Coalitional Stability in the NIMBY Problem: An Application of the Minimax Theorem

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1 Coalitional Stability in the NIMBY Problem: An Application of the Minimax Theorem Hirofumi Yamamura y First draft: January 21, 2013 This version: February 26, 2014 Abstract Because of the Fukushima Daiichi nuclear disaster, Japanese government has to construct an interim storage facility to control radioactive wastes, but any agent wants it to be located as far as possible. We consider this situation through an economic model in which agents choose the location of a public bad from a street according to a given voting mechanism. Agents have singledipped preferences over the set of feasible locations. We analyze coalitional behaviors for any given voting mechanism. We identify a necessary and su - cient condition for a voting mechanism to possess a strong Nash equilibrium by applying the minimax theorem (von Neumann and Morgenstern [44]). We introduce the class of core solutions and show that these solutions are characterized by strong Nash implementability. As a by-product of these result, we propose a simple voting mechanism that implements any core solution in strong Nash equilibria. Department of Social Engineering, Graduate School of Decision Science and Technology, Tokyo Institute of Technology, Ookayama, Meguro-ku, Tokyo Japan; Tel: ; yamamura-h@soc.titech.ac.jp y I am also grateful to Paulo Barelli, Srihari Govindan, Toshiyuki Hirai, William Thomson and Takehiko Yamato for their valuable advices and discussions. I also thank participants of the Meeting of Japanese Economic Association at Toyama University and the seminars at University of Rochester, Tokyo Institute of Technology, and Kobe University for helpful comments. 1

2 JEL Classi cation: D78, D72, C70. Key words: NIMBY problem, Single-dippedness, Strong Nash equilibrium, Minimax theorem, Core solution, Implementation, Manipulation. 1 Introduction Because of the Great East Japan Earthquake on 11 March 2011, Japanese goverment is still having many serious political issues. One of such issues derives from a lot of radioactive wastes caused by the Fukushima Daiichi nuclear disaster. The amount of radioactive wastes are so large that the ministry of the environment of Japan states that it still di cult to decide the way of nal disposal of radioactive wastes and that it is necessary to construct an interim storage facility to control radioactive wastes 1. However, no local goverment is eager to accept such a facility because of the possibile harm to residents. Anyone wants it to be located as far as possible. Such preferences cause a serious con ict among regions with regard to where to locate this facility. This problem is called a NIMBY (Not In My Back Yard) Problem. There are many other NIMBY problems in the real world; where to locate a base, a garbage disposal plant or a prison. A NIMBY problem can be represented by the following economic model. There is a set of possible alternatives proposed for a public facility. In the case of the location of the interim storage facility in Japan, since Japanese country is long and narrow, we can regard this set as one dimensional. Each agent has a single-dipped preference over the set of possible sites: there is a worst point, called a dip, and agent s welfare increases as the chosen location moves away from this point in either direction. That is, agents preferences are supposed to be concave. Theoretical studies on the NIMBY problems have focusud on strategy-proof voting schemes, in which any agent has no incentive to reveal false information (Manjunath [25]; Barbera, Berga and Moreno [4]). They show that (1) the range of a strategy-proof voting scheme contains at mots two sites (Barbera, Berga and Moreno [4]) and that the range of an e cient and strategy-proof voting schemes contains only the maximal alternative and the mimimal alternative (Manjunath [25]). 1 (in Japanese) 2

3 However, such strategy-proof voting schemes can be criticized for the following two issues. One issue comes from the standpoint of democratic aggregation. If the social planner wants to use a strtategy-proof voting scheme, he has to narrow down the possible alternatives to two without agents preferences. That is, the social planner has to choose two possible alternatives without a democratic procedure. It usually requires an oppressive power to choose two alternatives without information on agents. The other reason comes from the standpoint of protecting minorities. Suppose that k agents live at the maximal point and k 1 agents live at the minimal point 2. If the social planner persists in using an e cient and strategy-proof voting scheme, since the range contains only the maximal and the mimimal alternatives, the outcome must be the worst for either k agents or k 1 agents. If the social planner thinks that either the maximal alternative or the mimimal alternative is not good because of the worst welfare of minorities, he has to give up either strategyproofness or e ciency. The purpose of this paper is to consider what hapens under any given mechanism that may not be strategy-proof. Our study covers all feasible mechanisms containing indirect mechanisms. We focus on coalitional voting behaviors throughout this paper. In voting situations, agents with similar thoughts often form a coalition to in uence the outcome of the vote. For example, in the NIMBY problem, protest demonstrations against the construction of a public bad are usually done by groups. In a congress, most of its members belong to political parties, because an independent member usually has little power to in uence political decisions. When there is an election, some voters with a common interest form an interest group to in uence political decisions. The importance of analyzing coalitional behaviors in voting situations is explained by Sertel and Sanver [36], who studied coalitional manipulations in general voting games and it is implied by the fact that coalitional notions, such as the core, strong Nash implementation, and group strategy-proofness, are often applied to the studies of political situations. In order to analyze coalitional behaviors, we employ the notions of strong Nash 2 This example is pointed out by Renault and Trannoy ([31] [32]) in the context of choosing the location of public good when preferences are single-peaked. 3

4 equilibrium (Aumann [2]) and coalition-proof Nash equilibrium (Bernheim, Peleg and Whinston [8].) We obtain the following results. First, we identify a necessary and su cient condition for a voting mechanism to possess a non-empty set of strong Nash equilibria (Theorem 1). As far as this author knows, there has been no environment for which a necessary and su cient condition for the existence of strong Nash equilibria has been provided, other than the problem of choosing the location of one public good over a street when preferences are single-peaked [45]. Second, we analyze what happens in strong Nash equilibria. We deine a class of core solutions. A core solution assigns to each preference pro le a set of core outcomes of a certain coalitional voting game. We show that if a voting mechanism possesses a non-empty set of strong Nash equilibria, the set of strong Nash outcomes is characterized by a particular core solution (Theorem 2). Since this theorem covers all feasible voting mechanisms, it also provides the characterization of solutions which can be implemented in strong Nash equilibrium. In the proofs of Theorems 1 and 2, we use the well-known minimax theorem (von Neumann and Morgenstern [44]). Thanks to the minimax theorem, for each two-person zerosum game we can judge whether it has a Nash equilibrium and specify what happens in a Nash equilibrium. That is, (1) a two-person zerosum game has a Nash equilibrium if and only if one player s maximin payo is equal to his minimax payo ; and (2) in a Nash equilibrium of a two-person zerosum game, each player gains his maximin (or minimax) payo. Observe that a two-person zerosum game can also be regarded as a NIMBY problem, because in such a game the outcome space can be identi ed with one playes s payo space, which is a one-dimensional set, and two players have single-dipped preferences over his payo space, whose dips are its minimum and maximum, respectively. Such similarity between two-person zerosum games and NIMBY problems enables us to expand the results of the minimax theorem into NIMBY problems. Third, we study coalition-proof Nash equilibria. We show that though the set of coalition-proof Nash equilibria is not always equivalent to the set of strong Nash equilibria (Example 1), these two coincide under a large class of voting mecha- 4

5 nisms (Theorem 3). As long as the equivalence between strong Nash equilibria and coalition-proof Nash equilibria holds, we can also state that by Theorem 2 any coalition-proof Nash outcome result in the outcome a particular core soltion chooses. These theorems above help us analyze strategic manipulations of a direct revelation mechanism of a given social choice rule 3. Since we consider all indirect voting mechanisms containing direct mechanisms, the analysis of direct mechanisms can be obtained as corollaries. As by-products of these theorems, we provide a simple mechanism, called a binary mechanism, which implements any core solution in strong and coalitionproof Nash equilibria. In a binary mechanism, each agent has only two messages. The rest of this paper is organized as follows: in section 2, we introduce the model and de nitions; in section 3, we indicate our main result; in section 4, we propose applications for our theorems. 2 Notation 2.1 Basic De nitions Let N f1; 2; ; ng be the set of agents and A be the set of alternatives. The set A is a non-empty and closed subset of the extended real line R[ f 1; 1g 4. Each voter i2n has a complete, transitive and single-dipped preference R i (whose symmetric and asymmetric parts are denoted by I i and P i respectively) over A : there is an alternative in A, denoted by d(r i ) and called i s dip; such that for each pair a; b 2 A; b < a d(r i ) implies b P i a and d(r i ) a < b implies b P i a. To simplify notation let d i d(r i ): Let R be the set of single-dipped preferences. A mechanism is a pair = (fm i g i2n ; g); where M i denotes voter i s message space, and g : M! A; the outcome function which associates with each message 3 There have been some studies of the consequence of strategic manipulations in several economic models. See, for example, Hurwicz [18], Otani and Sicilian [29] and Kranich [23] for divisible goods economies, Tadenuma and Thomson [41] and Fujinaka and Sakai [15] [16] for economies of one indivisible good and money, Thomson [43] and Bochet, Sakai and Thomson [11] for division problems with single-peaked preferences, Ma [24], Alcade [1], Shin and Suh [37], Sönmez [35] and Takamiya [42] for matching problems and Sertel and Sanver [36] for general voting situations. 4 We allow A to be either nite or a closed interval. 5

6 pro le m (m i ) i2n 2 M Q M i a policy g(m) 2 A. i2n 2.2 Equilibrium Notions Under a mechanism, each voter reports his own messsage and a policy is decided according to the reported messages. We analyze coalitional behaviors under any given voting mechanism, based on strong Nash equilibrium (Aumann [2]) and coalition-proof Nash equilibrium (Bernheim, Peleg and Whinston [8]) These two concepts take coalitional deviations into consideration. First, let us de ne a strong Nash equilibrium. Given a voting mechanism = ((M i ) i2n ; g) and a preference pro le R 2 R N Q R, a coalition S N has a pro table deviation m 0 S 2 M S Q j6=im j at a voting pro le m 2 M, if for each i 2 S; g (m 0 S ; m S) P i g(m) 5. A message pro le m is a strong Nash equilibrium if no coalition has a pro table deviation at m. Given a voting mechanism and a preference pro le R, let SN v ( ; R) fm 2 M j there is no S N that has a pro table deviation at m:g i2n be the set of strong Nash equilibria and SN( ; R) g(sn v ( ; R)) be the set of strong Nash outcomes of for R. Next, we de ne a coalition-proof Nash equilibrium. Given and R, a coalition S N has a credible deviation m 0 S 2 M S at a voting pro le m, if for each i 2 S, g (m 0 S ; m S) P i g(m) and no T S, T 6= S has a credible deviation at (m 0 S ; m S) 6. A message pro le m is a coalition-proof Nash equilibrium if no coalition has a credible deviation at m. Given a voting mechanism and a preference pro le R, 5 There is another de nition of pro table deviation, in which m 0 S is a pro table deviation if for each i 2 S; g (m 0 S ; m S) R i g(m), and for some j 2 S; g (m 0 S ; m S) R j g(m). However, the di erence between these two de nitions does not a ect any our results. 6 Similarly with strong Nash equilibrium, there is another de nition of credible deviation. However, the change of the de nition does not a ect any our results. 6

7 let CN v ( ; R) fm 2 M j there is no S N that has a credible deviation at m:g be the set of coalition-proof Nash equilibria and CN( ; R) g(cn v ( ; R)) be the set of coalition-proof Nash outcomes of for R. It follows from the de nitions that for each and each R 2 R N ; SN( ; R) CN( ; R): 2.3 Core Solution A solution is a mapping F : R N! 2 A n fg that associates with each preference pro le R 2 R N a nonempty subset of alternatives F (R) A: This is considered to be the set of socially desirable alternatives. (Delete) One of the main purpose of implementation theory is to characterize the class of solutions that can be implemented in a particular equilibrium notion. If we can succeed in specifying what happens in strong Nash equilibria under any given mechanism, as a by-product of this result, we can also succeed in characterizing the class of solutions that can be implemented in strong Nash equilibria. (Delete) We will show that only the following solutions can be implemented in strong Nash equilibria. We introduce some notation Let a fa S g SN 2 Q A be a list of policies such that each pair S; T N; S T implies a S a T : Given fa S g SN and R 2 R N ; de ne a dominance relation! R over fa S g SN such that for each pair a S ; a T 2 fa S g SN ; a S! R a T if and only if (1) S ) T and for each i 2 SnT; a S P i a T ; or (2) S T and for each i 2 T ns; a S P i a T : SN Core solution, C a : For each R 2 R N, C a (R) fa S 2 A : there is no T N such that a T! R a S g : 7

8 We need to check whether C a (R) is always nonempty. The following proposition guarantees nonemptiness of C a (R). Proposition 1. For each R 2 R N ; C a (R) 6= : Proof. Let S 0 ; S 1 ; : : : ; S l N be a nite sequence of coalitions such that (1) = S 0 S 1 : : : S l ; (2) a Sl! R a Sl 1! R : : :! R a S0 ; and (3) there is no S S l such that a S! R a Sl : Since N is nite, such a sequence exists. We will show that a Sl 2 C a (R): By the de nition of S l, there is no S N such that S S l and a S! R a Sl. Suppose there is S N such that S S l and a S! R a Sl. Suppose in addition that S\S 1 6= : Since a S1! R a S0 ; for each i 2 S 1 ; a S1 P i a S0 ; so that d(r i ) a S1 : Hence, for each i 2 S 1 ; and each a 2 [a S0 ; a Sl ) ; a Sl P i a; which contradicts for i 2 S 1 \ S l ; a S P i a Sl : Therefore, S \ S 1 = ; so that a S a S1 : We can similarly show by induction that S \ (S 2 ns 1 ) = ; S \ (S 3 ns 2 ) = ;..., and S \ (S l ns l 1 ) =, which contradicts S S l : In order to interpret a core solution, imagine the following scenario. First, social planners narrow down the alternatives to 2 n without information on agents preferences. Secondly, social planners assign each group S one of 2 n alternatitives a S, so that a higher alternative is assigned to a larger group. Thirdly, social planners say to agents "we will choose an alternative from 2 n alternatitives in the following way. If an alternative a S is going to be chosen, only subsets of S or subsets of NnS can make objections to a S. When T S makes an objection to a S ; we will change an alternative from a S to a SnT. When T 0 NnS makes an objection to a S ; we will change an alternative from a S to a S[T 0." A core solution assigns to each preference pro le the core of such a voting situation. 8

9 3 Results 3.1 Strong Nash Equilibrium The purpose of this paper is to analyze the consequence of strategic behaviors by coalitions. We need the following notation. For each S N, de ne S and S 2 A as S min m NnS sup m S g(m S ; m NnS ) and S max inf g(m S ; m NnS ): m NnS If the members of S attempt to maximize the policy and the others attempt minimize it, the result can be expected to be either S or S: Note that by the de nitions S and S ; for each pair S; T N; if S T, then m S S S ; S T and S T : In order to establish our results, we need the following "Minimax Theorem." Theorem (Minimax Theorem 7.) Let f : X Y! R [ f 1; 1g be an extended real-valued mapping. There are (x ; y ) 2 X Y such that for each x 2 X and each y 2 Y; if and only if the two quantities exist and are equal. f(x ; y) f(x ; y ) f(x; y ); maxinff(x; y) and min sup f(x; y) y x x We now provide our results concerning strategic votes. First, we provide a necessary and su cient condition for a mechanism to possess a nonempty set of strong Nash equilibria. We inroduce the following notation. Let be a permutation of N and be the set of permutations of N. y Theorem 1. For each R 2 R N, SN v ( ; R) 6= ; if and only if satis es the following three conditions; 7 For example, Petrosjan and Zenkevich [30] provide a proof of this statement. 9

10 Condition 1: for each S N, S and S exist and are equal; Condition 2: for each 2 ; there is m N 2 arg maxg(m N ); such that for each m N k 2 f1; : : : ; n 1g ; m f(1);:::;(k)g 2 arg max m f(1);:::;(k)g inf g(m f(1);:::;(k)g ; m f(k+1);:::;(n)g ); m f(k+1);:::;(n)g Condition 3: for each 2 ; there is m N k 2 f1; : : : ; n 1g ; 2 arg ming(m N ); such that for each m N m f(1);:::;(k)g 2 arg min m f(1);:::;(k)g sup g(m f(1);:::;(k)g ; m f(k+1);:::;(n)g ): m f(k+1);:::;(n)g Proof. Lemma 1. For each S N; let R 2 R N be such that for each i 2 S; d i inf g(m), and for each i 2 NnS and d i sup g(m). Then, SN v ( ; R) 6= only if S exist and are equal. Proof. When preferences are R, m is a strong Nash equiliblium of each m S 2 M S and each m NnS 2 M NnS ; g(m S ; m NnS) g(m ) g(m S; m NnS ): S and only if for Then, by the minimax theorem, there is a strong Nash equiliblium m 2 SN( ; R), only if S and S exist and are equal. Lemma 2. For each 2, let R 2 R N be such that for each (k) 2 N; d (k) = f(1);:::;(k 1)g and for each a < f(1);:::;(k)g ; a P (k) N: Then, SN v ( ; R ) 6= only if there is m N 2 arg maxg(m N ); such that for each m N 10

11 k 2 f1; : : : ; ng ; m f(1);:::;(k)g 2 arg max m f(1);:::;(k)g inf g(m f(1);:::;(k)g ; m f(k+1);:::;(n)g ): m f(k+1);:::;(n)g Proof. When preferences are R, since d (1) = only if m (1) 2 arg max m f(1)g g(m f(1)g ; m f(2);:::;(n)g): = inf m g(m); m 2 SN v ( ; R ) Hence, g(m ) f(1)g : Since d (2) = f(1)g and d (1) f(1)g ; m 2 SN v ( ; R ) only if m f(1);(2)g 2 arg max g(m f(1);(2)g ; m m f(1);(2)g f(3);:::;(n)g): Hence, g(m ) f(1);(2)g We can inductively show that for each k 2 f1; : : : ; ng ; if m 2 SN v ( ; R ), then m f(1);:::;(k)g 2 arg max g(m f(1);:::;(k)g ; m m f(1);:::;(k)g f(k+1);:::;(n)g): Therefore, m 2 SN v ( ; R ) only if m 2 arg max m N g(m N ): Let m N 2 arg max m N g(m N ): Suppose there is k 2 f1; : : : ; n 1g such that m f(1);:::;(k)g =2 arg max m f(1);:::;(k)g Then, there is there is em f(k+1);:::;(n)g such that g( em f(k+1);:::;(n)g ; m f(1);:::;(k)g ) < max inf g(m f(1);:::;(k)g ; m f(k+1);:::;(n)g ); m f(k+1);:::;(n)g m f(1);:::;(k)g = f(1);:::;(k)g : inf g(m f(1);:::;(k)g ; m f(k+1);:::;(n)g ) m f(k+1);:::;(n)g Since for each i 2 f(k + 1); : : : ; (n)g ; d i d (k+1) = f(1);:::;(k)g ; we have g( em f(k+1);:::;(n)g ; m f(1);:::;(k)g ) P i g(m N ): Hence, em f(k+1);:::;(n)g is a pro table deviation at m N for f(k + 1); : : : ; (n)g. Therefore, if m 2 SN v ( ; R ); then m 2 arg max m N g(m N ) and for each k 2 11

12 f1; : : : ; n 1g ; m f(1);:::;(k)g 2 arg max m f(1);:::;(k)g inf g(m f(1);:::;(k)g ; m f(k+1);:::;(n)g ): m f(k+1);:::;(n)g Lemma 3. For each 2, let R 2 R N be such that for each (k) 2 N; d (k) = Nnf(1);:::;(k 1)g and for each a > Nnf(1);:::;(k 1)g ; a P (k) : Then, SN v ( ; R ) 6= only if there is m N k 2 f1; : : : ; ng ; 2 arg ming(m N ); such that for each m N m f(1);:::;(k)g 2 arg min m f(1);:::;(k)g sup g(m f(1);:::;(k)g ; m f(k+1);:::;(n)g ): m f(k+1);:::;(n)g Proof. We can prove this lemma in a similar way with the proof of Lemma 2. Lemmas 1, 2 and 3 together establishes the proof of the necessity part of Theorem 1. Before we proceed to the proof of the su ciency part, we introduce some notation. Let be a mechanism that satis es all the conditions in Theorem 1. For each S N, de ne S 2 A as S S = S : Let a f S g SN and! R be a dominance relation with respect to the core solution C a. Lemma 4. Let be a mechanism that satis es all the conditions in Theorem 1. Then, for each S 2 C a (R); S 2 SN( ; R): Proof. Let jsj = k: be Let 2 be such that for each i k; (i) 2 S and S R (i) f(1);:::;(i 1)g ; for each i k: 12

13 Let 2 be such that for each i n k; (i) 2 NnS and S R (i) Nnf(1);:::;(i 1)g ; for each i n k Since S 2 C a (R); such permutations exist. Let m N 2 arg maxg(m N ) and m m N 2 N arg ming(m N ) be message pro les de ned as in Conditions 2 and 3, respectively. Let m N m m S; m NnS : Since m S 2 arg max inf g(m S ; m NnS ) and m m S m NnS 2 arg min supg(m S ; m NnS ); by NnS m NnS m ns the minimax therorem, g (m ) = S : Suppose that some T N has a profitable deviation m 0 T 2 M T at m : We assume without loss of generality that g m 0 T ; m NnT < g (m ). Let (i T ) arg min f(i) : (i) 2 T g : Then, since T f(i T ); (i T + 1); : : : ; (n)g ; and m f(1);:::;(i T 1)g 2 arg max inf g(m f(1);:::;(it m f(1);:::;(it 1)g m f(it );:::;(n)g 1)g; m f(it );:::;(n)g); we have g m 0 T ; m NnT max inf g(m f(1);:::;(it m f(1);:::;(it 1)g m f(it );:::;(n)g = f(1);:::;(it 1)g: 1)g; m f(it );:::;(n)g) By the de nition of ; S R (it ) f(1);:::;(i T 1)g: Since f(1);:::;(i T 1)g g m 0 T ; m NnT < S ; by single-dippedness of R (it ); we must have S R (it ) g m 0 T ; m NnT ; 13

14 which is a contradiction. As a next step, we identify the set of strong Nash outcomes. Theorem 2. Let be a mechanism that satis es all the conditions in Theorem 1. Then, for each R 2 R N, SN( ; R) = C a (R): Proof. Theorem 1 already showed that for each R 2 R N, SN( ; R) C a (R): Hence, it is su cient to show that SN( ; R) does not contain any other outcome. Lemma 5. For each R 2 R N, if m 2 SN v ( ; R); then there is S N such that m S 2 arg max inf g(m S ; m NnS ) and m m S m NnS 2 arg min supg(m S ; m NnS ): NnS m NnS m S Proof. For each R 2 R N and each m 2 SN v ( ; R); let S fi 2 Njd(R i ) g(m )g : Suppose that there is m S 2 M S, such that g m S ; m NnS > g(m ): Then since for each i 2 S; g m S ; m NnS > g(m ) d(r i ); we have g m S ; m NnS P i g (m ) ; which contradicts m 2 SN v ( ; R): Hence, for each m S 2 M S ; g m S ; m NnS g(m ): We can similarly show that for each m NnS 2 M NnS ; g m NnS ; ms g(m ): Hence, by the minimax theorem, m S arg min m NnS sup m S g(m S ; m NnS ): 2 arg max inf g(m S ; m NnS ) and m m NnS 2 NnS Lemma 6. For each R 2 R N, if m 2 SN v ( ; R); then g(m ) 2 C a (R). Proof. For each m 2 SN v ( ; R); let S fi 2 Njd(R i ) g(m )g : Then, by Lemma 5, m S 2 arg max inf g(m S ; m NnS ) and m m S m NnS NnS so that g(m ) = S : m S 2 arg min m NnS sup m S g(m S ; m NnS ); Suppose that g(m ) = S =2 C a (R). That is, there is T such that T! R S : We moreover suppose without a loss of generality that T > S : Then, for each i 2 T ns; T P i S : For each i 2 S; since d(r i ) S < T ; we also have T P i S : By the de nition of T ; there is m 0 T 2 M T such that g(m 0 T ; m NnT ) T : Hence, for each i 2 T; g(m 0 T ; m NnT ) R i T P i S ; so that m 0 T 14 is a pro table deviation for T,

15 which contradicts m 2 SN v ( ; R): Therefore, if m 2 SN v ( ; R); then g(m ) 2 C a (R). Theorems 1 and 2 together characterize the strong Nash outcomes. These theorems say that strategic votes with communication result in the outcome recommended by the core solution whose list of parameters is f S g SN. In the context of implementation theory, Theorems 1 and 2 characterize the class of solutions that can be implemented in strong Nash equilibrium. These theorems tells us that a solution can be implemented in strong Nash equilibrium if and only if it is a core solution. The way to establish this result is quite di erent from the "standard" approach to implementation theory, such as Maskin [26] [27], Dutta and Sen [14], and Suh [38], in which a particular mechanism is constructed to show the implemenbility of a solution. Unlike other environments, in one-dimensional voting situations, we can analyze the performances of all mechanisms by applying the minimax theorem. In the reminder of this section, we illustrate how to use Theorems 1 and 2. The following proposition exhibits a su cient condition for the existence of strong Nash equilibrium. The following condition requests that a voter wants to minimize (or maximize) a policy, he is able to do so by reporting a particular message. This condition enables a voter to minimize (or maximize) a policy without anticipating which message other voters reveal. Optimizability: For each i 2 N, there are m i ; m i 2 M i such that for each m i 2 M i and each m i 2 M i, g(m i ; m i ) g(m i ; m i ) g(m i ; m i ) Proposition 2. Let be a mechanism that satis es optimizability. For each S N; let a S g(m S ; m NnS ) and a a S SN : For each R 2 RN, SN( ; R) = C a (R) 6= : Proof. By Theorems 1 and 2, it is su cient to show that an optimizable voting mechanism satis es all the conditions in Theorem 1. If satis es optimizability, then for 15

16 each S N, g(m S ; m NnS ) = maxinfg(m S ; m NnS ) = min sup g(m S ; m NnS ): m NnS m S m S m NnS Hence, Condition (1) of Theorem 1 is satis ed. and For each and each k 2 f1; : : : ; n m f(1);:::;(k)g; 2 arg m f(1);:::;(k)g 2 arg max m f(1);:::;(k)g min m f(1);:::;(k)g 1g ; since Conditions (2) and (3) of Theorem 1 are also satis ed. inf g(m f(1);:::;(k)g ; m f(k+1);:::;(n)g ) m f(k+1);:::;(n)g sup g(m f(1);:::;(k)g ; m f(k+1);:::;(n)g ); m f(k+1);:::;(n)g Since for each S N, a S = maxinfg(m S ; m NnS ) = min sup g(m S ; m NnS ); by m NnS m S m S m NnS Theorem 2, for each R 2 R N, SN( ; R) = C a (R). 3.2 Coalition-proof Nash Equilibrium Next, we study coalition-proof Nash equilibria. Though the sets of coalition-proof Nash outcomes do not coincide in genral with the sets of strong Nash outcomes, we can show that the coincidence holds for a large class of voting mechanisms. Here we give an example showing that CN( ; R) may be di erent from SN( ; R). Example 1. Let N = f1; 2g, A = f1; 2; 3g. Let = (fm 1 ; M 2 g ; g) be such that M 1 = M 2 = fx; y; zg and g is as in Table 1. Table 1. 1n2 x y z x y z

17 Let R = (R 1 ; R 2 ) be such that 3 P 1 2 P 1 1 and 1 P 2 3 P 2 2. Since SN v ( ; R) = 6= fx; xg = CN v ( ; R); SN( ; R) = 6= f2g = CN( ; R): On the other hand, as long as satis es optimizability, we have the equivalence between coalition-proof Nash equilibrium and strong Nash equilibrium. Theorem 3. Let R 2 R N, be a mechanism that satis es optimizability. Then, for each SN v ( ; R) = CN v ( ; R) 6=. Hence, for each R 2 R N ; SN( ; R) = CN( ; R): Proof. It su ces to show that if some S N has a pro table deviation at m 2 M; some T N also has a credible deviation at m 2 M. Suppose there is some S N that has a deviation m 0 S 2 M S at m 2 M. We assume without loss of generality that g(m 0 S ; m S) < g(m). Then, for each i 2 S; d i g(m 0 S ; m NnS) g(m S ; m NnT ): If no T S has a pro table deviation at (m S ; m NnS ) 2 M; then m S is a credible deviation for S: If some T S has a pro table deviation m 00 T 2 M T at (m S ; m NnT ), then for each i 2 T g(m S ; m NnS ) d i g(m 00 T ; m SnT ; m NnS ) g(m T ; m NnT ): Since for each i 2 T; g(m T ; m NnT ) R i g(m 00 T ; m SnT ; m NnS ) R i g(m S ; m NnS ); for each i 2 T and each a 2 g(m S ; m NnS ); g(m T ; m NnT ) A ; g(m T ; m NnT ) R i a: 17

18 Since for each U T and each m 000 U 2 M U, g(m 000 U ; m T nu ; m NnT ) 2 g(m U ; m T nu ; m NnT ); g(m T ; m NnT ) A g(m S ; m NnS ); g(m T ; m NnT ) A ; no U T has a deviation at (m T ; m NnT ). Hence, m T is a credible deviation for T: Note that by Theorem 3, we have found a new su cient condition for the equivalence between strong Nash and coalition-proof Nash equilibria. Konishi, Le Breton, and Weber [22] showed su cient conditions for the equivalence between these two solutions, but their results cannot apply to our case. 4 Discussions 4.1 Simple Mechanism for Strong Nash Implementation Theorems 1 and 2 show that a solution that can be implemented in strong Nash equilibrium is a core solution, because these theorems consider all possible mechanisms. From the standpoint of mechanism design, it is better to construct a mechanism as simple as possible 8. In fact, in the single-dipped environment, any core solution can be implemented in strong Nash equilibria by a mechanism in which each agent has only two messages. Proposition 3. Any core solution can be implemented in strong Nash equilibrium by a mechanism in which for each i 2 N; jm i j = 2. Proof. For each core solution C a, let a be a mechanism such that for each i 2 N, M i = f0; 1g, and for each m 2 M, g(m) = a fi2n:mi =1g. Since for each i 2 N, 8 The study of natural implementation starts from this motivation. 18

19 and each m i 2 M i ; g(0; m i ) = a fj2nnfig:mj =1g a fj2nnfig:mj =1g[fig = g(1; m i ); a is optimizable. Hence for each R 2 R N ; SN( a ; R) 6= : Since for each S N; a S = a fi2n:mi =1g = a S ; by Theorems 1 and 2 SN( a ; R) = C a (R) = C a (R): 4.2 Single-peakedness and Single-dippedness Though the single-dipped environment looks similar to the well-known singlepeaked environment, these two environment have studied separately, because different results have been obtained by several studies. For example, the studies of strategy-proof rules in single-peaked and single-dipped envirionments have characterized augmented median rules and binary rules, respectively; the studies of the core of the majority aggregation have obtained the median voter rule and the majority rule between both sides, respectively. On the other hand, for the existence coalitional stability, we can apply the similar approach to these two environments. Consider the situation where the dips of S and those of NnS are min A and max A; respectively. In this situation, the preferences of S and N ns are also single-peaked, whose peaks are max A; min A; respectively. Since the interests of S perfectly con icts with those of NnS; if communication between agents are permitted, we can regard this situation as a twoperson zerosum game between those who want to increase the outcome and those who want to decrease it. This situation connects the single-peaked environment with the single-dipped environment and makes it possible to apply the minimax theorem to these two environments. 19

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