Content-dependent preferences in choice under risk: heuristic of relative probability comparisons

Transcription

1 Content-dependent preferences in choice under risk: heuristic of relative probability comparisons Pavlo Blavatskyy CERGE-EI December 2002 Abstract: Theory of content-dependent preferences explains individual choice decisions by underlying relative preference relation. Preferences are relative when individual preference between two particular choice options depends on other available alternatives (content of the choice set). In choice situations under risk relative preferences can emerge when individual makes decisions accordingly to the heuristic of relative probability comparisons. This heuristic assumes that individuals prefer the lottery which is most probable to bring them to higher utility level than any other available gamble. The obtained experimental results demonstrate that in specially designed choice situations the majority of subjects use heuristic of relative probability comparisons. Predictions of the theory of content-dependent preferences such as the existence of rational intransitive preferences, preference reversal and violation of WARP are also confirmed experimentally. Key words: Content-dependent, intransitive, preference reversal JEL Classification codes: C91, D11, D81 Author s blapaul@yahoo.com, pavlo.blavatsky@cerge.cuni.cz I am grateful to Andreas Ortmann and Anton Küberger for helpful comments. Also I would like to thank Ganna Pogrebna and Roman Blavatskyy for their help in organizing experiments. 1

2 Introduction Individual preference is content-dependent if elimination of irrelevant alternatives can cause its reversal. Thus, individual decision on the most preferred option from the choice set can change if some inferior options 1 are added or deleted from the menu. When preference is contingent on the choice set, different choice sets can generate different indifference curves and utility functions defined over the same common elements. Therefore, rational choice can be intransitive and violating WARP. In this paper I examine rational decisions of individuals with content-dependent preferences in choice situations under risk. The rich literature on decision making under risk produced many normative theories trying to explain the experimental data. All these theories can be clustered into two broad groups accordingly to their assumption about individual preference formation. The first group of theories assumes that individual preference relation over lotteries is absolute, which means that lotteries can be compared on the universal utility scale irrespective of the context of choice situation. When preference relation is absolute there is a mapping from each element of the lottery space to some common scale, which can be a set of real numbers or some other ordered topological space. Thus, the typical theory of decision making with assumed absolute preference relation would specify: 1) how to map the elements of the lottery space into the ordered topological space (evaluation stage), and 2) how to choose the most preferred element of the image of the choice set (some subset of lottery space) on the ordered topological space (comparison stage). The (set of) most preferred lottery is the lottery (lotteries) whose image on the ordered topological space was chosen in the comparison stage. The example of the theories assuming absolute individual preference is the conventional expected utility theory allowing evaluation of any lottery once the possible outcomes and their 1 Options that are not chosen anyway. 2

3 associated probabilities are known. The decision is then simply deduced to choosing the lottery yielding the highest expected utility. In other words, the lotteries are first evaluated on common scale and then compared to each other. In the evaluation stage the information only about one lottery is used, the payoff distribution of other available lotteries is irrelevant. The second group of theories of decision making under risk assumes that individual preference relation is relative, which implies that lotteries can be evaluated and then compared only in the context of the choice situation i.e. when the information about all available alternatives is known. When preference relation is relative there is a mapping of each element of lottery space contingent on its choice set (the subset of lottery space containing the element of lottery space under consideration) to the ordered topological space. In general each element of lottery space can be mapped to different elements of the ordered topological space if this lottery is viewed as the element of different choice sets. Therefore, the typical theory from the second group would specify: 1) how to map the elements of the choice set (some particular subset of lottery space) to the ordered topological space (evaluation stage), and 2) how to choose the most preferred element of the image of the choice set on the ordered topological space (comparison stage). Note that when preference is absolute, if the lottery was already evaluated in the past there is no need to repeat evaluation stage when the old lottery is available again as the element of new choice set. When preference is relative, the evaluation stage has to be repeated each time for the same lottery when the choice set changes. Because relative preference ordering implies that each lottery is evaluated relatively to the other available gambles. For lottery evaluation it is important to know how different the probability distribution of a particular lottery is from probability distributions of other feasible lotteries. Generally, individual can use two discriminating clues how distinct are lottery outcomes and how distinct are lottery probabilities. 3

4 The examples of theories based on the relative preference relation are the regret theory (Loomes and Sugden (1982)) and the similarity theory (Rubinstein (1988)). Both theories allow comparisons among lotteries already in the evaluation stage. During the evaluation stage full information about payoff distribution of all available lotteries is used. In this paper I propose another mechanism of decision making under risk that leads to relative preference ordering over the lotteries. Heuristic of relative probability comparisons allows the decision maker to evaluate a lottery through focusing on the probability differences of the feasible gambles. In the evaluation stage, the individual evaluates the probability that a particular lottery brings a higher outcome than the outcomes of other available lotteries. Technically, the decision maker estimates the ex ante probability of a lottery to bring the highest ex post outcome (among the ex post outcomes of all feasible lotteries). In the comparison stage the lottery with the highest such probability value is chosen as the most preferred. Thus, given a particular choice set, this decision procedure selects a lottery, which is most likely to bring the highest outcome. The remaining part of this paper is structured as follows. Part two presents the design, implementation and results of experiment on decision making that brings empirical evidence on existence of heuristic of relative probability comparisons. Part three analyses the choice situations employed in the experiment. Specifically, in this part I demonstrate that the obtained experimental results can be explained by (and only by) usage of heuristic of relative probability comparisons by the majority of subjects. Part four discusses the relevant literature. Part five concludes the discussion. 4

5 Apple-tree experiment The conventional microeconomic theory assigns utility values to individual elements of the choice set independently of other elements available in the set. The theory of contentdependent preferences allows comparisons between different elements of the choice set. The following example of decision making under risk contrasts the decision procedure based on the heuristic of relative probability comparisons against the conventional benchmark. Heuristic of relative probability comparisons leads to relative preferences over lotteries and thus it can explain intransitive choice and violation of WARP by the same set of reasons. Ortmann and Gigerenzer (1997) argue that the pioneering discovery of contextual effects in psychology has a direct effect on experimental economics. Although the economists do not fully understand the impact of context (and content) on choice decisions, it is clear that conducting abstract (context-free) experiments endangers the validity of their results. Subjects may seem to make irrational decisions in abstract context, which are nevertheless reasonable from some particular point of view (presumably brought to laboratory from real life). Thus in the following experiment the situations of choice under risk are presented not by means of abstract lotteries but by simple real-life example planting an apple-tree with uncertain future crop. The introduction of specific context to risky choice situations leaves no freedom for the subjects in choosing their own decision perspective. The choice set consists of three apple-trees: A, B and C. Apple-tree A produces either 2 apples with probability 2 3 or 5 apples with probability 1 3. Annual apple-crop of tree B is always 3 apples. Starmer and Sugden (1998) discovered that intransitive preferences are more likely to appear when one of the lotteries in the choice triple is riskless (sure thing). This observation motivated me to use apple-tree B, which always produces the same apple crop. Apple-tree C yields either 4 apples with probability2 3 or 1 apple with probability 1 3. Obviously, the expected apple crop of every tree is 3 apples (see figure 0). If the individuals care 5

6 only about the expected payoffs, they should be indifferent between the proposed apple-trees and there will be no systematic pattern in their choice decisions. 0 To assess the empirical foundations of the heuristic of relative probability comparisons I ran an experiment asking the subjects to reveal their choice in the following four situations: 1) choice between apple-tree A and B, 2) choice between apple-tree B and C, 3) choice between apple-tree A and C, and 4) choice among apple-trees A, B and C. Since representation by natural frequencies facilitates individual decision making under risk (Hoffrage at al. 2002), in the experiment the subjects were acquainted with the apple-trees without the explicit usage of word probability 2 (complete questionnaire is presented in Appendix I). The experiment was conducted in six subject groups. The details of each group are presented in table 1. The results of the experiment are presented in table 2. In general, 24 distinct choice patterns are theoretically possible in the experiment. It turned out that 5 choice patterns were 2 In fact the first pilot experiment contained the description of choice situations with the usage of word probability. For example apple-tree A was described as the tree giving either 2 apples with probability 1/3 or 2 apples with probability 1/3 or 5 apples with probability 1/3. Nearly all subjects declared that they do not understand the word probability and the additional verbal explanations were required. The results from this experiment were qualitatively similar to those reported in the paper for situations 1) and 2). 94.7% of subjects have chosen apple-tree B in the first choice situation, 68.4% apple-tree C in the second situation, 52.6% apple-tree A in the third situation and 21% of the subjects preferred apple-tree C in the fourth situation. 78.9% of the subjects demonstrated preference reversal (violation of WARP). 36.8% of the subjects had intransitive preferences of the pattern A f C f B f A. 6

7 never chosen by any of 411 subjects and another 12 choice patterns were chosen by 4 or less than 4 subjects across all groups (less than 1% of subject pool). Therefore, in table 2 I present only the number of subjects in each group answering accordingly to one of the remaining 7 most frequent choice patterns. Table 2 also shows how many subjects revealed preference ABCC 3, which was the only intransitive pattern of type A f B f C f A ever selected, although this pattern was chosen by less than 1% of subject pool. Table 2 demonstrates that except for group 5 all groups gave qualitatively similar responses. Group 5 turned out to be extremely risk averse. The overwhelming majority of subjects has chosen apple-tree B in the first situation, around 70% of subjects have chosen apple-tree C in the second situation and apple-tree A in the third situation. Nearly half of subjects preferred apple-tree C in the fourth situation. Around two thirds of subjects violated WARP and more than half of subjects revealed intransitive preferences of the pattern A f C f B f A. Only around 1% of subjects had intransitive preferences of the opposite pattern. In all subject groups except group 5 we observe violation of weak stochastic transitivity more than 50% of subjects prefer B to A, C to B and A to C. Rieskamp et at. (unpublished) argue that the violations of weak stochastic transitivity are rarely reported in the literature and they are mostly explainable by neglect of just perceivable differences. However, in the apple-tree example the differences in probability are substantial (at least 33.3%) just like the differences in outcomes (at least 20%). This makes explanation of violation by just noticeable differences hardly convincing. 3 Pattern ABCC corresponds to the choice of apple-tree A in first situation, choice of apple-tree B in second situation, choice of apple-tree C in third situation and choice of apple-tree C in fourth situation. 7

8 Group Population Date of experiment Place of experiment Prague, Czech Republic Dnipropetrovs k, Ukraine Dnipropetrovs k, Ukraine Language of experiment English Russian 19 Russian Lviv, Ukraine Ukrainian Lviv, Ukraine Ukrainian Lviv, Ukraine Ukrainian Table 1 Subject pool Subject age Subject background Method of interviewing American, Dutch, German, Russian tourists Undergraduate students majoring in physics and technical science Undergraduate students majoring in finance Undergraduate students majoring in international relations Undergraduate students majoring in mass media Employees of Lviv Bus Plant specialized in technical engineering Individually, on Charles Bridge in Prague downtown In class In class In class In class During internal seminal meeting Group Population ABCC Number of subjects with response pattern Percentage of subjects answering BCAC BCAB BCCB BBAB BBCB ACCC BCCC 1) B 2) C 3) A 4) C WARP Intransitively % 78.9% 57.9% 50% 78.9% 52.6% % 77.2% 80.7% 54.4% 71.9% 64.9% % 71.7% 72.8% 47.8% 66.3% 55.4% % 64.5% 66.4% 27.3% 64.5% 39% % 38.6% 64.3% 24.3% 31.4% 21.4% % 71.4% 85.7% 39.7% 76.2% 63.5% intransitive Violation of WARP Table 2 Results of the experiment 8

9 Predictions from heuristic of relative probability comparisons Assuming individual non-satiation in apples let us consider the following situations: Situation 1) The individual decides between apple-tree A and apple-tree B, which is equivalent to playing a game against the nature. Figure 1 shows the normal form of this Nature p=2/3 p=1/3 Individual A 2 5 B 3 3 Figure 1 Choosing apple-tree A vs. apple-tree B game with numbers in the cells standing for individual s payoff. Individual knows that nature always plays a mixing strategy: either assigning 2 apples to apple-tree A with probability 2/3 or assigning 5 apples to apple-tree A with probability 1/3. Therefore, his best choice should be strategy B because the probability of collecting larger apple crop while playing B is twice as large as while playing A. Situation 2) The individual chooses between apple-tree B and apple-tree C. Similarly, the normal form of this game is presented by figure 2. In this case the rational individual should play C because this strategy is more likely to yield a higher payoff than the alternative. Nature p=1/3 p=2/3 Individual B 3 3 C 1 4 Figure 2 Choosing apple-tree A vs. apple-tree B Situation 3) The individual decides between the apple-tree A and apple-tree C. The normal form of this game is illustrated by figure 3. In this case the nature plays the following mixing strategy: a) to assign 2 apples to tree A and 4 apples to tree C with joint probability 4/9, b) to assign 2 apples to tree A and 1 apple to tree C with joint probability 2/9, 9

10 c) to assign 5 apples to tree A and 4 apples to tree C with joint probability 2/9, and d) to assign 5 apples to tree A and 1 apple to tree C with joint probability 1/9. The individual faces a non-trivial task of choosing between strategy A and strategy Nature Individual p=4/9 p=2/9 p=2/9 p=1/9 A C Figure 3 Choosing apple-tree A vs. apple-tree C C because of the increased number of possible outcomes. While strategy C gives higher payoff with probability 4/5 (case a) ), strategy A is optimal because it promises a higher yield with probability 5/9 (cases b), c) and d) ). Situation 4) The individual chooses among three apple-trees: A, B and C. The normal form of this game is the following (figure 4). Strategy A gives the highest payoff with probability 1/3, strategy B yields the largest apple crop with probability 2/9 and strategy C promises more apples than the alternatives with probability 4/9. Therefore, it would be optimal for the individual to choose apple-tree C. Nature Individual p=4/9 p=2/9 p=2/9 p=1/9 A B C Figure 4 Choosing among apple-trees A, B and C Notice, that the choice pattern of the individual using heuristic of relative probability comparisons is intransitive: B is revealed preferred to A, C is revealed preferred to B, and A is revealed preferred to C. 10

11 In choice situations 1)-3) the response pattern predicted by heuristic of relative probability comparisons was indeed strongly supported by the experimental data. However, in choice situation 4) the aggregate percentage of subjects choosing apple-tree C (predicted by heuristic) in groups 4, 5 and 6 is not statistically significantly different from 33.3%--the percentage of subjects choosing apple-tree C if they were indifferent and made choice decision at random. It is suggestive that the heuristic of relative probability comparisons is employed in binary choice but less subjects use it when faced with choice among three or more lotteries. Another explanation can be the complexity of probability evaluation subjects tend to choose the tree yielding sure apple crop when evaluation of relative probabilities becomes cumbersome (as in the case when there are three or more lotteries). If this explanation is valid the increase of incentives should motivate the subjects to invest more effort in choice situation 4) and employ heuristic of relative probability comparisons. Finally, another reason why so many subjects choose apple-tree B in situation 4) might be the individual risk aversion that becomes a crucial factor for decision making once the comparison of relative probabilities is complicated 4. In the first and second choice situation I effectively replicate Kahneman and Tversky (1979) result that people are risk averse for gains and risk lovers for losses. On one hand, people dislike lotteries promising a small loss with high probability or lotteries yielding significant gain with small probability. On the other hand, individuals are fond of gambles giving a small chance to loose a lot or gambles with a very likely small gain. In the first choice situation apple-tree A promises a relatively high probability of small loss (-1) together with a small probability of big gain (+2). Not surprisingly, the individuals find this option relatively less attractive than the certain payoff promised by apple-tree B. However, in the second choice situation the apple-tree C promises a high probability of small gain (+1) accompanied by a small probability of huge loss (-2). Since people are risk-lovers for losses and risk-averse for gains, it 4 Inducing risk neutrality can cure this problem. For example, it is possible to repeat the experiment with appropriately changed probabilities or outcomes. However, this can be a disentangled move for the purpose of the experiment. Only when subject group is clearly risk-averse (like group 5) inducing risk neutrality can help in determining the threshold point beyond which the subjects start to use heuristic of relative probability comparisons. 11

12 is also predictable that they would prefer apple-tree C (the gain of +1 with probability 2/3 and the loss of 2 with probability 1/3) to zero status quo (apple-tree B). Obviously, apple-tree B second order stochastically dominates apple-tree A and appletree C since apple crops of trees A and C are mean preserving spreads of 3. Therefore, a riskaverse individual (with concave utility function) should make a consistent choice in situations 1) and 2), always choosing apple-tree B. Risk loving individual (with convex utility function) should also make a consistent choice in situations 1) and 2), never choosing apple-tree B. The choice of apple-tree B only in one situation but not in the other implies one of two possibilities: a) individual utility function changes its second moment i.e. the underlying preferences are sensitive to the content of the choice set, b) individual has state-dependent utility i.e. she/he cares about the states of nature causing different apple crops. Since we do not explicitly specify what are the states of nature occurring with probability 1/3 and 2/3 it is unlikely that individual would care about these anonymous states. The intransitive preferences that are revealed in the first three binary choice situations constitute in fact a well-known preference reversal phenomenon. If we label apple-tree A as a P bet and an apple-tree C as a $ bet 5, then by a direct binary choice (third situation) P bet is revealed preferred to $ bet. However, in the first choice situation the individual reveals that P bet is worth less than 3 apples for sure, which is the promised apple-crop of apple-tree B. In the second choice situation the $ bet is valued more than 3 apples since the individual is not willing to give up apple-tree C for as low as 3 apples (to choose apple-tree B). Therefore, we observe the failure of procedure invariance: P bet is preferred to $ bet in direct binary choice but $ bet is valued more than the P bet in monetary terms. 5 Apple-tree A gives a sure chance to collect good but not very high apple crop (either 2 or 5 apples). Apple-tree B yields some chance to collect a high apple crop (4 apples), but also it might bring intolerably low (in fact minimum) apple crop of 1. Therefore, subconsciously, people associate apple-tree A with a probability lottery yielding almost certain modest payoff and apple-tree B with a dollar lottery yielding high but unsure payoff. 12

13 Apple-crop collected Apple-crop collected with probability 1/3 with probability 1/3 choice set in 5 situation 1) 5 A B A B choice set in situation 2) 1 C 1 C Apple-crop collected Apple-crop collected with probability 2/3 with probability 2/ Apple-crop collected Apple-crop collected with probability 1/3 with probability 1/ A 4 3 B choice set in 3 2 situation 3) 2 A B choice set in situation 4) 1 CC 1 C Apple-crop collected Apple-crop collected with probability 2/3 with probability 2/ Figure 5 Indifference curves consistent with revealed choice functions in each situation This apple-tree example can be conveniently represented by the conventional microeconomic apparatus of indifference curves and choice sets. All three apple-trees are points on the two-dimensional plane, where the horizontal axis measures how many apples the tree yields with probability 2/3 and the vertical axis shows the apple crop of each tree, which is collected with probability 1/3. In other words, apple-tree A corresponds to point A(2;5), appletree B is represented by point B(3;3) situated on 45 certainty line and apple-tree C is plotted as point C(4;1). The pattern of the indifference curves consistent with the choice functions revealed in the situations 1)-4) is presented on figure 5. 13

14 In choice situation 1) apple-tree B lays on a higher indifference curve than apple-tree A and at the same time in situation 2) apple-tree C is situated on a higher indifference curve than B. However, in the third choice situation apple-tree C is situated on a lower indifference curve than apple-tree A, which demonstrates that individual preferences can change as the choice set changes i.e. preferences are content-dependent. If the preferences were content-independent, the indifference curves would not intersect when we combine all sections of figure 6 together on one graph. Additionally, the knowledge of the pattern of indifference curves revealed in choice situations 1)-3) does not help to predict the pattern of indifference curves in situation 4), although the choice set in situation 4) is the union of choice sets in situations 1)-3). Finally, in choice situation 1) and possibly 3) the family of indifference curves is convex (risk aversion) whereas in choice situation 2), 4) and possibly 3) the indifference curves are concave (risk love). By comparing choice situations 3) and 4), we observe a violation of WARP apple-tree A is revealed preferred to apple-tree C in choice situation 3), however apple-tree C is chosen in situation 4) whereas apple-tree A was available in the choice set as well. A is revealed preferred to C when the choice set is {A,C} and C is revealed preferred to A when the choice set is {A,B,C}. Therefore, the less preferred option was chosen as the choice set enlarged, which is violation of WARP (Arrow, 1959). Alternatively, choice situations 1)-4) can be represented by the conventional probability triangle a two dimensional simplex model for choice under uncertainty. Since we have three gambles defined over five distinct possible outcomes the usual application of the method would require a representation in four dimensional probability simplex. Fortunately, we can remain in the conventions of two dimensional probability triangle, noticing that apple-tree A can be alternatively viewed as a gamble yielding a 100% chance of collecting 2 apples and the additional 33.(3)% chance of collecting extra 3 apples. Apple-tree B promises 3 apples for sure. Apple-tree C gives 1 apple for certain plus a possibility to collect three apples with probability 2/3. Thus all three apple-trees are defined as the lotteries over three possible outcomes: 1 apple, 14

15 2 apples and 3 apples. The unusual twist in this representation is, however, the probability crop of 1 apple 1 C D 0?? 1 crop of 3 apples 0 1/3 2/3 B 1 A crop of 2 apples Figure 6 Apple-trees representation in 3-D space distribution over these outcomes, which has a cumulative density function larger than unity, implying that the outcomes are not mutually exclusive (realization of one outcome does not prohibit the realization of the other outcome too). Figure 6 presents three apple-trees in three dimensional space formed by vector basis of three possible distinct outcomes. Apple-tree A corresponds to point A(1/3,0,1) on figure 6, apple-tree B is represented by point B(1,0,0) and apple-tree C is shown as point C(2/3,1,0). All points lie in one two dimensional simplex (figure 7), which is situated above the conventional probability triangle (shadowed plane on figure 6). Since the outcomes can not occur with negative probability or with probability higher than 1, the gambles situated in the shadowed (leftward and upward parts) area of simplex on figure 7 are obviously not feasible. This leaves us with the probability parallelogram ABCD 6 containing all three lotteries of interest. If the individual has expected utility function of von Neumann-Morgenstern type than in the probability parallelogram ABCD her/his indifference curves are parallel lines. From choice situations 2) and 3) we can conclude that individual s direction of increasing desirability is 6 Point D is a gamble yielding a sure gain of 1 apple plus a sure gain of 2 apples. As it is obvious from figure 7 point D also belongs to the plane ABC. 15

16 similar to the pattern presented on figure 7. However, in choice situation 1) individual effectively reveals a preference for apple-tree B as opposed to apple-tree A. Together with the revealed choices in situations 2) and 3) this preference forms an intransitive cycle. But it is possible to present such intransitive preferences by one pattern of parallel indifference curves in parallelogram ABCD. Choice option D, located in the same probability parallelogram, gives a sure chance to win 1 apple and a sure chance to win 2 apples i.e. D yields a sure payoff of three apples, which is the definition of apple-tree B. Therefore, the indifference curves shown on figure 7 explain the observed intransitive preferences: a) apple-tree B when viewed as option D (a 100% chance of getting 1 apple plus a 100% of getting 2 apples) is preferred to apple-tree A 7 (choice situation 1) ); b) apple-tree A is preferred to apple-tree C (choice situation 3) ); and c) apple-tree C is preferred to apple-tree B when the last is viewed as option B 100% chance of getting 3 apples (choice situation 2) ). Finally, in situation 4) individual using heuristic of relative probability comparisons reveals a different pattern of indifference curves (figure 8) than in choice situations 1)-3), which denotes that her/his preferences have changed. Figure 7 Apple-trees representation in 2-D simplex Figure 8 Indifference curves revealed in situation 4) 7 When apple-tree B viewed as option D is compared with apple-tree A, this is a decision over two lotteries: lottery D giving 100% chance of getting 1 apple plus a 100% of getting 2 apples and lottery A yielding 2 apples for sure plus 3 apples with probability 1/3. Eliminating the common consequence of getting 2 apples for sure, a gamble D yielding 1 apple for sure is compared with gamble A yielding 3 apples with probability 1/3. This is obviously an easier decision to make than comparing A vs. B. Thus, in situation 1) it s easier for individual to make choice when apple-tree B is viewed as option D. However, this representation is not helpful in choice situation 3). 16

17 Related Literature The theory of content-dependent preference postulates that the content of the choice set influences individual s binary preference relation over the elements of this choice set. In other words, individual s choice decision between two particular alternatives depends on the other options available in the choice set, that might be potentially chosen as well. The idea of contentdependent preference is related to several streams of microeconomic literature. Firstly, the notion of content-independent preference in individual choice parallels Independence of Irrelevant Alternatives (IIA) axiom in public choice. Kalai and Smorodinsky (1975) argued that IIA axiom does not necessarily hold and suggested an alternative solution to Nash bargaining problem the ration of two corner points of the bargaining set. The corner point is irrelevant alternative since it will be never chosen, however if it changes, the outcome of the bargaining process, accordingly to Kalai and Smorodinsky, becomes different too. Similarly, in the problem of individual choice if preference is content-dependent, the change in menu of available inferior alternatives influences individual choice of most preferred option. Kalai et al. (2002) gives a number of real-life choice procedures violating IIA that can be rationalized by several preference orderings. The revealed preferred element of each choice set is consistent with maximization with respect to one of those orderings. Kalai et al. (2002) approach is context-free. The decision maker has a set of preference orderings and each ordering is applied in some non-empty disjoint subset of universe of all choice situations. In contrast, the theory of content-dependent preference explains IIA violations by means of one preference ordering which is applied to all choice situations. However, this preference ordering is relative and not absolute. Formally, Kalai et al. (2002) assumes { f, f... 2 f } i [ 1, n] S, S i ( I S ) =, A S f A R i i i 1 n i 0 :, where R is ordered topological space and S I is a subset of the space I of all subsets of universe of possible choice alternatives Ω with at least two elements. In contrast, theory of content-dependent preferences assumes g : Ω I R. i 17

18 Secondly, the theory of content-dependent preference is another normative theory explaining the observed violations of revealed preference axioms. The literature on the axiomatisation process deals first of all with the consistency of individual choice decision as the choice set expands and contracts (Bandyopadhyay and Sengupta, 1991). For instance, Andreoni and Miller (2002) demonstrate the rationality of altruistic behavior by finding a surprisingly small number of revealed preference axioms violations when subjects were confronted with different choice sets. In this paper I keep the same consistency-checking methodology asking subjects to choose the most preferred element from the set and its subset. The binary preference relation between two elements of choice set is content-dependent if addition or subtraction of other third elements from the choice set can cause its reversal. Therefore, content-dependent preference always violates WARP and also its stronger (SARP) and weaker forms (asyclic rationality as defined by Bandyopadhyay and Sengupta, 1991). By adding more elements to the initial two-element choice set, it is possible to detect contentdependent preference if WARP is violated. Since WARP is a necessary and sufficient condition for existence of generating preference ordering (Arrow, 1959), by violating WARP content-dependent preferences must violate one of rationality axioms: either completeness or transitivity axiom. Thus, the idea of content-dependent preference fits into another stream of literature on intransitive preferences. Starmer (2000) notices that the transitive preference theory has to allow comparisons between choice options unlike the conventional microeconomic theory with single-argument utility function. From this point of view the proposed theory of content-dependent preference is a generalisation of regret theory, first introduced by Loomes and Sugden (1982) to justify intransitive behaviour by means of retrospective preference. The feeling of regret/rejoicing is only one particular set of reasons that might induce the individual to evaluate choice options relatively to the other (foregone) elements of the choice set). 18

19 Content-dependent preference is possible in choice situations both under certainty and risk. In choice under certainty, the literature on asymmetric dominance effect and attraction effect documented that the experimental manipulation of the choice set results in contextually induced preference reversals. The introduction of asymmetrically dominated alternative (that is dominated by one choice option but not the other) is inducing preference for dominant option. The phenomenon persists even when the added (decoy) option is not strictly dominated by any of the initial choice alternatives (attraction effect). Simonson and Tversky (1992) showed that inclusion of a low quality pen into the choice set already consisting of a high quality pen and a fixed amount of cash induces more people to choose high quality pen as opposed to monetary payoff. Wedell (1991) experimentally confirmed the existence of significant asymmetric dominance effect in choice under risk. More generally, the addition of a new choice option, highlighting the already available choice option in a more favourable light, reinforces individual s preference for this attractive option. For example, in the choice situation 3) of the apple-tree example tree A is revealed preferred to tree C but when apple-tree B is added to the choice set (situation 4) ) more subjects choose apple-tree C because the newly added tree B highlights positively apple-tree C and negatively apple-tree A. In choice situations under risk the idea of content dependent preference relates to the literature on the failure of expected utility theory to satisfy the procedure invariance assumption (Starmer, 2000). If individual prefers option A to B and preference is content independent, we cannot expect the same individual to prefer B to A if, additionally, some third option C might be also chosen. Loomes and Sugden (1983) showed that intransitive preferences could in fact be linked to the classical literature on preference reversal phenomenon a particular failure of procedure invariance when subjects prefer higher probability of winning less (P bet) to lower probability of winning more ($ bet), although their valuation of the second lottery is higher (Lichtenstein and Slovic 1971). The same phenomenon can be alternatively deduced from three pairwise choice decisions: P vs. $, P vs. some certain monetary gain M and $ vs. M. If the 19

20 individual reveals intransitive preferences P f $, Mf P and $ f M we can conclude that P is revealed preferred to $ (by direct pairwise choice) although $ has a higher monetary valuation ($ is valued more than M and P is valued less than M). However, Chu and Chu (1990) showed that incidence of preference reversals is lower a) in marketlike environment, where reversers are confronted with repeated arbitrage transactions causing them to loose money, and b) with subjects having had already the experience of marketlike environment. Similarly, Cox and Grether (1996) argue that repetitive market environments with feedback mechanisms reduce the rate of preference reversals and the asymmetry between predicted and unpredicted reversals. Nevertheless, intransitive preference may persist for example in individual consumer problem, where the market feedback is low. Mitchell (1912) argued that the markets do not provide a satisfactory feedback to the purchasers of goods and services. The consumer cannot even make objectively valid comparisons between the various gratifications which she mat secure for ten dollars. Subjective experiences of various purchases of different individuals are only roughly comparable and this limits the effectiveness of any feedback mechanism in consumer problem. But it is feedback rather than repetition which is responsible for the elimination of intransitive preferences. Humphrey (2001) found that intransitive preferences do not disappear when the choice triple is repeated for the second time. The consumer faces repetitive task but the feedback is low and thus intransitive preferences can easily persist. In the situation of individual decision making under risk, when the choice options are univariate probability distributions, the idea of content dependent preference is related to the prospect theory with a changing reference point. Kahneman and Tversky (1979) introduced the prospect theory by discriminating between the lottery outcomes relative to some reference point. However, with the content dependent preferences the reference point is not fixed, but it changes 20

21 every time when the content of the choice set varies. For instance in the apple-tree example the reference point was the highest apple crop promised by the remaining trees in the choice set. This paper extends the area of research in decision theory under risk, where the majority of papers conventionally focused on individual choice among risky alternatives with at most two positive (or negative) payoffs and non-trivial probability to receive zero payoff. Wu and Gonzalez (1996) remark that the established empirical regularities in these simple risky decisions do not directly apply to mixed gambles involving both gains and losses. In my appletree example I reshaped the conventional decision problem under risk such that all lotteries give positive payoffs but some of payoffs are actually losses and some of outcomes are de facto gains relative to the status quo apple crop of 3 apples. This representation mirrors closely the real life risky situations when rarely any project gives strictly zero payoff. It is interesting to note that Brandstätter and Küberger (2002) argue that standard experimentation when lottery is presented as a list of payoffs with associated probabilities induces outcome focused reasoning and not the probability based thinking. In other words, outcomes but not the probabilities affect the choice. Brandstätter and Küberger (2002) propose a simple heuristic how to discriminate between risky (high payoff with low probability) and safe (small payoff for sure) option. If there is a significant difference between the minimal payoffs of two gambles the lottery with higher minimal outcome is chosen; otherwise the risky option is selected. This heuristic would imply that apple-tree A is chosen in situation 1) and apple-tree B is chosen in situation 2) but the model is inconclusive in situation 3) where two risky gambles are involved. However, when probability information is made salient (like representation with natural frequencies), Brandstätter and Küberger (2002) observe the opposite individuals prefer sure option when high risky outcome is unlikely (choice of apple-tree B in situation 1) ) and tend to prefer risky option when the probability is high (choice of apple-tree C in situation 2) ). Yet, Brandstätter and Küberger (2002) also predict that probability focused reasoning will lead to 21

22 different choices than outcome focused reasoning only when low probabilities are involved. Since in the apple-tree example all probabilities are at least moderate we cannot attribute the obtained results to the representation in form of natural frequencies 8. Together Brandstätter and Küberger (2002) and this study result in a puzzle: why individuals prefer risky 1/3 chance of getting 9 apples to the sure crop of 3 apples and at the same time 3 apples for sure are preferred to 2/3 chance of getting 2 and 1/3 chance of getting 5 apples? Using lotteries with three distinct non-zero equally likely outcomes, Birnbaum and McIntosh (1996) find systematic violations of branch independence (but not comonotonic branch independence) that are inconsistent with the inverse S-shaped decision weight function observed in choice between two outcome gambles. Birnbaum and McIntosh (1996) conclude that decision weight function derived from choice between three outcome gambles is S-shaped, the conclusion being reconfirmed by the apple-tree example of this study. Using lotteries with four equally likely outcomes Birnbaum and Chavez (1997) experimentally test branch and distribution independence in choice situations where two common outcome-probability pairs are replaced so that their rank changes from the lowest to the highest. Applying Tversky and Kahneman (1992) cumulative prospect theory inverse S-shaped decision weight function w 1 ( ) γ γ γ γ ( p) p p + ( 1 p) =, γ < 1 to four outcome gambles, Birnbaum and Chavez (1997) find that pattern of violations of branch and distribution independence in choice between four outcome lotteries is consistent with S-shaped decision weight function ( γ > 1). One of the interpretations of the experimental results of Birnbaum and McIntosh (1996), Birnbaum and Chavez (1997) and this study is that the decision weight function induced by the context of 8 This is confirmed in fact by another pilot experiment when the choice situations 1)-4) were presented to subjects in the traditional form with the explicit usage of the word probability. 94.7% of the subjects have chosen apple-tree B in situation 1) and 68.4% of the subjects have chosen apple-tree C in situation 2). Therefore, in the apple-tree example the difference between probability focused thinking and outcome focused thinking is not significant and it is attributed to the fact that in the traditional representation the individuals do not understand properly the notion of probability. The subjects usually comprehend the statement apple-tree A gives 2 apples with probability 2/3 and 5 apples with probability 1/3 as apple-tree A may give either 2 apples or 5 apples. Therefore, probability information is not properly used and outcome focused thinking induces underestimation of large probabilities and overweighting of small probabilities (Brandstätter and Küberger (2002). 22

23 choice among three and four outcome lotteries is different from that induced in choice among two outcome gambles. Hertwig et al (2002) find that decisions based on the direct experience underweight rare events and in particular in repeated decision making this underweighting is due to a greater chance of facing fewer (rather than more) realizations of rare event than expected. Harbaugh et al (2002) report their surprising finding that the majority of subjects are risk lovers over highly likely gains and risk averse over high payoffs with small probability. They also find children to underweight low probabilities and overweight high probabilities much more persistently than the adults. This discovery should not be surprising given than the lotteries were presented to children by the visual model of spinning wheel (that induces probability focused reasoning) and not the conventional description of outcomes and their associated probabilities (that induces outcome based reasoning). Hertwig et al (2002) and Harbaugh et al (2002) use simple lotteries with at most two positive payoffs and usually low associated probabilities and their experiments induce probability focused thinking, thus their results are consistent with Brandstätter and Küberger (2002) finding. Finally, Brandstätter et al (2002) justify the inverse S-shaped decision weight function by making appeal to psychological notion of elation and disappointment. This approach is in contrast to this study, which advocates the binary comparison of lotteries during evaluation stage in the spirit of regret theory. Barandstätter et al (2002) argue that people like small chances to win a lot due to the expected elation from highly unlikely win (attractiveness of apple-tree A) and people dislike high chances to win less because the unlikely loss is disappointing (unattractiveness of apple-tree C). 23

24 Conclusion In this paper addressing individual decision making under risk, I attempt to demonstrate that the rational choice is done in a fundamentally different way than the neoclassical microeconomic theory assumes. The proposed theory of content-dependent preference postulates that individual preference is not absolute binary preference relation between two choice options is contingent on the choice set, the options are drawn from. I study the particular application of content-dependent preference in choice situations under risk when individuals first estimate the ex ante probability of each lottery to give the higher ex post payoff than any other feasible lottery and then choose the lottery with the highest such probability. This heuristic of relative probability comparisons obviously leads to content-dependent preference since the likelihood than a particular lottery yields the highest payoff depends on the payoff distribution of available alternatives. The experimental results confirm the predictions from the proposed normative theory rational individual preferences may be intransitive if they are aggregated over different choice sets, WARP does not hold and inclusion or exclusion of irrelevant alternatives can influence the choice. It is interesting to note that the heuristic of relative probability comparisons implies inter alia the S-shaped decision weight function that is supported strongly by the empirical results, both from the pilot experiments with outcome focussed reasoning and experiment with probability focused reasoning. Therefore, this paper also contributes to the recent literature on the curvature of decision weight function. The ongoing studies discovered that individual preference over lotteries changes in repetitive experiments vs. single shot experiments and in experiments with probability focussed reasoning vs. outcome focused reasoning. What this study confirms is that individual preference changes in experiments with simple lotteries involving at most two positive payoffs and zero outcome with nontrivial probability vs. experiments with more complicated lotteries involving three and more positive payoffs. In case of simple lotteries decision weight function is inverse S-shaped in experiments with outcome focussed reasoning, whereas in case of complicated lotteries decision weight function appears to be always S-shaped. 24

25 References 1. Andreoni, J. and Miller, J. (2002) Giving accordingly to GARP: an experimental test of consistency of preferences for altruism Econometrica 70, Arrow, K. (1959) Rational choice functions and orderings Economica 26, Bandyopadhyay, T. and Sengupta, K. (1991) Revealed preference axioms for rational choice Economic Journal 101, Brandstätter, E. and Küberger, A. (2002) Adaptive decisions under risk and uncertainty, unpublished manuscript 5. Birnbaum, M. and Chavez, A. (1997) Testing of theories of decision making: violations of branch independence and distribution independence Organizational Behavior and Human Decision Processes 71, Birnbaum, M. and McIntosh, W. (1996) Violations of branch independence in choice between gambles Organizational Behavior and Human Decision Processes 67, Brandstätter, E., Küberger, A. And Schneider, F. (2002) A cognitive-emotional account of the shape of the probability weighting function Journal of Behavioral Decision Making 15, Chu, Y.-P. and Chu, R.-L. (1990) The subsidence of preference reversals in simplified and marketlike experimental settings: a note American Economic Review 80, Cox, J. and Grether, D. (1996) The preference reversal phenomenon: response mode, markets and incentives Economic Theory 7, Harbaugh, W., Krause, K. and Vesterlund, L. (2002) Risk attitudes of children and adults: choices over small and large probability gains and losses Experimental Economics 5,