An advantage model of choice

Journal of Behavioral Decision Making, Vol. 2 , 1-23 (1989)

An Advantage Model of Choice

ELDAR B. SHAFIR AND DANIEL N. OSHERSON M.I.T., USA

EDWARD E. SMITH University of Michigan, USA

ABSTRACT A descriptive model of choice between simple lotteries is proposed. According to the model, people combine ‘absolute’ and ‘comparative’ strategies in choice situations. Three experimental tests of the model are reported. In its limited domain, the model appears superior to Prospect Theory on qualitative grounds, and to I-parameter versions of Expected Utility Theory on both qualitative and quantitative grounds.

KEY WORDS Utility theory Prospect theory Risky choice Monetary lottery Preference reversal Intransitivity

1. INTRODUCTION

Consider a choice between one lottery that offers a 50 percent chance to win $2000 and another that offers a 75 percent chance to win $1000. We may distinguish two approaches to understanding the preferences exhibited by naive subjects faced with such choice problems (Tversky, 1969). These approaches will be called ‘absolute’ and ‘comparative’. Both proceed by assigning a hypothetical attractiveness coefficient to each lottery in a given choice problem, and predicting that the lottery assigned the numerically higher coefficient will be preferred. Within the absolute approach the attractiveness of a lottery is assumed to be independent of the alternative with which it is paired. In contrast, theories within the comparative approach evaluate the attractiveness of lotteries only in the context of a specified choice problem, the attractiveness of a given lottery depending on the alternative to which it is compared. The present paper offers a model of choice that combines the two approaches. Before presenting the model we provide some background.

The most influential example of an absolute theory is the Theory of Expected Utility (von Neumann and Morgenstern, 1947; henceforth, ‘Utility Theory’). Within Utility Theory the attractiveness of a lottery (d,p) that offers a less-than-certain chance p to win a specified sum of money d is given as u(d)*p, where u is an empirically estimated function from monetary assets to real numbers (called ‘utilities’). It is to be emphasized that Utility Theory must be supplemented with external assumptions about the form of the function u before it can be used to make predictions about subjects’choices between a given pair of lotteries. Thus, u in Utility Theory may be conceived as a free function-parameter. Notwithstanding this freedom in the specification of u, it is widely appreciated that Utility Theory provides an inadequate description of subjects’ choices in risky situations. The descriptive shortcomings of the theory have been exposed by Allais (1953), Kahneman and Tversky (1979), Tversky and

0894-3257/89/010001-23$11.50 0 1989 by John Wiley & Sons, Ltd.

Received 19 June 1987 Revised 15 June 1988

2 Journal of Behavioral Decision Making Vol. 2 Iss. No. I

Kahneman (1981, 1986), Lichtenstein and Slovic (1971), and Slovic and Lichtenstein (1983), among others. Slovic, Lichtenstein and Fischoff (1988) provide a guide to the relevant literature.

Another example of an absolute theory is Kahneman and Tversky’s (1979) Prospect Theory. Within Prospect Theory, the attractiveness of (d,p) is given as u(d)*n(p), where now u is defined on monetary gains or losses and n is an empirically estimated function from probabilities to non-negative real numbers (called ‘decision weights’). Kahneman and Tversky (1979) show that Prospect Theory avoids many of the false predictions of Utility Theory, and leads to new and surprising predictions confirmed by experiment. Despite these successes, and largely due to its absolute character, Prospect Theory seems open to certain descriptive difficulties to be discussed in Section 3 below.

The absolute approach has also been brought into question by recent experimental evidence. Russo and Dosher (1983) present eye-movement data and verbal protocols to demonstrate that subjects first evaluate differences between lotteries on the separate dimensions of probability and payoff and then combine these estimates to yield a choice. These findings agree with other studies that use decision tracing methods and report evidence for comparative heuristics in preference judgments. Reviews of the relevant literature are provided by Russo and Dosher (1983) and by Schoemaker (1982). Russo and Dosher (1983, p. 694) conclude that the data are ‘difficult or impossible to explain with a holistic [i.e., absolute] strategy.’ They go on to sketch plausible hypotheses within the comparative framework but stop short of advancing an explicit rule for calculating the attractiveness of lotteries in a choice problem.

Other theoretical alternatives to Utility Theory have incorporated a comparative framework. Fore- most among these is Expected Regret theory, developed independently by Bell (1982, 1983), Fishburn (l982), and Loomes and Sugden (1982). Although there are important differences between the formulations offered by these authors, we believe that the following remarks apply to all three versions of the theory.

Regret theory replaces the absolute preference function Q L ) that attributes attactiveness to single lotteries by a function Q L I . L ~ ) defined over pairs of lotteries. An individual’s level of satisfaction is thus viewed as determined not only by what is obtained as a consequence of having made a choice, but also by what might have been obtained had another choice been made. In effect, Regret Theory combines an absolute measure (viz., the utility of what was obtained) with a comparative measure (viz., the difference between what was and what could have been obtained). The result is a mathematically sophisticated theory that avoids many of the false predictions of Utility Theory. On the other hand, the various versions of Expected Regret Theory share the disadvantage of being even more elusive than Utility Theory about predicting what decisions people will make when faced with specific choice problems. Predictions of this latter sort follow from the theory only in the presence of strong assumptions that go beyond its general picture of the subject’s bivariate (Fishburn) or multiattribute (Bell and Loomes and Sugden) utility function. The space of candidates for these functions is even less constrained than the space of unary functions within Utility Theory.

Regret Theory captures an appealing psychological intuition concerning the regret or rejoice that the decision maker experiences upon considering the outcome relevant to his choice. Apart from this intuition, however, the theory embodies no hypotheses about the mental processes and representations that underlie peoples’ choices between lotteries. In this respect, it remains within the Utility Theory tradition because the latter is equally reluctant to specify underlying mental processes.

The foregoing discussion cites only a fraction of the theories of choice under risk proposed in recent years. Each theory succeeds in deducing some important choice phenomena while leaving others unexplained. No theory appears to explain the entire spectrum of choice phenomena brought to light thus far. For more thorough review of the explanatory adequacy and inherent limitations of various choice theories, see Machina (1983, 1987), Battalio, Kagel, and Komain (1988), and Camerer (1988).

E. B. Shafir, D. N. Osherson and E. E. Smith An advantage model of choice 3

The present paper advances a descriptive model of choice between lotteries that invokes people’s perception of the advantages accruing to each of the alternatives they face. The model - called the Advantage Model of Choice - construes the attractiveness of a lottery as a multiplicative combination of absolute and comparative components which result from judgmental processes of an elementary character. The Advantage Model of Choice (henceforth, ‘the Advantage Model’) applies only to simply monetary lotteries, Consequently, in its present form, the model is incapable of predicting some important phenomena that have been shown to characterize risky choice in more complex situations. Furthermore, even in its limited domain, the model will be seen to face some unresolved difficulties. On the other hand, the model is highly parsimonious, posits a choice strategy of clear comparative character, and is well-supported by experimental test. I t may thus provide insight into fundamental processes underlying human choice behavior. In sum, we present the model not as a general theory of risky choice, but as an illustration of how much theoretical work can be performed by a simple and psychologically-motivated model.

In the present paper, the model is restricted to simple lotteries of the form: an x% chance to win (or lose) m dollars (and consequently, a I-x% chance to win (or lose) nothing). Work in progress extends the model to more complex lotteries. Although the present restriction limits the generality of the model, it allows us to formulate a simple and psychologically explicit hypothesis, and to test that hypothesis both qualitatively and quantitatively. The Advantage Model will be seen to include no more than a single real parameter.

Our presentation proceeds as follows. Section 2 introduces the Advantage Model. In Section 3 we discuss the model from a qualitative point of view. Among other topics, we show how the Advantage Model helps to make sense of choice phenomena like intransitivity that are inexplicable within an absolute framework. Psychological interpretation of the model’s single parameter occupies Section 4. After a brief summary in Section 5 , Section 6 turns to quantitative evaluation of the Advantage Model. The results of three experiments are shown to favor the model over one-parameter versions of what has been the most popular theory to date, namely, Expected Utility Theory. Quantitative comparison of the model to Prospect and Regret Theory was not attempted since clear candidates for these theories’ curve-parameters do not sugget themselves. Section 7 is devoted to a discussion of possible extensions of the Advantage Model to more complex choice situations.

2. THEORY

The Advantage Model is motivated by the intuition that when choosing between lotteries, people employ both absolute and comparative strategies, the results of which are subsequently combined. According to the model, a person’s absolute strategy consists of a rough evaluation of each lottery’s ‘size’, to be captured by Expected Monetary Value, s*p. His comparative strategy consists of an evaluation of (a) the difference in payoff between the two lotteries, and (b) their difference in probabilities.

2.1 Terminology We adopt the following terminology. By a ‘positive lottery’is meant a less-than-certain chancep to win a specified sum of money d. Such a lottery will be represented by the pair (+d,p). By a ‘negative lottery’ is meant a less-than-certain chance p to lose a specified sum of money d. Such a lottery will be represented by the pair (-d,p). Positive and negative lotteries in the sense just defined will be called ‘simple’ lotteries. To designate a simple lottery without specifying whether it is positive or negative, we use: (d,p). By a ‘simple choice problem’ is meant an invitation to choose between a pair of simple


di pi (2500, .33)* (4o00, .20)* (6000, .45) (m, .001)* (-4o00, .20) (-6oO0, .45)* (-m, .001)

(45, .20)*

d2 p2 (2400, .34) (3000, .25) (3000, .90)*

(3000, .002) (-3000, .25)* (-3000, .90)

(-3000, .002)* (30, .25)

?The payoffs in problems 1-7 refer to Israeli currency. To appreciate the amounts involved: the median net monthly income for a family at the time was about 3,000 Israeli pounds. Problem 8 is in dollars.

Table I . All simple choice problems from Kahneman and Tversky (1979; 1981)t (Asterisk indicates lottery preferred by a significant majority of subjects)

lotteries. Such a pair will be denoted: [(&PI), (d2,pz)l. By convention, all choice problems are represented so that PI I p2. Table 1 summarizes every simple choice problem appearing in Kahneman and Tversky’s (1979), Tversky and Kahneman (1981) well-known discussion of risky choice.

2.2 Formulation of the model To formulate the model precisely, let choice problem [(dl,pl), (dz,p2)] be given (where PI 5 p2). To represent the absolute component of a subject’s judgment, we define: EMVI = dlpl; EMV2 = d2p~. For the comparative component, we let p2-p1 represent the ‘probability advantage’ of ( d ~ , p ~ ) , and we let (dl-dz)/dl represent the ‘payoff advantage’ of (dlgl). The payoff advantage has been normalized by dl because in these situations people seem to be more sensitive to relative rather than absolute payoff magnitudes. (Other means of normalizing are possible; the present scheme was selected on the kind of qualitative grounds to be discussed shortly.) Observe that the probability advantage is determined by the difference between probabilities while the payoff advantage is ultimately determined by the payoffs’ ratio. It should be noted as well that the expressionsp2-pl and (dl-d2)/dl are positive numbers for both positive and negative lotteries. As a consequence, these two ‘advantages’ are not always desirable aspects of a lottery. In the case of negative lotteries, they are undesirable (and thus might best be conceived as ‘disadvantages’).

Finally, as a means for comparing the two qualitatively different advantages, we introduce a unitless parameter into our model to represent the relative weight of payoffs and probabilities. This factor takes the form of a multiplicative coefficient, k, attached to the payoff advantage (dl-d2)/d1 of the lottery ( d ~ , p ~ ) . According to our model, the weighting factor may differ from person to person, but each person has exactly one such factor, applicable to all simple choice problems. For other decision situations (e.g., assessing the monetary value of a lottery) the value of this weighting factor is assumed to vary in a systematic manner to be discussed presently.

Assembling the absolute and comparative components of the model, we arrive at the following formulation, which captures our motivating intuitions.

E. B. Shafir, D. N . Osherson and E. E. Smith An advantage model of choice 5

The advantage model of choice: For every person S, there is k > 0 such that for any problem

[ (dlgl) , (d292)I (where P I 5 P Z ) , S chooses (d ig , ) if EMVI [(dl-h)/dl]k > EMV2(p2-p1), S chooses (d292) if EMVl[(dl-dz)/dl]k < EMV2 @2-p1), S is indifferent if EMVl[dl-dz)/dl]k EMV2@2-pl).

The reader may verify that the Advantage Model correctly predicts various trivial cases of simple choice problems. For example, if pl =p2 and dl=d2, both left- and right-hand attractiveness coefficients equal zero and the person is indifferent. Similarly, the model makes the obvious predictions for cases where: I ) pl=p2, or 2) dl=d2, or 3) dl and d2 are of opposite signs. Less trivially, using k .40 the Advantage Model correctly predicts all eight simple choice problems of Table 1. We illustrate the calculations involved in problem (1):

Problem 1: [(2500, .33), (2400, .34)] EMV2 (2400) (.34) p2-p1 = .01 EMV2(pt-p1) 8.16

EMVi (2500) (.33)

EMV~[(dl-dz)/dl]k = 13.2 (dl-dz)/dl 100/2500

Hence, the Advantage Model (with k = .40) predicts that (2500, .33) will be preferred to (2400, .34) -which agrees with a significant majority of Kahneman and Tversky’s subjects. The same choice of k = .40 applies in similar fashion to the remaining problems in Table 1. It will be seen that this value of k is close to an empirical estimate derived from three experiments to be reported later.

By rearrangement of terms, the three clauses of the Advantage Model may be cast equivalently as follows.

S chooses (dl,pl) if [EMVI/ EMVzIk > @2-pi)/[(dl-dz)/dl], S chooses (d292) if [EMVII EMVzIk < @2-~1) / [ (d1 -dz ) /d1 ] , S is indifferent if [ E M V I / EMVzIk = @z-p~)/[(d~-dz)/d~].

Observe that EMVi /EMVz may be conceived as the advantage of (dl,pl) over (d2,pz) within the absolute perspective. On the other hand, @z-p~)/[(d~-dz)/d~] is the advantage of (d2,pz) over (dl,pl) within the comparative perspective. Choice between [ ( d ~ , p ~ ) , (dz,pz)] thus depends upon the relative sizes of these two kinds of advantages. In this framework, the parameter k may be interpreted as the relative importance for S of absolute versus comparative considerations. In what follows, we shall rely on the original formulation of the Advantage Model - where k represents the relative weight of payoffs and probabilities - because it is more faithful to choice protocols we have collected, and because the relative weight of payoffs and probabilities has been shown previously to affect preference (e.g., Slovic and Lichtenstein, 1968).

Another rearrangement of terms is also worth mentioning. The Advantage Model may be formulated as a purely comparative model: [(dl-dz)/d2]k and @2/p1) (Pz-pl) being the left- and right-hand attractiveness coefficients, respectively. Notice, however, that according to this formulation the independent evaluation of each option plays no role in the choice process. This not only contradicts the assumptions that have guided absolute theories, but it also conflicts with empirical studies that suggest both ‘inter- dimensional’ and ‘intradimensional’ strategies. (See, e.g., Payne (1976), Payne and Braunstein (1978), and Russo and Dosher (1983)) For this reason, we prefer to formulate the algebra of the Advantage Model in such a way as to exhibit absolute as well as comparative terms. It is this formulation that represents the intuition that choice typically involves a compromise between comparative and absolute strategies.


3. QUALITATIVE EVALUATION

The present section provides a qualitative evaluation of the Advantage Model. To help situate our model within the field of descriptive choice theories, we shall compare the Advantage Model to the two absolute theories discussed above, Utility Theory and Prospect Theory. In what follows, we describe six well documented phenomena that characterize people’s choices between simple lotteries. It will be seen that only the Advantage Model is consistent with all six phenomena. (It is worth pointing out, however, that Regret Theory, if supplemented with judicious assumptions about its binary utility function, is also consistent with the six phenomena.) Throughout our discussion we continue to employ the convention that choice problems [(dl,pl), (dzpz)] are ordered so that PI I pz.

3.1 Limited solvability Given a choice problem [(dl,pl), (d2,p2)] in which three of the four terms are determined, it is often possible to set the fourth term at some value so as to make either one of the lotteries preferred over the other. For example, given [( 1000, .40), (dz, .80)] there is a dz (e.g., 5 ) that will make the left-hand lottery preferred over the right (for most people), and another dz (e.g., 995) that will make the right-hand lottery preferred over the left. In the same fashion, given [(lOOO, .40), (600, pz)] there is ap2 (e.g., .95) that will make the right-hand lottery preferred over the left, and anotherpz (e.g., .45) that will make the left-hand lottery preferred over the right. Similar examples can be generated for undetermined dl orpl . Solvability is limited, however, because of choice problems for which the undetermined term cannot be ‘solved’. Thus, given [( 1000, .70), (10,pz)I there is nopz that will make the right-hand lottery preferred over the left.

Utility Theory and Prospect Theory both predict solvability in appropriate cases. Given [( 1000, .40), (dz, .go)], for example, both theories predict the existence of a dz (close enought to 1000) that will raise the value of the right-hand lottery above that of the left-hand lottery (i.e., n(.8O)*u(d~) > ~(.40)*u( lOOO), where for Utility Theory r i s the identity function). The Advantage Model also predicts solvability. To see this, consider choice problems of the form [(+dl,pl), (+d~,p2)]. As dz approaches zero, so does EMVz(pz-pl), whereas EMVl[(dl-d~)/d~]k increases. Hence, preference must shift to ( d ~ , p ~ ) . On the other hand, as dz approaches dl, the term EMV~[(d~-dz)/dl]k approaches zero, whereas EMV2@2- PI) increases. Hence, preference must shift to (d2,pz). A similar logic applies to the other cases. It may also be seen that all three theories are consistent with the limited nature of solvability.

Observe, on the other hand, that a purely comparative version of the Advantage Model - stripped of the terms EMVI and EMVZ - fails to predict patent cases of solvability. To see this, suppose that k .6 and consider the problem [( 17OO,p1), (500, .4)]. No value ofpl - even 0 - renders [(1700-500)/ 1700lk less than .4-p1. Hence, a purely comparative version of the Advantage Model yields the false prediction that ( 1700,pl) is preferred to (600, .4) for all probabilities PI. Similar examples can be constructed for any chosen value of k.

3.2 Constant difference If person S prefers (+dl,pl) in [ ( + d ~ , p ~ ) , (+d292)], it is possible to find a sum m of money such that S prefers (m+dz,pz) in [ (rn+d~,p~), (m+dzg~)]. For example, if S prefers the left-hand lottery in [( 100, .20), (30, .70)], he is likely to prefer the right-hand lottery in [(1100, .20), (1030, .70)]. A similar shift of preference - but in the opposite direction - can be shown for problems involving negative lotteries.

Prospect Theory and Utility Theory are able to predict the constant difference phenomenon by assuming that the value function of payoffs is generally concave for gains and convex for losses. Given such a function, as payoffs increase by the same amount their difference in values decreases. Since the probabilities are left unchanged, this yields a change in the relative attractiveness of the two lotteries.


The Advantage Model also predicts the constant difference phenomenon because it is based on payoff ratios rather than payoff differences. As we keep the difference between the two payoffs constant and increase their size, the ratio (dl-d?)/dl decreases. At the same time, the absolute value of EMV2 increases faster than that of EMVI since p2 1 PI. Hence, as payoffs are increased preference must eventually shift to the right-hand lottery in a positive problem, and to a left-hand lottery in a negative problem. We illustrate with the example above. According to the Advantage Model, [(loo, .20), (30, .70)] gives rise to a comparison between (100) (.20) [I-(30/ 100)lk and (30) (.70) (.70-.20), i.e., between 14k and 10.5. On the other hand, [(I 100, .20), (1030, .70)] gives rise to a comparison between (1100) (.20) [I-( 1030/ 1100)lk and (1030) (.70) (.70-.20), i.e., between 14k and 360.5. For a large range of k’s the left-hand lottery will be preferred in the first problem but the right-hand lottery in the second, which predicts the reversal.

3.3 Reflection The third phenomenon is called the ‘reflection effect’ by Kahneman and Tversky. It is illustrated by the following pair of problems (taken from Table I):

[(4000, .20)*, (3000, .25)] [(-4OOO, .20), (-3000, .25)*]

These problems are identical except that one involves positive and the other negative lotteries. The reflection phenomenon refers to the fact that people shift their preferences between these problems.

Utility Theory and Prospect Theory can account for this finding by means of their hypothesized value function, which is assumed to be symmetrical around the zero, or ‘reference’ point of the curve. A preference, say, for the left-hand lottery in [(4000, .20), (3000, .25)] entails that 4 .20)*~(4000) > n(.25)*u(3000) (for Utility Theory 7~ is the identity function). Thus, u(4000)/u(3000) > n(.25)/ 4 . 2 0 ) . Now, symmetry around the zero point entails that u(-4000)/u(-3000) > n(.25)/ 4 . 2 0 ) . But since the values of u are now negative, n(.20)*u(4000) < 4.25)*~(-3000) , thus predicting the shift of preference.

The Advantage Model also predicts the reflection effect. For, in the transition from [(+dl,pl) , (+& p2)] to [(-dl,pl), (-d2,p2)], the attractiveness coefficients EMV~[(dl-dz) /dl]k and EMV2 @2-p1) both shift from positive to negative sign. The direction of the inequality between these coefficients thus reverses as well.

3.4 Common ratio The common ratio phenomenon is illustrated by problems 3 and 4 of Table 1. While most people prefer the right-hand lottery in [(6000, .45), (3000, .90)], they prefer the left-hand lottery in [(6000, .001), (3000, .002)]. Notice that the probability ratios in the two problems are the same (namely, 2: 1).

The foregoing preference shift cannot be predicted by Utility Theory since in both problems 43000) is multiplied by twice as much as is u (6000). Prospect Theory, on the other hand, does account for the common ratio phenomenon. The introduction of ‘decision weights’ via the function 7r allows n(.90)/n(.45) to be greater than 4.002)/74.001). This fact allows derivation of the common ratio phenomenon. (The common ratio phenomenon is predicted by certain other absolute theories, which may also be conceived as weakenings of Utility Theory; see, e.g., Machina, 1982).

The Advantage Model predicts the common ratio phenomenon because it is based on probability differences rather than probability ratios. While everything else remains essentially the same, the probability advantage of the right-hand lottery changes from (.90-.45) in the first problem to (.002- .001) in the second. Thus, a wide range of k-values predicts a corresponding shift in preference. In their discussion of the common ratio effect, Kahneman and Tversky (1979) advance the following generalization (here presented for positive lotteries; a similar generalization - in the opposite direction - applies to negative lotteries). If (d1,pq) is preferentially equivalent to (dzp), then (dl,pqr) is preferred

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to (dzpr), 0 <p,q,r < 1. This property is incorporated into Prospect Theory. It can be shown formally that the above generalization is a necessary outcome of the Advantage Model.'

3.5 Intransitivity Consistent intransitivity of preferences can be demonstrated in people's choice among lotteries. One such intransitivity, discovered by Tversky (1969), involves the following five lotteries.

a) (5.00, 7/24) b) (4.75, 8/24) c) (4.50,9/24) d) (4.25, 10/24) e) (4.00, 1 1 / 24)

Many subjects prefer (a) to (b), (b) to (c), (c) to (d), (d) to (e), but (e) to (a). Because Utility Theory and Prospect Theory assign attractiveness coefficients to each lottery

independently of the alternatives to which it is compared, and because coefficients are then compared numerically, neither theory is able to predict intransitivity.2 In contrast, the Advantage Model - in virtue of its comparative component - predicts intransitivity in certain cases. According to the model, for example, the relative attractiveness of lottery (a) above differs when it is compared with lottery (b) from when it is compared with lottery (e). The reader may verify that the Tversky-intransitivity follows from the Advantage Model with k = .91.

3.6 Preference reversal Preference reversal occurs when subjects indicate a preference for one lottery in a choice problem, but then assign a larger monetary value to the other. The monetary value of a lottery is defined as that amount of money which, if received for sure, is as attractive as playing the lottery. To illustrate, if receiving $4 (for certain) is equally attractive to you as playing the lottery (10, SO), than the monetary value of this lottery for you is $4. An example of preference reversal that we have repeatedly observed is as follows. Given [(lo, .60), (5, .80)] subjects often prefer the right-hand lottery but assign higher monetary value to the left-hand lottery. Numerous experimental studies have revealed consistent preference reversals in a majority of subjects. (See Slovic and Lichtenstein, 1983, for a review).

Because of their absolute character neither Utility Theory nor Prospect Theory can predict preference reversal. Both theories assume that each lottery has an inherent attractiveness coefficient with respect to a given person. The lottery with the larger coefficient should both be preferred and have a larger monetary value. In contrast, we shall now see that preference reversal follows from the Advantage Model by considering the monetary value of a lottery (d,p) for a person S to be that sum x of money that renders S indifferent between (d.p) and (x,Z). Consider a person S for whom k = .25. The Advantage Model predicts that S prefers the right-hand lottery in the problem [( 10, .60), (5, .80)]. The following calculations demonstrate that the Advantage Model also predicts that S s monetary value will be larger for the left-hand lottery than for the right.

(10, .60) (x,l) (5, .80) = (x,l) 6 (1-~/ 10) (.25) = x(1-.60)

1.5 - . 1 5 ~ = .4x x = 2.73

4(1-~/5) (.25) x(1-.80) 1 - .2x = .2x

x = 2.5

It will be shown below that the foregoing account is incomplete, inasmuch as it does not predict all known examples of robust preference reversal. To supplement our account it is necessary to consider again the psychologial interpretation of the parameter k in the context of the Advantage Model. This is the topic of the section that follows.


4. THE PARAMETER k

The Advantage Model asserts that a person’s choices among simple lotteries are all governed by a single, fixed k. Consider, however, decision paradigms other than simple choice problems; for example, that of determining a subject’s monetary value for a simple lottery as is done at the end of the preceding section. As can be seen from those calculations, it is natural, within the perspective of the Advantage Model, to consider the monetary value of a lottery (d,p) for a person S to be that sum x of money that renders S indifferent between (d,p) and ( x , f ) . According to the model, x may be calculated from the equation dp[(d-x)/dk = x(f-p). We are not inclined to assume, however, that the value of k in this equation is the same as that used by S in the context of simple choice problems. Rather, since the procedure of determining monetary value focuses attention on payoffs, we may assume that the importance of payoffs (relative to probabilities) is enhanced in this context. In terms of the Advantage Model, this enhanced importance of payoffs means that k has increased in the monetary value paradigm compared to the simple choice paradigm.

Similar reasoning applies to what is known as the probability equivalent paradigm, in which a person is asked to determine a probability y such that a given lottery (dl,pl) is equivalent for him to lottery (dz,y). To illustrate, given (10,.50) and (5,y), you may decide that for you the probability equivalent y is .80, meaning that for you an 80% chance to win $5 is as attractive as a 50% chance to win $10. Here attention is focused on the probabilities, thus enhancing their importance relative to payoffs. As a result, in terms of the Advantage Model, the value of k is expected to decrease.

We may summarize this discussion by the following principle (which may be seen as a modular component of the Advantage Model):

(*) In a choice context in which payoffs (either positive or negative) are made particularly salient, the value of a person’s k typically increases (thereby increasing the relative weight of payoffs); conversely, if probabilities are salient, the value of k typically decreases (thereby increasing the relative weight of probabilities).

Principle (*) is consistent with the context dependency of feature-weighting as demonstrated, for example, by Tversky and Gati (1978; see also Tversky, Sattath and Slovic, 1988, as discussed below). Within the latter demonstrations, the importance that subjects attach to a given feature in a stimulus is shown to vary with the kind of judgment required and with the nature of the other stimuli in view.

Consider now the monetary value estimations in the preference reversal of the last section. As discussed above, such estimation is assumed to focus the subject’s attention on payoffs and thus -according to principle (*) - to increase the value of his k. It is easy to verify that a larger k in the monetary value calculations of the last section produces more pronounced preference reversal than that produced by the Advantage Model without principle (*). For example, suppose that k rises from its original value of .25 in the choice problem [(lo, .60), (5, .80)] to .80 in the associated monetary estimation task. Then, calculations like those in the last section yield monetary values of 4.74 and 3.53 for the left- and right-hand lotteries, respectively. These differ from each other more than the monetary values predicted without principle (*) and thus yield a more pronounced reversal. More generally, it follows from the Advantage Model that as a person’s k increases, his monetary value, x , for lottery (d,p) approaches d from below.3

It should be observed that principle (*) is crucial for the Advantage Model’s explanation of preference reversal. For, without this principle, the model is forced into false predictions. Thus, it follows from the arithmetic of the Advantage Model (when stripped of principle (*)) that for any choice problem [ ( d ~ , p ~ ) , (dz,pz)] where EMVI = EMV2, if k < I , the model cannot assign a higher monetary value to ( d ~ , p ~ ) , and, therefore, will not allow a preference reversal once (dz,pz) is chosen. Invoking principle (*) allows reversals to occur in this kind of case.4

10 Journal of Behavioral Decision Making Vol. 2 Iss. No. 1

Our account of preference reversals may be partially tested with the help of data reported by Goldstein and Einhorn (1987). Among other tasks, these investigators had subjects choose between lotteries in simple choice problems, and also determine the monetary value of each lottery appearing in a problem. We now list the simple choice problems appearing in Goldstein and Einhorn (1987).

[( 16.00, 1 1 / 36), (4.00, 35/ 36)] [(9.00,7/ 36), (2.00,29/ 36)] [(6.50, 18/36), (3.00, 34/36)] [(40.00,4/ 36), (4.00, 32/ 36)] [(8.50, 14/36), (2.50, 34/36)] [ (5 .00 , 18/36), (2.00, 33/36)]

Over all six problems, 40% of the subjects’ responses yielded preference reversals of kind (#):

(#) choice of the right-hand lottery coupled with a higher monetary value for the left-hand lottery,

and only 2% of the subjects manifested preference reversals of the opposite sort (##>:

(##) choice of the left-hand lottery coupled with a higher monetary value for the right-hand lottery.

These results are consistent with the Advantage Model as supplemented by principle (*). Recall that principle (*) requires a higher k for monetary value estimation than for choice. And indeed, according to the Advantage Model, a person with k 2 . 9 for monetary value estimation and k I .5 for choice must manifest preference reversal of kind (#) in all six of the simple choice problems above. We leave to the reader the arithmetic needed to verify this claim.

In addition, there is a stronger sense in which the Advantage Model as supplemented by principle (*) is consistent with Goldstein and Einhorn’s findings. Data will be presented in Section 6.2.2 to suggest that people’s typical k values for choice fall below .5. And it is easy to verify that no subject with a k below .5 can manifest preference reversals of kind (##) on any of Goldstein and Einhorn’s simple choice problems. This fact concords with the low frequency of this kind of preference reversals observed by Goldstein and Einhorn.

Notice, finally, that an increase in the relative weight of payoffs during monetary value estimation, as proposed by principle (*), is consonant with mechanisms proposed by other researchers. Thus, Lichten- stein and Slovic (1971) suggest that greater anchoring on payoffs occurs when monetary values are assigned than when choice between lotteries are made. In a similar vein, Tversky, Sattath and Slovic (1988) call for ‘contingent preference models’, in which the weight given to an attribute (e.g., payoffs or probabilities) is partly determined by the context.

To further illustrate the use of principle (*) in explaining decision phenomena, consider the following experimental demonstration. When first asked to determine the monetary equivalent x that renders him indifferent between, e.g., (5000, .25) and (x , .75), a person S decides that for him x is, say, 2000. But when later asked to determine the probability equivalent y that renders him indifferent between (5000, y ) and (2000, .75), S decides that his y is 30. A systematic pattern of preferences of this kind - where the probabilities estimated in the latter stage are larger than those which figured in the former - is reported by Delquie, de Neufville and Mangnan (1987), who gave people the two tasks separated by a two week interval. The indifferences indicated by S above yield the following inconsistent equivalences:

(5000, .25) - (2000, .75) and (5000, 50) - (2000, .75),

which, according to the Advantage Model, indicate k values of 1 .O in the first judgment and .25 in the second.

The foregoing shift in k-values is predicted by principle (*), according to which the value of S’s k in the first stage - where S focuses on payoffs - should be greater than that of the second stage - where

E. 6. Shafr, D. N . Osherson and E. E. Smith An advantage model of choice 11

S focuses on probabilities. The Advantage Model supplemented by principle (*) thus predicts the inconsistent indifference judgments exhibited by Delquie et a1.k subjects.

Finally, we consider a phenomenon related to the common ratio effect. People seem to overweigh outcomes that are considered certain compared to outcomes that are merely probable. Kahneman and Tversky (1979) dub this tendency ‘the certainty effect’ and illustrate it with the following pair of problems, in both of which pl is 80% of pz.

[(4000, .20)*, (3000, .25)] [(4000, .80), (3000, I)*]

Notice that these problems also instance the common ratio phenomenon (cf. Section 3.4) since the probabilities in the first problem are each one-quarter of their counterparts in the second problem. It is consequently not clear whether the certainty effect is a separate feature of choice behaviour or just a special case of the common ratio phenomenon. Following Kahneman and Tversky, let us assume that the certainty effect is a separate phenomenon and consider how the Advantage Model can account for it.

By definition, the certainty effect occurs in cases where one of the alternatives is not a simple lottery but rather an outcome at certainty (recall from Section 2. I that simple lotteries involve less-than-certain chances). Moreover, unlike monetary value estimation, here the task of simple choice does not focus the subject’s attention on payoffs. The context, therefore, is one where - due to the certainty offered by one of the alternatives - probabilities have been made salient. In this context, principle (*) states that the value of a person’s k typically diminishes (thereby increasing the relative weight of probabilities). Thus, a choice problem [(d,,p~), (d~, l ) ] is evaluated on the basis of a comparison between EMVI[dl- dz)/dl](k’) and EMVz( 1-pi), where k‘ < k , whereas problems with p2 < 1 are evaluated - as usual -using k . The result is that preference is biased towards (d2,p2) whenpz = 1.

The foregoing use of principle (*) has another satisfying consequence. Kahneman and Tversky (1979) show that the certainty effect is not a special case of risk-aversion. For, in negative lotteries subjects tend to avoid certain losses. This is illustrated in the following pattern of choices, which is the ‘reflection’ of the previous one.

[(4000, .20), (-3000, .25)*] [(-4OOO, .80)*, (-3000, I ) ]

This reflection/ certainty phenomenon is predicted by the Advantage Model supplemented by principle (*) since the value of k is predicted to decrease in both the positive and the negative case.

5. SUMMARY

In Section 3 we discussed six phenomena that characterize choice between simple lotteries. Utility Theory qualitatively predicts three of the phenomena, namely, limited solvability, constant difference, and reflection. Prospect Theory predicts four, namely, the previous three plus common ratio. The Advantage Model predicts all six, including intransitivity and preference reversal. (Of course, the Advantage Model cannot claim to provide an exhaustive account of the foregoing phenomena, since any one of them might rest upon multiple mechanisms, cf. note 2.)

The need to refine certain predictions of the Advantage Model led us, in Section 4, to postulate an additional component of the model, called principle (*). This principle recognizes the context- dependency inherent in the value of the parameter k. The Advantage Model as supplemented by principle (*) was shown to be consistent with a number of empirical findings and theoretical claims made in recent years.

Finally, we draw attention to the parameter-space of the theories. Utility Theory allows post-hoc estimation of a real-valued function, u. Prospect Theory allows post-hoc estimation of two such

12 Vol. 2 Iss. No. 1

functions, u and r. Regret Theory estimates a bivariate, or multiattribute function. The Advantage Model includes but a single, real parameter, k, that applies to all simple choice problems and whose value shows systematic change in certain other decision situations.

Journal of Behavioral Decision Making

6. QUANTITATIVE EVALUATION

We turn now to a quantitative evaluation of the Advantage Model. Three experiments will be reported, designed to compare our model against two comparable versions of Utility Theory. First, we consider the following exponential utility function, employing a single, real parameter, c:

u(x) = sign(x)[ 1 -e-""].

Apart from the use of sign and absolute value, this function represents a familiar version of Utility Theory (e.g., Keeney and Raiffa, 1976; Raiffa, 1968). Sign and absolute value have been introduced in order to produce a symmetric utility curve around zero, thereby allowing Utility Theory to predict reflection phenomena as well as constant difference effects for negative lotteries. (It should be noted that these changes yield a variant of Utility Theory that incorporates an element of Prospect Theory, namely, a reference point.) We then consider a second familiar version of Utility Theory (e.g., Kahneman and Tversky, 1982; Coombs, Dawes, and Tversky, 1970) that involves the following power function with a single real parameter:

u(x) sign(x)[lxl7.

Of course, these are only two of many possible versions of Utility Theory. They are intended mainly to provide a standard against which to compare the predictive capabilities of the Advantage Model. Other Utility functions (including expected monetary value and various logarithmic models) were also examined, but they fared less well empirically then the exponential and power functions discussed here.

The Advantage Model yields strong predictions about indifference judgments. For example, if person S is indifferent between lotteries (dl,pl) and (d2,p2), then he should be indifferent between any pair of lotteries (&,PI) and (d4,p2) such that (d3-d4)/d3 = (dl-dz)/dl. For another example, the Advantage Model predicts that a person will not exhibit indifference among any three simple lotteries with equal expected values. Given lotteries (dlgl) , (d2,p2), and (d,p3), with dlpl=dp2=d3p3 and p&-p2>p1, if person S is indifferent between (d l , p~) and (dz,p2) then, according to the Advantage Model, S must prefer (d3,p~) over either of the former. (These two predictions are easily deduced from the arithmetic of the Advantage Model.) The intuition of indifference, however, is unstable for even the most cooperative subject. For this reason we have designed our experiments around the sturdier judgment of strict preference.

6.1 Experimental method

6. I . 1 Design and materials Each of the three experiments consisted of 24 simple choice problems. No problem figured in more than one experiment. All 72 problems are listed, by experiment, in Table 2.

In experiment 1 payoffs range from $750-$1750. Twelve problems (numbers 1-12 in Table 2) were constructed so as to include all combinations of 30%, 4096, and 50% payoff differences and .lo, .20, .30, and .40 probability differences between lotteries. To illustrate, problem 1 - viz., [( 1600, .25), (800, .35)] - represents a 50% payoff difference and a .10 probability difference. The remaining 12 problems (numbered 13-24) are variants of problems 1-12, respectively. Thus, problem 13 is identical to problem 1 except that it involves negative rather than positive lotteries; problem 14 is identical to problem 2 except that the probabilities are halved; problem 16 is the negative and halved version of problem 4, etc.

E. B. Shajir, D. N . Osherson and E. E. Smith An advantage model of choice 13

I ) (1600, .25) (800, .35) 2) ( I 500, .30) (750, SO) 3) (1700, .35) (850, .65) 4) (1500, .40) (750, .80) 5 ) (1500, .30) (900, .40) 6) (1250, 5 5 ) (750, .75) 7) (1300, 50) (800, .80) 8) (1750, .30) (1050, .70) 9) (1650, .50)(1150, .60)

10) (1350, .40) (950, .60) 11) (1700, .20) (1200, 30) 12) (1400, .40) (IOOO, .80)

1) (160, .25) (80, .35) 2) (150, .30) (75, S O ) 3) (170, .35) (85 , .65) 4) (150, .40) (75, .80) 5 ) (150, .30 (90, .40) 6) (125, 3 5 ) (75, .75) 7) (130, 50) (80, 30) 8) (175, .30) (105, .70) 9) (165, 30) ( 1 15, .60)

10) (135, .40) (95, .60) 1 1 ) (170, .20) (120, S O ) 12) (140, .40) (100, .80)

1) (13, SO) (8, .70) 2) ( I 5 , .60) (9, 30) 3) (10, .60) ( 5 , .80) 4) ( I 1, .40) (7, .60) 5 ) (14, S O ) (9, 3 5 ) 6) (17, 50) (9, .70) 7) (16, .30) ( 5 , 50) 8) (16, .40) (10, .70) 9) ( 1 3, .30) (7, .60)

10) (15, .40) ( I I , .60) I I ) ( 1 1, .20) (5 , 50) 12) (17,.70)(11,.80)

Experiment 1 (N=53)

[52]

[I21 [9]

13) (-1600, .25) (-800, .35)

15) (-1700, .45) (-850, .85) 16) (-1500, .20) (-750, .40)

[28] 14) (1500, .15) (750, .25)

[49]

[7]

[43] [I71

17) (1500, .60) (900, .80)

19) (1300, .IS) (800, .25)

21) (1650, .25) ( 1 150, .30) 22) (1350, .lo) (950, .15)

[I91

[ I ]

18) (-1250, .55) (-750, .75)

20) (-1750, .30) (-1050, .70)

[2] [ I ]

23) (-1700, .20) (-1200, .50) 24) (-1400, .40) (-IOOO, .80)

Experiment 2 (N=57)

[57]

[I81 [8]

[26]

13) (-160, .25) (-80, .35) [33] 14) (150, .15) (75, .25)

15) (-170, .45) (-85, .85) 16) (-150, .20) (-75, .40)

[54]

[lo]

[Sl] [21] 22) (135, .10)(95, .15)

17) (150, .60) (90, 30)

19) (130, .15) (80, .25)

21) (165, .25) (1 15, .30)

18) (-125, .55) (-75, .75)

[3] 20) (-175, .30) (-105, .70)

[ I ] [O]

23) (-170, .20) (-120, .50) 24) (-140, .40) (-100, .80)

Experiment 3 (N=50) [25] [28] [37] [I61 16) (17, .40)(13, .60) [7]

13) ( 1 3, .30) (8, .40) 14) (15, .30) (9, .40) 15) (10, .30) (5 , .40)

17) (14, .20) (9, .35) [39] [43]

18) (-17, .50) (-9, .70) 19) (-16, .30) (-5, .SO)

[5] 20) (-16, .40) (-10,70) [9] [5 ]

21) (-13, .30) (-7, .60) 22) (-15, .40) (-1 1, .60)

[8] [43]

23) (16, .20) (10, SO) 24) ( I I , .70) (5 , .80)

Table 2. Simple choice problems used in experiments (In brackets: number of subjects who chose left-hand lottery)

Experiment 2 consists of the same problems as experiment I except that payoffs are uniformly decreased by a factor of 10. Note that both the Advantage Model and the present versions of Utility Theory claim a single parameter per subject, regardless of payoff-size.

Experiment 3 ranges over payoffs from $5-$17. Problems 1-12 were designed to require different k-values for the Advantage Model to predict indifference. These k’s ranged from . I to 1.07. Problems 13-24 were, as before, variants of the first 12.

The range of probabilities in the three experiments is .lo-35. We did not include extreme probabilities (such as .05 or .95) in order to avoid “editing” effects. Presented with a probability of .05, for

14 Journal of Behavioral Decision Making Vol. 2 Iss. No. 1

example, the subject might decide that the chance of winning is ‘essentially zero’ and base her choice entirely on that. We return to the topic of editing in the concluding section.

For each experiment, the 24 choice problems appeared on separate pages, assembled into a 24-page booklet. The problems (e.g., problem 1 of experiment 1) appeared in the following format:

Choose between: 25% chance to win $1600 ~

35% chance to win $800 - The original 12 problems formed an uninterrupted half of the booklet and their 12 variants formed the other half. This was done so as to avoid juxtaposition of two variants of the same problem. The order of the two halves was counterbalanced across booklets. Within each half, the order of problems was randomized for each subject. Finally, within each problem (i.e., on a single page), the order of the two competing lotteries was counterbalanced.

6.1.2 Subjects The subjects were 160 M.I.T. undergraduate volunteers, recruited from a variety of classes and paid for their participation.

6.1.3 Procedure The three experiments followed the same procedure. Subjects were presented with the following, written instructions:

This is an experiment on how people make decisions in uncertain situations. We will represent such situations as chances to win or lose sums of money. For example, one such situation might be:

You will be presented with pairs of situations of this kind and asked to choose the one you would prefer to be in. For example, you might be asked to choose between a situation with a 58% chance to win $30 and another situation with a 32% chance to win $60. This problem would be represented as follows:

36% chance to win $60

58% chance to win $30 - 32% chance to win $60 -

You are to place an ‘X’ in the blank corresponding to the situation you would prefer to be in. Sometimes the choice will be between two possible gains (as above); other times it will be between two possible losses, illustrated as follows:

28% chance to lose $50 - 37% chance to lose $40 -

For each problem, please choose exactly one of the two situations indicated (no ties allowed).

Next, each subject was handed a single booklet and asked to work through it at his own speed without referring back to earlier problems. Typically, subjects worked for 10-15 minutes. No bets were actually played. After completing their booklets, subjects were asked to indicate if they had used any predeter- mined, mechanical procedure to arrive at their choices, rather than responding intuitively. Data from subjects who had decided at the outset to use some mechanical procedure (e.g., ‘always choose higher probabilities’, ‘always choose higher payoff, etc.) were discarded. (These subjects, when later presented with particular examples, all agreed that the strategy they had adopted does not reflect their true preferences.) Three to five subjects were thereby eliminated from each experiment. Apart from those eliminated, 53 subjects participated in Experiment 1, 57 in Experiment 2, and 50 in Experiment 3.

E. B. Shajir, D. N . Osherson and E. E. Smith An advantage model of choice 15

6.2 Results and discussion

6.2. I Preliminary analysis The bracketed number next to each problem in Table 2 indicates the number of subjects who chose the left-hand lottery. Since a forced-choice procedure was employed, the remaining subjects chose the right-hand lottery. Substantial numbers of subjects exhibited the reflection and common ratio effects. This can be seen in Table 2 by comparing, for example, problems 1 and 1 3 , 5 and 17,8 and 20, and 12 and 24 in Experiment 1. Note further that decreasing all payoffs by a factor of 10 (Experiments 1 vs. 2) had no significant effect on subjects’ preferences between lotteries. This imples what is known in the economic literature as ‘proportional risk aversion’ (e.g., Keeney and Raiffa, 1976), and is predicted by the Advantage Model.

6.2.2 Within-subject tests of theories Consider first the Advantage Model. Each value assigned to the parameter k leads the Advantage Model to make a definite number of true predictions about the 24 choices of an individual subject. Call a k-value ‘optimizing’ with respect to a given subject if no other value of k leads the Advantage Model to a greater number of true predictions about that subject’s 24 choices. Preliminary searches of the parameter space indicated that a subject’s optimizing k is almost certain to fall in the interval [0,1]. (Since k represents the relative weight of payoffs and probabilities, this is consistent with findings that people generally give more weight to probability than to payoffs in choice problems; see Slovic and Lichtenstein, 1968). Consequently, for each subject we computed the lowest optimizing k in the interval [O,l], proceeding by increments of .005. The number of true predictions made by the Advantage Model for each subject, relative to his optimizing k was recorded. Table 3 presents the average number of true predictions per subject thereby obtained in each experiment, along with the average, lowest, optimizing k. Across all three experiments, the average number of correct predictions per subject was 20.71. The average, lowest, optimizing value of k across all three experiments was .380.

To gain some idea of the range of optimizing k’s for a given subject, the foregoing procedure was repeated with respect to the highest optimizing k. (By definition, of course, the highest and lowest optimizing k’s yield the same number of true predictions per subject.) Again we searched in the interval [0,1], using increments of .005. For Experiment 1, the average, highest, optimizing k, was .476 (SD: .230). For Experiments 2 and 3, the average, highest, optimizing k’s were 570. (SD: .250) and 321 (SD: .199), respectively. Thus, if the Advantage Model is correct, average k-values tend to fall in the interval from .35 to .55.5

Finally, we randomly generated 65 fictitious subjects and repeated the analysis for Experiment 1 on these random data (searching the interval [O,I] by steps of .005 for an optimizing k for each subject). The average number of correct predictions per randomly generated subject was 14.89 (standard

Average number of Average, lowest correct predictions optimizing k

Exp. 1 20.72(SD: 1.54) .364 (SD: .200) Exp. 2 20.67 (SD: 1.67) .378 (SD: .186) Exp. 3 20.74 (SD: 1.77) .398 (SD: .185)

Table 3. Correct predictions of the advantage model This table gives the average number of correct predictions per subject (out of 24) obtained by the Advantage Model, and the average, lowest optimizing k

16 Journal of Behavioral Decision Making VOl. 2 Iss. No. 1

deviation: 1.72). This figure is significantly lower than the average obtained in Experiment 1 for the 53 real subjects (p < .01, t-test).

Consider now the first version of Utility Theory (which involves the exponential function). Parallel to the Advantage Model, each value assigned to the parameter c leads the present version of Utility Theory to make a definite number of true predictions about the 24 choices of an individual subject. Call a c-value ‘optimizing’ with respect to a given subject if no other value of c leads Utility Theory to a greater number of true predictions about the subject’s 24 choices. A rough sketch of the resulting utility curves indicates that a subject’s optimizing c is almost certain to fall in the interval [0, .05]. Consequently, for each subject we computed the lowest optimizing c in the interval [0, .05], proceeding by increments of .ooOOl. The number of true predictions made by the present version of Utility Theory for each subject, relative to his optimizing c was recorded. Table 4a presents the average number of true predictions per subject thereby obtained in each experiment, along with the average, lowest, optimizing c. Across all three experiments, the average number of correct predictions per subject was 19.16. The average, lowest, optimizing value of c across all three experiments was .0062. Subsequent analyses established that the highesr optimizing c for any subject in experiment 1 was strictly less than the lowest optimizing c for any subject in experiment 2, and similarly for experiments 2 and 3. In other words, the c-values for the three experiments fall in three non-overlapping intervals.

a) Version 1 : u(x) = sign(x)[ 1 -c~’’’] Average number of Average, lowest correct predictions optimizing c

Exp. I 19.15 (SD: 1.59) .000293 (SD: .00029) Exp. 2 19.11 (SD: 1.46) .00264 (SD: .00292) Exp. 3 19.24 (SD: 2.56) .01649 (SD: .0114)

b) Version 2: u(x) = sign(x)[ I x 1‘3

Average number of Average, lowest correct predictions optimizing c

Exp. I 19.55 (SD: 1.65) .643 (SD: .193) Exp. 2 19.51 (SD: 1.56) .688 (SD: .182) Exp. 3 19.98 (SD: 2.27) .606 (SD: .179)

Table 4. Correct predictions of expected utility theory This Table gives the average number of correct predictions per subject (out of 24) obtained by the 1- parameter versions of Expected Utility Theory, and the average, lowest optimizing c’s

The Advantage Model’s average number of true predictions per subject was significantly higher than that of Utility Theory in every experiment ( r 5.73, 5.55,6.35, for Experiments 1-3, respectively; p < .01). Furthermore, in every experiment, for a significant majority of subjects, the Advantage Model predicted these subjects’ choices more successfully than did Utility Theory. In Experiment 1, the Advantage Model made more true predictions for 38 subjects, whereas Utility Theory made more true

E. B. Shafir, D. N. Osherson and E. E. Smith 17

predictions for 10 subjects. In Experiments 2 and 3, these figures were 40 versus 9 and 40 versus 3, respectively (p < .01 in the three cases).

For the second version of Utility Theory (the power function version), as before, each value assigned to the parameter c leads Utility Theory to make a definite number of true predictions about the 24 choices of an individual subject. Again, call a c-value ‘optimizing’ with respect to a given subject if no other value of c leads Utility Theory to a greater number of true predictions about that subject’s 24 choices. For each subject we computed the lowest optimizing c in the interval [0, 13, proceeding by increments of .005. The number of true predictions made by this version of Utility Theory for each subject, relative to his optimizing c was recorded. Table 4b presents the average number of true predictions per subject thereby obtained in each experiment, along with the average, lowest, optimizing c. Across all three experiments, the average number of correct predictions per subject was 19.67. The average, lowest, optimizing value of c across all three experiments was .647.

The Advantage Model’s average number of true predictions per subject was again significantly higher than that of Utility Theory in every experiment (t = 4.35,4.54,4.89, for Experiments 1-3, respective1y;p < .01). Furthermore, for a significant majority of subjects, the Advantage Model predicted their choices more successfully than did Utility Theory. In Experiment 1, the Advantage Model made more true predictions for 33 subjects, whereas Utility Theory made more true predictions for 13 subjects. In Experiments 2 and 3, these figures were 36 versus 10 and 29 versus 4, respectively (p < .01 in the three cases).

An advantage model of choice

6.2.3 Group tests of theories We proceed now to a test of the Advantage Model as a description of group preference. We observe at the outset that the ability of a theory to predict group preferences ought not be confounded with its ability to predict the preferences of any individual in the group. (For discussion, see Luce, 1959.) Consequently, the present group analysis does not reinforce the earlier, within-subject analysis, but rather bears on an independent characteristic of the Advantage Model, namely, its ability to predict group data. For the group analysis, we shall attempt to use the Advantage Model to predict the proportion of subjects opting for one or another lottery in a simple choice problem. In this use of the model, the k-value attributed to a group of subjects is the average lowest optimizing k obtained from the within-subject analysis of the experiment in question. This k-value will be denoted k(av). Thus, k(av) for Experiments 1-3 is .364, .378, and .398, respectively (cf. Table 3). The details of the group analysis are as follows.

The observed advantage of a given lottery in a given choice problem is defined to be the proportion of subjects who chose that lottery in that problem. For example, the observed advantage of (1600, .25) in the first problem of Experiment 1 is 52/53 since all but one subject chose that lottery. Because of the binary-choice nature of our procedure, it is sufficient to focus attention on the observed advantage of left-hand lotteries (the observed advantage of the right-hand lottery being 1 minus that of the left). To carry out our group test of the Advantage Model, we correlated the observed advantage of each left-hand lottery against its predicted advantage. With respect to the Advantage Model, predicted advantage is defined in the following way.

Recall from Section 2 that according to the Advantage Model, the theoretical attractiveness of the left-hand lottery in a choice problem of from [(dl,pl), (d2,p2)] (where P I I p2) is given by EMV,[(dl- d2)/dl]kr whereas the theoretical attractiveness of the right-hand lottery is given by EMV2 (p2-p). We define the predicted advantage of the left-hand lottery in a positive choice problem to be the theoretical attractiveness of the left-hand lottery divided by the sum of the theoretical attractiveness of the two lotteries figuring in the problem. As noted above, the value of k that figures in these calculations is the average lowest optimizing k for the experiment in question, denoted k(av). Thus:


the predicted advantage (according to the Advantage Model) of (+dlgl) in [(+dlgl), ( + d ~ g ~ ) ]

EMVI [(dl-dz)/ d l ] k ( ~ v )

EM VI [ (dl -dz)/ dl] k(av) + EMVz(P2-p I ) .

Predicted advantage must be defined differently in the case of negative lotteries because of the arithmetic of the Advantage Model. For, the greater (the absolute value of) the attractiveness cogfficient for ( - d ~ , p ~ ) , the more likely is the subject to be driven to the choice of ( 4 2 , ~ ~ ) . Thus:

the predicted advantage (according to the Advantage Model) of (-dlgl) in [(-dlpl), (-d292)] =

EMVZ (Pz-PI)

EMVI [(-dl+dz)/ -dl]k(av) + EMVZ(PZ-~I) .

By this method of calculation, given any k(av), the predicted advantage of (+dl,pl) in [(+dl,pl), (+dz,p2)] is always I minus the predicted, advantage of ( -dlgl) in [(-dlgl), (-d292)]. To illustrate, the predicted advantage of (1600, .25) in problem 1 of Experiment 1 (where k(av) = .364) is [((400) (3) (.364)) / ((400) (3 (.364) + (280) (.lo))] .72, whereas the predicted advantage of (-1600, .25) in the homologous problem 13 of Experiment 1 is [((-280) (.lo)) / ((-400) (3) (.364) + (-280) ( . lo))] = .28 = 1-.72.

On the basis we calculated the correlation between the observed and predicted advantage of left-hand lotteries in the three experiments separately. Each correlation, therefore, involves 24 paired numbers. The obtained correlations were .95 for Experiment 1 , .96 for Experiment 2, and .97 for Experiment 3. We next pooled all 72 choice problems and correlated observed and predicted advantage using the overall, average lowest optimizing k across all subjects (i.e., .380, as reported in Section 6.2.2). This overall correlation was .96.

Predicted advantage for Utility Theory was calculated just as for the Advantage Model. To simplify notation, we define the function U(d,p) = sign (d) [l-e-c(av)’d’lp on lotteries (d,p) for the first version of Utility Theory, and U(d,p) = sign (d) [Idlc(av)lp for the second version - where in each case c(av) is the average lowest optimizing c obtained from the within-subject analysis of the experiment in question (see Table 4). Then:

the predicted advantage (according to Utility Theory) of (+dlgl) in [ ( + d ~ g ~ ) , (+d292)] =

U(d1YPl)

U(dig1) + U(d292),

the predicted advantage (according to Utility Theory) of (+PI) in [ ( - d ~ , p ~ ) , (-dzgz)] =

U(d2,P2)

U(d1g1) + U(d292).

The obtained correlations for the first version of Utility Theory were .83 for Experiment 1, .82 for Experiment 2, and .85 for Experiment 3. We next pooled all 72 choice problems and correlated observed and predicted advantage using the overall, average lowest optimizing c across all subjects (i.e., .0062, as reported in Section 6.2.2). This overall correlation was .67. The obtained correlations for the second version of Utility Theory were .81 for Experiment 1, .81 for Experiment 2, and .89 for

E. B. Shafir, D. N. Osherson and E. E. Smith 19

Experiment 3. The overall correlation for all 72 choice problems (using c(av)=.647 as reported in Section 6.2.2) was 3 3 . The second version did better than the first on the overall correlation due to its more stable optimizing c across the three experiments.

Table 5 summarizes the correlational analyses performed. In all four comparisons, the Advantage Model does significantly better than either version of Utility Theory (p < .01 in all cases, significance test between dependent correlations, Bruning and Kintz, 1977, p.215).


Exp. I Exp. 2 Exp. 3 Overall ~

The Advantage Model of Choice .95 .96 .97 .96

Expected Utility Theory

u(x) = sign(x~I-e~""/l 3 3 32 .85 .67 4 x ) = signfx) [I x I 3 .81 .8 1 39 .a3

Table 5 . Correlations obtained between lotteries' observed and predicted advantage

7. CONCLUDING REMARKS

I t was seen in Sections 3 and 4 that, in the domain of simple lotteries, the Advantage Model provides a better account of certain qualitative choice phenomena than do either Utility Theory or Prospect Theory. The data presented in the last section indicate that the Advantage Model is quantitatively superior to plausible, 1 -parameter versions of Utility Theory. Moreover, the Advantage Model incorporates fewer degrees of freedom than either Regret Theory, Prospect Theory or the general version of Utility Theory. Though its successes have been demonstrated in only a limited domain, the underlying conception of the Advantage Model (viz., a compromise between absolute and comparative choice strategies, as discussed in Section 2) seems well supported. In this section we discuss some needed refinements of the model. We also provide a suggestive example of its extension to other domains.

7.1 Small probabilities and editing In its present form, the Advantage Model appears too crude to handle choice problems in which probability differences are small. This is highlighted by the following example communicated to us by Amos Tversky. Consider [(101, .49), (99, .51)]. Most people, it seems, are indifferent between the two lotteries. For the Advantage Model to predict this, k must be approximately equal to 1. Now consider [(6000, .001), (5000, .002)]. Most people, it seems, prefer the right-hand lottery to the left, which implies a k smaller than .01. Not only are the two required k-values outside most people's predicted range, but they also conflict with the claim of a single k per person.

We believe that the psychological mechanism underlying this kind of example is that people misrepresent small probabilities and probability differences. Such misrepresentations might take two forms. First, editing processes of the kind discussed by Kahneman and Tversky (1979) may intervene to change the character of the lottery internally represented by the subject. Thus, subjects may edit the problem [ ( I O I , .49), (99, .51)] to yield a pair of lotteries with essentially identical payoffs and probabilities. (Cf. Russo and Dosher's (1983) findings that small differences between alternatives are often ignored). Given this, the Advantage Model predicts a 'zero' advantage for each lottery and, hence, indifference between them, regardless of a subject's k. Second, we concur with Kahneman and Tversky

20 Vol. 2 Iss. No. I

(1979, p. 281) that ‘very low probabilities are generally overweighted’ by most subjects. Thus, the probabilities in a choice problem like [(6000, .001), (5000, .002)] might be perceived as considerably larger than .001 and .002. The reader may verify that such an increase in probabilities is compatible with a larger value of k for choice of the right-hand lottery. It may thus be seen that the existence of these two forms of probability-misrepresentation mitigates or eliminates the difficulty raised by the example discussed above.

Related to this example is a more general difficulty for the Advantage Model. Given a choice between two positive lotteries, (&pi) and (d2,p2), whose expected monetary values are equal, and wherepz >pi , the Advantage Model can predict choice of (d l , p~) only when k > ~ 2 . ~ For example, in order to predict choice of the left-hand option in the problem [(SOOO, .001), ( 5 , I)], k must be greater than 1. Similar results - but in the opposite direction - obtain for negative lotteries. Thus, in its present form, the Advantage Model is incapable of predicting extreme cases of gambling and insurance, which typically consist of a choice between a small gain/loss at certainty versus a very large gain/loss at a very low probability.

These considerations motivate refinement of the Advantage Model in order to allow it to handle lotteries with extreme probabilities. The needed refinements might involve both editing mechanisms and a ‘decision-weight’ function like the ninvoked in Kahneman and Tversky’s Prospect Theory. Even a utility transformation of dollars might be incorporated into our model. Such revisions do not compromise the underlying psychological tenets of the model, namely, that choice in risky situations involves a combination of absolute and comparative strategies. The latter revisions affect only the manner in which choice problems are represented. The revised Advantage Model would process the resulting representations according to the same principles as before.


7.2 An extension We conclude with a brief illustration of how the Advantage Model may be extended to non-monetary domains. Tversky and Kahneman (1981, p. 453) offered subjects a choice between two possible programs to combat a disease:

Imagine that the U.S. is preparing for the outbreak of an unusual Asian disease, which is expected to kill 600 people. Two alternative programs to combat the disease have been proposed. Assume that the exact scientific estimates of the consequences of the programs are as follows:

If Program A is adopted, 200 people will be saved.

If Program B is adopted, there is K probability that 600 people will be saved, and % probability that no people will be saved.

A second group of subjects was given the same cover story with the following descriptions of the alternative programs:

If Program C is adopted, 400 people will die.

If Program D is adopted, there is a y3 probability that nobody will die, and y3 probability that 600 people will die.

The outcomes presented to the two groups are essentially identical. They differ only in that the former are framed in terms of the number of lives saved, whereas the latter are framed in terms of the number of lives lost. This difference in framing, however, was shown by Kahneman and Tversky to have a decisive effect on people’s choices, since a significant majority in the first group chose Program A, whereas a sigdiciant majority in the second group opted for Program D.

In an attempt to apply the Advantage Model to the present context, it is natural to replace negative and positive monetary payoffs with human lives lost and saved. For example, under the present

E, B. Shajir, D. N. Osherson and E. E. Smith 2 I

interpretation, the alternative (-100, .30) signifies a 30% chance that 100 people will die. Now let us use the Advantage Model, interpreted in this fashion, to compare the alternative programs offered in the Tversky-Kahneman problem above. Programs A and B yield the following comparison (where, as usual, we abide by the convention that alternatives in a choice problem are ordered so that PI I p z . ) :


Program B: (600, 1/3) Program A: (200, 1)

[(m, 113)s (200, 01 200(400/600)k vs.. 200(2/ 3)

133.33k vs. 133.33

On the other hand, Programs C and D yield the following comparison:

Program D: (-600,2/3) Program C: (-400, 1)

[(-600,2/3), (-400, 01 -400(-200/ -600)k VS. -400( 1 / 3)

-133.33k VS. -133.33

For any k < I it follows from the Advantage Model that Program A is preferred to Program B, and that Program D is preferred to Program C. This is exactly the pattern of preferences exhibited by the majority of Tversky and Kahneman’s subjects.

NOTES

1. The proof is as follows: if (dl ,pq) is preferentially equivalent to (d2,p), then (1) d i @ d [(di-d2)ldiIk = d2@) @-Pq). A comparison of (d1,pqr) with (d2,pr) yields: dl@qr) [(dl-d2)/dl]k vs. h @ r ) $w-pqr), or: (2) dlr(pq) [(dl-d2)/dl]k vs. d2r @I @-pq). The right-hand term in statement (2) is equal to the right-hand term in equation ( I ) multiplied by r2 . The left-hand term in statement (2) is equal to the left-hand term in equation ( I ) multipled by r. Since r < I , r2 < r , and the left-hand term of statement (2) is larger than the right. Thus, (d1,pqr) is preferred over (d2,pr). 2. Tversky and Kahneman (1986) invoke ‘editing’ strategies to explain intransitivity. Thus, for the example cited above, it might be assumed that the probabilities of adjacent lotteries are considered -via editing - to be identical, whereas the probabilities of lotteries (a) and (e) differ enough to affect evaluation and choice. Of course, as Tversky and Kahneman (1986, p. S273) point out, intransitivity of preference may result from more than one psychological mechanism. 3. Let us show that, according to the model, no matter how large a person’s k may be, his monetary value, x, for lottery (d,p) will always remain smaller than d. The proof is as follows: Since x is the monetary value of (d,p), (d ,p) = (x,]). Therefore, according to the Advantage Model, dp[d-x)/d]k = ~ ( 1 - p ) . pdk-pxk X - X P , pdk x( I-p + pk) , and thus

Now, for any k > 0, since 0 < p < I , ( I -p) must be positive and [pk/ (( I -p) + p k ) ] < I . Therefore, d > x. 4. The proof is as follows. The monetary equivalents of (d1,pi) and (d2,p2) are X I and X I , respectively, so: (dl,pl) = (XI,]) and (d2,p2) (x2,l). The arithmetic of the Advantage Model yields: p d k x ~ ( l - p ~ ) + x p l k andpldzk = x 4 - p ~ ) + x2p2k. Since EMVI 2 EMV2, pldlk =pzdzk. Therefore, xl(l-pl) + xlplk = x2( ILp2) + mpzk, and

x = d* [ p k / ( l - p + p k ) ] .

X I / X I = [(1-p2) + ~ 2 k ] I [ ( ] - p i ) + p l k ] .

22 Vol. 2 Iss. No. I

Now, sincep2 >PI > 0, for any 0 < k < 1 the ratio on the right side of the equality must be < 1. Therefore, XI must be less than x2. To show that when supplemented with principle (*) the model allows reversals in the formerly-denied direction, we provide the following example, involving the choice problem [(40, .4), (20, .8)]. Notice that EMVI = EMVZ. Now, assume that k = .7 for simple choice and is doubled when assigning monetary value. Then the calculations for simple choice and for monetary values are as follows:


Choice: Monetary value: Monetary value: [W, .4), (20, .8)*1 (40, .4) = (X1,l) (20, .8) = (X2,I)

16(20/40)k VS. 1q.8-.4) 16(1-~/40) (1.4) = .6x I6( 1 -x/ 20) ( 1.4) = .2x 8(.7) vs. 6.4 x = 19.31 x = 16.97

Thus, while (d2,p2) is preferred, (dl,pl) is assigned a higher monetary value. 5 . Recall (Section 3.5) that a significantly larger k-value, namely, .91, was required to predict the Tversky-intransitivity. This is compatible with the fact that Tversky’s subjects were preselected for being ‘potentially intransitive’, i.e., were an unusual group. Tversky and Kahneman’s results in Table I , on the other hand, characterize majority choice and are predicted by a k-value within our obtained interval of average k’s. 6. The proof is as follows. Assume a choice between (dl,pl) and (d2,pz), where d1p2 = d2p2 and p2>p1. We know that:

Now, according to the Advantage Model, (dl,pl) will be preferred only if dlplk[(dl-d2)/dl)]>d~p2@*-1)l). Cancelling dl on the left-hand side of the inequality and substituting for the remaining dl using ( I ) , yields:

k >p2 then follows from (2) by elementary algebra. We are indebted to A. Tversky for this observation.

(1) dl = (d2P2)IPI.

(2) kpl[(d2~z/p1)-dz] > d2~2@2-~1).

ACKNOWLEDGEMENTS

We thank Philippe Delquie, Richard de Neufville, Daniel Kahneman, Amos Tversky, J. Frank Yates, and three anonymous reviewers for helpful discussion. The research reported herein was supported by NSF Grants #8609201 and #8705444. Comments may be addressed to Edward E. Smith, Department of Psychology, University of Michigan, 330 Packard Road, Ann Arbor, MI 48104-2994, USA.

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Authors ’ biographies: Eldar Shafir received his Ph.D. from MIT’s Department of Brain and Cognitive Sciences in 1988. He is currently a postdoctoral scholar in the Psychology Department at Stanford University. As of 1989, he will be Assistant Professor of Psychology at Princeton University. His interests include judgment, choice and inductive reasoning.

Daniel N. Osherson grew up in New York City, earned a Bachelor of Arts degree at the University of Chicago, and a Ph.D in Psychology at the University of Pennsylvania. After teaching at Stanford University and the University of Pennsylvania, he joined the faculty at MIT in 1979 where he is currently Professor of Brain and Cognitive Sciences.

Edward E. Smith is Professor of psychology at the University of Michigan. He received his Ph.D. from Michigan and has held positions at the University of Wisconsin, Stanford University, BBN Labs (Cambridge, Mass.), and MIT. He is interested in categorization, induction and other areas of cognition.

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An advantage model of choice