Luce Rule with Attraction System

Transcription

1 Luce Rule with Attraction System Jiangtao Li Rui Tang April 26, 206 Abstract We develop a two-stage extension of Luce s random choice model that incorporates an attraction parameter. Each option is associated with an attraction value and a Luce value. Attraction value measures the option s ability to attract the agent and Luce value measures the option s desirability. In the first stage, the agent is randomly attracted by some alternative according to an attraction-valued Luce-type formula. The agent then chooses among alternatives that are comparable with this option according to another Luce-valued Luce-type formula. While retaining the simplicity of the Luce model, the theory can explain the attraction (violations of regularity), compromise, similarity effects, and violations of stochastic transitivity, which are well-documented behavioral phenomena in individual choice. We are grateful to Soo Hong Chew, Faruk Gul, Shaowei Ke, Matthew Kovach, John Quah and Satoru Takahashi for helpful discussions. Part of this paper was written while Jiangtao Li was visiting the Economics Department at University of Michigan and he would like to thank the institution for its hospitality and support. All remaining errors are our own. Preliminary and Incomplete. Comments are Welcome. Department of Economics, National University of Singapore, jasonli07@gmail.com Department of Economics, National University of Singapore, a0292@u.nus.edu

2 Introduction Deterministic behavior is often assumed in models of economic choice. However, in both experimental and market settings, individual choice responses typically exhibit variability; see, for example, Sippel (997), McFadden (200) and Manzini, Mariotti, and Mittone (200). In this paper, we study a random choice model and assume choice responses to be given by a probability distribution ρ that indicates the probability ρ(a, A) that when the possible options faced by the agent are the alternatives in A, some alternative in A is chosen, as in the seminal work of Luce (959), Tversky (972b) and more recently, Gul, Natenzon, and Pesendorfer (204) (henceforth, GNP), Manzini and Mariotti (204), Echenique, Saito, and Tserenjigmid (204) and Brady and Rehbeck (205). In the celebrated Luce model, each option x is associated with a Luce value v x that measures the option s desirability. The probability of choosing x from an option set A containing x is ρ({x}, A) : v x / v y. y A Despite many attractive features, there are well-documented violations. We develop a twostage extension of Luce s random choice model to address such violations. The novel aspect of the paper is that we incorporate an attraction parameter into the Luce model. Each option x is associated with an attraction value χ x and a Luce value v x. Attraction value measures the option s ability to attract the agent. As in the Luce model, Luce value measures the option s desirability. In the first stage, the agent is randomly attracted by some alternative according to an attraction-valued Luce-type formula. The agent then chooses among alternatives that are comparable with this option according to another Luce-valued Luce-type formula. This type of decision procedure is ubiquitous in daily life. An investor is attracted by a particular car manufacturer (possibly due to prior knowledge). But when she compares within the car manufacturing industry, she might eventually invest in another car manufacturer, which has stronger fundamentals. A shopper is attracted by a recently launched handbag of a luxury brand (possibly through advertisement). But when she visits the store and compares the various handbags, she might eventually purchase a different handbag, which fits her even better. A foodie is attracted by a spicy dish at a Michelin restaurant (possibly recommended by a friend). But when she dines there and compares the various choices from the menu, she might eventually order a non-spicy dish, which suits her flavor even better. 2

3 While retaining the simplicity of the Luce rule, the theory can explain the attraction (violations of regularity), compromise, similarity effects, and violations of stochastic transitivity, which are well-documented behavioral phenomena in individual choice. what follows, we shall illustrate how we apply the theory to explain the attraction effect (violations of regularity). A fuller discussion of the attraction effect and the other behavioral phenomena can be found in Section 4. Most random choice models including Luce rule (Luce (959)), elimination-by-aspects rule (Tversky (972b)), attribute rule (GNP (204)) and random consideration set rule (Manzini and Mariotti (204)) obey the regularity property (or monotonicity). That is, when an alternative is added to the option set, the choice probability of the original members of the choice set cannot increase. Although regularity seems innocuous, this property conflicts with empirical evidence from the marketing literature, most notably Huber, Payne, and Puto (982) and Simonson (989). The following is a slight modification of an example proposed by Huber, Payne, and Puto (982). Example. Let S {x, y, z}. Assume that x, y and z are three brands of beer. Subjects know only the price and the average quality ratings. See Figure. Brand Dimension : Price Dimension 2: Quality x Low Medium y High High z Medium Low We might have ρ({x}, {x, y}) and ρ({x}, {x, y, z}) 3. Thus, we have a violation of 3 8 regularity. In Example, since z is worse than x in both dimensions, it is likely that x and z are comparable. Also assume that y and z are not comparable, since y is better in one dimension but worse in the other dimension. On the one hand, the addition of z decreases the probability that the agent is attracted by x and y in the first stage. On the other hand, when the agent is attracted by z, she still considers x, but not y in the second stage. Furthermore, it is plausible that x has a higher Luce value than z, since it is more desirable in both dimensions. If z has a sufficiently high attraction value and x has a sufficiently high Luce value relative to z, then the addition of z increases the probability of x being chosen. Hence, our model allows for violations of the regularity property. 3 In

4 Preference x Dimension : Price z y 0 Dimension 2: Quality Preference Figure : Placement of asymmetrically dominated decoys. Luce model can be viewed as a special case of Luce rule with attraction system, where all options are comparable. No matter which option the agent is attracted by in the first stage, she still considers all options in the second stage and eventually chooses an option using the Luce-valued Luce-type formula. This paper joins a growing literature that studies random choice. The closest to our model in terms of representation is the attribute rule (GNP (204)). GNP reinterpret the options as bundles of attributes. In the attribute rule, the agent is randomly attracted by some attribute according to a Luce-type formula and only options that has that attribute are considered in the second stage. Consider an investor who is choosing from car manufacturer x and several other companies. If we add another car manufacturer y in the option set, this never increases the probability of x being considered in the second stage. At best, y does not introduce attributes that are non-existent in the original option set, and the probability of x being considered in the second stage remains unchanged. In our model, the agent is randomly attracted by some option rather than some attribute. For example, suppose there are two car manufacturers x, y and other companies, an investor need not be fixated to investing in any particular industry. Instead, she is attracted by each company with certain probability, possibly due to prior knowledge of the companies, transaction history of the company stocks, or recommendations by a financial advisor. When she is attracted by either x or y, she compares all car manufacturers when she eventually makes the investment decision. Therefore, the addition of y increases the probability of x being considered in the second stage. 4

5 The rest of the paper is organized as follows. Section 2 introduces the axiom of additivity of relative probabilities and shows that Luce rule is the only choice rule that satisfies additivity of relative probabilities. Section 3 presents Luce rule with attraction system. Luce rule with attraction system weakens additivity of relative probabilities. Theorem 2, our main result, shows that Luce rule with attraction system is the only choice rule that satisfies the weak version of additivity of relative probabilities and an additional transitivity axiom. Section 4 applies the theory to explain the attraction (violations of regularity), compromise, similarity effects, and violations of stochastic transitivity. 2 Preliminaries Let X be a nonempty finite set of choice options and denote by A 2 X its power set. Let A + A\. We use A, B, C,... to denote option sets and x, y, z,... to denote options. With slight abuse of notation, sometimes we write x for {x}. To simplify the statements below, we use the following notational convention: AB : A B, xa : {x} A, xy : {x} {y}. Definition. A random choice rule is a map ρ : A A + [0, ] such that for all A A +, i) ρ(, A) is additive; ii) ρ(a, A) ; and iii) ρ({x}, A) (0, ] for all x A. The interpretation is that ρ(a, A) denotes the probability that when the possible options faced by the agent are the alternatives in A, some alternative in A is chosen. Additivity is the requirement that ρ(, A) is a probability. Equation ρ(a, A) is the feasibility constraint; ρ must choose among options available in A. We write ρ(x A) rather than ρ({x}, A) and ρ(a A) rather than ρ(a, A). Luce rule. The Luce rule (Luce (959)) is a well-known behavioral optimization model that can be described as follows. Let v : A R ++ and v x : v({x}). Such a function v is a 5

6 Luce value if it is additive. Hence, v( ) 0, and for all A A +, v(a) x A v x. Call the choice rule ρ a Luce rule if there exists a Luce value v such that ρ(x A) v x /v(a) whenever x A A. Clearly, every Luce value v represents a unique choice rule. Definition 2. In an option set C that contains A and B, we define the relative probability of choosing options in A and B conditional on C as follows: For notational simplicity, we write β(a, B C) ρ(a C) ρ(b C). β(a, B) β(a, B AB) ρ(a AB) ρ(b AB). We introduce the following axiom of additivity of relative probabilities, which asserts that for option x and option set A disjoint, the relative probability of choosing options in xa and B conditional on xab is the sum of two components: the relative probability of choosing x and B conditional on xb and the relative probability of choosing options in A and B conditional on AB. Axiom A - Additivity of Relative Probabilities : If x / A, then β(xa, B) β(x, B)+ β(a, B). It is easy to see that additivity of relative probabilities is another probabilistic form of independence from irrelevant alternatives. The following theorem shows that Luce rule is the only choice rule that satisfies additivity of relative probabilities. Theorem. A choice rule satisfies additivity of relative probabilities if and only if it is a Luce rule. 6

7 3 Luce rule with attraction system Each option is associated with an attraction value and a Luce value. An attraction value is a function χ : A R ++ and measures the option s ability to attract the agent. Again, we write χ x rather than χ({x}). Like Luce value, we require χ to be additive. Hence, χ( ) 0, and for all A A +, χ(a) : χ x. x A For a partition of X and x X, denote by X x the element of the partition that contains x. Call the choice rule ρ a Luce rule with attraction system if there exists a partition (X i ) n i of X, an attraction value χ and a Luce value v such that ρ(x A) χ(a X x) χ(a) v(x) v(a X x ) (a) whenever x A A. We call ((X i ) n i, χ, v) an attraction system. We say that the attraction system ((X i ) n i, χ, v) represents ρ if equation (a) holds for all such x, A. Clearly, every attraction system represents a unique choice rule. We say that two options x and y are comparable if the relative probability of choosing x and y is menu independent; that is, the relative probability of choosing x and y is the same in any option set that contains x and y. As in GNP (204), whether two options are comparable is subjective and must be derived from behavior. It cannot be decided based on physical characteristics of the options and therefore, it is a property of the choice rule and not of the options. More formally, Definition 3. Two options x and y are comparable if in any option set A that contains x and y, we have β(x, y A) β(x, y). We write x y if x and y are comparable and x y if otherwise. The relation is symmetric and reflexive. Also write A B if x A, y B, we have x y. Say that an option set A is a collection of comparable options if x, y A x y. In this case, we write A L. Note that even if A L and B L, it need not be the case that AB L. As the next example demonstrates, the relation is not necessarily transitive. Example 2. Let X {x, y, z} and the random choice rule ρ is given by ρ(x xy) ρ(y xy) 2, 7

8 ρ(y yz) ρ(z yz) 2, ρ(x xyz) ρ(y xyz) ρ(z xyz) 3, ρ(x xz) 3, ρ(z xz) 2 3. It is straightforward to verify that x y, y z and yet x z. Our first axiom asserts that the relation is transitive; that is, if x and y are comparable, y and z are comparable, then x and z are comparable. Axiom T - Transitivity : The relation is transitive. Next we define the notion of overlap of two options sets: when A and B have elements in common, they overlap. Even if A and B have no elements in common, they overlap if there are elements of A and B that are comparable. Definition 4. A and B are non-overlapping if x A, y B, x y. We write A B if A and B are non-overlapping. We introduce the following weak version of additivity of relative probabilities. It requires additivity of relative probability only for the case where B is a collection of comparable options, and xa and B are non-overlapping. More formally, Axiom WA - Weak Additivity of Relative Probabilities : If x / A and xa B L, then β(xa, B) β(x, B) + β(a, B). We show that Luce rule with attraction system is the only choice rule that satisfies transitivity and weak additivity of relative probabilities. Theorem 2. A choice rule satisfies transitivity and weak additivity of relative probabilities if and only if it is a Luce rule with attraction system. 4 Discussion Despite many attractive features of the Luce model, there are well-documented violations; see Rieskamp, Busemeyer, and Mellers (2006) for an excellent survey. In this section, we We provide further discussions on this axiom in Appendix A.4. 8

9 apply our theory to explain the attraction (violations of regularity), compromise, similarity effects, and violations of stochastic transitivity. When all options are comparable, Luce rule with attraction system reduces to the Luce model. Consequently, the data cannot exhibit any violations of the Luce model. For simplicity of exposition, in what follows, we shall exclude the discussion of such case Attraction effect (violations of regularity) As discussed in the introduction, the attraction effect can be described as follows: ρ(x xyz) > ρ(x xy). In this subsection, we show that Luce rule with attraction system allows for attraction effect (violation of regularity). Let us first revisit Example. Example 3 (Example Continued). In this case, x and z are comparable; that is, x z, x y. The attraction values are χ x χ z, χ y 2. The Luce values are v x 3, v y v z. We have exhibiting attraction effect. ρ(x xyz) and ρ(x xy) χ x + χ z v x 3 χ x + χ y + χ z v x + v z 8, χ x χ x + χ y 3, Proposition below provides a necessary and sufficient condition on the attraction system for attraction effect. Proposition. The following two statements are equivalent: i) ρ(x xyz) > ρ(x xy); ii) x z, x y and v x χ y χ z > χ x v z (χ x + χ y + χ z ). Proof. ii) i) is straightforward. For i) ii), it suffices to consider the following cases. Case i) x y, x z. We have a contradiction, since ρ(x xyz) v x χ x + χ y χ x + χ y + χ z v x + v y 2 Note that it is never revealed from the choice data that for any two options x and y, x y. Suppose to the contrary, x y for any x and y. Then for any option set A that contains options z and w, we have β(z, w A) β(z, w) χz χ w ; that is, z w. We have a contradiction. 9

10 Case ii) x z, x y. In this case, < v x v x + v y ρ(x xy). ρ(x xyz) > ρ(x xy) χ x + χ z v x > χ x + χ y + χ z v x + v z χ x χ x + χ y v x χ y χ z > χ x v z (χ x + χ y + χ z ). Case iii) x y, y z. We have a contradiction, since ρ(x xyz) χ x χ x + χ y + χ z < χ x χ x + χ y ρ(x xy). 4.2 Compromise effect The compromise effect is another observed violation of the Luce model; see, for example, Simonson (989) and Simonson and Tversky (992). Consider three options x, y and z. x and z are extreme options where x is the highest in both quality and price, z is the lowest in both quality and price, whereas y has moderate values on both dimensions. The agent s option set is {x, y} in Experiment and {x, y, z} in Experiment 2. The experimental data shows that the probability of choosing y increases when moving from Experiment to 2. This pattern is called the compromise effect because y may be viewed as a compromise between the two extreme options x and z. As in Rieskamp, Busemeyer, and Mellers (2006), compromise effect could be described as follows: β(x, y xyz) < β(x, y). In this subsection, we show that Luce rule with attraction system allows for compromise effect. Consider the following example. 0

11 Example 4. Let v x v y v z χ z and χ x χ y 2 where x z, x y. We have exhibiting compromise effect. β(x, y xyz) (χ x + χ z ) vx v x+v z χ y 3 4, and β(x, y) χ x χ y, Proposition 2 below provides a necessary and sufficient condition on the attraction system for compromise effect. Proposition 2. The following two statements are equivalent: i) β(x, y xyz) < β(x, y); ii) x y, x z, χ x χ y and v x (χ x + χ z ) < χ y (v x + v z ); or x y, y z, χ x χ y and χ x (v y + v z ) < v y (χ y + χ z ). Proof. ii) i) is straightforward. For i) ii), it suffices to consider the following cases. and and Case i) x y, x z. We have a contradiction, since Case ii) x z, x y. In this case, x y β(x, y xyz) β(x, y). β(x, y xyz) < Case iii) x y, y z. In this case, β(x, y) χ x χ y, (χ x + χ z ) vx v x+v z χ y < v x (χ x + χ z ) < χ y (v x + v z ). β(x, y) χ x χ y, β(x, y xyz) < χ x < (χ y + χ z ) vy v y+v z χ x (v y + v z ) < v y (χ y + χ z ).

12 4.3 Similarity effect In Luce model, the addition of a third option z will take from options x and y in proportion to their original shares. More formally, β(x, y xyz) β(x, y). Similarity effect says that an additional option takes disproportionately more share from those similar to it than from dissimilar options. Suppose that three alternatives are such that x and z are somehow very similar to each other, and clearly distinct from y. This setup is discussed by Tversky (972a), building on an example of Debreu (960). Similarity effect can be formalized as follows: β(x, y xyz) < β(x, y). In this subsection, we show that the Luce rule with attraction system allows for similarity effect. Consider the following example. Example 5. Let v x v y v z χ y χ z and χ x 2 where x z, x y. We have exhibiting similarity effect. β(x, y xyz) (χ x + χ z ) vx v x+v z χ y 3 2, and β(x, y) χ x χ y 2, There are two possible explanations for the similarity effect. We provide the intuition for one explanation here. Assume that x and z are comparable, and x and y are non-overlapping. The addition of z increases the probability of x being considered in the second stage. But if z has a sufficiently high Luce value, it may be that x is eventually chosen with a lower probabily. The exact effect on the relative probability of x and y when we add a third option z depends on the comparison of two values: the attraction value of z relative to x, and the Luce value of z relative to x. Proposition 3 below provides a necessary and sufficient condition on the attraction system for similarity effect. Proposition 3. The following two statements are equivalent: i) β(x, y xyz) < β(x, y); ii) x z, x y and χz χ x or x y, y z and χz χ y < vz v x ; > vz v y. Proof. ii) i) is straightforward. For i) ii), it suffices to consider the following cases. 2

13 Case i) x y, y z. We have a contradiction, since Case ii) x z, x y. In this case, x y β(x, y xyz) β(x, y). β(x, y xyz) < β(x, y) Case iii) x y, y z. In this case, (χ x + χ z ) vx v x+v z χ y χ z χ x < v z v x. < χ x χ y β(x, y xyz) < β(x, y) χ x (χ y + χ z ) vy v y+v z χ z χ y > v z v y. < χ x χ y If similar options must have the same attraction values and Luce values, it follows from Proposition 3 that choices do not exhibit similarity effect in our model. 4.4 Stochastic intransitivity Several psychologists, starting from Tversky (969), have argued that choices may fail to be transitive. When choice is stochastic, there are many ways to define analogues of transitive behavior in deterministic models. A weak such analogue is the following: Definition 5 (Weak Stochastic Transitivity). For any x, y, z A, β(x, y) and β(y, z) imply β(x, z). While the Luce rule must necessarily satisfy weak stochastic transitivity, we show that the Luce rule with attraction system allows for violations of weak stochastic transitivity. Consider the following example. Example 6. Let v x v y v z χ x and χ y χ z 2 where x y, y z. We have β(x, y) v x v y, 3

14 but also violating weak stochastic transitivity. β(y, z) χ y χ z, β(x, z) χ x χ z 2, Proposition 4 below provides a necessary and sufficient condition on the attraction system for the violations of weak stochastic transitivity. Proposition 4. The following statements are equivalent: i) β(x, y), β(y, z), β(x, z) < ; ii) x y, y z, v x v y and χ y χ z > χ x ; or x z, x y, v x < v z and χ x χ y χ z ; or x y, y z, v y v z and χ z > χ x χ y. Proof. ii) i) is straightforward. For i) ii), it suffices to consider the following cases. Case i) x y, y z. In this case, Case ii) x z, x y. In this case, Case iii) x y, y z. In this case, β(x, y) v x v y, β(y, z) χ y χ z. β(x, z) < χ x < χ z. β(x, y) χ x χ y, β(y, z) χ y χ z. β(x, z) < v x < v z. β(x, y) χ x χ y, β(y, z) v y v z. β(x, z) < χ x < χ z. 4

15 Under reasonable conditions, the Luce rule with attraction system satisfies weak stochastic transitivity. For example, if the ordering of the attraction values χ weakly agrees with the ordering of the Luce values v. 3 In this case, it follows from Proposition 4 that choices are weakly stochastically transitive. A Appendix A. Preparatory Lemmas Lemma. If A B, then for any C, we have β(a, B ABC) β(a, B). Proof. Since A B, for any x A, y B, we have x y. For any C and x A, we have β(b, x ABC) β(y, x ABC) y B y B β(y, x AB) β(b, x AB), () where the first equality follows from the definition of β and the second equality follows from the definition of the relation. Furthermore, we have β(a, B ABC) β(x, B ABC) x A x A β(x, B AB) where the second equality follows from (). β(a, B), Lemma 2. If A B L, then β(a, B) x A β(x, B). Proof. Without loss of generality, write A {x, x 2,..., x k }. Let A A, A 2 A \{x },..., A k A k \{x k } {x k }. Repeatedly applying Axiom WA, we have β(a, B) β(x, B) + β(a 2, B) 3 More formally, v x v y v z and χ x χ y χ z. 5

16 β(x, B) + β(x 2, B) + β(a 3, B) x A β(x, B). Lemma 3. If A B and AB C L, then β(ab, C) β(a, C) + β(b, C). Proof. By Lemma 2, β(ab, C) β(x, C) x AB β(x, C) + β(x, C) x A x B β(a, C) + β(b, C). Lemma 4. If a, b, c (0, ) and a + b a b a a c c + b b c c, then Proof. Rearranging gives a b c c. a 2 c 2 + 2abc 2 + b 2 c 2 2a 2 c 2abc + a 2 (c(a + b) a)) 2 0. Therefore, a b c c. 6

17 Lemma 5. For A, B, C L and pairwise disjoint, if A B, A C and B C, then β(a, B ABC) β(a, B). Proof. By Lemma 3, we have Rearranging gives β(ab, C) β(a, C) + β(b, C); β(ac, B) β(a, B) + β(c, B); β(bc, A) β(b, A) + β(c, A). β(ab, C) β(a, C) + β(b, C) β(c, A) + β(c, B) β(bc, A) β(b, A) + β(ac, B) β(a, B). (2) By Lemma 4, we have 4 β(a, B ABC) β(a, B). Lemma 6. If a, b, c, d > 0, and then Proof. Rearranging gives Simplifying further, we have and ad bc. ( a + b + c + d )( a + c ad bc. + + ), b d (a + b + c + d)(abc + abd + abd + bcd) (a + b)(c + d)(a + c)(b + d). a 2 d 2 2abcd + b 2 c 2 (ad bc) Let ρ(a ABC) a, ρ(b ABC) b, ρ(a AB) c. 7

18 A.2 Proof of Theorem It is easy to see that Luce rule satisfies additivity of relative probabilities. In what follows, we show that if a choice rule ρ satisfies additivity of relative probabilities, then it is a Luce rule. For x / A, β(xa, B) β(x, B) + β(a, B). Using similar techniques as in the proof of Lemma 2, Lemma 3 and Lemma 5, we can show that for any A, B, C A + where A B, β(ab, C) β(a, C) + β(b, C) and β(a, B ABC) β(a, B). Therefore, it is a Luce rule. A.3 Proof of Theorem 2 We first show that if a choice rule ρ satisfies transitivity and weak additivity of relative probabilities, then ρ is a two-stage Luce rule with attraction system. With transitivity, we can partition the set X in the following way: {X i } n i, such that () X i L, for any i, 2,..., n ; and (2) X i X j for any i, j, 2,..., n and i j. Without loss of generality, we assume that there are at least two elements in the partition. 5 Consider any option set A A + and option x A. Let A i A X i, we have A n i(a X i ) A A 2 A n. It is easy to see that () A i L, for any i, 2,..., n ; and (2) A i A j for any i, j, 2,..., n and i j. Without loss of generality, we assume that x A. By definition of the relation, {x} A. Therefore, ρ(x A) ρ(x A A 2 A n ) ρ(a A A 2 A n )β(x, A A A 2 A n ) ρ(a A A 2 A n )β(x, A A ) ρ(a A A 2 A n )ρ(x A ), (3) where the third equality follows from Lemma. Now we show that β(a i, A A A 2 A n ) β(a i, A ) in a series of lemmas. Say an n n matrix M is symmetrically reciprocal (SR) if a for all i, 2,..., n and a ij a ji for all i, j, 2,..., n and i j. Say an n n matrix M is transitive if all entry of M is positive and a ij a ik a kj for all i, j, k, 2,..., n. 5 For the case where X L, it is easy to see that our model reduces to the standard Luce rule. 8

19 Lemma 7. For any SR and transitive matrix M (a ij ) n n, there exists w i, i, 2,..., n such that a ij w i w j and n w i. Proof. We explicitly construct w i for i, 2,..., n. Let w i i a i n i a, i, 2,..., n. i It is easy to see that For i > and j >, a i w i w and a j w wj. a i,j a i a j w i w w wj w i wj. Let N (,,..., ) T denote the n vector of s. Lemma 8. For any SR and transitive matrices M (a ij ) n n and M (ā ij ) n n, if N (M M) 0, we have a ij b ij for all i, j, 2,..., n. Proof. By Lemma 7, we can choose w i for M and w i for M such that a ij w i w j, ā ij w i w j, n w i, i n i w i. Since N (M M 2 ) 0, we have (N (M M)) j (N M) j (N M)j n (a kj ā kj ) k n k ( w k w j w k w j ) Therefore, w j w j and a ij ā ij. w j w j 0. 9

20 Lemma 9. For distinct partition elements A i, A j, A k, we have β(a i, A j A i A j A k ) β(a i, A j ). Proof. This follows from Lemma 5. Lemma 0. For distinct partition elements A i, A j, A k, we have β(a i, A j )β(a j, A k ) β(a i, A k ). Proof. By Lemma 9, we have β(a i, A j A i A j A k ) β(a i, A j ), β(a j, A k A i A j A k ) β(a j, A k ), β(a i, A k A i A j A k ) β(a i, A k ). By the definition of β, β(a i, A k A i A j A k ) β(a i, A j A i A j A k )β(a j, A k A i A j A k ), we have β(a i, A k ) β(a i, A j )β(a j, A k ). Lemma. β(a i, A j ) β(a i, A j A). Proof. By Lemma 2, we have β(a i, A i ) β(x, A i ) β(a j, A i ). (4) x A i By the definition of β, Therefore, it follows from (4) and (5) that j:j i β(a i, A i ) j:j i β(a j, A i A). (5) j:j i β(a j, A i ) j:j i β(a j, A i A). (6) Now let a ji β(a j, A i ) and ā ji β(a j, A i A) if j i. Let a ii ā ii. Therefore, it follows from the (6) that a ji j j ā ji. (7) Furthermore, by Lemma 0, a ji a ik a jk. By the definition of β, ā ji ā ik ā jk. Therefore, by Lemma 8, we have a ij ā ij. Equivalently, β(a i, A j ) β(a i, A j A). 20

21 Therefore, ρ(x A) ρ(a A A 2 A n )ρ(x A ) ρ(a A A 2 A n ) n i ρ(a i A A 2 A n ) ρ(x A ) n i β(a i, A A A 2 A n ) ρ(x A ) n i β(a i, A ) ρ(x A ). (8) Construction of the attraction value χ. Fix an arbitrary x X and let χ( x). Define χ(y) β(y, x)χ( x) β(y, x) for any y / X. Fix an arbitrary ȳ X 2. Define χ(x) β(x, ȳ)χ(ȳ) β(x, ȳ)β(ȳ, x) for any x X and x x. Lemma 2. If x X i and ỹ X j, i j, then β( x, ỹ) χ( x) χ(ỹ). Proof. By symmetry, it suffices to consider the following cases. Case i) If x / X and ỹ / X, then β( x, ỹ) β( x, ỹ x xỹ) β( x, x x xỹ)β( x, ỹ x xỹ) β( x, x)β( x, ỹ) χ( x) χ(ỹ), where the first line and third line follow from Lemma 5. Case ii) If x x X, then ỹ / X. By construction, β( x, ỹ) χ(ỹ) χ( x) χ(ỹ). Case iii) If x X, x x and ỹ / X 2, then β( x, ỹ) β( x, ỹ xỹȳ) β( x, ȳ xỹȳ)β(ȳ, ỹ xỹȳ) β( x, ȳ)β(ȳ, ỹ) χ( x) χ(ȳ) χ(ȳ) χ(ỹ) χ( x) χ(ỹ), where the first line and third line follow from Lemma 5. 2

22 Case iv) If x X, x x and ỹ ȳ X 2, then by construction, β( x, ȳ) β( x,ȳ)χ(ȳ) χ(ȳ) χ( x). χ(ȳ) Case v) If x X, x x and ỹ X 2, ỹ ȳ. Repeatedly applying Axiom WA, we have β( x x, ỹȳ) β( x, ỹȳ) + β( x, ỹȳ) β(ỹȳ, x) + β(ỹȳ, x) β(ỹ, x) + β(ȳ, x) + β(ỹ, x) + β(ȳ, x). Similarly, Since β(ỹȳ, x x) β( x, ỹ) + β( x, ỹ) + β( x, ȳ) + β( x, ȳ). β( x x, ỹȳ)β(ỹȳ, x x) ( β(ỹ, x) + β(ȳ, x) + β(ỹ, x) + β(ȳ, x) )( β( x, ỹ) + β( x, ỹ) + β( x, ȳ) + β( x, ȳ) ), by Lemma 6, we have β(ỹ, x)β(ȳ, x) β(ȳ, x)β(ỹ, x). Therefore, β( x, ỹ) β( x, ȳ)β( x, ỹ)β(ȳ, x) χ( x) χ( x) χ(ȳ) χ(ȳ) χ(ỹ) χ( x) χ( x) χ(ỹ). Lemma 3. If A X i, B X j, i j, then Proof. By Lemma 2, β(a, B) χ(a) χ(b). β(a, B) x A β(x, B) 22

23 β(b, x) x A x A y B β(y, x) χ(y) x A y B χ(x) χ(a) χ(b). By Lemma 3, ρ(x A) n i β(a i, A ) ρ(x A ) n i2 β(a i, A ) + ρ(x A ) n i2 χ(a i ) + ρ(x A ) χ(a ) χ(a ) χ(a) ρ(x A ). (9) Construction of the Luce value v. For each partition element X i, fix an arbitrary option ˆx i X i. Define v(x i ) β(x i, ˆx i ), for any x i X i. Lemma 4. If x i, x i X i and x i x i, then β(x i, x i) v(x i) v(x i). Proof. By construction, v(x i ) β(x i, ˆx i ) and v(x i) β(x i, ˆx i ). Therefore, β(x i, x i) β(x i, x i x i x iˆx i ) β(x i, ˆx i x i x iˆx i )β(ˆx i, x i x i x iˆx i ) β(x i, ˆx i )β(ˆx i, x i) v(x i) v (x i ). By Lemma 4, ρ(x A ) β(x, A ) 23

24 β(a, x) y A β(y, x) y A v(y) v(x) v(x) v(a ). Continuing from (9), ρ(x A) χ(a ) χ(a) ρ(x A ) χ(a ) χ(a) This concludes the proof for this direction. v(x) v(a ). Next we show that if a choice rule ρ is a two-stage Luce rule with attraction system, it satisfies transitivity and weak additivity of relative probabilities. If a choice rule ρ is a two-stage Luce rule with attraction system, there exists an attraction system ((X i ) n i, χ, v) such that ρ(x A) χ(a X x) χ(a) v(x) v(a X x ) (a) whenever x A A. Consider the following coarsening of the partition (X i ) n i. For each element X i, if X i >, X i remains an element of the new partition. 6 Let B 0 denote the union of the partition elements whose cardinality is one; that is, B 0 j: Xj X j. After the coarsening, relabel the elements of new partition as B 0, B, B 2,..., B m. x, y B i, Consider the new partition (B i ) n i0. Now for any partition element B i, i, if v(x) v(y) χ(x) χ(y), we union all these elements together with B 0. After such coarsening, relabel the elements of new partition as C 0, C, C 2,..., C m. For any x C 0, we define χ (x) v (x) χ(x). Otherwise, define χ (x) χ(x) and v (x) v(x). 6 For any set S, let S denote its cardinality. 24

25 We show that the attraction system ((C i ) m i0, χ, v ) also represents the choice data. Take any arbitrary x and A containing x, we need to show that ρ(x A) χ (A C x ) χ (A) v (x) v (A C x ). (0) If x C i for some i, then it is trivial to see that equality (0) holds. If x C 0, then χ (A C x ) χ (A) χ(a C x) χ(a) χ(a X x) χ(a) χ(a X x) χ(a) ρ(x A), v (x) v (A C x ) χ(x) χ(a C x ) χ(x) χ(a X x ) v(x) v(a X x ) where the first equality follows from definition of χ and v, and the third equality follows from the property of C 0. Therefore, the attraction system ((C i ) m i0, χ, v ) also represents the choice data. We show that the attraction system ((C i ) m i0, χ, v ) satisfies transitivity and weak additivity of relative probabilities. Note that if x, y C i, then x y. Therefore, to show transitivity, it suffices to show that for some x C i, whenever x y for some y, y C i. Suppose not, there exists some y x and yet y C j for some j i. Either x or y belongs to C k for some k. Without loss of generality, we assume y C k for some k. Our construction ensures that there exists z C k such that z y and Therefore, v (y) v (z) χ (y) χ (z). () β(x, y) and β(x, y xyz) χ (x) v (x) χ (x)+χ (y) v (x) χ (y) v (y) χ (x)+χ (y) v (y) χ (x) χ (y), χ (x) v (x) χ (x)+χ (y)+χ (z) v (x) χ (y)+χ (z) v (y) χ (x)+χ (y)+χ (z) v (y)+v (z) χ (x) (χ (y) + χ (z)) v (y) v (y)+v (z) 25

26 χ (x) v (y) + v (z) χ (y) + χ (z) v (y) χ (x) x (y) + x (z). χ (y) + χ (z) x (y) χ (x) χ (y) β(x, y), where the inequality follows from (). But since y x, β(x, y) β(x, y xyz). We have a contradiction. This concludes the proof that transitivity is satisfied. For weak additivity of relative probabilities, if x / A and xa B L, β(xa, B) χ(xa) χ(xa)+χ(b) χ(b) χ(xa)+χ(b) χ(x) χ(x)+χ(b) χ(b) χ(x)+χ(b) + χ(a) χ(a)+χ(b) χ(b) χ(a)+χ(b) β(x, B) + β(a, B). A.4 Weak additivity of relative probabilities Axiom WA asserts that if x / A and xa B L, then β(xa, B) β(x, B) + β(a, B). Here we consider possible weakening of this axiom, namely Axiom WA* and Axiom WA**. We present examples where the choice data satisfies Axiom T and Axiom WA* (Axiom WA** respectively), but cannot be represented by a two-stage Luce rule with attraction system. Axiom WA* : If x y and xy B L, then β(xy, B) β(x, B) + β(y, B). Axiom WA** : If x / A and xa y L, then β(xa, y) β(x, y) + β(a, y). Example 7 discusses choice data that satisfies both Axiom T and Axiom WA*, but cannot be represented by a two-stage Luce rule with attraction system. Example 7. Let X {a, b, c, d} and the random choice rule ρ is given by ρ(a abcd) ρ(b abcd) ρ(c abcd) 3 0, ρ(d abcd) 0 ρ(x xyz) 3 ρ(x xy) 2 for distinct x, y, z {a, b, c, d} for distinct x, y {a, b, c, d} 26

27 From the choice data, a b, b c, c a and abc d. It is easy to verify that the choice data satisfies Axiom T and Axiom WA*. However, 9 β(abc, d) β(a, d) + β(bc, d) 3. The choice data does not satisfy weak additivity of relative probability and hence cannot represented by a two-stage Luce rule with attraction system. Example 8 discusses choice data that satisfies both Axiom T and Axiom WA**, but cannot be represented by a two stage Luce rule with attraction system. Example 8. Let X {a, b, c, d} and the random choice rule ρ is given by ρ(a abcd) ρ(b abcd) 6, ρ(c abcd) ρ(d abcd) 3 ρ(x xyz) 3 ρ(x xy) 2 for distinct x, y, z {a, b, c, d} for distinct x, y {a, b, c, d} From the choice data, a b, c d and ab cd. It is easy to verify that the choice data satisfies Axiom T and Axiom WA**. However, 2 β(ab, cd) β(a, cd) + β(b, cd). The choice data does not satisfy weak additivity of relative probability and hence cannot represented by a two-stage Luce rule with attraction system. References Brady, R. L., and J. Rehbeck (205): Menu-Dependent Stochastic Feasibility, mimeo. University of California San Diego. Debreu, G. (960): Review of R.D. Luce, Individual Choice Behavior: A Theoretical Analysis, American Economic Review, 50(), Echenique, F., K. Saito, and G. Tserenjigmid (204): The Perception-Adjusted Luce Model, mimeo. California Institute of Technology. Gul, F., P. Natenzon, and W. Pesendorfer (204): Random Choice as Behavioral Optimization, Econometrica, 82(5),

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