ON STRATEGY-PROOF SOCIAL CHOICE BETWEEN TWO ALTERNATIVES

Transcription

1 Discussion Paper No ON STRATEGY-PROOF SOCIAL CHOICE BETWEEN TWO ALTERNATIVES Ahinaa Lahiri Anup Pramanik Octoer 2017 The Institute o Social and Economic Research Osaka University 6-1 Mihogaoka, Iaraki, Osaka , Japan

2 On Strategy-proo Social Choice etween Two Alternatives Ahinaa Lahiri and Anup Pramanik Octoer 4, 2017 Astract We study strategy-proo rules or choosing etween two alternatives. We consider the ull preerence domain which allows or indierence. In this ramework, or strategyproo rules, ontoness does not imply eiciency. We weaken the requirement o eiciency to ontoness and characterizes the class o strategy-proo rules. We argue that the notion o eiciency is not desirale always. Our main result provides a simple characterization o the class o onto, anonymous and strategy-proo rules in this ramework. Our analysis can help policy makers choose among these rules. Keywords. Strategy-prooness, Anonymity, Generalized voting y committees, Quota rules, Welare dominance under preerence replacement. JEL Codes D71. We would like to thank Salvador Barera, Jordi Masso, Deasis Mishra, Shin Sato, Arunava Sen, Shigehiro Serizawa, participants o the Delhi Economic Theory Workshop, participants o the 2017 Society or Economic Design meeting and seminar participants at the Nanjing Audit University or their comments and suggestions. Anup Pramanik acknowledges inancial assistance rom JSPS KAKENHI - 15K ISBF, New Delhi, India. Address: 15A, Ring Road Lajpat Nagar IV, New Delhi ahinaalahiri@gmail.com. ISER, Osaka University, Japan. Address: 6-1 Mihogaoka, Iaraki, Osaka, anup.isid@gmail.com. 1

3 1 Introduction In this paper, we study social choice prolem where a inite set o individuals/agents have to choose one etween two alternatives. Let a and e two alternatives. We assume that individuals can report one among the ollowing three preerences over these two alternatives: (1) a is strictly preerred to, (2) is strictly preerred to a and (3) a is indierent to. Based on individuals reported preerences, a Social Choice Function (or simply a rule) selects an alternative. Choosing etween two alternatives has many important applications - such as, two candidate elections, up-down votes on legislation, choosing one out o two locations or locating a pulic acility, yes-no decisions aout uilding a new pulic acility or any situation with a status-quo alternative and a new alternative. Throughout this paper we consider non-constant rules i.e. onto rules. Ontoness implies eiciency (or unanimity) or strategy-proo rules deined over a suitaly rich domain o strict preerences ( see Dogan and Sanver (2007)). However, ontoness does not imply eiciency i preerence domain includes indierence (see Examples in section 4) 1. We do not impose eiciency criteria on rules and characterize the class o strategy-proo rules in this ramework. Our main result provides a simple characterization o the class o anonymous and strategyproo rules. A natural ojection could e why one might compromise eiciency. In election, it is quite oten that a signiicant proportion o voters express their opinion as indierence. For instance, astaining rom voting can e interpreted as indierence. Moreover, in India voters are allowed to vote or none o the aove (NOTA) - which can also e interpreted as indierence. In this scenario, i we only look at eicient and strategy-proo voting rules, the outcome is simply ased on voters who do not express their opinion as indierence. We elieve that this is not desirale in particular when the numer o indierent voters is very large. However i we relax the requirement o eiciency, the outcome depends on oth the voters who are indierent and who are not. O course, the class o eicient and strategyproo rules is contained in the class o onto and strategy-proo rules. Since our main result provides a simple description o the class o anonymous and strategy-proo rules, we elieve that our analysis can help policy makers choose among these rules. We otain ollowing two results: Theorem 1: A rule is onto and strategy-proo i and only i it is generalized voting y committees. Theorem 2: A rule is onto, anonymous and strategy-proo i and only i it is either 1 I we restrict our attention to group strategy-proo rules, then ontoness implies eiciency (see Barerà et al. (2012), Manjunath (2012) and Harless (2015)). 2

4 a quota rule with indierence deault a or a quota rule with indierence deault. A generalized voting y committees (GVC) rule is descried y two sets. The irst one is a nonempty set o susets o N satisying monotonicity condition and we say it as a committee or indierence deault d {a, }. The second one is a set containing or each M N 2, a committee or the alternative a at M. Moreover, the second set, not only depends on the irst set, ut also satisies urther properties (see section 5 or details). For any preerence proile, i the set o agents who are indierent etween two alternatives, elongs to the committee or indierence deault d, then the rule selects d at that proile. Otherwise, consider the committee or a at the set o agents with strict preerences over a and - i the set o agents who preers a to, elongs the that committee, then the outcome is a or i it does not elong the that committee, then the outcome is. Larsson and Svensson (2006) characterizes the class o eicient and strategy-proo rules in this ramework. These rules are known as voting y extended committees (see susection 3.1 or details). These rules are contained in the class o GVC rules - in act, eicient GVC rules are voting y extended committees rules. To e precise, i we impose eiciency on GVC rules, then the irst set oils down to a singleton set including N itsel. Moreover, the second set does not depend on the irst set anymore. Further we show that an anonymous GVC rule can e descried as either a quota rule with indierence deault a or a quota rule with indierence deault. A quota rule with indierence deault a is descried simply y a vector o integers o length k, k {1, 2,..., n}, x = (x 1, x 2,..., x k ) {1} {1, 2}... {1, 2,..., k}, where x i+1 1 x i x i+1 or all i {1, 2,..., k 1}. Note that x i is the i th component o the vector x, where i {1, 2,..., k}. The rule works as ollows. For any preerence proile, i the numer o agents who are indierent etween two alternatives, is at least k, then the rule selects the indierence deault a at that proile. Suppose that the numer is less than k, i.e. the numer o agents with strict preerences elongs to {n k + 1,..., n}. In particular, we assume that the numer o agents with strict preerences is n k + l where l {1, 2,..., k}. Then the outcome is a i the numer o agents who preers a to is at least x l and the outcome is i the numer is less than x l. Here, k is the quota or indierence deault a i.e. whenever the the numer o indierent agents is at least k, the outcome is a. Also, x l is the quota or a when the numer o strict agents is n k + l, l {1, 2,..., k} i.e when n k + l is the numer o strict agents, the outcome is a i the numer o agents who preers a to is atleast x l and the outcome is i the numer is less than x l. Quota rule or a with indierence deault can also e descried in similar ashion (see section 5.2 or details). Theorem 2 characterizes the class o anonymous and strategy-proo rules in terms o quota rules in this ramework. 2 N is the set o individuals and N = n. 3

5 Further, we study solidarity property in this ramework. We consider the ollowing solidarity property: Welare dominance under preerence replacement (WDPR), which says that when the preerences o one agent change, the other agents all weakly gain or all weakly lose. We characterize the class o rules satisying WDPR among the class o quota rules. Ju (2003) considers the prolem o choosing a suset o a inite set o alternatives and characterizes strategy-proo rules or separale weak orderings. It is important to mention that our ramework appears as a special case o Ju s ramework. However, Ju (2003) provides an implicit characterization o strategy-proo rules. In this paper we consider the speciic prolem o choosing etween two alternatives and provide an explicit characterization o strategy-proo rules. Marchant and Mishra (2015) studies the same prolem and characterizes eicient and strategy-proo rules using transers in quasi-linear private values environments. This paper is organized as ollows. Section 2 descries the asic notation and deinitions. Section 3 discusses the relationship etween ontoness and unanimity (or eiciency). Section 4 provides some rules which are strategy-proo ut not eicient. The main results are presented in section 5. Proos o Theorem 1 and Theorem 2 are relegated to the Appendix. Section 6 discusses rules satisying WDPR. We conclude the the paper in section 7. 2 Basic Notation and Deinitions Let A = {a, } denote the set o two alternatives and N = {1,..., n}, n 2, a inite set o agents/individuals. Each individual in N has a preerence relation over A: she either preers a, preers, or is indierent etween them. Let R e the set o these three preerence relations. For each i N, let R i R denote individual i s preerence relation. I a is at least as good as according to individual i, we write ar i. I she preers a to, we write ap i and i she is indierent etween the two, ai i. Let P e the set o two strict preerence relations deined over A. A preerence proile is a list R = (R 1, R 2,..., R n ) R n o individuals preerences. For any coalition S N and any proile R R, R S denotes the restriction o the proile R to the coalition S i.e. R S = (R i ) i S. A proile R R n is deined to e a i deviation rom another proile R R n i R N\{i} = R N\{i}. For each R R n, let N a (R) e the set o individuals who preer a to at R. Similarly, let N (R) e the set o individuals who preer to a, and let N A (R) e the set o individuals who are indierent etween a and at R. Finally, let ς e the set o permutations o N. For each R R n and each σ ς, let σ(r) = (R σ(i) ) i N. Deinition 1. A SCF is a mapping rom R n to A i.e. : R n A. 4

6 A SCF is sometimes called a voting rule (or simply a rule). Deinition 2. A SCF is onto i or every alternative x A there exists a proile R R n such that (R) = x. Note that, as A = 2, i is not onto, then it must e a constant rule i.e. a rule that selects the same alternative at each proile. We list some well-known properties o SCFs elow. Deinition 3. A SCF satisies unanimity, i or all proile R R n ; (R) = a whenever N a (R) and N (R) =, and (R) = whenever N a (R) = and N (R). I x A is at least as good as A \ x y all individuals and at least one individual preers x, then y unanimity, the SCF must select x. Unanimity is also known as eiciency in this model. The next property imposes a weaker requirement than unanimity. I all individuals preer x A, then the SCF must select x. Deinition 4. A SCF satisies weak unanimity, i or all proile R R n ; (R) = a whenever N a (R) = N, and (R) = whenever N (R) = N. Deinition 5. A SCF is strategy-proo i, or any i N, or any R R n and or any i deviation R R n o R, we have (R)R i (R ). A SCF is strategy-proo i no individual can otain a preerred alternative y misrepresenting her preerences or any announcement o the preerences o the other individuals. Strategy-prooness ensures that that or every agent truth-telling is a weakly dominant strategy. Anonymity requires that the names o the agents should not matter. In particular, when the identities o the agents are shuled, the rule must select the same alternative. Deinition 6. A SCF is anonymous i or any R R n and or any σ ς, we have (R) = (σ(r)). 3 Unanimity versus Weak Unanimity It is important to mention that unanimity implies weak unanimity and weak unanimity implies ontoness. However, ontoness does not imply weak unanimity and weak unanimity does not imply unanimity. I we restrict our attention to strategy-proo SCFs, then ontoness implies weak unanimity. In the ollowing, we show this. 5

7 Proposition 1. Let : R n A e a strategy-proo SCF. I is onto, then it satisies weak unanimity. Proo. Suppose not. We assume that (R) = where ap i or all i N. Since is onto, there exists R R n such that (R ) = a. Applying strategy-prooness repeatedly, it ollows that (R ) = (R 1, R 2,..., R n) = (R 1, R 2, R 3,..., R n). = (R 1,..., R n ) = a This contradicts the assumption (R) =. A similar argument will lead to a contradiction i we assume that (R) = a where P i a or all i N. Thereore satisies weak unanimity. A natural question arises - i a SCF satisies strategy-prooness and weak unanimity, does it satisy unanimity? In Section 4, we provide SCFs which satisy strategy-prooness and weak unanimity, ut not unanimity. In the ollowing, we irst introduce the class o unanimous and strategy-proo rules known in the literature. 3.1 Unanimous and strategy-proo rules To introduce the class o unanimous and strategy-proo rules on R n, we need ollowing notations and deinitions. For each M N, a committee or alternative a at M, F M, is a set o susets o M, satisying the ollowing two properties: 1. Non-emptyness: I M, then F M and / F M. I M =, then F M =. 2. Monotonicity: For each S F M and T M, I S T, then T F M. A collection o committees or a, F {F M } M N, is a set containing or each M N a committee or a i.e. F M, satisying the ollowing properties: For each M N and each i M 1. I S F M and i / S, then S F M\{i}. 2. I S {i} / F M, then S / F M\{i}. 6

8 Deinition 7. A SCF is voting y extended committees, denoted y V EC a,t, i there exists a collection o committees or a (i.e. F) and a tie-reaker t A such that or all R R n ; t i N A (R) = N V EC a,t (R) = a i N a (R) F N\NA (R) otherwise Larsson and Svensson (2006) shows that the only unanimous and strategy-proo rules are V EC a,t. 4 Rules In this section, we provide some rules which are strategy-proo and onto ut not unanimous. Example 1. Consider the ollowing SCF : R n A: { a i ar 1 (R) = i P 1 a Note that satisies strategy-prooness and ontoness (see susection 5.1). However, it does not satisy unanimity. To see this, consider a preerence proile R where ai 1 and or all j N \ {1}, P ja. Unanimity implies that must select at R. However, (R ) = a. Thereore is not unanimous. Note that the rule in Example 1 is not anonymous. However, there are anonymous, onto and strategy-proo rules which are not unanimous. Example 2. Consider the status-quo rule with respect to the status-quo alternative a, a : R n A: { a i is preerred y all agents (R) = a otherwise It is straightorward that a is strategy-proo, anonymous and onto (see susection 5.2). However, a is not unanimous. Consider a preerence proile R where ai i or some i N and or all j N \ {i}, P j a. Unanimity implies that a must select at R. However, a (R) = a. Thereore a is not unanimous. The status-quo rule with respect to the status-quo alternative, is deined as ollows: { a a i a is preerred y all agents (R) = otherwise It can e seen that is strategy-proo, anonymous and onto ut not unanimous. 7

9 The ollowing class o rules can e ound in Chapter 2 o Fishurn (2015). Example 3. Let s : R {1, 0, 1} such that 1 i ap i s(r i ) = 0 i ai i 1 i P i a For each R R n, we denote s(r) = n i=1 s(r i). We ix an integer h ( n, n] Z and deine the SCF h, as ollows: For all R R n h (R) = First, we make ollowing remarks on these rules. { a i s(r) h Otherwise 1. I h = 1, we get the simple majority rule i.e. a eats whenever more individuals preer a to than preer to a and eats a whenever the converse holds. 2. The case where a wins i the numer o individuals preer a to exceeds the numer o individuals preer to a y atleast a positive integer r, and wins otherwise, is descried y h = r. 3. I h = n, then we get the status-quo rule with respect to status quo alternative. Similarly, i h = (n 1), then we get the status-quo rule with respect to status-quo alternative a. In susection 5.2, we show that h is strategy-proo, anonymous and onto. Whether h is unanimous or not, that depends on the value o h. In particular, it can e seen that h is unanimous i h {0, 1}. However i h > 1 or h 1, then h is not unanimous. To see this, we irst assume that h > 1. Let R R n e a preerence proile where ap i and ai j or all j N \i. By unanimity, we should select a at R. However h (R) =, ecause s(r) = 1 < h. Similarly, i h 1, at R R n where P i a and ai j or all j N \ i, h (R) = a, ecause s(r) = 1 h - violates unanimity. We can think o a rule where the numer o individuals who are indierent etween two alternatives, can determine the outcome. For instance, consider a rule which selects an alternative x A i the numer o indierent individuals is atleast a positive integer r {1, 2,..., n}. Otherwise i the numer is less than r, then ased upon the preerences o strict individuals, the rule selects x or the other alternative A \ x. Below, we introduce a class o such rules. 8

10 Example 4. We ix a positive integer r {1, 2,..., n} and deine the SCF r, as ollows: For all R R n r (R) = a i N A (R) r We make ollowing remarks on these rules. i N A (R) < r and N (R) i N A (R) < r and N (R) = 1. I r = 1, then we get the status-quo rule with respect to status quo alternative. 2. I r = n, then we get the consensus rule with disagreement-deault and indierencedeault (Manjunath (2012)). In susection 5.2, we show that r is strategy-proo, anonymous and onto. However, whether r is unanimous or not depends on r. In particular, i r = n, then it is straightorward to show that r is unanimous. However, i r < n, r is not unanimous. To see this, consider R R n where ap i and ai j or all j N \ i. By unanimity, we should select a at R. However r (R) =, ecause N A (R) = n 1 r. 5 Results 5.1 Generalized voting y committees In this section, we characterize onto and strategy-proo rules. For this, we need to introduce additional notation and deinitions. A committee or indierence deault d {a, }, denoted y I d, is a set o susets o N, satisying the ollowing two properties: 1. Non-emptyness: I d and / I d. 2. Monotonicity: For each S I d and T N, I S T, then T I d. Since d {a, }, I a denotes a committee or indierence deault a. Similarly, a committee or indierence deault is denoted y I. Let M N and I d e a committee or indierence deault d. A committee or a at M with respect to I d, denoted y F M,I d, is a set o susets o M, satisying the ollowing two properties: 1. Non-emptiness with respect to I d : I N \ M / I d, then F M,I d and / F M,I d. I N \ M I d, then F M,I d =. 9

11 2. Monotonicity: For each S F M,I d and T M, I S T, then T F M,I d. A collection o committees or a with respect to I a, denoted y F I a {F M,I a} M N, is a set containing or each M N a committee or a with respect to I a i.e. F M,I a, satisying the ollowing properties: For each M N and each i M 1. I N \ M / I a and {N \ M} {i} I a, then or all S M such that i S, S F M,I a. 2. I S F M,I a, i / S and {N \ M} {i} / I a, then S F M\{i},I a. 3. I N \ M / I a, S {i} / F M,I a and {N \ M} {i} / I a, then S / F M\{i},I a. Similarly, a collection o committees or a with respect to I, F I {F M,I } M N, is a set containing or each M N a committee or a with respect to I i.e. F M,I, satisying the ollowing properties: For each M N and each i M 1. I N \ M / I and {N \ M} {i} I, then or all S F M,I, i S. 2. I S F M,I, i / S and {N \ M} {i} / I, then S F M\{i},I. 3. I N \ M / I, S {i} / F M,I and {N \ M} {i} / I, then S / F M\{i},I. Given a committee or indierence deault d, I d and a collection o committees or a with respect to I d, we deine generalized voting y committees (GVC), as ollows. Deinition 8. A SCF is GVC, denoted y F Id, i there exists a committee or indierence I d deault d, I d where d A and a collection o committees or a with respect to I d, F I d, such that or all R R n ; F Id (R) = I d d i N A (R) I d a i N a (R) F N\NA (R),I d and N A(R) / I d otherwise Now we state the main result o this section. Theorem 1. Let : R n A e an onto SCF. is strategy-proo i and only i is GVC. The proo o Theorem 1 is in the Appendix. Theorem 1 in the ollowing: However, we make several remarks on 10

12 1. Larsson and Svensson (2006) characterizes unanimous (or eicient) and strategy-proo rules in this ramework. In particular, they show that the only unanimous and strategyproo rules are V EC a,t (see susection 3.1). We do not impose unanimity property on rules. We consider much weaker requirement o ontoness and characterize strategyproo rules in this ramework. The class o V EC a,t rules elongs to the the class o GVC rules. In particular, a GVC rule, Id F I d is unanimous i and only i Id = {N}. 2. It can e seen that the rule in Example 1 is a GVC rule where I a = {S N : 1 S} and F I a {F M,I a} M N is as descried elow: F M,I a = { S M : 1 S } i 1 M i 1 / M 3. We must coness that GVC rules are not simple to descrie. However, the rules that are anonymous, can e descried in much simpler way. We talk aout this in details in the next section. 5.2 Anonymous rules Theorem 1 provides a characterization o onto and strategy-proo rules in our model. In this section we provide a characterization o onto, strategy-proo and anonymous rules. First we deine the ollowing class o rules. Deinition 9. A SCF is a quota rule with indierence deault a, denoted y a k,x, i there exists a vector o natural numers o length k, x = (x 1, x 2,..., x k ) {1} {1, 2}... {1, 2,..., k}, where k {1, 2,..., n} and x i+1 1 x i x i+1 or all i {1, 2,..., k 1} such that or all R R n a i N A (R) k k,x a (R) = a i N A (R) < k and N a (R) N (R) = n k + l or some l {1, 2,..., k} and N a (R) x l otherwise Next we deine another class o rules as ollows. 11

13 Deinition 10. A SCF is a quota rule with indierence deault, denoted y k,y, i there exists a vector o natural numers o length k, y = (y 1, y 2,..., y k ) {n k + 1} {n k + 1, n k + 2}... {n k + 1, n k + 2,..., n}, where k {1, 2,..., n} and y i+1 1 y i y i+1 or all i {1, 2,..., k 1} such that or all R R n i N A (R) k k,y (R) = a i N A (R) < k and N a (R) N (R) = n k + l or some l {1, 2,..., k} and N a (R) y l otherwise Next we state the main theorem o this paper. Theorem 2. Let : R n A e a SCF. is strategy-proo, anonymous and onto i and only i it is either a quota rule with indierence deault a or a quota rule with indierence deault. The proo o Theorem 2 is in the Appendix. In the ollowing, we make several remarks on Theorem 2: 1. An anonymous and strategy-proo rule can e descried simply y a vector o natural numers o length k, where k {1, 2,..., n}. In particular, a quota rule with indierence deault a, k,x a, is descried y a vector o natural numers o length k, x = (x 1, x 2,..., x k ) {1} {1, 2}... {1, 2,..., k}, where k {1, 2,..., n} and x i+1 1 x i x i+1 or all i {1, 2,..., k 1}. For any R R n, k,x a works as ollows. I the numer o individuals who are indierent etween two alternatives at R, is atleast k, i.e. N A (R) k, then the rule selects the indierence deault a i.e. k,x a (R) = a. Here, k is the quota or indierence deault a. I N A (R) < k, then note that N a (R) N (R) = n k+l or some l {1, 2,..., k} and we consider x l which represents the quota or alternative a. I the numer o individuals who vote or a is atleast x l, i.e. N a (R) x l, then k,x a (R) = a; otherwise k,x a (R) =. A quota rule with indierence deault, can e descried in a similar way as well. 2. Note that a k,x only i k = n. is unanimous i and only i k = n. Similarly, k,y 12 is unanimous i and

14 3. Rules in Example 2: The status-quo rule with respect to the status-quo alternative a, a is a quota rule with indierence deault a, a k,x where x is a vector o natural numers o length 1 i.e. k = 1 and x (x 1 ) = (1). The status-quo rule with respect to the status-quo alternative, is a quota rule with indierence deault, k,y where y is a vector o natural numers o length 1 i.e. k = 1 and y (y 1 ) = (n). 4. Rules in Example 3: I h > 0, the h is a quota rule with indierence deault, k,y a where y is a vector o natural numers o length n h + 1 i.e. k = n h + 1 and y (y 1,..., y n h+1 ) = (h, h + 1, h + 1, h + 2, h + 2, h + 3,...). I h 0, the h is a quota rule with indierence deault a, a k,x where x is a vector o natural numers o length n+h i.e. k = n+h and x (x 1,..., x n+h ) = (1, 1, 2, 2, 3, 3,...). 5. The rule in Example 4: It can e seen that the rule in Example 4 is a quota rule with indierence deault, k,y where y is a vector o natural numers o length r i.e. k = r and y (y 1,..., y r ) = (n r + 1, n r + 2,..., n). 5.3 Weak strategy-prooness In this section, we introduce a weaker notion o strategy-prooness as ollows. Deinition 11. A SCF is weakly strategy-proo i, or any i N, or any R R n and or any i deviation R R n o R such that R i P and ai i, we have (R)R i (R ). Next we show that in our model, strategy-prooness and weak strategy-prooness are equivalent 3. Lemma 1. Let : R n strategy-proo. A e a SCF. is strategy-proo i and only i is weakly Proo. Note that i is strategy-proo then it is weakly strategy-proo. So suppose that is weakly strategy-proo, ut to the contrary is not strategy-proo. Then there exist an agent i N and a proile R R n and an i deviation R R o R such that R i, R i P and (R )P i (R). So it ollows that R i R i. Without loss o generality, assume that ap i and P i a. So it ollows that (R) = and (R ) = a. Now consider the proile R R n such that R N\{i} = R N\{i} = R N\{i}, and ai i. Now weak strategy-prooness or the deviation rom R to R implies that (R ) =. On the other hand weak strategy-prooness or the deviation rom R to R implies that (R ) = a, which contradicts the act that (R ) = and concludes the proo. 3 Weak strategy-prooness is also known as participation property o an SCF in this ramework (see Núñez and Sanver (2017) or details.) 13

15 We present our main results as the ollowing corollaries. Corollary 1. Let : R n A e an onto SCF. is weakly strategy-proo i and only i it is GVC. Proo. Follows rom Theorem 1 and Lemma 1. Corollary 2. Let : R n A e an onto SCF. is anonymous and weakly strategyproo i and only i it is either a quota rule with indierence deault a or a quota rule with indierence deault. Proo. Follows rom Theorem 2 and Lemma 1. 6 Solidarity and quota rules Among the class o anonymous and strategy-proo rules, the rules satisy solidarity property, is studied in this section. We consider the ollowing solidarity property: welare dominance under preerence replacement, which says that when the preerences o one agent change, the other agents all weakly gain or all weakly lose. Deinition 12. A SCF satisies welare dominance under preerence replacement (WDPR) i or any R R n, or any i N and or any R i R, either (i) or each j N \ {i}, we have (R)R j (R i, R i ) or (ii) or each j N \ {i}, we have (R i, R i )R j (R). Beore presenting the main results o this section, we state the ollowing lemma. Lemma 2. Let : R n A satisies WDPR. Then or all R, R R n such that N a (R), N (R), N a (R ), N (R ), we have (R) = (R ). Proo. The proo can e ound in lemma 1 o Harless (2015). Hence, it is omitted. According to Lemma 2, i a rule satisies WDPR, then it selects the same alternative in each disagreement proile 4. Now we are ready to state our results. The ollowing proposition characterizes the class o rules satisying WDPR among the class o quota rules with indierence deault a. Proposition 2. Let n 3 and a k,x : R n A e a quota rule with indierence deault a. a k,x satisies WDPR i and only i either (i) x is a vector o natural numers o length n and x = (x 1,..., x n ) {(1, 1,..., 1), (1, 2,..., n)} or (ii) x is a vector o natural numers o length k, k {1, 2,..., n 1} and x = (x 1,..., x k ) = (1, 1,..., 1). 4 A proile R R n is called disagreement proile i at R, N a (R) and N (R). 14

16 Proo. Only i part. Let a k,x e a quota rule with indierence deault a and it satisies WDPR. Thereore, the length o x is either (i) k = n or (ii) k {1, 2,..., n 1}. First we assume that k = n. I x n = 1 or n then we are done. We assume or contradiction that x n {2, 3,..., n 1}. Let R e a preerence proile such that N a (R) = and N a (R) = x n. Since a k,x e a quota rule with indierence deault a, a k,x (R) = a. Let R e a preerence proile such that N a (R ) = and N a (R ) = x n 1. Note that N a (R), N (R), N a (R ), N (R ). Thereore, y lemma 2 a k,x (R) = a k,x (R ). However, since a k,x e a quota rule with indierence deault a, a k,x (R ) = - a contradiction. Thereore x n = 1 or n, which in turn imply that x = (x 1,..., x n ) {(1, 1,..., 1), (1, 2,..., n)}. Finally we assume that k {1, 2,..., n 1}. Note that i we can show x i = x i+1 or all i {1, 2,..., k 1}, then we are done. I k = 1, we are done trivially. Thereore we assume that k > 1. We assume or contradiction that there exists i {1, 2,..., k 1} such that x i x i+1. Let i e he minimum among all i {1, 2,..., k 1} such that x i x i+1. Note that x i = 1 and x i +1 = 2. Let R and R e preerence proiles such that N a (R) N (R) = N a (R ) N (R ) = n k + i + 1. Moreover we assume that N a (R) = 2 and N a (R ) = 1. Since N a (R), N (R), N a (R ), N (R ) ; y lemma 2 a k,x (R) = a k,x (R ). However, since a k,x e a quota rule with indierence deault a, a k,x (R) = a = a k,x (R ) - a contradiction. Thereore, x i = x i+1 or all i {1, 2,..., k 1}, which in turn imply that x = (x 1,..., x n ) = (1, 1,..., 1). I part. We irst prove the ollowing claim. Claim 1. Let : R n A selects the same alternative in each disagreement proile. Then satisies WDPR. Proo. Let R R n, i N and R i R. I oth R and (R i, R i ) are disagreement proile then (R) = (R i, R i ). Suppose this is not the case. Then either (i) or each j N \ {i}, we have (R)R j (R i, R i ) or (ii) or each j N \ {i}, we have (R i, R i )R j (R). In either case WDPR is satisied. Let a k,x : R n A e a quota rule with indierence deault a. I x is a vector o natural numers o length n and x = (x 1,..., x n ) = (1, 1,..., 1), then a k,x selects a in each disagreement proile. I x is a vector o natural numers o length n and x = (x 1,..., x n ) = (1, 2,..., n), then k,x a selects in each disagreement proile. I x is a vector o natural numers o length k, k {1, 2,..., n 1} and x = (x 1,..., x k ) = (1, 1,..., 1), then k,x a selects a in each disagreement proile. Thereore, y claim 1, all these rules satisy WDPR. Next we characterize the class o rules satisying WDPR among the class o quota rules with indierence deault. 15

17 Proposition 3. Let n 3 and k,y : R n A e a quota rule with indierence deault. k,y satisies WDPR i and only i either (i) y is a vector o natural numers o length n and y = (y 1,..., y n ) {(1, 1,..., 1), (1, 2,..., n)} or (ii) y is a vector o natural numers o length k, k {1, 2,..., n 1} and y = (y 1,..., y k ) = (n k + 1, n k + 2,..., n). Proo. Only i part. Let k,y e a quota rule with indierence deault and it satisies WDPR. Thereore, the length o y is either (i) k = n or (ii) k {1, 2,..., n 1}. First we assume that k = n. I Y n = 1 or n then we are done. We assume or contradiction that y n {2, 3,..., n 1}. Let R e a preerence proile such that N a (R) = and N a (R) = y n. Since k,y e a quota rule with indierence deault, k,y (R) = a. Let R e a preerence proile such that N a (R ) = and N a (R ) = y n 1. Note that N a (R), N (R), N a (R ), N (R ). Thereore, y lemma 2 k,y (R) = k,y (R ). However, since k,y e a quota rule with (R ) = - a contradiction. Thereore y n = 1 or n, which in turn indierence deault, k,y imply that y = (y 1,..., y n ) {(1, 1,..., 1), (1, 2,..., n)}. Finally we assume that k {1, 2,..., n 1}. Note that i we can show y i y i+1 or all i {1, 2,..., k 1}, then we are done. I k = 1, we are done trivially. Thereore we assume that k > 1. We assume or contradiction that there exists i {1, 2,..., k 1} such that y i = y i+1. Let i e he minimum among all i {1, 2,..., k 1} such that y i = y i+1. Thereore y i = y i +1 = n k + i. Let R and R e preerence proiles such that N a (R) N (R) = N a (R ) N (R ) = n k + i + 1. Moreover we assume that N a (R) = n k + i and N a (R ) = n k + i 1. Since N a (R), N (R), N a (R ), N (R ) ; y lemma 2 k,y (R) = k,y (R ). However, since k,y e a quota rule with indierence deault, k,y (R) = a = k,y (R ) - a contradiction. Thereore, y i y i+1 or all i {1, 2,..., k 1}, which in turn imply that y = (y 1,..., y k ) = (n k + 1, n k + 2,..., n). I part. Let k,y : R n A e a quota rule with indierence deault. I y is a vector o natural numers o length n and y = (y 1,..., y n ) = (1, 1,..., 1), then k,y selects a in each disagreement proile. I y is a vector o natural numers o length n and y = (y 1,..., y n ) = (1, 2,..., n), then k,y selects in each disagreement proile. I y is a vector o natural numers o length k, k {1, 2,..., n 1} and y = (y 1,..., y k ) = (n k + 1, n k + 2,..., n), then k,y WDPR. selects in each disagreement proile. Thereore, y claim 1, all these rules satisy We conclude this section y making ollowing remarks on Proposition 2 and I n = 2, then quota rules with indierence deault a and quota rules with indierence deault, satisy WDPR. For n > 2, this is not true. 2. For unanimous rules, WDPR implies strategy-prooness and anonymity. However, or onto rules, WDPR does not imply strategy-prooness and anonymity (see Harless 16

18 (2015) or details). By proposition 2 and 3, the comination o strategy-prooness and anonymity does not imply WDPR or n > 2. In particular, proposition 2 and 3 characterize the class o rules satisying WDPR among the class o anonymous and strategy-proo rules. 7 Conclusion We study social choice prolem where a inite set individuals have to choose one etween two alternatives. We consider the ull preerence domain which allows or indierence. We weaken the requirement o eiciency to ontoness and analyze strategy-proo rules in this ramework. Our main result provides a simple description o the class o anonymous and strategy-proo rules in this ramework. These rules can e descried simply y a vector o integers. We elieve that our analysis can help policy makers choose among these rules. Reerences Barerà, S., D. Berga, and B. Moreno (2012): Group strategy-proo social choice unctions with inary ranges and aritrary domains: Characterization results, International Journal o Game Theory, 41, Barerà, S., H. Sonnenschein, and L. Zhou (1991): Voting y committees, Econometrica: Journal o the Econometric Society, 59, Dogan, E. and M. R. Sanver (2007): On the alternating use o unanimity and surjectivity in the Giard Satterthwaite theorem, Economics Letters, 96, Fishurn, P. C. (2015): The Theory o Social Choice, Princeton University Press. Harless, P. (2015): Reaching consensus: solidarity and strategic properties in inary social choice, Social Choice and Welare, 45, Ju, B.-G. (2003): A characterization o strategy-proo voting rules or separale weak orderings, Social Choice and Welare, 21, Larsson, B. and L.-G. Svensson (2006): Strategy-proo voting on the ull preerence domain, Mathematical Social Sciences, 52, Manjunath, V. (2012): Group strategy-prooness and voting etween two alternatives, Mathematical Social Sciences, 63,

19 Marchant, T. and D. Mishra (2015): Mechanism design with two alternatives in quasilinear environments, Social Choice and Welare, 44, Núñez, M. and M. R. Sanver (2017): Revisiting the Connection etween the No-Show Paradox and Monotonicity, Mathematical Social Sciences, Forthcoming. Appendix 1. The Proo o Theorem 1 Proo. I part. Let F Id e a GV C rule. Let Id e the committee or indierence deault I d d {a, } and F I d, the collection o committees or a with respect to I d. We show that F Id I d is onto and strategy-proo. To prove that Id F I d is onto, we show that there exist R, R R n such that Id F I d (R ) = a and Id F I d (R ) =. Let R and R e such that N a (R ) = N and N (R ) = N respectively. Note that N A (R ) = N (R ) =, N a (R ) F N\Na (R ),I d and N A(R ) / I d. Thereore, F Id (R ) = a. Again, since N I d A (R ) = N a (R ) =, N a (R ) / F N\Na (R ),I and N A(R ) / I d, d we have F Id (R ) =. I d Next we show that F Id satisies strategy-prooness. We consider R Rn and R I d i R. First we assume that F Id (R) = a. I ar I d i, then i can not manipulate at R via R i. I P i a then we show that F Id (R I d i, R N\{i} ) = a. The ollowing two cases arise : (i) ai i and (ii) ap i. (i) Suppose ai i. Let d = a. I N A (R) I a, then N A (R i, R N\{i} ) I a. Thereore, F Ia I a(r i, R N\{i} ) = a. I N A (R) / I a, then F Ia I a(r) = a implies that N a (R) F N\NA (R),I a. Now we consider the set N A (R i, R N\{i} ). I N A (R i, R N\{i} ) I a, then F Ia I a(r i, R N\{i} ) = a. I N A (R i, R N\{i} ) / I a, then the property 2 o F I a would imply that N a (R i, R N\{i} ) F N\NA (R i,r Ia a. N\{i}),I Thereore, F I a(r i, R N\{i} ) = a. Let d =. Since F I (R) = a, N I A(R) / I and N a (R) F N\NA (R),I. Now we consider the set N A (R i, R N\{i} ). I N A (R i, R N\{i} ) I, then y property 1 o F I, i N a (R) which is not possile. Thereore, N A (R i, R N\{i} ) / I. Since N a (R) F N\NA (R),I and N a(r i, R N\{i} ) = N a (R), y the property 2 o F I we have N a (R i, R N\{i} ) F N\NA (R i,r. Thereore, N\{i}),I F I I i, R N\{i} ) = a. (ii) Suppose ap i. Let d = a. Note that N A (R i, R N\{i} ) = N A (R). I N A (R) I a, then F Ia I a(r i, R N\{i} ) = a. I N A (R) / I a, then F Ia I a(r) = a implies that N a (R) F N\NA (R),I a. By monotonicity property o F N\N A (R),I a, N a(r i, R N\{i} ) F N\NA (R),Ia. Since N A (R i, R N\{i} ) = N A (R), F Ia I a(r i, R N\{i} ) = a. 18

20 Let d =. Since F I (R) = a, N I A(R) / I and N a (R) F N\NA (R),I. Also, since N A (R i, R N\{i} ) = N A (R), N A (R i, R N\{i} ) / I. By monotonicity property o F N\NA (R),I, N a (R i, I R N\{i} ) F N\NA (R),I. Thereore, F (R I i, R N\{i} ) = a. Now we assume that F Id (R) =. I R I d ia, then i can not manipulate. I ap i then we show that F Id (R I d i, R N\{i} ) =. The ollowing two cases arise : (i) ai i and (ii) P i a. (i) Suppose ai i. Let d = a. Since F Ia I a(r) =, N A (R) / I a and N a (R) / F N\NA (R),I a. Now we consider the set N A (R i, R N\{i} ). I N A (R i, R N\{i} ) I a, then y property 1 o F I a, N a (R) F N\NA (R),I a which is not possile. Thereore, N A(R i, R N\{i} ) / I a. Since N a (R) / F N\NA (R),I a and N A(R i, R N\{i} ) / I a, property 3 o F I a would imply that N a (R i, R N\{i} ) / F N\NA (R i,r Ia a. N\{i}),I Thereore, F I a(r i, R N\{i} ) =. Let d =. I N A (R) I, then (y monotonicity property o I ) N A (R i, R N\{i} ) I. Thereore, F I (R I i, R N\{i} ) =. I N A (R) / I, then F I (R) = implies that I N a (R) / F N\NA (R),I. Now we consider the set N A(R i, R N\{i} ). I N A (R i, R N\{i} ) I, then F I (R I i, R N\{i} ) =. I N A (R i, R N\{i} ) / I, then property 3 o F I would imply that N a (R i, R N\{i} ) / F N\NA (R i,r I a. N\{i}),I Thereore, F (R I i, R N\{i} ) =. (ii) Suppose P i a. Note that N A (R i, R N\{i} ) = N A (R). Let d = a. Since F Ia I a(r) =, N A (R) / I a and N a (R) / F N\NA (R),I a. Since N A(R i, R N\{i} ) = N A (R), we have N A (R i, R N\{i} ) / I a and N a (R) / F N\NA (R i,r N\{i}),I a. Note that N a(r i, R N\{i} ) / F N\NA (R i,r a, N\{i}),I otherwise y monotonicity property o F N\NA (R i,r N\{i}),I a, N a(r) F N\NA (R i,r N\{i}),I a which is not possile. Thereore, F Ia I a(r i, R N\{i} ) =. Let d =. I N A (R) I, then N A (R i, R N\{i} ) I. Thereore, F I (R I i, R N\{i} ) =. So, we consider that N A (R) / I. since F I (R) =, N I a(r) / F N\NA (R),I. Note that since N A (R i, R N\{i} ) = N A (R), we have N a (R i, R N\{i} ) / F N\NA (R i,r a, N\{i}),I otherwise y monotonicity property o F N\NA (R i,r N\{i}),I a, N a(r) F N\NA (R i,r N\{i}),I a which is not possile. Thereore, F I (R I i, R N\{i} ) =. Only i part. Let e an onto and strategy-proo SCF. Let R R n denotes the preerence proile where all agents are indierent etween a and. We show that i ( R) = a, then there exists a committee or indierence deault a, I a and a collection o committees or a with respect to I a, F I a, such that or all R R n ; (R) = Ia F I a(r). Similarly, i ( R) =, then there exists a committee or indierence deault, I and a collection o committees or a with respect to I, F I, such that or all R R n ; (R) = I F I (R). In the ollowing, we consider these two cases. 19

21 Case 1: ( R) = a. For each M N, let g M e the restriction o to {R R n : ai i i i / M}. In other words, let g M : {R R n : ai i i i / M} A e a unction deined as g M(R) = (R) or all R {R Rn : ai i i i / M}. First we show the ollowing claim. Claim 2. For each M N, either g M is a constant rule that picks a or g M is onto. Proo. I M =, then it is trivial that g M is a constant rule that picks a. Thereore, or contradiction, we assume that there exists M N such that g M is a constant rule that picks. W.o.l.g. let M = {1, 2,..., k}, k n. Let R e such that ai i i i {k + 1,..., n} and ap i i i {1,..., k}. Since g M strategy-prooness, we have This contradicts ( R) = a. is a constant rule that picks, (R ) =. Applying (R 1, R 2,..., R n) = (R 1,..., R k 1, R k, R k+1,..., R n) = (R 1,..., R k 2, R k 1, R k, R k+1,..., R n). = ( R 1,..., R k, R k+1,..., R n) = ( R) = Let I a () = {S N : g N\S act. is constant rule that picks a }. Next we show the ollowing Fact 1. I a () is a committee or indierence deault a. Proo. We show that I a () satisies ollowing two properties. 1. Non-emptiness: Since N I a (), I a (). Since is onto and strategy-proo, g N is onto. Thereore, / I a (). 2. Monotonicity: Let S I a (), T N and S T. We show that T I a (). Since g N\S is a constant rule that picks a, g N\T is a constant rule that picks a. Thereore, T I a (). The ollowing claim is a direct implication o Theorem 1 o Barerà et al. (1991). Hence, we omit the proo. Claim 3. For each M N, i g M is onto, then it is a voting y committee or a at M. 20

22 For each M N such that g M is onto, Claim 3 implies that g M is a voting y committee or a at M. Let F gm M e the committee or a at M associated with gm. Now, or each M N, we deine the set F M,I a () as ollows. I g M is onto, then F M,I a () = F gm M. I gm not onto i.e. g M is a constant rule that picks a, then F M,I a () =. First we show the ollowing act. Fact 2. For each M N, F M,I a () is a committee or a at M with respect to I a (). Proo. We show that or each M N, F M,I a () satisies ollowing two properties. 1. Non-emptiness with respect to I a (): I N \ M / I a (), then g M is onto. Thereore, F M,I a () = F gm M. Since F gm M and / F gm M, we have F M,I a () and / F M,I a (). This ollows rom Claim 3 and Barerà et al. (1991). I N \ M I a (), then g M is a constant rule that picks a. Thereore, F M,I a () = y deinition. 2. Monotonicity: W.o.l.o.g. we assume that F M,I a (). Thereore F M,I a () = F gm M. Since F gm M satisies monotonicity (rom Claim 3 and Barerà et al. (1991)), we have that or each S F M,I a () and T M, i S T, then T F M,I a (). is Next we show that F I a () {F M,I a ()} M N satisies the properties o a collection o committees or a with respect to I a (). Fact 3. F I a () {F M,I a ()} M N satisies the properties o a collection o committees or a with respect to I a (). Proo. We show that or each M N and each i M: 1. I N \ M / I a () and {N \ M} {i} I a (), then or all S M such that i S, S F M,I a (). Suppose not. There exist M N and S M such that N \ M / I a (), {N \ M} {i} I a () and i S / F M,I a (). Let R R n e a preerence proile such that ai k or all k {N \ M}, ap k or all k S and P k a or all k M \ S. Note that g M (R) =. Thereore (R) =. Since is strategy-proo, (R i, R N\{i} ) = where R i = ai i. This contradicts with the act that {N \ M} {i} I a (). 2. I S F M,I a (), i / S and {N \ M} {i} / I a (), then S F M\{i},I a (). Let R R n e a preerence proile such that ai k or all k {N \ M} {i}, ap k or all k S and P k a or all k M \ {S i}. Let R = (R i, R N\{i} ) where R i = P i a. Since S F M,I a (), g M(R ) = a. Thereore (R ) = a. Then y strategy-prooness, (R) = a, i.e g M\i (R) = a. Since {N \ M} i / I a (), S F M\i,I a (). 21

23 3. I N \M / I a (), S {i} / F M,I a () and {N \M} {i} / I a (), then S / F M\{i},I a (). Let R R n e e a preerence proile such that ai k or all k N \ M, ap k or all k S i and P k a or all k M \ {S i}. Let R = (R i, R N\{i} ) where R i = I ia. Since S i / F M,I a (), g M (R) =. Thereore (R) =. Then y strategy-prooness, (R ) =, i.e g M\i (R ) =. Since {N \ M} i / I a (), S / F M\i,I a (). We complete this case y showing that or all R R n ; (R) = h Ia () F I a () (R). Consider any proile R. By Claim 2, g N\N A(R) is an onto rule. Let g N\N A(R) is either a constant rule that picks a or it e a constant rule that picks a. Thereore, g N\N A(R) (R) = a implies that (R) = a. Which, in turn, implies that N A (R) I a (); i.e.; h Ia () F I a () (R) = a. Now we assume that g N\N A(R) is an onto rule. Thereore, N A (R) / I a (). By Claim 3, g N\N A(R) is a voting y committee or a at N \ N A (R). Let F g N\N A (R) N\N A (R) e the committee or a at N \ N A (R) associated with g N\N A(R). Thereore, g N\N A(R) (R) = a i N a (R) F g and g N\N A(R) N\N A (R) (R) = i N a (R) / F g N\N A (R). Since gn\n A(R) (R) = (R) and gn\n A(R) F N\NA (R),I a () = F, we are done. N\N A (R) N\N A (R) N\N A (R) Case 2: ( R) =. A similar argument (as in case 1) shows that there exists a committee or indierence deault, I and a collection o committees or a with respect to I, F I, such that or all R R n ; (R) = I F I (R). 2. The Proo o Theorem 2 We prove this theorem with the help o the ollowing propositions. The irst proposition is a direct implication o adding anonymity on Theorem 1. For this purpose, we introduce the ollowing deinitions. A committee or indierence deault d {a, }, I d, is anonymous, i S I d implies S I d or any S N such that S = S. I a committee or indierence deault d is anonymous, then we reer it as anonymous committee or indierence deault d. Let M N and I d e a committee or indierence deault d. A committee or a at M with respect to I d, F M,I d, is anonymous, i S F M,I d implies S F M,I d or any S M 22

24 such that S = S. I a committee or a at M with respect to I d is anonymous, we reer it as anonymous committee or a at M with respect to I d. A collection o anonymous committees or a with respect to I d, is a collection o committees or a with respect to I d, satisying ollowing properties 1. For any M N, F M,I d is a anonymous committees or a at M with respect to I d. 2. For any M, M N, S M and S M where M = M and S = S, i S F M,I d then S F M,I d. We deine generalized voting y anonymous committees (GVAC), as ollows. Deinition 13. A SCF is GVAC, denoted y F Id, i there exists a anonymous committee I d or indierence deault d, I d where d A and a collection o anonymous committees or a with respect to I d, F I d, such that or all R R n ; d i N A (R) I d F Id (R) = I a i N d a (R) F N\NA (R),I and N A(R) / I d d otherwise This rings us to the ollowing proposition. Proposition 4. Let : R n A e an onto SCF. I is anonymous and strategy-proo, then is GVAC. Proo. Let e an onto, anonymous and strategy-proo SCF. Let R R n denotes the preerence proile where all agents are indierent etween a and. We show that i ( R) = a, then there exists a anonymous committee or indierence deault a, I a and a collection o anonymous committees or a with respect to I a, F I a, such that or all R R n ; (R) = Ia F I a(r). Similarly, i ( R) =, then there exists a anonymous committee or indierence deault, I and a collection o anonymous committees or a with respect to I, F I, such that or all R R n ; (R) = I F I (R). In the ollowing, we consider these two cases. Case 1: ( R) = a : As is strategy-proo and onto, we have the ollowing. For any M N, let g M : {R R n : ai i i i / M} A e a unction deined as g M(R) = (R) or all R {R Rn : ai i i i / M}. Then either g M is a constant rule that picks a or g M is onto. This ollows rom Claim 2 in the proo o Theorem 1. 23

25 I a () = {S N : g N\S is constant rule that picks a} is a committee or indierence deault a. This ollows rom Fact 1 in the proo o Theorem 1. F I a () {F M,I a ()} M N, where { R {R R n : ai i i i / M} S M : with S = N a (R) M and g M (R) = a F M,I a () = } i i g M is an onto unction g M is a constant rule that picks a is a collection o committees or a with respect to I a (). This ollows rom Claim 3 and Facts 2 and 3 in the proo o Theorem 1. a. Next we are going to show that I a () is an anonymous committee or indierence deault Claim 4. I a () is an anonymous committee or indierence deault a. Proo. Consider S, S N such that S = S. Suppose S I a (), ut to the contrary S / I a (). This implies that g N\S is a constant rule that selects a, ut g N\S is onto. So there exists a R {R R n : ai i i i / N \ S } such that g N\S (R) =. As S = S, we have N \ S = N \ S. As N a (R) N \ S, there exists T N \ S such that N a (R) = T and (N \S )\N a (R) = (N \S)\T. So we can deine the ollowing unctions; σ 1 : S S, σ 2 : N a (R) T and σ 3 : (N \ S ) \ N a (R) (N \ S) \ T ; which are all one-to-one and onto. Next, we deine a permutation σ : N N as ollows. σ 1 (i) i i S σ(i) = σ 2 (i) i i N a (R) σ 3 (i) i i (N \ S ) \ N a (R) Note that σ is a well-deined permutation and σ(r) {R R n : ai i i i / N \ S}. This implies that g N\S (σ(r)) = a; i.e.; (σ(r)) = a. But this contradicts anonymity o as g N\S (R) = implies (R) =. This concludes the proo o Claim 4. I a (). Next we show that F I a () is a collection o anonymous committees or a with respect to Claim 5. F I a () is a collection o anonymous committees or a with respect to I a (). 24

26 Proo. First we show that or every M N, F M,I a () is a anonymous committees or a at M with respect to I a (). First note that as is anonymous, so or any M N, it ollows that g M is also anonymous. Now or any M N, consider S, S M such that S = S. Suppose or contradiction that S F M,I a (), ut S / F M,I a (). This implies that there exists a proile R {R R n : ai i i i / M} such that N a (R) = S and g M (R) = a. As S = S, we have M \S = M \S. So we can deine the ollowing unctions; σ 1 : S S and σ 2 : M \S M \S ; which are all one-to-one and onto. Next, we deine a permutation σ : N N as ollows. i σ(i) = σ 1 (i) σ 2 (i) i i N \ M i i S i i M \ S Note that σ is a well-deined permutation and σ(r) {R R n : ai i i i / M} and N a (R) = S. Now S / F M,I a () implies that g M (σ(r)) = ecause gm is onto. But this contradicts anonymity o g M, as gm (R) = a. This shows that or every M N, F M,I a () is a anonymous committee or a at M with respect to I a (). Next, consider any M, M N and S M and S M such that M = M and S = S. We are going to show that i S F M,I a (), then S F M,I a (). So suppose or contradiction that S F M,I a (), ut S / F M,I a (). As S F M,I a (), there exists a proile R {R R n : ai i i i / M} such that N a (R) = S and g M (R) = a. As M = M and S = S, so it ollows that M \ S = M \ S and N \ M = N \ M. So we can deine the ollowing unctions; σ 4 : N \ M N \ M, σ 5 : S S and σ 6 : M \ S M \ S ; which are all one-to-one and onto. Next, we deine a permutation σ : N N as ollows. σ 4 (i) σ (i) = σ 5 (i) σ 6 (i) i i N \ M i i S i i M \ S Note that σ is a well-deined permutation and σ (R) {R R n : ai i i i / M } and N a (R) = S. As g M is onto, so it ollows that g M is also onto. Otherwise there would e a violation o Claim 4 as N \M = N \M. Then S / F M,I a () implies that g M (σ (R)) = ; i.e.; (σ (R)) =. This contradicts anonymity o as g M (R) = a implies that (R) = a. This concludes the proo o Claim 5. We complete this case y showing that or all R R n ; (R) = Ia () F I a () (R). This ollows rom the deinition o I a () and F I a () as shown at the end o case 1 in the proo o the only i part o Theorem 1. 25