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A new necessary condition for Nash implementation
Journal of Mathematical Economics 29 Ž1998. 381–387
A new necessary condition for Nash implementation Abderrahmane Ziad Centre de Recherches en Economie Mathematique et Econometrie, UniÕersite´ de Caen, 14032 Caen, ´ ´ Cedex, France Received 1 February 1997
Abstract In this paper we formulate a family of conditions called ‘Bk-monotonicity’ that are necessary for Nash implementation, where k is a natural number that indexes a particular condition, and where the condition only becomes more restrictive as k increases. Bk-monotonicity is in general a stricter condition than Maskin monotonicity, and can be used to show that certain social choice correspondences that satisfy Maskin monotonicity cannot be Nash implemented. q 1998 Elsevier Science S.A. JEL classification: D78
1. Introduction In his seminal paper, Maskin Ž1977. proved that any Nash implementable social choice correspondence ŽSCC. must satisfy a certain monotonicity condition. Roughly speaking, this means that if the choice rule selects a certain outcome in one preference profile then the rule must continue to select that outcome in any other profile under which the outcome has moved up in everyone’s rankings. Furthermore, if there are at least three agents, then any SCC satisfying monotonicity and an additional condition termed ‘no veto power’ is Nash implementable. Maskin’s result has been refined by Williams Ž1984., Maskin Ž1985., Repullo Ž1987., Saigo Ž1988., McKelvey Ž1989., Moore and Repullo Ž1990., Danilov Ž1990. and Yamato Ž1992.. 0304-4068r98r$19.00 q 1998 Elsevier Science S.A. All rights reserved. PII S 0 3 0 4 - 4 0 6 8 Ž 9 7 . 0 0 0 1 4 - 1
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A. Ziad r Journal of Mathematical Economics 29 (1998) 381–387
These refinements aim to come up with a condition that together with Maskin monotonicity is sufficient for Nash implementation, but with both of which being distinct from other such conditions in the literature. However, there are few methods for proving that a social choice correspondence that is Maskin monotonic cannot be Nash implemented. In this paper, we establish a new condition termed Bk-monotonicity, where k is any positive integer. We prove that Bk-monotonicity implies both Bky1-monotonicity and Maskin monotonicity. However, Bk-monotonicity is necessary for Nash implementation for each k. The result obtained can be used to detect the possible non-implementation of a social choice correspondence that is Maskin monotonic. The plan of this paper is as follows. In Section 2, we introduce some notation and definitions. In Section 3, we formulate our main result and prove it. We compare our result with those in the existing literature, in Section 4.
2. Notation and definitions Let A denote the set of alternatives Žsocial states. and I s 1, 2, . . . , n4 be the set of agents, with generic element i. Each agent i is characterized by a preference relation R i defined over A, which is complete and reflexive in some class R i of admissible preference relations. Let R s Ł i g I R i . An element R s Ž R 1 , R 2 , . . . , R n . of R is a preference profile. A social choice correspondence ŽSCC. is a correspondence F from R into A, that associates with every R a non-empty subset of A. For any R i g R i , and a g A, let us denote the lower contour set for agent i with preference R i at a by LŽ a, R i . s x x g A:aR i x 4 . A mechanism Žor game form. is a pair G s Ž S, g . where S s Ł i g I Si , Si denotes agent i’s strategy set and g is a function from S into A. Elements of S are written s s Ž s1 , s2 , . . . , sn . s Ž si , syi ., where sy1 s Ž s1 , . . . , siy1 , siq1 , . . . , sn .. Given s g S and x i g Si , Ž x i , syi . s Ž s1 , . . . , siy1 , x i , siq1 , . . . , sn . is obtained after replacing si by x i , and g Ž Si , si . is the set of outcomes that agent i can induce when the other agents select syi from Syi s Ł j g I, j/ i S j . A Nash equilibrium of the game Ž g, R . is a vector of strategies s g S such that for all i and for all x i g Si , g Ž s . R i g Ž x i , syi ., i.e. for all i, LŽ g Ž r ., R i . = g Ž Si , syi .. Let N Ž g, R, S . denote the set of Nash equilibria of the game Ž g, R .. A mechanism G s Ž S, g . implements an SCC F in Nash equilibria if for all R g R, F Ž R . s g Ž N Ž g, R, S ... We say that an SCC F is implementable in Nash equilibria if there exists a mechanism implementing F in Nash equilibria. The notation x g A y B means x g A and x f B. To characterize the SCC that can be Nash implemented, Maskin Ž1977. defined the following conditions.
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An SCC F is monotonic if for all R, R ) g R, for all a g F Ž R ., if for all i g I, LŽ a, R )i . = LŽ a, R i ., then a g F Ž R ) .. An SCC F satisfies no veto power if for i, R g R, and a g A, if LŽ a, R j . s A for all j / i, then a g F Ž R .. No veto power means that if an outcome is at the top of all but one of the agents’ rankings, then it must be chosen by the choice rule. An SCC F satisfies unanimity if for all a g A, R g R, LŽ a, R i . s A for all i g I, then a g F Ž R .. A detailed discussion of monotonicity and the no veto power condition can be found in Maskin Ž1977, 1985..
3. The main result In this section we formulate our main result and prove it. We define the Bk-monotonicity which is a stronger condition than Maskin monotonicity but is necessary for implementation. Let R g R be any preference profile. Then F Ž R . / B. Fix a g F Ž R . and define the sets Bk Ž a, R . by: B0 Ž a, R . s B B1Ž a, R . s j RX g R b g A y a4 : LŽ b, RXi . s A for all i g I and b f F Ž RX .4 . Since Bk Ž a, R . has been defined, we define Bkq1Ž a, R . by Bkq1Ž a, R . s j RX g R b g A y a4 : LŽ b, RXi . = A y Bk Ž a, R . for all i g I and b f F Ž RX .4 . In the following we get some properties of the sequence Ž Bk Ž a, R .. k . Proposition 1. The sequence Ž Bk Ž a, R .. k satisfies Bkq 1 Ž a, R . = Bk Ž a, R . for all k Proof . By induction on k. Let b g B1Ž a, R .. Then LŽ b, RXi . s A for all i g I and b f F Ž RX . for some X R g R. But A = A y B1Ž a, R .. Then LŽ b, RXi . = A y B1Ž a, R . for all i g I, hence, we get b g B2 Ž a, R .. The rest is obvious. I Proposition 2. Ži. A y a4 = Bk Ž a, R . for all k. Žii. If for some k 0 G 1, Bk Ž a, R . s B, then Bk Ž a, R . s B for all k. 0 Proof . Use the definition of Bk Ž a, R ., and Proposition 1. If Bk 0Ž a, R . s B for some k 0 ) 1, then by Proposition 1, we get B1Ž a, R . s B. If B1Ž a, R . s B, then by definition we get Bk Ž a, R . s B1Ž a, R . s B for all k. I Definition 1. An SCC F satisfies Bk-monotonicity if for all R, R ) g R, for all a g F Ž R . and LŽ a, R )i . = LŽ a, R i . y Bk Ž a, R . for all i g I, then a g F Ž R ) ..
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Remark 1. B0-monotonicity is simply Maskin monotonicity. Proposition 3. If an SCC F satisfies Bk-monotonicity, then F satisfies Bky1-monotonicity. Proof . Using the fact Bk Ž a, R . = Bky1Ž a, R . for all k, we get LŽ a, R i . y Bky1Ž a, R . = LŽ a, R i . y Bk Ž a, R .. The rest is obvious. I Remark 2. The Bk-monotonicity is stronger than Bp-monotonicity for all p F k, especially when p s 0, i.e. Maskin monotonicity. But this strengthening of the monotonicity condition need not worry us since it is still a necessary condition for implementation as it will be shown in the following results. Proposition 4. If an SCC F is implementable in Nash equilibria by a game form G s Ž S, g . then for each s g S, g Ž s . f Bk Ž a, R . for each k. Proof . By induction on k, k s 1, suppose that there exists s g S such that g Ž s . g B1Ž a, R .. By definition there exists a preference profile R ) such that L Ž g Ž s . , R )i . s A for all i g I
Ž 1.
and g Ž s . f F Ž R ) .. From Eq. Ž1. we get s g N Ž g, R ) , S ., and by implementation g Ž s . g F Ž R ) . which is absurd. Suppose that the proposition is true for each p F k we prove it for Ž k q 1.. Suppose the contrary. Then there exists s g S such that g Ž s . g Bkq 1Ž a, R .. Then there exists a preference profile R ) such that LŽ g Ž S ., R )i . = A y Bk Ž a, . R for each i g I, and that g Ž s . f F Ž R ) .. But for each i g I, x i g Si g Ž x i , syi . f Bk Ž a, R ., then g Ž x i , syi . g LŽ g Ž s ., ). R i , i.e. s g N Ž g, R ) , S ., and by implementation g Ž s . g F Ž R ) ., which is a contradiction. I Theorem 1. If an SCC F is implementable, then F is a Bk-monotone for all k. Proof . Since F is implementable, there exists a game form G s Ž S, g ., with F Ž R . s g Ž N Ž g, R, S .. for all R g R. Let a g F Ž R ., then there exists s g N Ž g, R, S . such that a s g Ž s ., i.e. for all i g I, for all x i g Si , we have g Ž s . R i g Ž x i , syi .. From Proposition 4 we know that for each s g S, g Ž s . f Bk Ž a, R . for each k. From LŽ a, R )i . = LŽ a, R i . y Bk Ž a, R . for all i g I, we get g Ž x i , syi . g LŽ a, R i) ., i.e. s g N Ž g, R ) , S . and by implementation a g F Ž R ) .. I
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Remark 3. In the literature, most papers haÕe concentrated on some conditions that together with Maskin monotonicity are sufficient for Nash implementation. These conditions are distinct, and are also applicable to examples of social choice correspondences. Of course ‘the best’ sufficient condition is a condition which is also necessary. A number of authors have since tried to find the precise necessary and sufficient conditions on F for Nash implementation Žsee Moore and Repullo, 1990, Sjostrom, ¨ ¨ 1991.. It is very difficult to apply their conditions to determine whether or not a given social correspondence can be implemented, because their conditions were stated in terms of unknown sets. We have proposed a new condition which is stronger than Maskin monotonicity. However, our condition is necessary for implementation. It is also interesting that the necessity of the Bk-monotonicity allows us to prove that some social choice correspondences are not Nash implementable even though they are Maskin monotonic Žsee Example 1..
4. Example and discussion Let us review some results in the literature. McKelvey Ž1989. proposed the following conditions. For R g R, and a s A, we say that R is F-minimal for a if a g F Ž R ., and for any R ) g R such that LŽ a, R )i . = LŽ a, R i . for all i g I and LŽ a, R )i . / LŽ a, R i . for some i, a f F Ž R ) .. We say that an SCC F is F-closedness if for all R g R, and a g F Ž R ., there exists R ) g R such that Ži. R ) is F-minimal for a, and Žii. LŽ a, R i . = LŽ a, R )i . for all i g I. An SCC F satisfies weak no veto power if for all a g A, R g R, and i g I, if LŽ a, R j . s A for all j / i, and there exist b g A and R ) g R such that Ži. R ) g R is F-minimal for b, and Žii. LŽ a, R i . = LŽ b, R i) ., a g LŽ b, R )i ., then a g F Ž R .. Clearly, no veto power implies both weak no veto power and unanimity. McKelvey proved that if n G 3, and an SCC F satisfies weak no veto power, unanimity, monotonicity and F-closedness, then F is implementable in Nash equilibria. Danilov Ž1990. and Yamato Ž1992. proposed the following conditions. Let i g I, B a subset of A, Essi Ž B, F . s b g B: b g F Ž R . and B = LŽ b, R i .4 . An SCC F is strongly monotonic if for all R, R ) g R, for all a g F Ž R ., if for all i g I, LŽ a, R )i . = Essi Ž LŽ a, R i ., F ., then a g F Ž R ) .. They proved that if n G 3, and an SCC F satisfies strong monotonicity, then F is implementable in Nash equilibria. In the following example we construct an SCC F which fails to satisfy no veto power, unanimity, strong monotonicity and weak no veto power. However, F is Maskin monotonic. Then none of the existing results in the literature can show whether F can or cannot be implemented in Nash equilibria.
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Example 1. Let A s a, b, c4 , n s 3, I s 1, 2, 34 , R s R, R ) 4 defined by R1 a b c
R2 a c b
R3 a b c
R 1,) b a c
R 2 ,) b a c
R 3,) b a c
and F Ž R . s a, c4 , F Ž R ) . s c4 . Then LŽ a, R i . s A, LŽ a, R i,) . s a, c4 , LŽ b, R i,) . s A, LŽ c, R 1 . s LŽ c, R 3 . s LŽ c, R i,) . s c4 for all i g I, LŽ c, R 2 . s LŽ b, R 1 . s LŽ b, R 3 . s b, c4 and LŽ b, R 2 . s b4 . And Essi Ž LŽ a, R i ., F . s a, c4 for all i g I. Observe that LŽ b, R i,) . s A for all i g I, b f F Ž R ) ., then F fails to satisfy unanimity and no veto power. In the other hand a g F Ž R . and LŽ a, R i,) . = Essi Ž LŽ a, R i ., F . for all i g I, a f F Ž R ) ., then F fails to satisfy strong monotonicity. The preference profile R is F-minimal for a, LŽ b, R i,) . s A and LŽ b, i,) . R = LŽ a, R i ., b g LŽ a, R i . for all i g I, but b f F Ž R ) .. Then F fails to satisfy weak no veto power. It is obvious that F is Maskinmonotonic. By simple calculation we get B1Ž a, R . s b4 , LŽ a, R i . y B1Ž a, R . s a, c4 , LŽ a, R )i . s a, c4 for all i. Then LŽ a, R )i . = LŽ a, R i . y B1Ž a, R . for all i, but a g F Ž R . and a f F Ž R ) .. F is not B1-monotone, thus F is not implementable in Nash equilibria. Remark 4. McKelÕey has proÕed that weak no Õeto power is necessary for Nash implementation if R is ‘complete’: R is ‘complete’ if for all a g A, i g I, and for all subsets B of A with a g B, there exists R ) g R such that L(a, R )i ) s B and L(a, R j ) s L(a, R )j ) for all j / i. Consider a g A, i s 1, R g R in Example 1 and B s LŽ a, R )i . s a, c4 . However, LŽ a, R j . / LŽ a, R )j . for all j / i. Then R is not ‘complete’. Yamato also proved that strong monotonicity is necessary for Nash implementation if R satisfies condition D, i.e. if for all a g A, R g R, i g I, and b g LŽ a, R i ., there exists R ) g R such that LŽ a, R i . s LŽ b, R )i . and for all j / i LŽ b, R )j . s A. Take a g A, i s 1, and c g LŽ a, R 1 . in Example 1. For all R ) g R we have LŽ a, R 1 . / LŽ c, R 1) ., R fails to satisfy condition D.
Acknowledgements We are indebted to anonymous referees for comments. Their remarks contributed to a better presentation of the results. Thanks also to Maurice Salles.
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References Danilov, V., 1990. Implementation via Nash equilibrium. Central Economic-Mathematical Institute, Moscow Žmimeo... Maskin, E., 1977. Nash equilibrium and welfare optimality. M.I.T. Žmimeo... Maskin, E., 1985. The theory of implementation in Nash equilibrium: a survey. In: Hurwicz, L., Schmeidler, D., Sonnenschein, H. ŽEds.., Social Goals and Social Organisation. Cambridge University Press, Cambridge, pp. 173–204. McKelvey, R.D., 1989. Game forms for Nash implementation of general social choice correspondence. Social Choice and Welfare 6, 139–156. Moore, J., Repullo, R., 1990. Nash implementation: a full characterization. Econometrica 58 Ž3., 1083–1099. Repullo, R., 1987. A simple proof of Maskin’s theorem on Nash implementation. Social Choice and Welfare 4, 39–41. Saigo, T., 1988. Strategy space reduction in Maskin’s theorem: sufficient conditions for Nash implementation. Econometrica 56 Ž3., 693–700. Sjostrom, ¨ ¨ T., 1991. On the necessary and sufficient conditions for Nash implementation. Social Choice and Welfare 8, 333–340. Williams, S., 1984. Realization and Nash implementation of mechanism design. Institute for Mathematic and its Applications Preprint Series a69, University of Minnesota Žrevised version in Econometrica 54, 139–151.. Yamato, T., 1992. On Nash implementation of social choice correspondences. Games and Economic Behavior 4, 484–492.
A new necessary condition for Nash implementation Abderrahmane Ziad Centre de Recherches en Economie Mathematique et Econometrie, UniÕersite´ de Caen, 14032 Caen, ´ ´ Cedex, France Received 1 February 1997
Abstract In this paper we formulate a family of conditions called ‘Bk-monotonicity’ that are necessary for Nash implementation, where k is a natural number that indexes a particular condition, and where the condition only becomes more restrictive as k increases. Bk-monotonicity is in general a stricter condition than Maskin monotonicity, and can be used to show that certain social choice correspondences that satisfy Maskin monotonicity cannot be Nash implemented. q 1998 Elsevier Science S.A. JEL classification: D78
1. Introduction In his seminal paper, Maskin Ž1977. proved that any Nash implementable social choice correspondence ŽSCC. must satisfy a certain monotonicity condition. Roughly speaking, this means that if the choice rule selects a certain outcome in one preference profile then the rule must continue to select that outcome in any other profile under which the outcome has moved up in everyone’s rankings. Furthermore, if there are at least three agents, then any SCC satisfying monotonicity and an additional condition termed ‘no veto power’ is Nash implementable. Maskin’s result has been refined by Williams Ž1984., Maskin Ž1985., Repullo Ž1987., Saigo Ž1988., McKelvey Ž1989., Moore and Repullo Ž1990., Danilov Ž1990. and Yamato Ž1992.. 0304-4068r98r$19.00 q 1998 Elsevier Science S.A. All rights reserved. PII S 0 3 0 4 - 4 0 6 8 Ž 9 7 . 0 0 0 1 4 - 1
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A. Ziad r Journal of Mathematical Economics 29 (1998) 381–387
These refinements aim to come up with a condition that together with Maskin monotonicity is sufficient for Nash implementation, but with both of which being distinct from other such conditions in the literature. However, there are few methods for proving that a social choice correspondence that is Maskin monotonic cannot be Nash implemented. In this paper, we establish a new condition termed Bk-monotonicity, where k is any positive integer. We prove that Bk-monotonicity implies both Bky1-monotonicity and Maskin monotonicity. However, Bk-monotonicity is necessary for Nash implementation for each k. The result obtained can be used to detect the possible non-implementation of a social choice correspondence that is Maskin monotonic. The plan of this paper is as follows. In Section 2, we introduce some notation and definitions. In Section 3, we formulate our main result and prove it. We compare our result with those in the existing literature, in Section 4.
2. Notation and definitions Let A denote the set of alternatives Žsocial states. and I s 1, 2, . . . , n4 be the set of agents, with generic element i. Each agent i is characterized by a preference relation R i defined over A, which is complete and reflexive in some class R i of admissible preference relations. Let R s Ł i g I R i . An element R s Ž R 1 , R 2 , . . . , R n . of R is a preference profile. A social choice correspondence ŽSCC. is a correspondence F from R into A, that associates with every R a non-empty subset of A. For any R i g R i , and a g A, let us denote the lower contour set for agent i with preference R i at a by LŽ a, R i . s x x g A:aR i x 4 . A mechanism Žor game form. is a pair G s Ž S, g . where S s Ł i g I Si , Si denotes agent i’s strategy set and g is a function from S into A. Elements of S are written s s Ž s1 , s2 , . . . , sn . s Ž si , syi ., where sy1 s Ž s1 , . . . , siy1 , siq1 , . . . , sn .. Given s g S and x i g Si , Ž x i , syi . s Ž s1 , . . . , siy1 , x i , siq1 , . . . , sn . is obtained after replacing si by x i , and g Ž Si , si . is the set of outcomes that agent i can induce when the other agents select syi from Syi s Ł j g I, j/ i S j . A Nash equilibrium of the game Ž g, R . is a vector of strategies s g S such that for all i and for all x i g Si , g Ž s . R i g Ž x i , syi ., i.e. for all i, LŽ g Ž r ., R i . = g Ž Si , syi .. Let N Ž g, R, S . denote the set of Nash equilibria of the game Ž g, R .. A mechanism G s Ž S, g . implements an SCC F in Nash equilibria if for all R g R, F Ž R . s g Ž N Ž g, R, S ... We say that an SCC F is implementable in Nash equilibria if there exists a mechanism implementing F in Nash equilibria. The notation x g A y B means x g A and x f B. To characterize the SCC that can be Nash implemented, Maskin Ž1977. defined the following conditions.
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An SCC F is monotonic if for all R, R ) g R, for all a g F Ž R ., if for all i g I, LŽ a, R )i . = LŽ a, R i ., then a g F Ž R ) .. An SCC F satisfies no veto power if for i, R g R, and a g A, if LŽ a, R j . s A for all j / i, then a g F Ž R .. No veto power means that if an outcome is at the top of all but one of the agents’ rankings, then it must be chosen by the choice rule. An SCC F satisfies unanimity if for all a g A, R g R, LŽ a, R i . s A for all i g I, then a g F Ž R .. A detailed discussion of monotonicity and the no veto power condition can be found in Maskin Ž1977, 1985..
3. The main result In this section we formulate our main result and prove it. We define the Bk-monotonicity which is a stronger condition than Maskin monotonicity but is necessary for implementation. Let R g R be any preference profile. Then F Ž R . / B. Fix a g F Ž R . and define the sets Bk Ž a, R . by: B0 Ž a, R . s B B1Ž a, R . s j RX g R b g A y a4 : LŽ b, RXi . s A for all i g I and b f F Ž RX .4 . Since Bk Ž a, R . has been defined, we define Bkq1Ž a, R . by Bkq1Ž a, R . s j RX g R b g A y a4 : LŽ b, RXi . = A y Bk Ž a, R . for all i g I and b f F Ž RX .4 . In the following we get some properties of the sequence Ž Bk Ž a, R .. k . Proposition 1. The sequence Ž Bk Ž a, R .. k satisfies Bkq 1 Ž a, R . = Bk Ž a, R . for all k Proof . By induction on k. Let b g B1Ž a, R .. Then LŽ b, RXi . s A for all i g I and b f F Ž RX . for some X R g R. But A = A y B1Ž a, R .. Then LŽ b, RXi . = A y B1Ž a, R . for all i g I, hence, we get b g B2 Ž a, R .. The rest is obvious. I Proposition 2. Ži. A y a4 = Bk Ž a, R . for all k. Žii. If for some k 0 G 1, Bk Ž a, R . s B, then Bk Ž a, R . s B for all k. 0 Proof . Use the definition of Bk Ž a, R ., and Proposition 1. If Bk 0Ž a, R . s B for some k 0 ) 1, then by Proposition 1, we get B1Ž a, R . s B. If B1Ž a, R . s B, then by definition we get Bk Ž a, R . s B1Ž a, R . s B for all k. I Definition 1. An SCC F satisfies Bk-monotonicity if for all R, R ) g R, for all a g F Ž R . and LŽ a, R )i . = LŽ a, R i . y Bk Ž a, R . for all i g I, then a g F Ž R ) ..
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Remark 1. B0-monotonicity is simply Maskin monotonicity. Proposition 3. If an SCC F satisfies Bk-monotonicity, then F satisfies Bky1-monotonicity. Proof . Using the fact Bk Ž a, R . = Bky1Ž a, R . for all k, we get LŽ a, R i . y Bky1Ž a, R . = LŽ a, R i . y Bk Ž a, R .. The rest is obvious. I Remark 2. The Bk-monotonicity is stronger than Bp-monotonicity for all p F k, especially when p s 0, i.e. Maskin monotonicity. But this strengthening of the monotonicity condition need not worry us since it is still a necessary condition for implementation as it will be shown in the following results. Proposition 4. If an SCC F is implementable in Nash equilibria by a game form G s Ž S, g . then for each s g S, g Ž s . f Bk Ž a, R . for each k. Proof . By induction on k, k s 1, suppose that there exists s g S such that g Ž s . g B1Ž a, R .. By definition there exists a preference profile R ) such that L Ž g Ž s . , R )i . s A for all i g I
Ž 1.
and g Ž s . f F Ž R ) .. From Eq. Ž1. we get s g N Ž g, R ) , S ., and by implementation g Ž s . g F Ž R ) . which is absurd. Suppose that the proposition is true for each p F k we prove it for Ž k q 1.. Suppose the contrary. Then there exists s g S such that g Ž s . g Bkq 1Ž a, R .. Then there exists a preference profile R ) such that LŽ g Ž S ., R )i . = A y Bk Ž a, . R for each i g I, and that g Ž s . f F Ž R ) .. But for each i g I, x i g Si g Ž x i , syi . f Bk Ž a, R ., then g Ž x i , syi . g LŽ g Ž s ., ). R i , i.e. s g N Ž g, R ) , S ., and by implementation g Ž s . g F Ž R ) ., which is a contradiction. I Theorem 1. If an SCC F is implementable, then F is a Bk-monotone for all k. Proof . Since F is implementable, there exists a game form G s Ž S, g ., with F Ž R . s g Ž N Ž g, R, S .. for all R g R. Let a g F Ž R ., then there exists s g N Ž g, R, S . such that a s g Ž s ., i.e. for all i g I, for all x i g Si , we have g Ž s . R i g Ž x i , syi .. From Proposition 4 we know that for each s g S, g Ž s . f Bk Ž a, R . for each k. From LŽ a, R )i . = LŽ a, R i . y Bk Ž a, R . for all i g I, we get g Ž x i , syi . g LŽ a, R i) ., i.e. s g N Ž g, R ) , S . and by implementation a g F Ž R ) .. I
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Remark 3. In the literature, most papers haÕe concentrated on some conditions that together with Maskin monotonicity are sufficient for Nash implementation. These conditions are distinct, and are also applicable to examples of social choice correspondences. Of course ‘the best’ sufficient condition is a condition which is also necessary. A number of authors have since tried to find the precise necessary and sufficient conditions on F for Nash implementation Žsee Moore and Repullo, 1990, Sjostrom, ¨ ¨ 1991.. It is very difficult to apply their conditions to determine whether or not a given social correspondence can be implemented, because their conditions were stated in terms of unknown sets. We have proposed a new condition which is stronger than Maskin monotonicity. However, our condition is necessary for implementation. It is also interesting that the necessity of the Bk-monotonicity allows us to prove that some social choice correspondences are not Nash implementable even though they are Maskin monotonic Žsee Example 1..
4. Example and discussion Let us review some results in the literature. McKelvey Ž1989. proposed the following conditions. For R g R, and a s A, we say that R is F-minimal for a if a g F Ž R ., and for any R ) g R such that LŽ a, R )i . = LŽ a, R i . for all i g I and LŽ a, R )i . / LŽ a, R i . for some i, a f F Ž R ) .. We say that an SCC F is F-closedness if for all R g R, and a g F Ž R ., there exists R ) g R such that Ži. R ) is F-minimal for a, and Žii. LŽ a, R i . = LŽ a, R )i . for all i g I. An SCC F satisfies weak no veto power if for all a g A, R g R, and i g I, if LŽ a, R j . s A for all j / i, and there exist b g A and R ) g R such that Ži. R ) g R is F-minimal for b, and Žii. LŽ a, R i . = LŽ b, R i) ., a g LŽ b, R )i ., then a g F Ž R .. Clearly, no veto power implies both weak no veto power and unanimity. McKelvey proved that if n G 3, and an SCC F satisfies weak no veto power, unanimity, monotonicity and F-closedness, then F is implementable in Nash equilibria. Danilov Ž1990. and Yamato Ž1992. proposed the following conditions. Let i g I, B a subset of A, Essi Ž B, F . s b g B: b g F Ž R . and B = LŽ b, R i .4 . An SCC F is strongly monotonic if for all R, R ) g R, for all a g F Ž R ., if for all i g I, LŽ a, R )i . = Essi Ž LŽ a, R i ., F ., then a g F Ž R ) .. They proved that if n G 3, and an SCC F satisfies strong monotonicity, then F is implementable in Nash equilibria. In the following example we construct an SCC F which fails to satisfy no veto power, unanimity, strong monotonicity and weak no veto power. However, F is Maskin monotonic. Then none of the existing results in the literature can show whether F can or cannot be implemented in Nash equilibria.
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Example 1. Let A s a, b, c4 , n s 3, I s 1, 2, 34 , R s R, R ) 4 defined by R1 a b c
R2 a c b
R3 a b c
R 1,) b a c
R 2 ,) b a c
R 3,) b a c
and F Ž R . s a, c4 , F Ž R ) . s c4 . Then LŽ a, R i . s A, LŽ a, R i,) . s a, c4 , LŽ b, R i,) . s A, LŽ c, R 1 . s LŽ c, R 3 . s LŽ c, R i,) . s c4 for all i g I, LŽ c, R 2 . s LŽ b, R 1 . s LŽ b, R 3 . s b, c4 and LŽ b, R 2 . s b4 . And Essi Ž LŽ a, R i ., F . s a, c4 for all i g I. Observe that LŽ b, R i,) . s A for all i g I, b f F Ž R ) ., then F fails to satisfy unanimity and no veto power. In the other hand a g F Ž R . and LŽ a, R i,) . = Essi Ž LŽ a, R i ., F . for all i g I, a f F Ž R ) ., then F fails to satisfy strong monotonicity. The preference profile R is F-minimal for a, LŽ b, R i,) . s A and LŽ b, i,) . R = LŽ a, R i ., b g LŽ a, R i . for all i g I, but b f F Ž R ) .. Then F fails to satisfy weak no veto power. It is obvious that F is Maskinmonotonic. By simple calculation we get B1Ž a, R . s b4 , LŽ a, R i . y B1Ž a, R . s a, c4 , LŽ a, R )i . s a, c4 for all i. Then LŽ a, R )i . = LŽ a, R i . y B1Ž a, R . for all i, but a g F Ž R . and a f F Ž R ) .. F is not B1-monotone, thus F is not implementable in Nash equilibria. Remark 4. McKelÕey has proÕed that weak no Õeto power is necessary for Nash implementation if R is ‘complete’: R is ‘complete’ if for all a g A, i g I, and for all subsets B of A with a g B, there exists R ) g R such that L(a, R )i ) s B and L(a, R j ) s L(a, R )j ) for all j / i. Consider a g A, i s 1, R g R in Example 1 and B s LŽ a, R )i . s a, c4 . However, LŽ a, R j . / LŽ a, R )j . for all j / i. Then R is not ‘complete’. Yamato also proved that strong monotonicity is necessary for Nash implementation if R satisfies condition D, i.e. if for all a g A, R g R, i g I, and b g LŽ a, R i ., there exists R ) g R such that LŽ a, R i . s LŽ b, R )i . and for all j / i LŽ b, R )j . s A. Take a g A, i s 1, and c g LŽ a, R 1 . in Example 1. For all R ) g R we have LŽ a, R 1 . / LŽ c, R 1) ., R fails to satisfy condition D.
Acknowledgements We are indebted to anonymous referees for comments. Their remarks contributed to a better presentation of the results. Thanks also to Maurice Salles.
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