Rational Contextual Choices under Imperfect Perception of Attributes

Transcription

1 Rational Contextual Choices under Imperfect Perception of Attributes Junnan He Washington University in St. Louis January, 018 Abstract Classical rational choice theory proposes that preferences do not depend on context, i.e. are independent of irrelevant alternatives. Empirical choice data, however, display several contextual choice effects that seem inconsistent with rational preferences. This paper studies a choice model in the classical rational paradigm with a novel information friction: the agent s perception of the options is affected by an attributespecific noise. Under this friction, the agent obtains useful information when additional options are introduced. Therefore, the agent chooses contextually, exhibiting intransitivity, joint-separate evaluation reversal, attraction effect, compromise effect, similarity effect, and the phantom decoy effect. Our model also provides a connection between random utility models and reference-dependent models. It approximates the conditional probit model and a reference-dependent model at different parameter values. Finally, our framework offers a possible reconciliation between two pieces of seemingly contradicting empirical evidence on phantom decoy effects. JEL Codes: D01, D81, D83 Keywords: compromise effect, context effect, imperfect perception, intransitive choices, joint-separate evaluation reversal, phantom decoy effect, rational preferences. The author is grateful to Paulo Natenzon, Jonathan Weinstein, John Nachbar, and Liang Guo for the inspiring discussions and comments. The author also thanks participants at the Spring 016 Midwest Economic Theory Meeting, 016 SAET Conference, 017 Econometric Society AMES, 017 Econometric Society NASM, and Xiamen University micro BBS seminar for their comments. All errors are the author s responsibility. The author acknowledges support from the Center for Research in Economics and Strategy CRES, in the Olin Business School, Washington University in St. Louis, and the Economic Theory Center at Washington University in St. Louis. Washington University in St. Louis, Campus Box 108, One Brookings Drive, St. Louis, MO contact: junnan.he@wustl.edu. 1

2 1 Introduction The classical theory of rational choice assumes the existence of a complete preference that is independent of the choice set. Such independence is formalized in axioms such as Sen s property α in deterministic choice models and Luce s independence of irrelevant alternatives IIA in random choice models. According to these models, the revealed preference between two objects should not be affected by the presence of other objects in a choice set. However, this prediction from classical theory is often found to be systematically violated in empirical research. One of the most widely studied classes of violations is contextual choice effects, whereby in a choice problem involving an object x, changing the other objects in the choice problem changes the choice probability of x in a way that implies that the decision making agent evaluates x differently. For instance, in Huber et al. 198 and Pratkanis and Farquhar 199, experimenters offer the subjects two choice problems. One involves only two options x and y and the other includes a third choice z. They find that the inclusion of z can reverse the relative frequency of choosing x or y, even though z itself is rarely chosen attraction effect or listed as unavailable phantom decoy effect. Another example is joint-separate valuation reversal. Hsee 1996 documented that when the willingness to pay for each of x and y is elicited separately, x can be valued higher than y, but when both items are elicited simultaneously, x becomes inferior to y. All of these experiments indicate that when the context changes, it is as if the utilities of x and y change ordinally. Within the rational choice literature, some contextual choice effects, such as the similarity effect Tversky and Russo 1969, attraction effect Huber et al. 198 and compromise effect Simonson 1989, can be explained see e.g. Hausman and Wise 1978, Kamenica 008, Guo 016, Natenzon 016. However, other contextual choice effects such as the phantom decoy effect, joint-separate evaluation reversal Hsee 1996 and stochastic intransitivity Tversky 1969 have not yet been explained in a classically rational framework. 1 1 The term phantom decoy effect is used differently here than in Natenzon 016. Here, it refers to a situation where an unavailable third option that is asymmetrically dominating, i.e. better than the target in all attributes but worse than the competitor in some attributes, increases the attractiveness of the target. Such results are found in Pratkanis and Farquhar 199 and later in Highhouse 1996, Pettibone and

3 Our paper proposes a rational choice model that systematically predicts the aforementioned experimental findings. With a novel informational friction, our model generically exhibits both the violation of stochastic transitivity and the joint-separate valuation reversal when there are trade-offs between attributes in the options, as is observed in data. Moreover, we show that our general model, without parametric assumptions on the utility functions and with a quite general noise structure, predicts the decoy choice pattern, by which we mean the data pattern that captures the attraction effect, the phantom decoy effect and the compromise effect. Therefore, in contrast to many other general random choice models, our model is not a generalization of the Luce model, since where there are trade-offs between attributes among the options, Luce s IIA will not hold. Our general model also predicts that classical rational choice holds for a class of choice problems when there is no trade-off between attributes among the alternatives. In other words, in our model, violation of the classical model does not occur for a subclass of choice problems. Our novel information friction is that the decision making agent suffers from a systematic perception error in the attributes. Each option x has precise attribute levels x about which the agent s utility function is defined. However the agent cannot observe these precise attributes, but a noisy signal X x. The noise is systematic in the sense that when there are multiple alternatives, the noisy signals conditional on the true attributes are correlated. Because of this correlation, the agent learns different information about an object when she is presented with different alternatives. For example, in the choice problem {x, y }, the agent observes the signals X, Y but not the actual attribute levels x and y. She forms a posterior belief, say about x, conditional on the signals X, Y. When she faces the choice problem {x, z }, the posterior belief about x is conditional on the signals X, Z. These two posterior beliefs about x are generally different distributions. Because the agent s utility function is defined on the actual attributes, her posterior expected utility of x is in general different in the choice set {x, y } than in {x, z }. A similar analysis holds for y and z, and therefore, it can happen that x is preferred in the choice set {x, y }, and y is Wedell 000, Pettibone and Wedell 007 and Hedgcock et al. 009 etc. In our model, these three different effects share the same underlying mechanism. This is similar in vein to the suggestion in Highhouse 1996, that the attraction effect and the phantom decoy effect may have the same cause. 3

4 preferred in the choice set {y, z } but z is preferred in {x, z }, causing intransitivity even though the preference over the attributes x, y and z is transitive. For the rest of the analysis in this paper, we impose a specific type of correlated noise termed imperfect perception of attributes, namely that the perception error is idiosyncratic over attributes and consistant across objects. If the agent over-perceives an attribute in an object, she over-perceives the same attribute in other objects. We model imperfect perception as an attribute-specific error term common across items, perturbing the perceived attribute levels of each item while keeping the relative differences unchanged. Although this error term directly induces a change in the utility levels, it is not equivalent to adding a common error term directly to the utilities. A common shock to the utilities would result in the classical rational choice model. A detailed discussion about the general framework and the empirical motivation of the information friction is provided in section. This imperfect perception causes a contrast effect in the perception of each attributes by a Bayesian agent. 3 Take apartment choice as an example. An agent prefers an apartment with abundant natural light. She visits two apartments on the same day, and sees that apartment x is brighter than y. Although she can compare them relatively, she does not know how bright the apartments typically are i.e. she does not observe x, y. There is a common noise term in her signals X, Y e.g. she is seeing both apartments at roughly the same time under the same weather. This particular type of correlation not only provides the agent with extra information about the typical brightness if she sees additional apartments, but it also induces a contrast effect in perception as Bayesian updating filters out the common noise. For instance, suppose that the agent on the same day also visited another apartment z that is much brighter than both x and y. In the presence of the apartment z, the agent revises downwards her perception of brightness of x and y. Similarly, one s judgements about other attributes of the apartments are also affected by contrast effects. For example, an apartment can be perceived as quieter in the presence of a really noisy apartment. In general, with our specific type of noise, a change in context can simultaneously affect the perception of two attributes differently increase one and decrease another. 3 A contrast effect is the strengthening or weakening of the perception about an attribute when the object is contrasted with surrounding objects of different levels in that attribute. 4

5 When the agent s preference is determined by a single attribute monotonically, the effect just discussed is inconsequential: she always chooses to maximize or minimize that attribute in the model. 4 However, if her preference involves at least two attributes, the contrast effect from changing context can alter the relative posterior expected utility levels between options. Therefore, in the previous example, even if the agent doesn t choose z maybe because it is too expensive, having seen option z can affect her choice between x and y. Despite the reliance of our model on Bayesian rationality, we do not claim that in reality people perform sophisticated Bayesian updating and calculates posterior expectations. Instead, we interpret the model as an as-if representation of the decision process. Nonetheless, the detailed calculation of the Bayesian updating of this as-if procedure does parallel some intuitive explanations of contextual choices. In section 3, we apply a parametric special case of the model to explain intransitive choices, joint-separate valuation reversal, and the compromise effect in detail. To illustrate the compromise effect, take the apartment choice example. Suppose the agent faces a trade-off between natural lighting and quietness in choosing an apartment. She prefers better lighting as well as a quieter living place. As before, suppose there are two apartments {x, y } available. Her noisy signals X and Y comes from the same-day inspection of the apartments. 5 According to the signals, x has good natural lighting but some sound of cars from the street can be heard, whereas y has a gloomy interior but it is very quiet. Suppose the agent is tempted to choose y between the two options. Now introduce a third option z that has an even better lighting than x but is also much noisier. Since the agent can not observe x, y and z the actual brightness levels in a typical day and the typical noise-levels in the apartments, she infers them using the correlated signals. Conditional on X, Y and Z, the posterior means of the perceived lighting levels for both x and y are lower than those means conditional on only X and Y the contrast effect in perception of light. And similarly, conditional on Z in additional to X and Y increases the posterior means of perceived quietness for both x and y the contrast effect in perception 4 I.e, if the agent only cares about lighting, she always chooses the brightest apartment with certainty. 5 As before, the agent is seeing the apartments at roughly the same time of the day, and the same day of the week, the quietness in the apartments are also correlated. 5

6 sound. However, since the agent s marginal utility in lighting diminishes, reducing the perceived lighting level of x and y affects both apartments negatively, but more so for y. And since the marginal utility in the dimension of quietness diminishes, increasing the perceived quietness of x and y affects both apartments positively, but more so for x. Consequently, x has a higher utility level after z is introduced. The analysis of the general model is presented in section 4, where the decoy choice pattern and the choice problems for which the agent exhibit classical rational choice are studied. Our paper is also related to the reference-dependent theory of choice and random utility models. In section 5, we show that at some parameter values, a classical rational agent under our friction behaves as if she has a certain reference dependent preference, where the reference point is the average attribute levels of the choice set. At some other parameter values, our model can be approximated by a random utility model called the conditional probit model Hausman and Wise Towards the end of the section, we provide an explanation for why the experiments in Soltani et al. 01 do not necessarily contradict those in Pratkanis and Farquhar 199. A detailed review of the literature is postponed to the section 6. All proofs are contained in the appendix. The Model of Imperfect Perception.1 The Motivations, the Primitives and Notations In the experiments where the contextual choice effects were discovered, choice problems consists of several options, each with a description in two or more different attributes. Therefore, we take the primitives of our model to be the attributes of each object. In particular, we use R n for n to represent the attribute space. The attributes of each item x is represented as a vector x := x 1,..., x n in the attribute space, with each of its coordinate given by the corresponding attribute level. In our model they are not directly observed by the agent. However, because each object can be uniquely identified by its attribute vector, from now on, we will simplify the notation and refer to both the object x and its attributes as x. In many of the experiments, contextual choices are observed as long as there are two different attributes. Therefore we restrict our discussion to R in this 6

7 paper for mathematical simplicity. 6 In accordance with the classical theory, the agent is assumed to be rational in two senses. Firstly, she has a preference over the attribute space that can be represented by a vnm utility function u : R R. Secondly, she is Bayesian rational with a prior belief over R. The prior distribution represents the agent s anticipation about the attribute levels before she observes any choice alternatives. When she observes noisy signals, she chooses the object that maximizes posterior expected utility conditional on the signal. The main assumption of imperfect perception of attributes proposes that some noise affects the agent s perception of each attribute. Since the noise is only specific to the attribute s perception but not to the objects, the perception of each object s attribute location is affected by a common noise term. Capital letters i.e. X = X 1, X denote the initial noisy perception by the agent. For example, the items x, y are perceived as X = x + ɛ and Y = y + ɛ with the same vector ɛ. 7 This implies that it is easier to perceive relative differences in attributes among the items, i.e. X Y = x y, but more difficult to perceive the absolute locations x and y in the attribute space. This resembles the findings in some experiments. For example, in Ariely et al. 003, the participants underestimated the price of several different objects or overestimated all of them if they were anchored. As is summarized in their paper we show that consumers absolute valuation of experience goods is surprisingly arbitrary... we also show that consumers relative valuations of different amounts of the good appear orderly.... They also found a similar pattern of coherent arbitrariness in attributes such as duration and drink size. In some experiments, perception errors seem impossible because the subjects are shown the choice objects each with an accurate numeric description of the attributes. However, experimental evidence suggests otherwise, that even when these descriptions are displayed in standard units, the subjects may not be able to perceive numerical information precisely. For example, Ariely et al. 003 suggest that stating the volume of noise in units does not provide more information than actually hearing the noise. One interpretation of the perception noise in these experiments is as follows. When an agent needs to choose among 6 The mechanisms for the main theorems can be extended to higher attribute dimensions. 7 Our model predictions only change marginally if we relax the assumption so that X = x + ɛ + ɛ x where ɛ x is a small i.i.d. noise for each object x. 7

8 the alternatives, she needs to first have an accurate perception of the units of measurements before she translates the numbers into the perception of attributes. For instance, in choosing apartments, the agent is also concerned with the safety of the respective neighborhoods. She can obtain a signal of this attribute by consulting the last-year crime statistics published by the same authority. 8 Even though this attribute can be measured in numbers such as number of crimes per thousand of people for each neighborhood, it may still be a correlated noisy signal in the perspective of the agent. For example, suppose the statistic for each neighborhood is calculated by the same city council. Since the city council s definition for crimes can be stricter or broader than the agent thinks, the published crime rate for every neighborhood would be under or over stated in the perspective of the agent. In general, the agent can have an inaccurate perception of maybe the units of measurements in which the attributes are described. Under these circumstances, even when the agent is looking at the numerical description of attributes for each alternative, her perception is still affected by the imperfect perception. We now summarize the notation used in the paper. Letters with an asterisk denote both the object and the true attribute levels of an object in R. Different letters denote different alternatives. When there are more than 3 alternatives the superscripts i.e. x 1, x,... etc. are used. Capital letters denote the initial noisy perception by the agent. Lower case letters i.e. x = x 1, x are reserved for the agent s posterior belief about the true attributes. Subscripts distinguish the respective attribute-dimensions for a given vector. Choice behavior is a function that specifies the choice probability of an object when it is presented in a set of alternatives for which a subset is not available. We use the notation Cx, {x 1, x,..., x i, x i+1,..., x i+j } to denote the choice probability of x from the set {x 1... x i+j } in which {x i+1,..., x i+j } are unavailable. A C.,. that assigns a probability for any x in every nonempty finite set of alternatives S, with any S S specifying the unavailable objects, is called the choice behavior of an agent. The choice behavior satisfies i Cx k, {x 1, x,..., x i, x i+1,..., x i+j } = 1 k=1 8 E.g. local police department and city websites, or the Uniform Crime Reports by FBI in US. 8

9 . Definitions Formally, our imperfect perception assumption is as follows: Definition.1 An agent suffers from imperfect perception if when a set of n alternatives {x 1, x,..., x n } R are presented, the agent receives signal {X 1,..., X n } R such that X i x i = ɛ for all i. The noise term ɛ N 0, T 1 is normal where T 1 = 1 1 R for some t > 0, R 1, 1. t R 1 We follow a conventional assumption that the noise ɛ is normally distributed. Another simplifying assumption is that the noise in the signals is perfectly correlated across objects. However, because the choice probability is continuous in the noise covariances, the change in model prediction is only marginal if we relax the perfect correlation to high correlations. Moreover, in the definition, the variances of the shock in each attribute are the same, because the diagonal entries are all one. It will not change our model qualitatively if we allow these numbers to differ. The definition allows the noise across attributes to have a non-zero correlation in R. 9 For conjugacy, we will endow the agent with a normal prior distribution. Without loss of generality, we can translate and scale the attribute space and let the prior mean be the origin and the prior variance be Ω := 1 r for some correlation coefficient r 1, r 1 Definition. An agent is normal-bayesian rational if she is Bayesian with normal prior N 0, Ω and maximizes posterior expected utility. Finally, we define the preference over the attributes. Following classical consumer theory, we assume that the agent has a monotonic utility function over R, and the marginal utilities are decreasing. In other words, the two attributes are both goods in the positive direction such that the utility function displays insatiability along each axis. Other standard 9 Such a correlation can arise when attributes are closely related, such as the sugar content and calories in a soft drink, one might expect a correlation in the noise across these attributes. 10 Such a correlation can arise when, for example, the two attributes are price and quality. One can interpret r < 0 as the agent having a prior belief that a good price is associated with low quality. 9

10 assumptions from consumer theory include diminishing returns in both attributes and weak complementarity between attributes. properties. We call a preference standard if it displays these Definition.3 A preference over distributions on R is standard if it can be represented by a vnm utility function u : R R that is differentiable, increasing i.e. u 1 > 0, u > 0, and exhibits decreasing marginal sensitivity i.e. u 11 < 0, u < 0 and weak complementarity i.e. u 1 0. Any utility function representing a standard preference is called a standard utility function. 3 A Parametric Special Case In this section, we use the following specific utility function u : R R, ux := ux 1, x = e γx 1 e ρx, for any x := x 1, x R. Moreover, for this section, the agent s prior distribution is the bivariate standard normal N 0, I, where I is the identity matrix; and the noise follows ɛ N 0, 1 I t. Through this parametric special case, we first illustrate the stochastic intransitive choices and joint-separate evaluation reversal among non-dominating choice objects. We then show that parametric model can predict the compromise effect found in experiments Simonson We discuss how the model also predicts both the attraction and phantom decoy effect, and compare with Koszegi and Szeidl 013, Bordalo et al. 013 and Natenzon 016. In a problem of binary choice, there are two types of positionings of the objects. One is when an alternative strictly dominates another i.e. x > y, both entries of x dominate those of y. We will show in later sections that the dominated alternative receives probability 0 under more general assumptions. The second type of positioning is when neither object dominates the other, i.e. there is a trade-off between the objects. In this parametric model, the choice probabilities for such binary choice problems can be calculated explicitly. Lemma 3.1 For any x, y where x 1 > y 1 and y > x, the parametric model gives Cx, {x, y } = Φ θγ, ρ, x, y, t, where Φ is the standard normal c.d.f. function and 10

11 θγ, ρ, x, y, t is defined as θ := γ ρ t γ + ρ + ρx + y γx 1 + y1 t + + t exp γy γ + ρ t ρ + γ ln 1 exp γx 1. exp ρx exp ρy Remark 3.1 The purpose of this Lemma is to show that one can explicitly calculate the choice probabilities in the parametric model. It can potentially be used by a modeler or an experimenter to fit the choice data. Here, we need to point out that an implicit assumption similar to that in Koszegi and Szeidl 013 is adopted. To maintain empirical identifiability and avoid excessive degrees of freedom, the definitions and measurements of the attributes must be determined before fitting the model to data. They should not be free parameters but part of the data that the model seeks to explain. 11 Although the complex expression can be useful for experimenters, the agent in the model does not evaluate this complicated algebra before making the choice. She simply chooses the choice item that maximizes her expected utility while being completely unaware of the choice probabilities her actions generate. For the ease of interpretation, in the remainder of this section, we will only state the propositions for parameter values γ = ρ = 3 and t = 1. The intuition is similar for general parameter values but the expressions become complicated. 3.1 Violation of Weak Stochastic Transitivity Weak stochastic transitivity refers to the proposition that if Cx, {x, y } > 0.5 and Cy, {y, z } > 0.5, then Cx, {x, z } > 0.5. Early evidence about intransitive choice can be found in Tversky 1969, and more recently Rieskamp et al Some evidence discussed in the two papers suggests that weak transitivity may be violated when there is no clear domination among x, y, z. In this subsection, we will calculate the stochastic indifference curve for the parametric model and identify a necessary and sufficient condition for a violation. 11 While it is easier to satisfy this procedure in marketing experiments where the attributes of each object are specified by the experimenter, it is sometimes difficult to include other relevant attributes in real life decision-making processes. For example, when shopping online or in person, individuals may base their decisions on attributes that are not listed on the product descriptions. For example, decisions may be made based on the retailer s customer service, which is usually not listed in the product labels. Hence it is difficult to account for these influences. 11

12 Since ɛ is random, given x, y fixed, the actual choice depends on the realization of ɛ. Nonetheless, we can define indifference between true attributes in binary choices stochastically. We say x is stochastically indifferent to y i.e. x y if Cx, {x, y } = 0.5. Similarly, a stochastic indifference curve through x, means the set of true attribute vectors in R that are stochastically indifferent to x when presented together with x, i.e. {y R x y }. Proposition 3.1 Under the given parameter values, for any x, y R, they are stochastically indifferent to each other, x y, if and only if exp y 1 + x 1 exp x 1 + y 1 = exp x + y exp y + x. With x fixed, this equation implicitly defines the stochastic indifference curve through x as we vary y1 and let y be determined by the equation. Now that we have defined stochastic indifference, we can proceed to explain violations of stochastic transitivity. Suppose we have two alternatives x, y such that x y, and x y. In general, the indifference curves {z z x } and {z z y } have different slopes at y. In other words, indifference curves can cross, and in general, crossing indifference curves give rise to intransitive choices. Proposition 3. Under the given parameter values, for any x, y R such that x y, if y x i.e. equation holds and exp y1 + x 1 + exp x 1 + y1 exp x + y + exp y + x, then the stochastic indifference curve for x and that for y have different slopes at y. The above inequality holds generically even under the restriction of equation. Intuitively, this is because although x y, they are different objects x y. And thus the presence of either one of them leads to a different inference about z with which the agent evaluates. Therefore, even when z is indifferent to y, it need not be indifferent to x. We provide the following numerical example for better illustration. 1

13 Figure 1: Crossing stochastic indifference curves As in Figure 1, x = 3, 0 and y = 0, 3, the indifference curve through y is exp z 1 exp z 1 = exp z + 3 exp 6 + z shown in blue, and the indifference curve through x is exp z exp 6 + z 1 = exp z exp z shown in red. A plot of the two curves shows that they are distinct and cross at 3 points. The indifference curves cross at both x and y meaning they are indifferent to each other. There is space between the two curves and z = 5, 1/ lies in the lower-right region. The probability of choosing z over y is, by Lemma 3.1, Cz, {z, y } = Φ 5 1 e ln > 0.5. e 3/ e 9 Similarly, the probability of choosing x over z is Cx, {x, z } = 1 Cz, {x, z } 0.89 > 0.5. Therefore, it can be seen that x is chosen more frequently over z, and z more frequently over y. Although y is chosen fity-fifty against x, by continuity, we can change x slightly so that Cy, {x, y } > 0.5, and get a strict cycle. In fact, since >, there must exist some ɛ such that the agent with error ɛ prefers y to x, x to z and z to y strictly in binary choices. Remark 3. The formation of a cycle in this model crucially depends on not presenting 13

14 objects simultaneously. One empirically testable prediction of our model is that if the agent is asked to evaluate all three objects simultaneously before choosing from any pairs of them, then a choice cycle should not appear. 3. Joint-Separate Valuation Reversal Joint-separate valuation reversal involves the average willingness to pay for two items in different contexts. Evidence for this phenomenon can be found in Hsee 1996 and Hsee et al In one of the experiments in Hsee 1996, the subjects are asked about their hypothetical willingness to pay for two dictionaries. The first attribute is the preserved condition and the second is the size of entries. Dictionary x is brand new and has 10k entries, and dictionary y has a torn cover with 0k entries. When the subjects were asked to evaluate x alone, the average willingness to pay was about 4 dollars; when the subjects were asked to evaluate y alone, the average willingness to pay was about 0 dollars. Denote by $x and $y the average willingness to pay for x and y in dollars. In Hsee 1996, the experiment of separate evaluations, the inequality $x > $y is statistically significant. However, according to Hsee 1996, when the two options were presented together, the inequality between the amounts reversed. With abuse of notations, denote by $x y the average willingness to pay for x in the presence of y, and $y x for y in the presence of x. The experiment data shows $x y = 19 < 7 = $y x. In this subsection, the average posterior expected utility is used as a proxy for average willingness to pay. That is, the reversal is interpreted as follows: the average posterior expected utility of x is greater than that of y when they are evaluated separately, but when evaluated together, the average posterior expected utility of x becomes less than that of y. In our model, the effect appears because when there is only one object, Bayesian inference only takes into account the information of that object. When there are two objects, the inference tries to filter out the common noise. However, because the two objects have different attribute values, the adjustment from Bayes rule affects their utilities differently. In the dictionary example, x, y R where x 1 > y1 denote preserved condition of the dictionaries, and x < y denote the sizes of entries. Apparently, there is a trade-off between the two attribute dimensions. Intuitively, If x 1 is close to what the agent s a priori 14

15 expectation, and x is only moderately less than expected, the dictionary x would receive a reasonable evaluation when it is presented alone. On the other hand, if y 1 is a lot worse than expected, then even if y is better than expected, the dictionary y would not receive a decent evaluation when it is presented alone due to diminishing marginal utility. However, when y exceeds x by a significant amount, then during the joint evaluation, x appears particularly inadequate in x in the presence of y. And thus x is evaluated much lower than in the separate evaluation. To mathematically illustrate the reversal, we first calculate the mean posterior expected utility of a single object. Since there is noise in her perception, when the agent is presented with an item x, she receives the signal X = x + ɛ. Her posterior for x would be x X N 1 x + ɛ, 1I. The posterior expected utility is E[ux X] = E[ e 3x 1 e 3x 3x X] = exp 1 + ɛ x exp + ɛ Since ɛ is random, the posterior distribution as well as the posterior expected utility are random. The average posterior expected utility is obtained by simply taking the expectation over ɛ, 3x $x := exp x exp $y is defined in the same way. A similar calculation can be done for the joint valuation case and we have $x y = exp x 1 + y 1 + exp x + y +. The corresponding expression for $x y can be obtained by simply exchanging x with y in the last expression. Now the joint-separate valuation reversal is the two inequalities $x > $y and $x y < $y x, and they reduce to 3x exp 1 3x exp 3y > exp 1 3y exp exp y 1 x 1 exp y x < exp x 1 y 1 exp x y. The set of solutions to the above inequalities is non-empty. For example, x = 0, 1 and y = 1.5, 1. It can be seen that the previous intuitive explanation can be used to quantify a solution region of the above system of inequalities. The proof of the following result is tedious and is omitted 15

16 Corollary 3.1 Under the given parameter values, if y1 1.5 < 1 x < x 1 = 0 y. If y x y1, then the above system of inequality holds, and hence joint-separate valuation reversal occurs. Recall that the agent has prior distribution with mean 0 and standard deviation 1. According to this prior, y 1 < 1.5 < 1 < x < x 1 = 0 < y means x 1 is exactly as expected; x is within one standard deviation from expectation while y 1 is worse than 1.5 standard deviations. Even if y is better than expected, y is evaluated less than x in separate evaluations. However, because y exceed x significantly, y appears superior to x in a joint valuation. 3.3 Illustrating the Ternary Choices Through the Compromise Effect We proceed with a review of several effects summarized in Figure. Start with a binary choice problem with x, y where x is better than y in the first attribute but y is better in the second. The phantom decoy effect Pratkanis and Farquhar 199 occurs in the situation when z is positioned near the area P. Usually, the phantom alternative is better than x in first attribute and no worse than x in the second. Also, it should not be better than y in the second attribute. In experimental settings, the subjects are told that such z is unavailable to choose and hence the agent has to choose from {x, y }. Empirically, the phantom decoy increases the frequency of choosing x. The compromise effect Simonson 1989 refers to introducing a third z in or near the region C where z seems extremely favorable in the first attribute but extremely unfavorable in the second one. Empirically, given these options, people are generally led to choose the compromising option x, increasing its choice frequency. The attraction effect Huber et al. 198 corresponds to introducing a third option z in or near the region A in Figure. In general, z needs to be inferior to x in the second attribute, and no better in the first. In addition, z needs to be no worse than y in the first attribute. Empirically, such a third option itself is rarely chosen. However, it increases the choice probability of x over y. In all the effects discussed above, the frequency with which the target x is chosen increases when z is introduced. The next proposition illustrates that our parametric model predicts the 16

17 compromise effect. A more general result concerning all three effect will be presented in the next section. Figure : Areas for the phantom decoy effect P, the compromise effect C and the attraction effect A As in Figure, let the initial choice set be {x, y } and the expanded choice set be {x, y, z } where z1 > x 1 > y1 and y > x > z. Following Simonson 1989, the compromise effect refers to Cx, {x, y, z } > Cx, {x, y } for all z inferior enough. Proposition 3.3 The Compromise Effect Under the given parameter values, for any x 1 > y1 and x < y, let z = z1, z lie in the region that z1 > x 1 and z < x. There exists δ > 0 and D R such that Cx, {x, y, z } > Cx, {x, y } whenever z1 x 1 < δ and z < D. The intuition of the proof requires both Bayesian learning and decreasing marginal sensitivity in the utility function. As the agent receives more signals, she gradually learns the vector of the noise as more objects and therefore more signals become available. Since the noise shifts the perceived location of every item in the same direction, direct calculation reveals that the posterior inference of the noise is close to the average attributes of the 17

18 set of signals. In the case of compromise effect x, y, z, since the third alternative z is presented in the lower right half of the plane R, it draws the agent s posterior belief about the noise to the lower right half of the plane. Therefore, in the choice problem of {x, y, z }, the posterior locations of x and y move towards the upper left direction to adjust for the noise term. Insofar as the utility function is monotonic and has decreasing marginal sensitivity, moving up left in the attribute space benefits harms x more less than y. The attraction effect and the phantom decoy effect follow from a similar reasoning. In this sense, the three effects can be seen as parts of a more general pattern. Here, we omit their formal proofs because they will be covered under the decoy choice pattern in the next section. Nonetheless, the analysis of the attraction effect phantom decoy effect can be summarized by the following intuition. Let x, y be as before. x is inferior in the second attribute and y is inferior in the first attribute. Now introduce the third object z that is is extremely bad in the second attribute good in the first attribute. In comparison, z in A P makes both x and y seem better in the second attribute worse in the first attribute than before. Such a change makes x relatively more favorable less repulsive since y was already good enough in the second attribute barely acceptable in the first attribute at the outset. In terms of prediction and explanation of the attraction effect and the phantom decoy effect, our model is related to Koszegi and Szeidl 013, Bordalo et al. 013, and Natenzon 016. Our model mechanism of the phantom decoy effect resembles the intuition in Koszegi and Szeidl 013, that the attribute in which the alternatives differ most in this case the first attribute is more influential for the agent. However, we differ in the case of attraction effect, where the attribute with larger differences is not the more influential one in our model. We share a similar prediction of the attraction effect with the model in Natenzon 016. However, our prediction differs in the case of the phantom decoy effect. Overall with respect to these two effects, the predictions of our model are closest to the one in the salience theory Bordalo et al In the attraction effect, the second attribute becomes less salient for y, and in the phantom decoy effect, the first attribute becomes more salient for y. This resemblance comes from the Bayesian updating of the common error term in perception. In 18

19 the posterior distribution, the noise vector can be approximated by the average attribute of the set of objects. In Bordalo et al. 013, the average attribute vector is taken to be the reference point in determining the salient attribute for each choice object. A more detailed discussion of the relation between our model and the salience theory can be found in section 4.3. In our model, the agent learns about her noise with increasing accuracy as the number of different items increases. As the number of items grows large, presenting an extra object to the agent provides little additional information about the noise. Hence the model predicts that the agent conforms more to the rational choice paradigm after seeing many alternatives. For example, showing the agent a large variety of objects, and then restricting her to choosing from a subset of two to three alternatives would reduce the occurrence of the above contextual choice effects. On the other hand, in order for the agent to display contextual effects in future choice tasks, she needs to forget what she has learned from the objects that she was previously exposed to. For example, if the agent perfectly remembers all the items she has seen, then a removal of the third object in the attraction effect should not change the choice probability of the remaining ones. However, the experimental evidence in Sivakumar and Cherian 1995 shows that the choice probability of x the target is significantly reduced following the removal of z in the attraction effect, although it does not recover fully to what it was before the introduction of z. One way to model such an observation is to assume that the agent partially forgets what she has learned when the stimuli are removed. Since here we are proposing one simple mechanism that can cause several contextual choice effects, we do not discuss in detail the modeling of forgetfulness in this paper. 4 The General Results We have now shown that our simple parametric model can explain and predict several contextual choice effects. These results are not the outcomes of model flexibility. On the contrary, the model is quite rigid in the sense that these types of contextual effects have to occur even without the parametric assumptions. One can view these as testable implications of the model. We first define the term decoy choice pattern, as a unification of the at- 19

20 traction, compromise, and phantom decoy effect. We will then show that the general model predicts the decoy choice pattern under the general class of preferences and prior-signal distributions as described in section. We also show that a Bayesian rational agent under imperfect perception behaves as if she has a perfect perception of each item and maximizes some reference dependent preference. If we vary the prior variance parameter, our model connects the conditional probit model and the salience theory at extreme parameter values. Therefore our model can naturally accommodate the phenomena that can be explained by both of these models. At the end of the section, we discuss why the experiments in Soltani et al. 01 do not fulfill the assumptions of our model, and why they do not necessarily contradict Pratkanis and Farquhar 199 in our interpretation. 4.1 The Decoy Choice Pattern As shown in Figure, there is a pattern for the attraction, compromise and phantom decoy effects: the third option z, generally located in the lower-right half of the figure, is either infrequently chosen or not available, and its presence increases the choice probability of x. Because of the symmetry, we can draw a curve between x and y as in Figure, and reasonably infer that a reverse effect will hold to the upper-left of the curve. 1 It is also reasonable to predict that both the attraction effect and the compromise effect will remain qualitatively unchanged when the third option z becomes unavailable. We call this inferred pattern the decoy choice pattern. In other words, the term decoy choice pattern refers to the effect that an unavailable third option z increases the choice probability of x when z moves towards the lower right corner and the choice probability of y when it moves towards the upper left corner. Definition 4.1 The choice behavior is said to display the decoy choice pattern if there exists a vector R in the fourth quadrant, such that for any x, y, z R with x 1 > y 1, x < y, the inequality Cx, {x, y, z } Cx, {x, y, z + } holds. 1 There were also experiments studying symmetrically dominated third alternatives that are dominated by both x and y, i.e. z < x and z < y. It was found that such a third option does not significantly affect the choice probability of either one of the alternatives Wedell

21 The following theorem shows that our model predicts the decoy choice pattern, which is an empirically testable implication in two ways. First, when the set of all choice items is large enough e.g. variations in at least two attributes, the agent does not satisfy the Luce s IIA. Second, the agent violates Luce s IIA in a specific way. The reversal of the inequality in the above definition is not allowed by the model. Theorem 4.1 Any normal-bayesian rational agent with standard preference and imperfection perception displays the decoy-choice pattern. Observe that the theorem is a sufficiency result. Intuitively, it states that if z is moved towards the bottom-right direction, the posterior expected utility of x relative to y is improved. Due to symmetry, when lies in the diagonally opposite upper left quadrant, it increases the expected utility of y relative to x. The proof of the main result also identifies a sufficient region in which change z to z + improves the choice probability of x. The region is defined by the following inequalities < min{ 31 R +1 rrt r Rt r Rt 1, 31 R +1 rrt 1 } if r > R; 1 > 0, < 0 if r = R; 31 R +1 rrt r Rt 1 < < r Rt 31 R +1 rrt 1 if r < R with the prior correlation r, the signal variance t and signal correlation R. If moves outside this region, the result of the movement to the expected difference between x and y has two conflicting effects. When that happens, such a movement is beneficial for x in one attribute but beneficial for y in another. This is consistent with the evidence documented by Wedell 1991 which showed that there is a lack of significant preference reversal when z is dominated by both x and y in that z1 = y1 < x 1 and z < x < y. One interesting implication of the theorem is that the attraction effect and the compromise effects should still exist even when z is unavailable. Since z is infrequently chosen in experiments, such a prediction is reasonable to expect, but distinct for our model. Other choice models usually do not consider an unavailable option in the menu. 1

22 4. Rationality in the Model We have seen previously that when there is a trade-off between the alternatives, i.e. some alternatives are better in the first attribute while others are better in the second, contextual choices arise in the model. A natural question is what would the model predict when such a trade-off is absent. Intuitively, if we are given two alternatives x and z where z > x, a rational agent should always choose z due to the monotonicity of the utility function. 13 The prediction of our model fits this intuition. Since the error ɛ in perception is the same for each of x and z, the perturbed signal X = ɛ + x and Z = ɛ + z preserves the inequality: Z > X. A Bayesian rational agent can hence correctly infer the inequality and choose optimally. Theorem 4. For any x, z R, a normal-bayesian rational agent with standard preference and imperfect perception chooses z with probability 1 from the choice set {x, z } if z > x. A further implication of the above theorem is that transitivity holds deterministically for a set of choice objects such that each one is dominated or dominates one another. Therefore, violation of weak stochastic transitivity can only happen when complete domination does not occur. The above theorem can be trivially generalized to the following statement. When S = {x 1,..., x n } is the choice set involving multiple options, if x i is dominated in the set S, then Cx i, S = 0. In other words, objects are chosen with positive probability only when they are on the attribute possibility frontier. This is a rationality condition that the agent has to satisfy, and it rules out many other types of irrational choice behaviors. 14 It is clear that the above theorem predicts the following intuitive choice effect described and observed in Tversky 197 and Tversky and Russo Consider an individual that is choosing between a trip to Paris x and a trip to Rome y. If she is interested to see both places and doesn t have a strong preference for one over the other, let s say the choice 13 The vector inequality z > x means z 1 x 1 and z x with at least one inequality being strict. 14 It is also one of the distinctions between our model and Natenzon 016. In his model, an object x < {y, z... } is chosen with probability 0 only if it is also maximally similar to a dominator. In the paper, this means x = λy for some dominator y and some λ 0, 1.

23 probability for Paris x would be 1/. Now if we offer the individual a new choice problem with two alternatives, a trip to Paris x or a trip to Paris plus a $1 bonus z, he would probably not hesitate to choose the option with the extra dollar. In other words, choosing z over x has probability 1. However, if we offer him a third choice problem that consists of a trip to Paris plus $1 and a trip to Rome, it is intuitive that the choice probability should still be roughly 1/. 5 Further Discussions 5.1 Linking the Random Utility Models and the Reference-dependent Theories In this section, we will show that our model approximates the conditional probit model of Hausman and Wise 1978 and the salience theory of Bordalo et al. 013 at special parameter values. For this reason, and because the conditional probit model can explain the similarity effect Hausman and Wise 1978, our model can also explain the similarity effect. is Observe that given a finite choice set S = {x 1,... x n }, the posterior inference for x i x i X 1,... X n N T + nω 1 1 T X i + nω 1 X i n Ω 1 X j, T + nω 1 1. In this section, let us take the utility function from the parametric model so that ux := ux 1, x = e γx 1 e ρx. As before, our agent chooses x i S to maximize E[ux i X 1,... X n ]. To approximate the conditional probit model, we take the limit Ω. 15 In the limit, we have x i X 1,... X n N X i, T 1. It is clear that in this limit, the utility of x i do not depends on the presence of other objects. The posterior expected utility of an object x = x 1, x is now E[ux] where x N x + ɛ, T 1. If the variance of the noise term T 1 is small enough, the distribution would be concentrated around its mean. This gives the approximation E[ux] ux + ɛ = e γx 1 +ɛ 1 e ρx +ɛ. The Conditional Probit Model. In Hausman and Wise 1978, their random utility function takes the form ũx = u 1 x 1+u x +β 1 u 1 x 1+β u x where β 1, β are mean- 15 it corresponds the situation where the Bayesian agent adopts an improper uniform prior. j=1 3