and the graduate groups have educational backgrounds s i m i l a r to members of t h e i r group, attitudes are not homogeneous within any group. The median responses for the confidence in method items f a l l on the negative attitude half of the Lickert scale. The p r o f i l e for the p r o b a b i l i t y discrimination band indicates that the median responses (7=extremely s i g n i f i c a n t to 1=extremely i n s i g n i f i c a n t ) increases as the probability difference increases. However, t h i s relationship i s not compatible with the p r o f i l e for the day discrimination band. Thus, the median perceived significance of a probability s t i m u l i increases with increasing s t i m u l i differences while the same does not hold for day s t i m u l i . The median interpretations of " c e r t a i n " and "impossible" are close to 100% and 0% respectively. However, the p r o f i l e indicates that there were extreme responses of 70% and 35% for " c e r t a i n " and "impossible" respectively. 117 K=23 Unde N=14 C ed 0 e rgrad u c <rj •H •O 0> E iat<B 6 c u c g Gra N=9 c a to £ duateE c cs •H TJ 01 e 1 d ta c E Tot »=2 c as E a l 3 c IS •H •a o> E d range ACCEPTANCE OF "RATIONALITY" t r a n s i t i v i t y 5.0 5.5 1.7 2-7 5.8 6.1 •1.2 . 3-7 5.: 5.8 1.6 2-7 AXIOMS s u r e - t h i n g 5.2 6.0 2.0 1-7 5.8 5.8 1.0 5-7 5." 5.7 1.7 1-7 compound-gamble 5.5 6.0 1.7 1-7 5.0 5.0 1.8 2-7 5 . : 5.6 1.7 1-7 PROPENSITIES FOR JUDGEMENTAL MODES unemotional e v a l u a t i o n 4.8 5.1 1.5 2-7 5.0 5.2 1.4 • 3-7 4 . 5 5.1 1.3 2-7 p r i o r - t o - e i t u a t i o n e v a l u a t i on 4 .0 4 .0 1.7 2-7 3.5 4 . 0 1 ' 9 1-6 3.8 4.0 1.8 1-7 l o g i c a l s y s t e m a t i c e v a l u a t i o n 4.5 4.8 1.3 2-7 5.1 5.2 1.6 3-7 5.4 5.4 1.4 3-7 REALISM OF SCENARIOS r e a l i s m o f &T s c e n a r i o - 4 .8 5.3 1.5 1-6 4.5 4.6 1.5 2-7 4.7 5.0 *-5 1-7 r e a l i g n o f VM s c e n a r i o 3.0 3.0 1.2 1-5 4.6 4.5 1.4 3-7 3 . 6 3.4 1.4 1-7 " CONFIDENCE IN " METHOD el i c i t e d i n l i e u o i p e r s o n a l judgements (MT) 3.4 3.1 1.3 2-7 3.3 2.7 2 .0 1-7 3.4 3.0 1.6 1-7 r e v i s i o n oJ • " i n c o n s i s t e n t i e s " (CT) 3.3 3.1 1.4 1-6 3.6 3.7 1.6 1-6 3.4 3.2 1.5 1-6 e l i c i t e d i n l i e u oi personal judgements (VM) 3.7 3.5 1.6 1-3 3.3 2 .7 2 .0 1-7 3.5 3.1 3.7 3-7 r e v i s i o n o i " i n c o n s i s t e n c i e s " (VM) 3.4 3.1 1.6 1-6 3.8 4.0 1.7 2-7 3 . 6 3.4 1.6 1-6 DISCRIMINATION FOR PROBABILITY o.nj v s o . i 3.2 2 . 9 1.9 1-7 2.7 2 .0 2.1 1-7 3.0 2.7 3 .0 3-7 0.5 v s 0 . 5 5 3.1 3.2 1.2 '1-5 3.3 3.3 1.0 2-5 3.2 3.2 1.1 3-5 0.9 vs 0 . 9 5 3.7 3.5 1.8 1.6 3.8 4.0 1.7 1-6 3.8 3.7 1 .8 1-6 0.1 v s 0.2 4 . 6 4.8 1.1 2-6 3.7 3.2 1.8 2-7 4.3 4.5 1.4 2-7 0.3 vs 0.5 5.3 5.6 1.3 2-7 4 . 6 4.7 1.5 3-7 5.0 5.4 1.4 2-7 0.6 va 0 . 8 5.5 5.7 1.3 3-7 5.0 5.0 1.1 3-7 5.3 5.3 1.2 3-7 DISCRIMINATION FOR DAY 1/4 vs 1/2 1.9 1.3 1.6 1-7 2.8 2.2 2.2 1-7 2.3 1 .6 1.8 3-7 2 vs ?i 2 .0 2.1 0.9 1-4- 3.3 2.3 2.0 1-6 2.4 2.1 1.5 1-6 15 vs 14J 1.8 1.2 1.3 1-5- 2.2 2.0 1.2 j-4 2.0 1.4 1.2 3-5 4 V B 5 4.3 4.7 1.1 2-6 4.5 4 .8 1.0 3-6 4.4 4.7 1 . c 2-6 10 vs 9 4.0 4.0 1.4 2-6 3.6 4 . 0 1.0 2-5 4 . 0 4.0 1 •? 2-6 BIAS IN INTERPRETATION OP PROBABILITY a s s i g n p r o h a b i l i j y . t o " c e r t a i n * 94.5 99.4 1C.7 70-100 96.1 J8.7 6.9 ! lo-100 95.1 99-4 1 .9 70-100 a s s i g n p r o b a o i i i j y . t o " i m p o s s i b l e " 5.2 .0.1 9.4 1-35 0.5 0.3 1.6 o-5 2.2 0.1 7.5 o-35 ' Figure 17 - Questionnaire Responses - P r o f i l e of Results 118 Linear Associations_among the Questionnaire Responses Figure 18 displays the c o r r e l a t i o n matrix of the questionnaire item responses. The two s i q n i f i c a n t correlations among the questionnaire items characterizing acceptance of " r a t i o n a l i t y " axioms indicates that those who expressed strong agreement with the axiom of r i s k l e s s behavior ( t r a n s i t i v i t y ) also expressed strong agreement with the axioms of r i s k y behavior (sure-thing and compound-gamble). There i s no s i g n i f i c a n t l i n e a r c o r r e l a t i o n between acceptance of the sure-thing axiom and acceptance of the compound-gamble axiom. although the existence of a s i g n i f i c a n t non-linear r e l a t i o n s h i p i s possible, i t i s also quite conceivable that persons who related to the s i m p l i c i t y of the sure-thing axiom may not necessarily have been able to r e l a t e to the r e l a t i v e complexity of the compound-gamble axiom. I t i s i n t e r e s t i n g to note a s i g n i f i c a n t positive c o r r e l a t i o n between acceptance of the t r a n s i t i v i t y axiom and the need for revision of spontaneous choice (which i s found to be contradictory to e l i c i t e d preferences) for the case of the MacCrimmon-Toda method which i s based primarily upon the assumed t r a n s i t i v i t y of preferences. This association lends credence to Savage*s (1954) d e f i n i t i o n of r a t i o n a l i t y that people accept r a t i o n a l i t y i f they are w i l l i n g to revise choices which are " i r r a t i o n a l " . 119 The abundance of s i g n i f i c a n t positive correlations among s e n s i t i v i t i e s towards s t i m u l i of various probability differences may suggest that subjects maintain their r e l a t i v e (to other subjects) s e n s i t i v i t y over a l l s t i m u l i l e v e l s . The i n t e r c o r r e l a t i o n s among s e n s i t i v i t i e s towards s t i m u l i of various day differences suggests a similar result for day as for pr o b a b i l i t y s t i m u l i . This figure also shows s i g n i f i c a n t entries of the cor r e l a t i o n matrix for s e n s i t i v i t i e s toward day st i m u l i (column of matrix) with s e n s i t i v i t i e s toward probability s t i m u l i (row of matrix) which were previously hidden by the apparent nonassociation between the independent variables representing each stimulus. These correlations may have sig n i f i c a n c e for the MacCrimmon-Toda method i n which subjects were required to judge pairs of day and probability stimulus. Linear_Associations_-_Ques Responses^ ^ Dependent Variables The r e s u l t s displayed in Figure 19 do not indicate any s i g n i f i c a n t l i n e a r correlations between any pa r t i c u l a r " r a t i o n a l i t y " axioms with any inconsistency measures, although these r e s u l t s do not prelude the lack of any s i g n i f i c a n t r e l a t i o n s h i p (a non-linear association i s possible),, i t i s conceivable that subjects who agree with " r a t i o n a l i t y " may not necessarily make judgements conforming to " r a t i o n a l i t y " . I f , i n fact, t h i s explanation i s the 1 2 0 N«Z3 Only correlations slitnlficAntly noncoro at the P.10 confidence level are shown. A C " S F ~ A N C E .)F V A T I C N A L I T T " AXicrs t r . i n R i t i v i t y sure-thing conpoynfl-cnipble 0u« £22 S C O i. . . O 3 1 1 . 3 £ ^-*t c *— * r>—• tt t; >. L. > o 3 « l & *tt-t in a U1 til •a « u c (C d • c u " : l l . ' O V i> t | t3 t> t>, t l o a. » re i - ' — J •# re -t W « i« b « o « c -' t i B C * 1 > £ « tst Ml < ° CO > > o CN. © o o Si 1/1 o OO. it (Li a. »-CC f >- o K rHcnwsrits; FOR JViKr-TNTAI. »-pP2> u n e r p t i o p a l e v a l u a t i o n evaluat l^r^ p r i'c r 1 1 o - c r i v n t i o n t 7 7 f o V l r M syFteir -ATIC evp l i n t I on :5.? hi HEA11S>: OF SCENARIOS ronl imi nf Ik'neCrlmmon-fen 1 i:-nt oi" standard K12 \lOfill - *n i cit^d VP TF nlTil r e V J S I pn iieu nl r.S? e l T r i t * < i i n l i e u o f r o r ^ n r a l *uri t"ftT.pnt* (W.) 5is:.msJSATJCN FOR FSOSATIITY r e v i s i o n o i * i nprr**; r-l p n e i p a * (VV) C P S vs v.I Vs f.so •JS--Jf- >3l t i l -.30 m -3S ! -fl *2L KV 0 . 5 vp 0 . 5 5 675" 0 . 9 5 0.1 vs 0 . 2 cT.J vs 0 . 5 in Ml tit K>1 0ISCSIVINATI0K FOR DAI 1/2 vs l A 2 vs ?i 15 vs * vs 5 10 V9 9 assi£r.r.ent o f p r o b a b i l i t y HAS II: TI-TESPSETATION l -F r H O E A D I L l T Y asojpnxer.t o f p r o b a b i l i t y t e " ^ " ? [ ? f ^ ^ , * > " Figure 18 - Spearman Correlation Matrix for Questionnaire Responses actual case, the resul t demonstrates that persons who want 121 to act r a t i o n a l l y may not actually act r a t i o n a l l y . This r e s u l t lends credence to Raiffa's (1961) rationale for a normative u t i l i t y theory. B r i e f l y , Raiffa asserts that u t i l i t y theory should provide norms of behavior for those people who want to act r a t i o n a l l y but do not. The density of s i g n i f i c a n t entries i n the c o r r e l a t i o n matrix for s e n s i t i v i t i e s toward day s t i m u l i versus the r e l a t i v e sparsity of the corresponding matrix for pr o b a b i l i t y s t i m u l i may suggest that subjects allow consideration of the day s t i m u l i to dominate their choice when required to make judgements upon choices involving pr o b a b i l i t y and day, i . e . the outcome hospital days has more e f f e c t upon choice than does the probability of outcome. £21EgFi§2B-gl..ScoresMfor Inconsistency Measures To compare the methods used, we have tested for si g n i f i c a n c e of differences between scores for inconsistency measures (test-retest, intermethod, and averaged intermethod). In order to compare inconsistency scores, each subject was used as his own c o n t r o l ; the Wilcoxon matched-pairs signed-ranks test was employed to test for differences between: (1) MacCrimmon-Toda te s t - r e t e s t inconsistency and standard gamble t e s t - r e t e s t inconsistency, (2) test-retest inconsistency (for each method) and averaged intermethod inconsistency, and (3) intermethod inconsistency (for each session) and averaged intermethod inconsistency. To test for differences in inconsistency measures given 122 (NJ . rg r? H 1 1-H H CM <M > 1 rH > i — i > i rH H fM > i (M (av(MT) -av(VM) ACCEPTANCE OF "RATIONALITY" transitivity AXIOKS sure-thing compound-gamble PROPENSITIES FOR JUDGEMENTAL MODES unemotional evaluation -.44 -.28 pr.ior-to-situation evaluation + . 31 + .35 + .45 logical, systematic evaluation -.36 -.45 REALISK OF SCENARIOS realism of iVT scenarj o • realism of Vi'. scenario . -.46 -.38 COHFIUcNCE IK " VETHOD elicited in lieu of wrsor.il judircpents (VT) -.28 re v is 1 or. o; i • "incnnr.ipter.ties" (WT) i elicited in lieu ol• personal judgements (VK) -.25 -.23 -.46 revision ol " . "inconnistoncies" (VT") UlbCRi;.UKATIO!v' •• • • -FOR PROBABILITY O.Oj vs 0.1 — 0.5 vs 0.55 0.9 vs 0.95 -.38 0.1 vs 0.2 0.3 vs 0.5 -.47 -.60 0.6-vs 0.8 DISCRIMINATION FOR DAY 1/4 vs 1/2 -.29 _-_.J4 -.39 -.40 -.43 2 MS ?i -.63 -.30 15 vs 14J 4 v 6 5 + .31 -.38 -.38 10. -vs 9 BIAS IN INTERPRETATION 0? PROBABILITY . assign probr.biliiy. to "certain" + .49 + .42 assign prow;oiiity. to "ir.nossible" + .49 _+-.42„ -.68 LINEARITY B MT 1 -.50 -.82 RVM1 -.45 -,4C ) -.8( ) - . 3 : -.66 RMT2 -.5' -.4C ) -.41 RVM2 -.8: I Only correlations significantly nonzero at the 0.10 confidence level are shown. Figure 19 - Spearman Correlations - Questionnaire Responses S Dependent Variables p a r t i c u l a r p r o b a b i l i t y ranges and reference points, we have 123 sampled points from each of four probability ranges: (1) 0.00-0.25, (2) 0.25-0.50, (3) 0.50-0.75, and (U) 0.75-1.00, and evaluated respective differences between indifference curves associated with p a r t i c u l a r reference points. The terms (MT1-MT2) 2ij, (VM1-VM2)2ij, (BT 1-VM1)2ij# (MT2-VM2) 2ij, and (av (MT)-av (VM)) 2ij are defined as sum of sguared deviations between values of days associated to the two indifference curves with reference point j [ j=1 for (4,0.50) reference point and j=2 for (7,0.50) J and to sampled points from probability region i [i=1,2,3,4 correspond to p r o b a b i l i t y regions 0.00-0.25, 0.25-0.50, 0.50-0.75, and 0.75-1.00 r e s p e c t i v e l y ] . iJ§g££4gmPRz£o,3§..Yg-Standard Gamble .Test-retest Figure 20 displays the s t a t i s t i c a l r e sults of the one-t a i l e d Hilcoxon test of the n u l l hypothesis that (VM1-VM2) 2 i j= (MT1-MT2) 2 i j . In 5 of the 8 sample classes i j , the test indicates that the standard gamble method test-retest predicted preferences are more consistent than the test-retest preferences predicted by the MacCrimmon-Toda method. One suspects that differences in consistency could be due to differences i n d i f f i c u l t y of judgements reguired by each method. In f a c t , 57% of the t o t a l sample f e l t that judgements reguired from them in the standard gamble method were easier to make, 30% f e l t that the judgements i n the MacCrimmon-Toda method were easier to make while 13% f e l t that the judgements i n both methods were of equal d i f f i c u l t y . Furthermore, the r e s u l t s discussed previously show t h a t the average R 2, a measure of l i n e a r i t y , o f the standard gamble curves i s higher than t h a t of the HacCrinunon-Toda c u r v e s . These two r e s u l t s seem to suggest that ease of judgements and l i n e a r i t y c o n t r i b u t e to c o n s i s t e n c y . N=23 Only entries which indicate a significant difference at the .005 confidence level are shown. Each entry is a Wilcoxon sum of similar-signed ranks. Probability region i Reference Point j (KT1-MT2)2-. vs _ (VMl-VK2r 1 1 2 1 52 3 -1 22 k 1 61 1 2 2 2 40 3 2 45 k 2 (VH1-VM2)2< (MT1-KT2)2 for significant entries i n the column Figure 20 - MacCrimiBon-Toda vs Standard Gamble T e s t - r e t e s t 125 Test-retest (for_each^methodl_vs_Average d_ Intermethod Figure 21 displays the s t a t i s t i c a l r e s u l t s of the one-t a i l e d Wilcoxon test of the hypothesis that (MT1-MT2) 2ij=(av (HT)-av (VM)) 2ij. In the sample classes i j from the pr o b a b i l i t y ranges of 0.50-0.75 and 0.75-1.00, and the indifference curve associated with the f i r s t reference point, t e s t - r e t e s t differences i n preferences predicted from the MacCrimmon-Toda method are s i g n i f i c a n t l y smaller than the corresponding differences between indifference curves obtained for the same reference point using differences between average curves (each average curve i s obtained by applying one of the methods over two experimental sessions and averaging the two curves). The following figure also displays the s t a t i s t i c a l r e s u l t s of the one-tailed Wilcoxon test of the n u l l hypothesis that (VM1-VM2) 2 i j= (av (HT)-av (VM)) 2 i j . In 6 of the 8 sample classes i j , t e s t - r e t e s t preferences predicted from the standard gamble method are s i g n i f i c a n t l y more consistent than the averaged intermethod correspondence. One may conclude that the correspondence of predictions obtained from repeat application of each method i s generally higher than correspondence between predictions obtained using d i f f e r e n t methods. 1 2 6 N=23 Only entries which indicate a significant difference at the .005 confidence level are shown. Each entry is a Wilcoxon sum of similar-signed ranks. Probability Region i Reference > Point 2 (av(MT)-av(W) )2 V S (Wl-VT.^)2 (av(KT)-av(VM)) 2 vs -(KT1-KT2)4 1 1 2 1 34 3 1 32 67 k 1 67 1 2 66 2 2 24 3 2 28 k 2 58 (W1-VM2)2< (av(M!)-av(Viyi))2 for significant entries i n f i r s t column (KT1-KT2)2< (av(KT)-av(VM)) 2 for significant entries i n second column Figure 2 1 - Test-retest (for each method) vs Averaged Intermethod ZHi££12til2^ (for each_session) ys.Ayeraged Intermethod Figure 2 2 displays the s t a t i s t i c a l r e s u l t s of the one-t a i l e d Wilcoxon test of the n u l l hypothesis that ( M T 1 -V M 1 ) z i j= (av (HT)-av (VM)) 2 i j . Only in two sample classes i s averaged intermethod correspondence higher than intermethod preferences predicted from the f i r s t experimental session. The following figure also displays the s t a t i s t i c a l r e s u l t s of the one-tailed Wilcoxon test of the n u l l hypothesis-that (MT2-VM2) 2 i j= (av (MT) -av (VM)) 2 i j . Only i n one sample class i s averaged intermethod correspondence higher than intermethod preferences predicted from the second 127 exp e r i m e n t a l s e s s i o n . One may conclude t h a t a v e r a g i n g of repea t r e s u l t s f o r each method g e n e r a l l y c o n t r i b u t e m a r g i n a l l y to the correspondence of p r e d i c t i o n s obtained from each method. N=23 Only entries which indicate a significant difference at the .005 confidence level are shown. Each entry i s a Wilcoxon sum of similar-signed ranks. Probability region i Reference Point j (av(KT)-av(VM))2 V S (MT1-YK1)2 ' (avCCTj-avlW)) 2 vs (KT2-VE2) 2 1 1 2 1 3 1 47 4 1 73 68 l 2 2 2 3 2 4 2 (av(KT)-av(TO).)2<(KTl-VMl)2 for significant entries i n f i r s t column (av.(KT)-av(VK.))2<(r.7Dl-^l)2. for significant entries i n second colunu F i g u r e 22 -Intermethod Intermethod ( f o r each session) vs Averaged 128 Goodness of Prediction An average of k pairs of diammetrically opposed gambles were presented to each s u b j e c t " . The re s u l t s showed that an average of 72% of the preferences expressed by each subject were i n accord with predictions made by the MacCrimmon-Toda method. 2 5 From the day X pro b a b i l i t y choice space, we sampled points which l i e between the indifference area derived by the MacCrimmon-Toda method i n the second session and the indifference curve derived from the standard gamble method in the second session. 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Our r e s u l t s N=23 collonu.ial phrase c^ertain extrernety likely " "hi"grTy" procaine" ""probable ~ "uncertain'"" unlikely " i i n n b s s i b T e Results of Lichtenstein and Newman (I967) N=180 8p~_ • 90 7"r__{7_5"" Vo T5o~ 18 " "ffS" 0^0 0.1" :?0 -Q0 3 -oq 0.1 8 0.1 0 Figure 2 3 - Numerical Estimates of Probabil i t y Expressions 142 Appendix L B This Appendix groups together questionnaire items which are relevant to a p a r t i c u l a r independent variable and within each group provides a description for each item. The questionnaire items are denoted by b r i e f descriptive expressions. ^Sgg£-§Bc§,.,9l--Ig^-9S§-4^I?!^ &gJ9S?§ t r a n s i t i v i t y George prefers driving a Ford Pinto to a Toyota MK I I . Furthermore, he prefers driving a Toyota MK II to a Datsun 1600. Yet, from a rent-a-car which offers a Datsun 1600 or a Ford Pinto at the same rental rate, George rents a Datsun 1600 instead of a Ford Pinto. Realizing t h i s "inconsistency", George should change his choice to Ford Pinto. 7=strongly agree-6-5-4-3-2-1=strongly disagree (2) sure-thing Upon entering his l o c a l confectionery store. B i l l decides to spend a dime on either a bag of j e l l y beans or a grab bag which contains either j e l l y beans or chocolate. Although he prefers eating j e l l y beans to eating chocolate. B i l l spends the dime on a grab bag rather than on a bag of j e l l y beans. In l i g h t of his actual preferences, B i l l ought to reverse his decision and spend the dime on a bag of j e l l y beans instead. 7=strongly agree-6-5-'4-3-2-1 = strongly disagree J31 _comjJound-gamblg_ A sweepstake t i c k e t e n t i t l e s the holder to either a prize of $1.oo or a chance in the grand f i n a l draw. The grand f i n a l draw prize w i l l be either $100.00 or $1.00. Another sweepstake t i c k e t e n t i t l e s the holder to a prize of $1.00 or $100.00. Both sweepstake t i c k e t s s e l l for the same price. Taking the chances of winning into account, Dan calculates that the p r o b a b i l i t i e s of winning each prize are the same for both sweepstakes. In sp i t e of th i s information, Dan i n s i s t s upon buying the second sweepstake t i c k e t and i s even w i l l i n g to pay s l i g h t l y more for t h i s t i c k e t . Dan should stop favoring the second sweepstake. 7=strongly agree-6-5-4-3-2-1=strongly disagree Propensitiesfor_Judgemental, Modes of Evaluation _Q] unemotional evaluation In health matters, people ought to c a r e f u l l y evaluate t h e i r preferences among alternatives without being influenced by their mood or emotion at the moment of evaluation. 7=strongly agree -6-5-4-3-2-1=strongly disagree J 2 _ . p r i o r - t o - s i t u a t i o n evaluation In matter concerning i l l n e s s , people ought to evaluate t h e i r preferences among altern a t i v e s before the i l l n e s s actually occurs because under pain and discomfort they may not be cl e a r l y aware of the i r preferences. 7=strongly agree -6-5-4-3-2-1=strongly disagree .{31. JO-^SAl, s Y s t e m ^ ^ 3 ; S _ § y § J 7 U a ^ i 2 9 Suppose that in r e a l l i f e you had to make a decision in a choice s i t u a t i o n involving several a l t e r n a t i v e s . How important do you f e e l that i t i s for you to analyze your preferences in a l o g i c a l systematic manner (as was done in the experimental sessions) before making a decision? 7=extremely important-6-5-t-3-2-1=extremely unimportant 145 Realism of Scenario How r e a l i s t i c do you f e e l that the ... scenario i s ? 7=extremely realistic-6-5-4-3-2-1=extremely u n r e a l i s t i c Confidence in Method _ Q ] e l i c i t e d ^ i n l i e u of personal judgements. Suppose that with reference to a p a r t i c u l a r hypothetical scenario concerning a health matter, a trained health personnel derives your indifference (or u t i l i t y ) curve. If a s i t u a t i o n resembling the scenario a r i s e s i n r e a l l i f e , would you l e t a physician determine the decision for you from a c a r e f u l consideration of your indifference (or cardinal u t i l i t y ) curve? 7=without any doubt-6-5-4-3-2-1=with complete doubt _£2) r e v i s i o n of "inconsistencies^ Suppose that you were to actually encounter a s i t u a t i o n where you had to compare two alternatives each involving a domain of choices as presented i n the ... scenario? Furthermore suppose that the decision you actually make does not conform with your e l i c i t e d preferences. In l i g h t of t h i s information, how important do you f e e l that i t 146 i s for you to change your decision? 7=extremely important-6-5-4-3-2-1=extremely unimportant Rate the significance of the following differences between p r o b a b i l i t i e s on a 1 to 7 scale: (1) occurrence with probability 0.5 as opposed to 0.55 (2) occurrence with probability 0.9 as opposed to 0.95 (3) occurrence with probability 0.1 as opposed to 0.2 (4) occurrence with pr o b a b i l i t y 0.6 as opposed to 0.8 (5) occurrence with probability 0.3 as opposed to 0.5 (6) occurrence with probability 0.05 as opposed to 0. 1 7=extremely significant-6-5-4-3-2-1=extremely i n s i g n i f i c a n t fiiscrij^nation_for_Daj Rate the significance of the following differences between number of days of rest in bed on a 1 to 7 scale: (1) 1/2 day of rest i n bed as opposed to 1/4 day (2) 15 days of rest in bed as opposed to 14 147 1/2 days (3) 4 days of rest i n bed as opposed to 5 days (4) 2 days of rest in bed as opposed to 2 1/4 days ' (5) 10 days of rest in bed as opposed to 9 days 7=extremely significant-6-5-4-3-2-1=extremely i n s i g n i f i c a n t Bias i n Interpretation of Pr o b a b i l i t y What p r o b a b i l i t i e s do you associate with the following words (or phrases): (1) cert a i n (2) unlikely (3) highly probable (4) uncertain (5) probable (6) impossible (7) extremely l i k e l y 

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