Self-implementation of social choice correspondences in Nash equilibrium

A social choice correspondence is Nash self-implementable if it can be implemented in Nash equilibrium by a social choice function that selects from it as the game form. We provide a complete characterization of all unanimous and anonymous Nash self-implementable social choice correspondences when there are two agents or two alternatives. For the case of three agents and three alternatives, only the top correspondence is Nash self-implementable. In all other cases, every Nash self-implementable social choice correspondence contains the top correspondence and is contained in the Pareto correspondence. In particular, when the number of alternatives is at least four, every social choice correspondence containing the top correspondence plus the intersection of the Pareto correspondence with a fixed set of alternatives, is self-implementable.


Introduction
A social choice correspondence assigns a set of alternatives to a profile of agents' preferences. A specific social choice correspondence -for instance, a particular political election system -may have been adopted since it has some properties that are regarded as attractive, but these properties may be lost if agents behave strategically -do not vote according to their true preferences. Thus, agents should be provided with incentives in order to reach a truthful alternative, that is, an alternative assigned by the social choice correspondence to the true preference profile. This leads to the classical question of implementation, which dates back to Hurwicz (1972) and Maskin (1999): given a social choice correspondence, can we find a game form such that for every preference profile the equilibrium outcomes of the game form coincide with the outcomes of the social choice correspondence? Here, 'equilibrium' may refer to Nash equilibrium or to other equilibrium concepts, for instance strong equilibrium.
Game forms in the literature used for implementing social choice correspondences tend to be quite complicated. Typically, they may ask individuals to report complete strategy profiles, and they use integers or similar devices as part of strategies in order to endow individuals or coalitions -depending on the equilibrium concept -with sufficient possibilities to deviate, thus avoiding equilibria with outcomes that are not contained in the set assigned by the social choice correspondence. Indeed, the complexity of the message space or space of strategies is an issue that has gathered quite some attention in the literature (see already Mount and Reiter, 1974). Recently, Koray and Yildiz (2018) consider implementation via rights structures, based on an idea by Sertel (2001), which also avoids the use of integer devices. For (relatively) recent surveys on implementation see Jackson (2001) and Börgers (2015).
In the present paper we restrict our attention to direct game forms which are single-valued selections (social choice functions) from the social choice correspondence. This means, simply, that the strategy space of each agent consists of the set of all preferences -agents are not required to report complete profiles, integers, or any similar or other information -and the outcome function selects (at every preference profile) from the social choice correspondence to be implemented. Our main motivation for studying this case is twofold. First, considering that a social choice correspondence in vigor in a group or society is usually regarded as appropriate or desirable by its members, choosing according to a selection from it seems to be natural, both for simplicity and in order to convince agents to accept the game form (the rules of the game). Several papers have discussed the advantages of direct game forms. For example, Hammond (1996) shows that such game forms, in which individual strategies are direct reports of their preferences, are implicit in Sen's model of rights (Sen, 1970). Second, but related, it seems to be an obvious and natural question to establish which social choice correspondences can be implemented this way, i.e., by a direct game form selecting from the correspondence. One could call this kind of implementation 'natural', but this word already has different meanings in this context (e.g., Saijo et al, 1996). Therefore, we use the term 'self-implementation', an expression which has already been used informally, but with the same meaning, for social choice functions (see Remark 4.2.7 in Peleg, 2002). We will focus attention on self-implementation in Nash equilibrium. 1 Moreover, we consider only social choice correspondences which are unanimous and anonymous, and only anonymous selections (social choice functions). Under these restrictions our results are as follows. Let the 'top correspondence' T C assign to each preference profile all alternatives that occur at top for at least one agent, and let the 'Pareto correspondence' P C assign all Pareto optimal alternatives. Then every anonymous and unanimous Nash self-implementable social choice correspondence contains T C and is contained in P C. 2 3 If there are exactly two alternatives, then T C and P C of course coincide; and we show that T C is Nash self-implementable if and only if the number of agents is not equal to 2, 3, 4, or 6; moreover, we describe all implementing selections (Corollary 4.8).
If there are exactly two agents, then no anonymous and unanimous social choice correspondence is Nash self-implementable, a result which also follows from Hurwicz and Schmeidler (1978). If there are three agents and three alternatives then only T C is Nash self-implementable. If there are at least three agents and and at least four alternatives then T C and P C are Nash self-implementable but also some social choice correspondences in between these two; the characterization of all such correspondences is still open. The remaining case is the case with three alternatives and at least four agents: then T C is Nash self-implementable but it is an open question whether T C is uniquely Nash self-implementable.
To the best of our knowledge, this concept of Nash self-implementation has not been explicitly studied in the literature, although some of our results can be derived from existing results, for instance the two-agent case mentioned above. Nevertheless, the proofs below are self-contained. Also, the social choice correspondences prominent in this paper have occurred frequently in the literature, for instance the Pareto Correspondence only recently in Mukherjee et al (2019).
The outline of the paper is as follows. Section 2 presents the main definitions and a few basic results. Sections 3 and 4 deal with the two agents and two alternatives cases, respectively. The case with at least four alternatives is presented in Section 5, while Section 6 treats the three alternatives case. Section 7 concludes.

Definitions and preliminary results
Let A be the set of m alternatives, m ≥ 2, and let N = {1, . . . , n}, n ≥ 2, be the set of agents. Let L be the set of preferences, i.e., reflexive, complete, antisymmetric and transitive binary relations, on A. Then L N is the set of (preference) profiles. A social choice correspondence (SCC) is a function H : For a social choice function F and a profile R N we denote by (F, R N ) the (ordinal) game with the agents as players who each have strategy set L, and for a (strategy) (1) We call social choice correspondence H Nash self-implementable if (1) holds for some selection F from H.
For a preference R ∈ L the top alternative t(R) ∈ A is the alternative such that t(R)Rx for all x ∈ A. The social choice correspondence T C, defined by Throughout, we will focus on unanimous and anonymous social choice correspondences. Clearly, T C and P C are unanimous and anonymous.
If a social choice correspondence H is unanimous and anonymous and F is a selection from it, then clearly also F is unanimous, but F is not necessarily anonymous. In what follows we will always consider implementation by anonymous social choice functions.
The following lemmas collect some basic results which will be useful throughout. In these lemmas H is a unanimous and anonymous social choice correspondence which is Nash implemented by an anonymous selection F . The first lemma implies that for every preference profile and every alternative there is an agent who can make sure that this alternative will be chosen. Furthermore, H contains all top alternatives and is contained in the set of Pareto optimal alternatives.
Lemma 2.1. Let H be a unanimous and anonymous social choice correspondence and let the anonymous selection F implement H in Nash equilibrium. Let Proof. (a) Let y = F (R N ). If y = x we are done. Otherwise, consider a profile Z N ∈ L N with t(Z i ) = x for all i ∈ N and yZ i z for all z ∈ A \ {x} (i.e., x is ranked first and y second by every agent). Then H(Z N ) = {x} by unanimity, and therefore R N is not a Nash equilibrium of (F, Z N ). Hence, there is an i ∈ N and a Q i ∈ L such that F (R N \{i} , Q i )Z i y, which implies that (b) This follows directly from (a) and anonymity of F .
The second lemma says that for every alternative x there is a collection of n preferences with one of them different from the n − 1 other preferences, such that x is assigned by F to every profile containing that particular preference and at least n − 2 of the other preferences.

Lemma 2.2. Let H be a unanimous and anonymous social choice correspondence and let the anonymous selection
contradicting Lemma 2.1(a). The lemma follows by taking P x,i = Q i for every i ∈ N and anonymity.

Two agents
For the case of two agents we have the following impossibility result. We provide a proof, but the result also follows from a more general result in Hurwicz and Schmeidler (1978).
Theorem 3.1. Let n = 2 and let H be a unanimous and anonymous social choice correspondence. Then H is not Nash implementable by an anonymous selection from it.
Proof. Let F be an anonymous selection from H, and let (R 1 , We derive a contradiction, which completes the proof. By applying Lemma , which is the desired contradiction.

Two alternatives
Throughout this section A = {a, b}, and in view of Theorem 3.1 we assume n ≥ 3. We say that a profile R N is of the form a k b n−k if a occurs exactly k times on top at R N , where 0 ≤ k ≤ n. We also use this expression to denote a(ny) profile of that form. This notation is convenient since we only consider anonymous social choice functions and correspondences. We use the expressions ab and ba for the preferences with a and b on top, respectively.
In Lemmas 4.1-4.6 we assume that H is a unanimous and anonymous social choice correspondence and that the anonymous selection F from H implements H in Nash equilibrium. Proof. Follows directly from Lemma 2.1(c).
Proof. We only prove the first claim, the proof of the other claim is similar.
. . , n − 2, n}. By Lemma 2.1(a) and anonymity of F we must have Without loss of generality in our profile R N = a 1 b n−1 let R 1 = ab. If both (2) and (3) Then by Lemma 4.2, also Q 1 = ab, but Q N = a n is not a Nash equilibrium in (F, a 1 b n−1 ) again by Lemma 4.2. So we have that only (2) is true, which implies, again by the fact that Q N is a Nash Lemma 4.4. n = 3 and n = 4.
Proof. This follows from Lemma 2.1(a) by considering a profile a k b n−k . Lemmas 4.1-4.6 provide necessary conditions for a unanimous and anonymous social choice correspondence to be Nash implementable by an anonymous selection. We now drop our assumptions on H and F and proceed with the converse.
Lemma 4.7. Let social choice function F be an anonymous selection from P C satisfying Then F Nash implements P C.
Proof. Define the correspondence H : Hence Q N is not a Nash equilibrium in (F, R N ). Therefore, H(a n ) = {a} = P C(a n ).
Similarly one proves H(b n ) = {b} = P C(b n ). Now let 1 ≤ k ≤ n − 1 and consider a profile R N = a k b n−k . Let, without loss of generality, R i = ab for i = 1, . . . , k. Let Q N = a n−1 b 1 with Q n = ba.
which completes the proof of the lemma.
We summarize the results of this section as follows. Observe that the Pareto and top correspondences coincide in this case. Also, the implementing selection F in Corollary 4.8 is not necessarily unique. The conditions (i)-(iii) in Lemma 4.7 uniquely determine F (a k b n−k ) for k = n − 3, . . . , n and for k = 0, . . . , 3, but for other values of k there is freedom of choice subject to condition (iii). For n = 5 and n = 7 we have uniqueness of F , but for n = 8 and n = 9 there are each time two possibilities. For n = 8 we may have F (a 4 b 4 ) = a or F (a 4 b 4 ) = b. For n = 9 we may have F (a 5 b 4 ) = a and F (a 6 b 3 ) = b or F (a 5 b 4 ) = b and F (a 6 b 3 ) = a. The number of possible selections expands rapidly as n increases.

At least three agents and at least four alternatives
In game forms (mechanisms) that are used to implement social choice correspondences often strategies (messages) include mentioning an integer number, or sending a comparable message, in order to provide the agents with additional strategic power. Since in the case of self-implementation agents can exclusively report a preference, in Section 5.1 we will label these in a convenient way in order to replace the standard integer messaging. If there are at least four alternatives then there are many different preferences and therefore many different strategies or messages. This case is treated in Sections 5.2, 5.3, and 5.4. The case m = 3 is more subtle and will be studied in Section 6. Throughout this section we write A = {x 1 , . . . , x m } and M = {1, . . . , m}. Also, for a preference R ∈ L we use notations such as R = xyz . . .; R = x i1 . . . x im ; etc., meaning that xRyRzR . . .; x i1 R . . . Rx im ; etc.

Labelling preferences
In this section we describe a labelling of the preferences. In fact, this will just be alphabetic labelling, with x 1 , . . . , x m as the 'alphabet'. Just for completeness, we include the formula.
For ∅ = L ⊆ M and k ∈ L let π(k, L) be the number of elements of L smaller than or equal to k, formally: π(k, L) = |{ ∈ L | ≤ k}|. Observe that π(k, M ) = k for every k ∈ M . We define As illustrated in Example 5.1, this way of labelling has the property that all preferences that share the same top alternative(s), are labelled consecutively with no gaps. This is important later on.

The top correspondence
In this subsection we show that the top correspondence can be Nash selfimplemented for the case with at least three agents and at least four alternatives.
For each j ∈ M we define the preferences P j and P j by In words, P j is the preference with x j on top and all the other alternatives in the ordering x 1 . . . x m , while P j has x j on top and the remaining alternatives in reverse ordering. We define the selection F 1 from the top correspondence T C as follows. Let R N ∈ L N , and denote ν( In words, F 1 picks x j if one agent has preference P j and all other agents except at most one have preference P j , for some j ∈ M . In all other cases, F 1 chooses from the top alternatives by using modulo counting based on the labels of the preferences. Clearly, F 1 is an anonymous selection from T C. We have the following result.
Theorem 5.2. F 1 implements T C in Nash equilibrium.
Proof. (i) Let R N ∈ L N and x j ∈ T C(R N ). Take ∈ N with t(R ) = x j . Then Q N ∈ L N with Q = P j and Q i = P j for all i ∈ N \ { } is a Nash equilibrium in (F 1 , R N ) such that F 1 (Q N ) = x j .
(ii) Let R N ∈ L N and suppose that Q N ∈ L N is a Nash equilibrium in (F 1 , R N ) with, say, F 1 (Q N ) = x j . We have to show that x j ∈ T C(R N ). Suppose not.
If Q N is of the format as in (1.1) with Q = P j for some agent with, say, t(R ) = x k for some k = j, then by (1.2) there isQ ∈ L with t(Q ) = x k such that F 1 (Q N \{ } ,Q ) = x k : observe that T C(Q N \{ } ,Q ) contains at most three alternatives and that there are (m − 1)! ≥ 6 consecutive labels associated with preferences that have x k on top. This contradicts that Q N is a Nash equilibrium in (F 1 , R N ).
If Q N is of the format as in (1.2) then T C(Q N ) contains at most m elements and each agent i can obtain t(R i ) be deviating from Q N by a preferenceQ i ∈ L with t(Q i ) = t(R i ): there are (m − 1)! such preferences labelled consecutively, and (m − 1)! > m for m ≥ 4. Hence, F 1 (Q N \{i} ,Q i ) = t(R i ) for at least one suchQ i , contradicting again that Q N is a Nash equilibrium in (F 1 , R N ).

The Pareto correspondence
Here, we show that for at least three agents and at least four alternatives also the Pareto correspondence is Nash self-implementable, by a selection F 2 that can be seen as a refinement of F 1 . Before we can define F 2 we need a few notations and a definition. First, if a subset B of A is used in the notation of a preference R, e.g., R = . . . B . . ., this means that each alternative of A \ B is ranked either above or below the alternatives of B, and that the alternatives of B are ranked in the order x 1 . . . x m : that is, if x j , x k ∈ B and j < k then x j Rx k . The notation R = . . . B . . . has a similar meaning except that now the alternatives of B are ranked in reverse order: if x j , x k ∈ B and j < k then In words, for an agent i ∈ K, in R i the top alternative is x, next the alternatives that are not in B i are ranked in the order x 1 . . . x m , and finally the alternatives in B i are ranked in the same order. For an agent not in K also x is the top alternative, but all remaining alternatives are ranked in reverse order. Observe that a profile with agent ∈ N having a preference P j for some j ∈ M and all other agents having preference P j , as defined in Section 5.2, is a special case: it is an ( We first show that in an (x j , (B i ) i∈K )-profile the set K and the sets B i are uniquely determined.
Lemma 5.3. Let R N be an (x, (B i ) i∈K )-profile and also an (x, (C i ) i∈K )-profile.
Proof. First observe that, since m ≥ 4, by definition of the preferences we have N \ K = N \ K , and thus also K = K . Next, the only possible preference of an agent i ∈ K where B i and C i could be different would be a preference R i = x(A \ {x}), but then there can be only one such preference in R K (since such preferences share the same bottom alternative), so that We next define the social choice function F 2 , which is a selection from the Pareto correspondence (actually, from the top correspondence), by the following procedure. Let R N ∈ L N .
In words, if R N is the (x j , (B i ) i∈K )-profile except for the preference of exactly one agent outside K, then F 2 assigns x j . If R N is the (x j , (B i ) i∈K )-profile except for the preference of at most one agent inside K, then F 2 assigns x j unless agent ∈ K reports a preference with an alternative from B on top: this is case (2.2), which is well-defined in view of Lemma 5.3. Also observe that cases (2.1) and (2.2) are exclusive. In all remaining cases F 2 , like F 1 above, chooses from the top alternatives by using modulo counting based on the labels of the preferences: this is (2.3). Observe that if R N is the (x j , (B i ) i∈K )-profile then F (R N ) = x j : this follows from (2.2).
Clearly, F 2 is an anonymous selection from the top correspondence, hence also from P C. We will prove that F 2 Nash implements the Pareto correspondence and to this end first prove that it satisfies the necessary condition stated in Lemma 2.1(a).
Proof. We are done if F 2 (R N ) = y. Now suppose F 2 (R N ) = x j = y. We distinguish three cases.
(a) If (R N \{ } , P j ) is the (x j , (B i ) i∈K )-profile for some agent with R = P j , then by (2.3) every agent i ∈ N \ { } has a preference Q i ∈ L with t(Q i ) = y such that F 2 (R N \{i} , Q i ) = y (namely, such a profile has at most three top alternatives and there are at least six preferences with top alternative y that have consecutive labels).
(b) If R N is as in (2.2), with notations as there, then either y ∈ B and then F 2 (R N \{ } , Q ) for any Q ∈ L with t(Q ) = y, or y / ∈ B . In the latter case, if R = Q, then by (2.2) agent i ∈ K \ { } with y ∈ B i has a preference Q i ∈ L with t(Q i ) = y such that F 2 (R N \{i} , Q i ) = y; and if R = Q then by (2.3) each agent i ∈ N \ { } has a preference Q i ∈ L with t(Q i ) = y such that F 2 (R N \{i} , Q i ) = y (by an argument similar as in case (a)).
(c) If (2.3) applies to R N then there is an agent i ∈ N such that (2.3) still applies to every (R N \{i} , Q i ) with t(Q i ) = y. By a similar argument as in the last paragraph of the proof of Theorem 5.2, there is a Q i with t(Q i ) = y and F 2 (R N \{i} , Q i ) = y.
Theorem 5.5. F 2 implements P C in Nash equilibrium.
Proof. (a) First, let R N ∈ L N and x ∈ P C(R N ). For each agent i ∈ N let L(x, R i ) denote the strict lower contour set of x at R i , that is, It is sufficient to prove that Q N is not a Nash equilibrium in (F, R N ). Since x / ∈ P C(R N ) there is a y ∈ A \ {x} such that yR i x for every i ∈ N . By Lemma 5.4 there is an ∈ N and V ∈ L such that F 2 (Q N \{ } , V ) = y. This implies that Q N is not a Nash equilibrium in (F, R N ).

Between top and Pareto
In the case of four or more alternatives and more than two agents there are also other correspondences, apart from T C and P C, that are Nash self-implementable, as we will show now. For Then H B is unanimous and anonymous, and T C( In words, H B picks all top alternatives, and from the Pareto optimal alternatives only those that are in B. Fix B ⊆ A. We define the anonymous selection F B from H B as follows (notations as earlier).
(B.1) If there is j ∈ M and distinct , ∈ N such that R = P j and R i = P j for all i ∈ N \ { , }, then F B (R N ) = x j .
(B.2) If there is an ∈ N and x j ∈ B such that R = P j and (R N \{ } , P j ) is the Observe that (B. This theorem can be proved by similar arguments as in the proofs of Theorems 5.2 and 5.5: we omit the details.
It is an open question whether there are other Nash self-implementable anonymous and unanimous social choice correspondences in between the top and the Pareto correspondence.

Three alternatives
The case of three alternatives is more subtle, due to the fact that there are less different preferences and, thus, less different labels available for messaging.
As in the previous section, A = {x 1 , . . . , x m }, with now m = 3. Throughout, however, we write a = x 1 , b = x 2 , and c = x 3 . Instead of P 1 , P 1 , P 2 , P 2 , P 3 , P 3 (introduced in Section 5.2) we write P a , P a , P b , P b , P c , P c , respectively, and repeat the table in Example 5.1: We first consider the top correspondence and next the Pareto correspondence.

The top correspondence
Theorem 5.2 still holds for m = 3 but needs a different proof.
Theorem 6.1. Let m = 3 and n ≥ 3. Then F 1 implements T C in Nash equilibrium.
Proof. (i) Let R N ∈ L N and let x ∈ T C(R N ). Take an agent i ∈ N such that t(R i ) = x, and let Q N be the profile with Q i = P x and Q j = P x for all j ∈ N \ {i}. As in the proof of Theorem 5.2 it follows that Q N is a Nash equilibrium in (F 1 , R N ) with F 1 (Q N ) = x.
(ii) For the converse, suppose that Q N is a Nash equilibrium in (F 1 , R N ) for some Q N , R N ∈ L N . Without loss of generality let F 1 (Q N ) = a. We wish to show that a ∈ T C(R N ). Suppose not. Then without loss of generality a is ranked second everywhere at R N , hence for every agent i, R i = P b or R i = P c . We distinguish cases according to Q N and show that in each case we have a contradiction. In these cases we assign specific roles to agents, which is without loss of generality in view of Anonymity. As before, we denote ν(Q N ) = i∈N ν(Q i ), i.e., the sum of the labels of the preferences in Q N .
(2) Q N is not of the form as in (1). We consider the following subcases.
Otherwise, the same holds for every i ∈ N .
Suppose there are at least two agents at Q N who have b on top. Suppose Q i = P b for some i ∈ N . Then ν(Q N \{i} , P c ) mod 3 = (ν(Q N ) − 3 + 5) mod 3 = 2, so that F 1 (Q N \{i} , P c ) = c. Otherwise there is an agent i ∈ N with Q i = P b . In that case, ν(Q N \{i} , P c ) mod 3 = (ν(Q N ) − 4 + 6) mod 3 = 2, so that F 1 (Q N \{i} , P c ) = c.
(2c) T C(Q N ) = {a, c}. This case is almost identical to case (2b) by switching the roles of b and c.

The Pareto correspondence
We show that the Pareto correspondence cannot be Nash self-implemented if, besides m = 3, also n = 3. Proof. Without loss of generality let R N = (P a , P a , P b ) and let Q N ∈ L N such that F (Q N ) = c. In view of Lemma 2.1 it is sufficient to show that Q N is not a Nash equilibrium in (F, for every V 3 ∈ L. By Lemma 2.1(a) this implies that there is a V 1 ∈ L with F (V 1 , Q 2 , Q 3 ) = a or a V 2 ∈ L with F (Q 1 , V 2 , Q 3 ) = a, and in both cases Q N is not a Nash equilibrium in (F, R N ). If both Q 1 = P c,3 and Q 2 = P c,3 then F (V 1 , Q 2 , P c,3 ) = F (Q 1 , V 2 , P c,3 ) = c for all V 1 , V 2 ∈ L. By Lemma 2.1(a) this implies that there is a V 3 ∈ L such that F (Q 1 , Q 2 , V 3 ) = b, so that also in this case Q N is not a Nash equilibrium in (F, R N ).
The case of more than three agents is still open. In other words, for m = 3 and n ≥ 4, does there exist a unanimous and anonymous social choice correspondence H with H = T C and T C(R N ) ⊆ H(R N ) ⊆ P C(R N ) for all R N ∈ L N , which is implementable in Nash equilibrium by an anonymous selection?

Concluding remarks
The following table summarizes the results of the paper. A sign means nonexistence of a unanimous and anonymous social choice correspondence Nash implementable by an anonymous selection. The question marks indicate the open problems mentioned earlier.
n 2 3 ≥ 4 2 for n = 4, 6; P C = T C otherwise m 3 only T C T C, more? ≥ 4 T C, P C, and more T C, P C, F B (B ⊆ A), more?
We conclude with a further consideration of the requirement that the implementing social choice function be a selection from the social choice correspondence that is to be implemented, in other words, the self-implementation assumption. We do this by providing two examples. The first example is an implementation of T C by a social choice function that not always selects from T C, and thus shows the (not surprising) fact that a self-implementable social choice correspondence may be implementable by a direct mechanism that is not a selection. The second example shows that a subcorrespondence of T C may be implementable by a direct mechanism that is not a selection.
Example 7.1. Let n ≥ 3, m ≥ 4, and consider the social choice functionF 1 defined as follows.
(i)F 1 (R N ) = x 2 if R N is a profile such that there is one agent with preference x 1 x 2 x 3 x 4 . . . x m and all other agents have preference x 1 x 3 x 2 x 4 . . . x m .
ThenF 1 Nash-implements T C, which can be seen by adding to the proof of Theorem 5.2 the consideration that a profile R N as in (i) cannot be a Nash equilibrium in the game (F, Q N ) for any Q N ∈ L N . However,F 1 is not a selection from T C.
Example 7.2. Let n ≥ 3, m ≥ 4, and let the social choice correspondence H be defined by for all R N ∈ L N . We define the social choice function F 1 as follows. Let R N ∈ L N , and denote ν(R N ) = n i=1 ν(R i ).
(1a) If there is ∈ N such that R = P 1 and R i = P 1 for all i ∈ N \ { }, then F 1 (R N ) = x 1 .
(1b) If there is j ∈ M \ {1} and distinct , ∈ N such that R = P j and R i = P j for all i ∈ N \ { , }, then F 1 (R N ) = x j .
By adding to the proof of Theorem 5.2 the consideration that a profile as in case (1a) can only be a Nash equilibrium if every agent has x 1 as top alternative in the true profile, we see that F implements H. However, (2) implies that F is not a selection from H.