Applied Intelligence 20, 147–163, 2004 c 2004 Kluwer Academic Publishers. Manufactured in The United States.
A Behavioral Choice Model When Computational Ability Matters SHU LI∗ Office of the President, Hwa Nan Women’s College, China
[email protected]
Abstract. This paper presents a model of how humans choose between mutually exclusive alternatives. The model is based on the observation that human decision makers are unable or unwilling to compute the overall worth of the offered alternatives. This approach models much human choice behavior as a process in which people seek to equate a less significant difference between alternatives on one dimension, thus leaving the greater one-dimensional difference to be differentiated as the determinant of the final choice. These aspects of the equate-to-differentiate model are shown to be able to provide an alternative and seemingly better account of the prominence effect. The model is also able to provide an explanation and prediction regarding the empirical violation of independence and transitivity axioms. It is suggested that the model allows understanding perplexing decision phenomena better than alternative models. Keywords: intuitive abilities, behavioural choice model, prominence effect, independence axiom, transitivity axiom 1.
Introduction
Individuals are constantly faced with the need to choose one alternative from a mutually exclusive set of alternatives. Making our choices amongst competing alternatives is a comprehensive social and psychological process. For example, being a consumer in the Information Age is both exhilarating and mind-boggling. There are vast amounts of goods and services available to us, and the range is increasing day by day. We are faced with decisions every minute: To buy or not to buy? Which features to purchase? Which brand to use? Moreover, consumers are often asked to make difficult value tradeoffs, such as price versus safety in purchasing an automobile, or environmental protection versus convenience in a variety of goods. Various decision rules have been developed to help the decision-maker reach a unique decision outcome. Svenson [1] introduces a representation system for the description of decision alternatives and decision rules. The rules are classified according to ∗ Present
address: Office of the President, Hwa Nan Women’s College, Fuzhou, Fujian 350007, P.R. China. Email: lishu@hnwomen. com.cn;
[email protected]
their metric requirements (i.e., metric level, commensurability across dimensions, and lexicographic ordering) in the system. Stevenson et al. [2] outline the general rules that determine the characteristics of choice. They organize a variety of preferential choice rules by crossing three factors: compensatory versus noncompensatory, dimensional versus holistic, and deterministic versus probabilistic. The decision rules which are accessible for making the final choice are so liberal that studies on rule selection are much needed. A major conclusion drawn from relevant studies (see [1, 3–5]) is that individuals tend to use diverse decision rules or use a combination or sequence of different decision rules to make a choice, contingent on task and context. The present paper postulates a representational system for describing choice alternatives and uses this system to demonstrate that the human decision-maker may be coherently ruled by a single decision strategy across various situations. The decision rule to be presented for much human choice behavior suggests that, in the case of pairwise choice where each alternative is generally better than the other on a single dimension, individuals seek to equate offered differences between
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alternatives on one dimension, in order to differentiate the unequated one-dimensional difference as the determinant of the preferred alternative. That is, the offered alternatives are assumed to be equivalent on one dimension, and the alternative with the greater utility on the other dimension will then be chosen. Dominance detecting would then be a special case of the equateto-differentiate rule. The present model is not guided by either a philosophy of “aim for the best” or of “aim to avoid the worst” in a consequentialist manner. As a choice rule, this bi-choice model is a rule of approximate estimation. Its philosophy is simple: when unable to calculate the area of the circle by applying the formula πr 2 , just do it by utilizing the formula a × b. This is most easily seen by way of graphical illustrations shown in Fig. 1. Decision-makers who utilize the equate-to-differentiate rule would be clearly aware of its drawback in that the formula a × b is never “something” like πr 2 in calculating the area of the circle, and that this may have certain unwanted consequences with regard to consistency. In doing this, there will always be some loss left which will in fact cause perplexing paradoxical patterns of behavior or violations of normative decision axioms. However, an accurate quantitative approach represents cognitively complex and sophisticated strategies for information integration. Without the aid of Decision Support Systems (e.g., expert systems, artificial intelligence systems, and executive information systems) [6], it may be impossible
Figure 1(b). Rectangles inscribed within a circle (from Hsiao TaoTshun’s Hsiu Chen ThaiChi Hun Yuan Thu, c. +11th century). Adapted from Fig. 84, Needham [37, p. 144], with permission.
for alternatives to be ordered on some judged continuum by human decision-makers. To choose from nondominated alternatives, the present alternative way is to “equate” those conflicting dimensional differences (cf., to consciously round off the residual area) so as to make a weak dominance rule (cf., a × b) applicable. Such heuristic choice allows evaluation to proceed without facing the difficulties (computational and emotional) of making trade-offs. After being equated, the differentiated alternative is analogously seen as the greater inscribed rectangle that approaches the area of the circle. In proposing such a model the author has attempted to postulate a coherent model which is able to cope with empirical evidence and which also accounts for preferences that are considered anomalous by existing normative theories.
2.
Figure 1(a). Integration of the thin rectangles in circle area measurement, from the Kaisan-ki Komoku (+1687) of Mochinaga Toyotsugu and Ohashi Takusei, derived from Sawaguchi Kazuyuki’s Kokon Sampo-ki (c. +1670). Adapted from Fig. 83, Needham [37, p. 144], with permission.
The Model
Most choice problems encountered in the real world fall into the category of multiattribute evaluation, i.e., the alternatives to be chosen usually involve consequences measurable on several, often conflicting, attributes or traits. Each alternative is generally conceived of as having a subjective value, to be elicited by some appropriate means. The present model assumes that a
A Behavioral Choice Model
decision-maker’s cognitive representation of the choice alternatives can be described by reference to a number of dimensions. In order to state the problem more generally, consider a subjective space with M dimensions representing a binary choice between A and B. Define D j as the jth dimension in M-dimensional space and x j as the objective value of each alternative with respect to the jth dimension. The present model assumes that any given dimension D j undergoes a transformation which is a monotonic utility function u A j = U A j (x j ) or u B j = U B j (x j ) at a specific moment for a specific individual. It expresses the subjective evaluation of all objectively conceivable consequences along that dimension. Obviously, dominance and even weak dominance is not applicable in most cases. Weak dominance states that if alternative A is at least as good as alternative B on all attributes, and alternative A is definitely better than alternative B on at least one attribute, then alternative A dominates alternative B (cf. [7, 8]). When a dominant alternative exists, it is unambiguously the best alternative available and therefore no further analysis is required. Dominance serves as the cornerstone of the normative theory of choice, so any rational choice rule would always select the dominant alternative. Instead, in order to utilize all available information to reach a final choice decision the human decisionmakers must construct an overall ordinal utility function such as: v A = V (u A1 , u A2 , . . . , u A j , . . . u AM )
or
v B = V (u B1 , u B2 , . . . , u B j , . . . u B M ) The question arises as to whether we possess the cognitive apparatus required to evaluate such a function. If the answer is positive, then there exist two approaches to evaluating the alternatives. One is the holistic approach, which is to carefully examine each alternative and assign to it a number which reflects its overall value. The other is an approach which decomposes each alternative into its component dimensions (or attributes), valuing each dimension (or attribute) separately, and then combines them in order to map the individual utilities onto the overall valuation for each alternative. In doing this there are at least two decision rules: the additive rule (see [9], Ch. 3) and the averaging rule (see [10]), which both ultimately lead to the evaluation of a decision alternative. This means that with the construc-
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tion of an overall function v, a more thorough evaluation of the available information could be provided, and tradeoffs between different dimensions could be allowed. In short, if, in spite of the evidence to the contrary, the answer to the question concerning whether or not we possess the necessary cognitive ability is “yes”, then it will always be possible to find an overall value to function as an index of preference. The basic assumption of the present model is that the answer to the question, about whether or not humans possess the relevant cognitive abilities to cope with an overall ordinal utility function, is in the negative. Without computational aids, humans are assumed to be unable or unwilling to utilize all available information by aggregating all the individual utilities or utility differences to form an overall subjective value. However, those lacking a huge effort are assumed to be able to utilize all relevant information in an alternative way, as follows. The present model postulates that, in order to utilize the very intuitive or compelling rule of weak dominance to reach a binary choice between A and B in more general cases, the final decision is based on detecting A dominating B if there exists at least one j such that U A j (x j ) − U B j (x j ) > 0 having subjectively treated all U A j (x j ) − U B j (x j ) < 0 as U A j (x j ) − U B j (x j ) = 0, or, detecting B dominating A if there exists at least one j such that U B j (x j ) − U A j (x j ) > 0 having subjectively treated all U B j (x j ) − U A j (x j ) < 0 as U B j (x j ) − U A j (x j ) = 0, where x j ( j = 1, . . . , M) is the objective value of each alternative on Dimension j (for an axiomatic analysis, see [11]). Subsequent sections will address the question of how alternatives can be equated through utility transforming. In the equating process some information is lost in terms of utility differences on the remaining dimensions since, in the present model, not all utility differences are to be summed across dimensions or across alternatives to produce a cumulative preference. Thus, the resulting model differs from other normative decision models in that it does not necessarily require that the decision-maker obeys all the rational decision rules, such as independence, transitivity, invariance and others which have resulted assuming the truth of an overall ordinal utility function. It will then always be possible with special laboratory-constructed designs to observe so-called paradoxical patterns of decision behavior which are so labeled within the present choice models involving judgment-based criteria (for experimental demonstration see [12–15]).
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In summary, the present model is noncompensatory regarding overall judgment because it does not allow deficiencies on one dimension to be compensated for by high values on another dimension. It is not holistic but dimensional, because all of the alternatives are evaluated along one dimension before considering the next dimension. Finally, although the model allows for choice variability across repetitions, it is not necessarily a probabilistic but an “individual” deterministic model. It is individual for two possible reasons. One is that utility functions vary with the individual decisionmaker, while each person’s utility functions vary with occasion and situation. The second reason is that the strategy of deciding which dimensional difference is to be equated and which is to be differentiated is individual dependent. 3.
ordered from human life, which is the most important, then love, freedom, time, and money, in that order. For the Chinese, the order is human life, freedom, love, time and finally money. A slight difference in order
Equating Less Significant Differences or Basing a Decision Only on More Important Attribute
Among a wide range of decision rules explored, such as conjunctive, disjunctive, elimination by aspects, majority of confirming dimensions, satisficing, addition of utilities, addition of utility differences and random choice, one decision rule which is intuitively appealing is that individuals tend to choose the alternative that is superior in the more important attribute. The present experiment was designed to test the implications of the prominence hypothesis [16, 17] against the equateto-differentiate hypothesis. The Traffic Problem and Benefit Problem used by Tversky et al. [17] were employed in the present experiment as choice stimuli. It is reported by Tversky et al. [17] that most respondents choose according to the more important attribute offered by these two problems. The present experiment involved, in addition, a rating task and a matching task, which were designed to help test the prominence hypothesis and the equate-to-differentiate hypothesis in further detail. In this experiment, 33 mathematics students at Fujian Normal University, 47 psychology students at the University of New South Wales and 80 business students at Nanyang Technological University participated as subjects. Each subject responded, in this order, to a choice, a matching and a rating task. The choice, matching and rating tasks are shown in Appendix A exactly as they were posed to subjects. The rating results are shown in Fig. 2. For Australian and Singaporean, the average importance scores are
Figure 2.
Rating results.
A Behavioral Choice Model
(but vital in ideology) lies in that the Australians and Singaporeans rank love higher than freedom, while the Chinese rank freedom above love. The choice pattern for the Traffic problem reported by Tversky et al. [17] was that 67% of the 96 respondents favored Program A over Program B. A similar pattern of responses was observed in the present experiment. However, the credibility of the prominence account of these data is still open to question. If the prominence hypothesis provides an adequate explanation, it is both reasonable and plausible to expect that, among the minority who favor the more economical Program B that saves fewer lives, there should be a fair number of respondents who would rank the cost as more important than the casualties. In fact, 90% of the majority who favored Program A, which saves more lives at a higher cost per life saved, rated life more important, but only 24% of the minority who preferred Program B rated money as more important. This is shown in the 2 × 2 table generated by the choice and rating results in Table 1. Such a disequilibrium in the proportions does not provide sufficient evidence to support the prominence account. A variance analysis reveals that prominence significantly accounts for about 4% (eta squared) of the variance in choice. The present Benefits Problem was used in its original form [17, p. 374]. However, the treatment of the attributes raises some potential problems with the credibility of the prominence account. In their original analysis, Tversky et al. [17] assumed that the earlier payment (in 1 year) acts as the primary attribute, and the later payment (in 4 years) acts as the secondary at-
Table 1. 1–3.
Contingency table for choice and rating data for Choices Choice
Problem
A
B
Traffic (Choice 1)
Rating Life super (85) Money super 9
50 (16)
Benefit (Choice 2) (Treatment 1)
Rating 1 year super 4 year super
(78) 0
82 (0)
Benefit (Choice 2) (Treatment 2)
Rating Time super (51) Money super 27
56 (26)
Modified benefit (Choice 3) Rating Time super (71) Money super 50
26 (13)
Note. The data in brackets are numbers of respondents who choose according to prominence hypothesis.
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tribute. Here the problem is brought about by the ordering of the prominence attribute. Surely people will prefer to receive a payment sooner rather than later. Hence, if we follow Tversky et al.’s analysis, that the earlier payment is one attribute, while the later payment is another, it will be almost certain that the attribute which is assumed as the primary one will be accepted by subjects as that. It is therefore not necessary to verify that the attribute presented is actually perceived as the primary one, as we did before in the Traffic problem. Since the problem itself has almost certainly determined that all the respondents will accept the earlier payment (in 1 year) as the primary attribute, it is not possible for the prominence account to explain the observed behaviors of those who preferred Plan B (51%, N = 160). A 2 × 2 table was generated (see Table 1) by the choice data and the assumed rating data. These data allow us to deduce the proportion of the choice variance accounted for by prominence. If we do not accept Tversky, Sattath and Slovic’s analysis, but treat the “amount of payment” and the “time of payment” independently as two attributes, it will also result in difficulties for the prominence explanation. Suppose that the “amount of payment” acts as one dimension and the “time of payment” as another (see Fig. 3, Benefit Problem—Representation 1), and also suppose that the prominence effect would result in the modal pattern as suggested by Tversky et al. [17], i.e., that Plan A is more popular (their results: 59%, N = 36). If “time of payment” is the primary attribute, the prominence hypothesis, or even a probabilistic version of the lexicographic rule, postulates that the alternative with the best value on this attribute would be selected. However, no one alternative can be uniquely identified for selection because both alternatives A and B share the best value (1 year) on this attribute, thus not resolving the conflict among the acceptable alternatives. If “amount of payment” is the primary attribute, the prominence or lexicographic account requires that the alternative which is the most attractive on the “amount of payment” attribute ($4,000) should be selected, which contradicts the modal pattern data presented by Tversky et al. [17]. An analysis of the contingency table reveals that prominence accounts for only about 0.1% of the variance in choice based on the choice and rating results presented in Table 1. To skirt round these difficulties and make the prominence effect applicable and testable, the next experiment modifies the original Benefits Problem. The
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(a)
(b)
(c)
(d) Figure 3. (a) Traffic problem. (b) Benefit problem (Representation 1). (c) Modified benefit problem. (d) Benefit problem (Representation 2).
Modified Benefits Problem is based on the original scenario, but the profit-sharing plans are simplified. The ordering of the prominence attribute is no longer made mandatory, and the ordering of the levels within each attribute is no longer overlapped. As with the Traffic problem presented earlier, if the interpretation that the payment time acts as the primary attribute, and the payment amount acts as the secondary attribute, can account for the overwhelming earlier-payment preferred behavior (76%, N = 160), it is reasonable to predict that those who prefer to receive a payment earlier would rate time as more important, while those who prefer to receive a larger payment would rate money as more important. The observed result is presented in Table 1. This time prominence accounts for only 0.5% of the variance in choice. In conclusion, the foregoing analysis has indicated that the prominence effect does not provide an adequate explanation of the causes of choice. The present data also serve to provide a test of the equate-to-differentiate hypothesis. With the Traffic problem and the Modified Benefit problem, the two offered attributes may be treated independently as two dimensions representing the alternatives, where the two alternatives A and B can be represented as two unique points in the 2-dimensional space (see Fig. 3, Traffic problem and Modified Benefit problem). With the original Benefit problem, its structure is somewhat special despite the fact that all the objective values of the offered alternatives are with respect to only two attributes, “amount of payment” and “time of payment”. As can be seen from the above discussion, the conceivable treatment of the “amount” and “time” information will always be such as to question the viability of the prominence hypothesis. A cognitive representation is shown in Fig. 3 (Benefit Problem—Representation 2). In such a representation, the two dimensions representing the two profit-sharing plans are “payment in 1 year” (dimension 1) and “payment in 4 years” (dimension 2), where the outcomes within each dimension are monetary values which are weighted by payment due-time. Expressed in this way, all the choice alternatives of the three problems are represented by three 2dimensional spaces respectively, and all objective outcomes along each given dimension are expressed on at least an ordinal scale. Now, the equate-to-differentiate rule predicts that a choice is based on the differences between the utilities of different alternatives on only one exceptional dimension. To express this operationally,
A Behavioral Choice Model
the outcomes of alternatives on each dimension are paired, as in the earlier matching task. If the equate-todifferentiate’s one-dimensional difference account is correct, then the knowledge of the chosen pair will permit explanation or prediction of option preference. That is, if the subject’s utility functions make one pair of outcomes the “most equivalent”, then the alternative with the greater utility value on the “most different” dimension will be chosen. Take the Modified Benefit problem as an example. The two alternatives offered, Plan A and Plan B, were represented in a 2-dimensional space (Fig. 3, Modified Benefit problem). The outcomes of the two alternatives on the “time of payment” dimension were paired as “payment in 1 year” vs. “payment in 4 years”, whereas the outcomes of the two alternatives on the “amount of payment” dimension were paired as “payment of $2,500” vs. “payment of $6,000”. If the subject thinks that one of the two pairs is the “most equivalent” according to his or her utilities, he or she will choose the alternative with the better outcome in the “most different” pair. That is, if C (D) is circled as being most different, then B (A) will then be chosen, and vice versa. The observed results of these equate-to-differentiate predictions across all the three choice problems are shown in Table 2. An analysis of the contingency table reveals a large effect (eta squared) of the “matching” of paired outcomes on choice, that is, matching significantly accounted for 34, 18 and 19% of the choice variance in Choice 1, Choice 2 and Choice 3 respectively. Taken together, knowledge of the importance of all the attributes does not permit a satisfactory explanation or prediction of the observed choice preferences, but knowledge of paired “most different” outcomes chosen by subjects does.
Table 2. Contingency table for choice and matching data for Choices 1–3. Choice Problem
A
B
Traffic (Choice 1)
Matching
C D
(70) 24
10 (56)
Benefit (Choice 2)
Matching
C D
(44) 34
13 (69)
Modified benefit (Choice 3)
Matching
C D
23 (98)
(26) 13
Note. The data in brackets are numbers of respondents who choose according to equate-to-differentiate hypothesis.
4.
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Differentiating Greater One-Dimensional Difference or Computing Overall Worth
The optimizing decision-making model rests on the proof that each alternative chosen by the decisionmaker could be taken as evidence that preferred alternatives have higher numbers, if preferences obey several simple axioms. When the choice patterns that violate the axioms have been pointed out, the theoretical avenue taken has usually been to attempt to modify the shape of the value function so as to allow the chosen alternative to be a computational or judgmental greatest “something”. This is the case from linear value function where v(x) could be just x, to nonlinear diminishing marginal utility, to negative acceleration of gain function, to positive acceleration of loss function, to steeper loss function, to asynchronous shifts in reference point, to additional inflection points (see [18], Fig. 2) and so on. Unlike the generalized utility models, the equateto-differentiate model does not require further modification for the shape of utility function that was derived from psychophysical measurements. This is the case even if such a utility function fails to demonstrate that decision-makers are trying to maximize “something” in a process of utility integration. It is assumed that the role non-linear utility functions play in the binary choice procedure is mapping objective outcomes into their subjectively commensurate counterparts, thus leading the decision-maker to decide whether a given dimensional difference should be considered as the determinant of the final choice. The two choice problems (Choices 4 and 5), shown in Appendix B, illustrate how utility functions in the choice situation where the objective consequences along the dimension lie on a magnitude scale. The first problem was presented in questionnaire form to 66 students from Fujian Building Materials Engineering Institute. The second choice problem was presented to 175 students from the Northwest China Institute of Telecommunications Engineering, Min Jiang University and Fujian Building Materials Engineering Institute. In Choice 4, if decisions were based on the mission requirement, or the degree of short supply, or the amount of additional payment, the decision-makers should choose the two alternatives offered equally often. However, an even choice is in fact observed when the additional payment is 3, but not when the additional payment is 300. A 2 (additional payment) ×
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2 (response) chi-square test revealed a significant relationship between additional payment value and option choice, χ 2 (1) = 8.94, p < .01. The presence of such a modal choice amounts to a demonstration of a violation of “conjoint independence” (For a review of the independence condition, the sure-thing principle and the conjoint independence see [19]). In the context of the present problem, conjoint independence asserts that if Product A (or B) is preferred to Product B (or A) with an additional payment of 3, then Product A (or B) should also be preferred to Product B (or A) with an additional payment of 300. Changing the value of the additional payment is not expected to reverse the decision. The contrasting choices suggest that the respondents tended to evaluate the additional 300 in relation to the price of the products A and B. In relative terms, the common value 300 added to 2500 is apparently seen as more impressive than the common value 300 added to 7400, in spite of the fact that the two additional payments are exactly the same in terms of their physical amount (300) as well as their physical scaling unit (-yuan). This result is in full harmony with the psychophysical law that psychological response is a nonlinear function of the magnitude of physical change. Choice 4 and Choice 5 have one feature in common, that is, it seems that the two offered alternatives in Choice 5 should also be chosen equally often from a “rational” perspective. This is because, if we consider that the alternatives are represented by two separate attributes, namely the number of items awarded and the amount of money awarded, then alternatives A (2 items and 680 awarded) and B (2 items and 680 awarded) will coincide in such a representational system, so that each outcome of the two alternatives on each attribute is equally attractive, either when the evaluation is holistic or when the evaluation is performed by the additive decomposition of a series of attributes. However, of the respondents presented with this choice, the majority said they would prefer B, despite the fact that both the total amount of money and the basic nature of the two items are the same for both alternatives. This uneven choice suggests that what is chosen by the decision-maker may not necessarily be the computational or judgmental greatest overall value assigned to the alternatives. The present proposal for an explanation of these results rests on a representational system which is to be presented as an alternative to that of two separate attributes (see Fig. 4). In the present view, the two dimensions representing the alternatives are the monetary value of the bicycle and the monetary value of the
Figure 4.
Representation for Choice 5.
recorder. It is proposed that both dimensions undergo a unique transformation according to a utility function. Because the utility function for money is a negatively accelerated (concave) one-to-one mapping function over the range of concern, the difference between 660 and 640 is likely to be judged to as less than the difference between 40 and 20. Thus, what appear to be equivalent differences in terms of the same monetary scaling unit, turn out to represent different utility differences, thereby leading to resolution of conflict and ultimate selection. The results observed in the present section provide evidence that the utility function along a single dimension can make objectively identical values subjectively different. It is the present contention that the derivation of utility function from choice should be based on differentiating greater one-dimensional difference, but not on computing overall worth. Operationally, the logic would be not to assume respondents are trying to maximize “something” in a process of utility integration, but are trying to choose between monodimensional utilities. The utility measured by using this equate-to-differentiate deductive method should yield the same result as that measured by using a psychophysical approach. 5.
Equating or Differentiating: Planned Intransitivity of Choices
One of the core axioms in both risky and riskless decision theories is transitivity [20–23]. That is, preferences should be transitive. If Alternative A is preferred to Alternative B, and Alternative B is preferred to Alternative C, then this would necessarily indicate a preference for Alternative A over Alternative C. If this were not the case, intransitivity of choice could lead, for example,
A Behavioral Choice Model
to the exploitation of what has been called the money pump. Since individuals are not perfectly consistent in their choices, most theoretical accounts of binary choice imply some form of stochastic transitivity (see [24] for an exposition). In the following discussion, let the symbol denote the relation “is preferred to”. A consideration of the details of the equate-to-differentiate model suggests that transitivity will be obeyed if all alternatives are represented on a single dimension. Otherwise, we might not be surprised to observe a preference relation satisfying A B, B C, C A, or B A, A C, C B if we were able to construct a series of choice problems in which individuals’ equate-to-differentiate strategy is switched, say, from one in which the equated differences in the first two binary choices are all on the same dimension, but the equated differences in the last binary choice are on another. An attempt to produce such so-called circular triad preferences is demonstrated in the choice problem shown in Appendix C, which was given to 206 high school graduates in Fujian China, who volunteered to participate in the experiment. It can be seen from Appendix C that there are in fact three pairs of alternatives {A, B}, {B, C} and {C, A} that are presented to subjects (P = B; Q = U = C; V = A). The first pair is designed as a probe. According to the equate-to-differentiate process, people who prefer Alternative A (the better university) over Alternative B (the better speciality) would consider the outcome difference along the “university” dimension to be greater than that along the “speciality” dimension. In order to trigger a 3-cycle intransitivity: Alternative A is preferred to Alternative B, which is preferred to Alternative C, which is (paradoxically) preferred to Alternative A, the strategy which has been used is to select those subjects who prefer better universities in the first pair and then ask them to choose from the second and third pairs. The alternatives in the second pair allow them to use the same dimension they used in the first pair, but the alternatives in the third pair force them to choose on the other dimension. Operationally, the second pair of alternatives is designed so as to keep the difference on the “university” dimension sufficiently great, while enlarging the difference on the “speciality” dimension a little, but not sufficiently so as to trigger a choice based on that dimension. The third pair is then designed so that the difference on the “speciality” dimension is greater than that on the “university” dimension, thus leading the subject to choose the better speciality, which is Alternative C.
Table 3.
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Preference data for Admission A preferred subjects. 1st Choice A B (N = 75)
2nd Choice
3rd Choice
BC
CA
55
36
ABC
ABC A
55
21
(37.5)
(18.75)
Note. The main entries are numbers of subjects giving the choice indicated. The data in brackets are numbers of subjects who are expected to choose by chance.
The results are shown in Table 3. Among those who preferred Admission A, we see that 21 out of 75 subjects behaved in contradiction to the transitivity axiom. If those 75 subjects were assumed to choose by chance, a chi-square test for goodness of fit yields χ 2 (2) = 8.44, p < .025. An inference from the above analysis is that, if we can preliminarily select those whose utility will render the equating of difference on dimension D1 easier than that on dimension D2 to produce choices in one direction, such as A B, then we can at the same time select those whose utility will render the equating of difference on dimension D2 easier than that on dimension D1 to generate another way around intransitivity, such as B A. Guided by such thinking, Choice 7 (see Appendix C), which concerns the selection of the most suitable candidate for a position of Production Engineer, is designed not only to produce choices in one direction, such as A B, but also to allow preferences both ways, i.e., A B as well as B A. The scenario of Choice 7, with some changes, is borrowed from Tversky et al. [17]. The choice problem was distributed to 55 undergraduate students at the University of New South Wales. It can be seen that those subjects who chose Candidate X in fact indicated their preferences from within three pairs of alternatives: {X, Y }, {Y, Z } and {Z , X }, since F = Y , G = I = Z , J = X . On the other hand, those who chose Candidate Y indicated their preferences from within four pairs of alternatives: {Y, X }, {X, W }, {W, V } and {V, Y }, since P = X , M = Q = W , N = R = V , S = Y . In order to determine whether the following cycles exist: X Y , Y Z , Z X or the reverse cycle: Y X , X W , W V , V Y , the first step is to classify subjects into two categories: the
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“Technical Knowledge” maximizers and the “Human Relations” maximizers. As with the previous Choice 6, the first pair of alternatives in Choice 7 is designed as the starting point. After that, careful attempts can be made to create conditions where the “Technical Knowledge” maximizers are allowed to continue to maximize on the “Technical Knowledge” dimension, while the “Human Relations” maximizers can continue on the “Human Relations” dimension in the pairs to which they are directed. Finally, the last pair is constructed to stimulate a “Technical Knowledge” maximizer to become a “Human Relations” maximizer, and vice versa. Table 4 gives the experimental results. Of all the subjects who answered Choice 7, there remains a total of 10 out of 55 subjects who did not maintain the deterministic transitivity. Five of these did not maintain transitivity through a sequence of 3 steps, and another 5 though 4 steps. Thus the results do not support the transitivity principle of rational choice. A chi-square test for goodness of fit yields χ 2 (4) = 12.26, p < .02, if those 55 subjects were assumed to choose by chance all problems except the first one. It can be can seen that, despite the existing evidence of intransitivity (e.g., [24]), subsequent theories investigated by Tversky have assumed transitivity [17, 25, 26]. It also can be seen that the non-linear additive difference (NLAD) rule, proposed by Tversky [24] to explain intransitivity, assumes that people choose according to an ordering implied by an aggregate of multidimensional net difference so that intransitivity is originated. The NLAD rule is compensatory in that relative judgments of all dimensions are taken into account, such relative judgments being non-linear functions of intradimensional differences. In reality, however, the state we are left with is that the non-linear additive difference rule is merely studied under theo-
retical conditions, whereas the construction of the observed stochastic intransitivity is empirically guided by a non-NLAD model. The present notion does not address the soundness of its axiomatic justification for producing an intransitive ordering, or the question of whether elementwise processing is inconsistent even with prospect theory [25, 27], although there is evidence that the form of the difference functions applied to the various intradimensional differences cannot be deduced from the form of the value and weighting functions [28]. The problem raised is if an intradimensional evaluation is possible for an additive difference model, the possibility of forming an interdimensional evaluation through a simple additive model that guarantees transitivity should not be logically excluded. Be that as it may, it appears that the evidence that human cognitive ability is adequate to perform an overall multidimensional addition can be used not only to validate the NLAD, but also to falsify the equate-to-differentiate model.
6.
A Comparison with Other Comparable Rules
How people ought to make decisions has been an important subject of study for a long time. When a disparity between how people should behave and how they actually do behave exists, it raises issues of human rationality. However, cumulative empirical evidence indicates that normative theory, such as expected utility theory, fails as a general representation of actual behavior. Especially with the award of Nobel Prizes to Herbert Simon in 1978 and Maurice Allais in 1988, the assumption in mainstream decision literature, that all rational people conform to normative theory, is slowly beginning to change. In the present paper the disparity between actual and “rational” behavior has been seen
Three choices for Candidate X preferred subjects (N = 18) and four choices for Candidate Y preferred subjects (N = 37).
Table 4.
1st Choice Y ≺ X (N = 18)
1st Choice Y X (N = 37)
3rd Choice
2nd Choice
2nd Choice
3rd Choice
4th Choice
X≺Z
Z ≺Y
XW
W V
V Y
9
13
27
21
12
X ≺Z ≺Y ≺X
Z ≺Y ≺X
Y XW
Y XW V
Y X W V Y
5
13
27
17
5
(4.5)
(9)
(18.5)
(9.25)
(4.625)
Note. The main entries are numbers of subjects giving the choice indicated. The data in brackets are numbers of subjects who are expected to choose by chance.
A Behavioral Choice Model
to occur if “rationality” is defined in terms of compatibility between value and choice. The prime justification for proposing the present model lies in the wealth of evidence that human cognitive ability, if no special efforts or emphasis or support is involved, seems to be not always adequate to cope coherently with choices in line with either a philosophy of “aim for the best” or of “aim to avoid the worst”. The following section will address the question of why the present model might possibly be added to the family of descriptive approaches as a new member. The present model frequently employs the idea of cognitive limitations. The cognitive inadequacy has also been the justification for Simon’s [29] satisficing model, which suggests that humans are “bounded rational decision-makers.” For Simon, human limitations result in the acceptance of a “good enough” level lying somewhere between “the best” and “the worst.” However, for the present model those limitations result, in the context of a dimensional analysis of the choice task, in decisions being based on a subset of the available information. The “equate” step leaves the decision-maker with a lower cognitive load. The “differentiate” step then involves an “aim for the best” process if the determinant dimension is a “good” one, or an “aim to avoid the worst” process if the determinant dimension is a “bad” one (see Li [14] for an exposition of seeing risky choice behavior simply as a choice between the best possible outcomes or a choice between the worst possible outcomes). Thus, the difference mentioned will result in different decision outcomes. In searching for a “good enough” alternative, the decision strategy the satisficing model takes is a nonconsequentialist one, whereas the present model takes a consequentialist one. For the satisficing model, the set of possible alternatives may be large and complex, and the individual’s knowledge of this set may be incomplete and uncertain. It would be difficult to seek out the absolute best alternative and, considering the cost of such a search, it may be preferable to stop the search as soon as a satisfactory alternative is found. This is in contrast to the present model, which requires that an individual choose a course of action on the basis of the expected consequences of different actions. In determining the criterion for a “good enough” alternative, the satisficing model suggests that a satisfactory alternative is one with values that exceed an aspiration level on each dimension. Hence, a satisfactory alternative might not be a differentiated alternative, if all or none of the alternatives have values that exceed an aspiration level on each
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dimension. In such a case, the satisficing model would suggest that either all alternatives will be selected, or all alternatives will be rejected. According to the equateto-differentiate model, however, one and only one single alternative must be selected because it suggests that the chosen alternative is the one with a value that is superior to the other alternative on one dimension. The idea that the final decision should be based on a single dimension is not unique to the present model. If the way of seeing dimension left no doubt, the lexicographic rule (and its probabilistic version, the elimination by aspects rule) would also consistently require that the decision be based on one dimension. The lexicographic rule states that the most important dimension is considered first, and the alternative with the best value on this dimension is selected. Any alternative which is deficient on the most important dimension is eliminated, regardless of the values on the remaining dimensions [30]. The lexicographic rule, however, differs from the present model in two essential respects. The first difference is about which dimension is the determinant dimension. For the lexicographic rule, only the most important dimension can be considered as the determinant dimension on which the choice is based (if several alternatives tie for the best value on the most important dimension, then the second most important dimension is considered). For the equate-to-differentiate rule, however, any dimension can serve as the determinant dimension if the difference on that dimension is seen as the greatest. Becoming aware of importance is insufficient to make a choice (referring to the decision outcomes revealed in Choices 1–3). The second difference is about how to treat the values of the remaining dimensions. For the lexicographic rule, all differences between the alternatives on the remaining dimensions, either the group of dimensions on which the selected alternative is better than the rejected alternative, or the group of dimensions on which the selected alternative is worse than the rejected alternative, are completely “ignored”. This indifferent way of treating dimensional differences differs from the present model in that, if the selected alternative is better than the rejected alternative on all the remaining dimensions, in the present model it is not necessary to ignore, in the sense of mental efforts to be made, those differences on remaining dimensions; if the selected alternative is worse than the rejected alternative on all the remaining dimensions, it appears that simply to ignore is not enough. In the former case, no “equating” effort is needed. In the latter case, efforts should be made to equate these conflicting
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differences so as to make dominance accessible. That is, not only small differences should be equated but also great differences, which might be hard to passively ignore in an absolute sense, should be actively equated by means of a psychophysical mapping. Therefore, if any post-decision dissonance, such as regret, ambiguity and a tendency to reverse choice, is observed, the explanation for this should be quite different and testable. For the lexicographic rule, the switching of judgment of importance is the cause which is most likely to be responsible for the change of determinant dimension. For the equate-to-differentiate rule, however, it is not the judgment of importance but the difference judgment that is regarded as the cause which is most likely to be responsible for the change of determinant dimension. To make the dominance rule applicable is all the present model struggled for. However, the equate-todifferentiate rule, all in all, is not a dominance rule itself. Dominance is an optimizing rule. The rule always yields a unique solution. In most decision situations, however, we do not find an alternative which, strictly speaking, dominates over all other alternatives. This implies that its applicability is limited in contrast to that of the equate-to-differentiate rule, which can apply to both dominated and nondominated alternatives. One very comparable rule is the search for dominance structure [32].1 This decision rule proposes that the decision process is seen as a search for a dominance structure, that is, a cognitive structure in which one choice alternative can be seen as dominant over the others. In spite of the fact that both the search for dominance structure rule and the present rule by themselves are not pure dominance rules, and that both rules seek to make a dominance rule applicable, our views are divergent on some important aspects. First, the representing system, in which either dominated or nondominated alternatives are subjectively represented, is not exactly the same for the two rules. Thus, the use of the same heuristic procedure can lead to opposite predictions. Take, for example, the most controversial decision area, decision making under risk or uncertainty, the representing system used by Montgomery will describe alternatives in two simple risky dimensions [31]. What is seen as the two simple risky dimensions (the probability of winning and the payoff) will be considered as only one dimension (possible-outcome dimension) from the equate-to-differentiate rule’s point of view (e.g., the representation of risky problems involving the reflection and the framing effects in Li [14]). Second, what is a good dominance structure, operationally?
For the equate-to-differentiate rule, the answer is clear and unique: a good dominance structure is a cognitive structure where all the dominated alternative favoring dimensional differences have been equated. However, the search for dominance structure rule does not seem to provide a unique answer of what a good dominance structure is. A dominance structure is assumed to be a representation where one alternative has at least one advantage compared to other alternatives, and where all disadvantages associated with that alternative are neutralized or counterbalanced in one way or another. By doing so, various decision rules (e.g., a noncompensatory rule, a compensatory rule, or a combined usage of the two) can be seen as operations in the process of changing the representation. Third, the search for dominance structure rule suggests that a number of problems associated with either non-compensatory rules or compensatory rules could be avoided if the rules are seen as operators in search of a dominance structure. On the contrary, the present model neither intends to evade such problems nor renders different phases of the decision process where these two types of rule could apply jointly. Finally and most importantly, note that, although a lot of decision rules (five non-compensatory and three compensatory rules) are assumed to serve various local functions in each particular stage, Montgomery does not assume a rule like the equate-to-differentiate rule that could be coherently used in the searching operations. Therefore, if careful tests were designed, any of these eight rules would predict choices other than the present model would. 7.
Conclusion
An attempt has been made in this paper to present a choice model that can provide a credible alternative to other normative decision models used at present. In evaluating the present model against other comparable rules, the above analyses will make the model’s both “to be” and “not to be” testable, in the sense of falsifiablity. The data gathered in the present study shows fundamental limitations in people’s capacity to process information. Such a finding, together with those obtained in the domains of risky choice [14, 33, 34], uncertain choice [13, 35] and Prisoner’s Dilemma games [36], shows that what often appears to be “paradoxical” decision behavior, can, from the point of view of normative theories, be understood as emergent properties of the equate-to-differentiate process that individuals use to perform decision tasks.
A Behavioral Choice Model
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Appendix A Choice 1. Traffic Accident Problem (The original scenario was adapted by substituting “Australia” or “Fujian Province” or “Singapore” for “Israel” in the Australian version, the Chinese version and the Singaporean version respectively. The associated percentages of subjects choosing each option are listed on the right, with one asterisk denoting significance at the .05 level and two at the .01 level. The data for Australian, Chinese and Singaporean students are given as A =, C = and S = respectively. For Chinese subjects the dollar sign was replaced with the yuan sign)2 : About 600 people are killed each year in Australia (Fujian Province/Singapore) in traffic accidents. The ministry of transportation has investigated various programs to reduce the number of casualties. Consider the following two programs, described in terms of yearly costs and the number of casualties per year that are expected following the implementation of each program.
Program A Program B
Expected number of casualties
Cost
500 570
$55 million $12 million
A = 60%; C = 61%; S = 57.5% A = 40%; C = 39%; S = 42.5%
Choice task: Which program do you favor? Please circle your choice:
A
B
Matching task (Circle the one whose alternatives are most different) C. “500 lives lost” vs. “570 lives lost” D. “Yearly costs of $55M” vs. “yearly costs of $12M”
A = 34%; C = 52%; S = 59% A = 66%; C = 48%; S = 41%
Choice 2. Benefit plans problem: Imagine that, as a part of a profit-sharing program, your employer offers you a choice between the following plans. Each plan offers two payments, one due in one year and one in four years (For Chinese subjects the dollar sign was replaced with the yuan sign).
Plan A Plan B
Payment in 1 year
Payment in 4 years
$2,000 $1,000
$2,000 $4,000
A = 38%; C = 48%; S = 55% A = 62%; C = 52%; S = 45%
Choice task: Which plan do you prefer? Please circle your choice:
A
B
Matching Task (Circle the one whose alternatives are most different) C. “Payment of $2,000 in 1 year” vs. “Payment of $1,000 in 1 year” D. “Payment of $2,000 in 4 years” vs. “Payment of $4,000 in 4 years”
A = 30%; C = 42%; S = 36% A = 70%; C = 58%; S = 64%
Choice 3. Modified benefit problem: Imagine that, as part of a profit-sharing program, your employer offers you a choice between the following plans. Each plan offers a payment in one year’s time, or in four years (For Chinese subjects the dollar sign was replaced with the yuan sign).
Plan A Plan B
Amount of payment
Time of payment
$2,500 $6,000
in 1 year in 4 years
A = 81%∗∗ ; C = 82%∗∗ ; S = 70%∗∗ A = 19%; C = 18%; S = 30%
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Choice task: Which plan do you prefer? Please circle your choice:
A
B
Matching task (Circle the one whose alternatives are most different) C. “Payment of $2,500” vs. “Payment of $6,000” D. “Payment in 1 year” vs. “Payment in 4 years”
A = 24%; C = 40%; S = 32.5% A = 76%; C = 60%; S = 67.5%
Rating task: Please use the numbers 1–5 to indicate the order of importance of the following 5 items. 5 denotes the most important, while 1 denotes the least important. Write this number in the blank space to the left of each item. ( ) Time
( ) Human Life
( ) Love
( ) Money
Appendix B Choice 4. Imagine you are a purchasing agent and your mission is to buy Product A and Product B produced in a factory. The product salesperson in the factory tells you that both A and B are goods in great demand. If you want to buy either of them you must buy extra unsalable goods. In order to buy A and B, you agree to buy the extra unsalable goods which you would otherwise be unwilling to buy. Furthermore, the salesperson insists that both goods are in such short supply that you can only buy one of them: • Suppose the extra unsalable goods cost 3. Which product are you going to buy? Product A: cost 2500 Product B: cost 7400
50% 50%
N = 66
( ) Freedom
tastes, the person in charge of the award arranges for there to be two price combinations available for you to choose from. Which combination do you prefer? Please circle your choice. Combination A: Bicycle (660); Recorder (20)
33%∗∗
Combination B: Bicycle (640); Recorder (40)
67%
N = 175
Appendix C Choice 6. Imagine that, as a candidate for the National Entrance Examination, your personal rank for the academic reputation of universities is as follows: Academic reputation order
• Suppose the extra unsalable goods cost 300. Which product are you going to buy?
The actual questionnaires presented to subjects were presented in two different versions (3/300 and 300/3 versions). Approximately half the subjects were randomly assigned to respond to each of the two versions.
1. National key universities 2. Key universities under the jurisdiction of the State Education Commission 3. Local Universities under the jurisdiction of the State Education Commission 4. Local key universities under the jurisdiction of the Provincial Government 5. Local ordinary universities under the jurisdiction of the Provincial Government
Choice 5. Suppose at the end of year everyone in your work unit is to receive an award. Each person is to be given a bicycle and a cassette recorder, worth a total of 680. Realizing that it will be difficult to cater for all
If you simultaneously received two admission notices after the examination, in which the universities and specialities to which you were admitted were as follows, which of them would you accept?
Product A: cost 2500 Product B: cost 7400
29%∗∗ 71%
N = 66
A Behavioral Choice Model
Admission A: To a local University under the jurisdiction of the State Education Commission; 5th favourite speciality. Admission B: To a local key university under the jurisdiction of the Provincial Government; 4th favourite speciality. Now that you have made your choice, two of your best classmates are faced with a similar choice situation. They would like to ask you to make a choice imagining yourself in their positions. Details of the two classmates’ situations are as follows. Please circle your choice. Classmate X:
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Now that you have made your decision, if you prefer X please go to Problem X and choose from another two pairs of candidates. Otherwise please go to Problem Y and choose from another three pairs of candidates. Problem X. (Candidate X preferred) Technical knowledge
Human relations
First pair Candidate F
62
65
Candidate G
57
582
Difference
5
17
Candidate I
57
82
Candidate J
66
56
Difference
9
26
Second pair
Admission P: To a local key university under the jurisdiction of the Provincial Government; 4th favourite speciality. Admission Q: To a local ordinary university under the jurisdiction of the Provincial Government; 1st favourite speciality.
Problem Y. (Candidate Y preferred) Technical knowledge
Human relations
Candidate M
72
47
Candidate N
79
37
Difference
7
10
Candidate P
66
56
Candidate Q
72
47
Difference
6
9
Candidate R
79
37
Candidate S
62
65
Difference
17
28
Classmate Y: Admission U: To a local ordinary university under the jurisdiction of the Provincial Government; 1st favourite speciality. Admission V: To a local University under the jurisdiction of the State Education Commission; 5th favourite speciality. Choice 7. Imagine that, as an executive of a company, you have to select between two candidates for the position of Production Engineer. The candidates were interviewed by a committee who scored them on two attributes (technical knowledge and human relations) on a scale from 100 (superb) to 20 (very weak). Both attributes are important for the position in question, but technical knowledge is more important than human relations. On the basis of the following scores, which of the two candidates would you choose? Technical knowledge
Human relations
Candidate X
66
56
Candidate Y
62
65
4
9
Difference
First pair
Second pair
Third pair
Acknowledgments The author wishes to thank three anonymous referees of this journal for their helpful comments on the initial version. Notes 1. The author thanks Lola Lopes for her suggestion of including this rule in the comparisons.
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2. At the time that the Australian and Chinese data collection took place, U$1.00 ≈ 5.70 while U$1.00 ≈ A$1.50 and, one kg of rice cost about A$1.00 in Sydney, Australia while cost about 1.00 in Fuzhou, China. I am grateful to Ward Edwards for suggesting this addition.
19.
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20.
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A Behavioral Choice Model
Shu Li (PhD UNSW, M.Ed. Hangzhou U, B.Eng. Fuzhou U) is currently the President of Hwa Nan Women’s College, China. He was at one time a University of New South Wales Vice-Chancellor’s Post-
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Doctoral Research Fellow, an Australian Research Council Postdoctoral Research Fellow, an Assistant Professor at Nanyang Technological University and at Macau University of Science and Technology. His present research interests are in the area of behavioral decision making. His research has mainly appeared (or will appear) in such journals as Acta Psychologica Sinica, Australian Journal of Psychology, Chinese Journal of Psychology, Ergonomics, European Journal of Operational Research, Journal of Behavioral Decision Making, Journal of Economic Behavior and Organization, Journal of Economic Psychology, Journal of Risk Research, Organizational Behavior and Human Decision Processes, Psychology and Marketing, Psychologia, and others. He also has articles and presentations in several international conferences and serves as a guest reviewer for several international journals.
