The purpose of this paper is to provide a constructive way of checking whether or not a social choice correspondence can be implemented in Nash equilibria. The results apply when there are two or more players. The usefulness of this constructive approach is illustrated for the case of single-peaked preferences over IR, a two-person public good economy with monotonic preferences, and a two-person exchange economy with monotonic preferences. * I am grateful to Rajeev Bhattacharya, William Thomson, Takehiko Yamato, and an anonymous referee, for helpful comments. This research was supported by grants from the Bank of Sweden Tercentenary Foundation and the Royal Swedish Academy of Sciences. i For the special case of an unrestricted domain of preferences, additional results are contained in Danilov [1] andYamato [7]. 336 T. Sj6str6m Conversely, suppose F satisfies condition M. Put B=B* and C~(a,R) = C* (a, R/) in the definition of condition p. Clearly a e C/* (a, R~) c L (a, R i ) n B*. Conditions (i), (ii) and (iii) follow from the definition of condition M, Lemma 1 and Lemma 2. Therefore, F satisfies condition p. [] Lemma 4. Condition M2 and condition p 2 are equ&alent.
Proof Suppose F satisfies condition p2. For any (i, R, a) e N × ~ × A such that a e F(R), let B and Cg(a,R) be as in the statement of condition p2. By the argument of the first part of the proof of Lemma 3, a e Ci (a, R) ~ C/* (a, R~) and C* (a, R~) satisfies (i).
For any ((1,1~,a,R) e A × 3 , ~× A × ~
with fieF(l~) and a e F ( R ) , let ~b (& /~, a, R) be as in the statement of condition (iv). Then ~b(fi,/~,a,R)
e C~ (d,t~)c~C;(a,R)c_C*(d,z~)~C*(a, R2). Suppose, for some R * e ~, C~*(d,K~)c_L(~b (d,l~,a,R), R~) and C*(a, R2)c__L(c~(~,t~,a,R),R~). Then C~(gt, l~)~C~(gt,_Rt)cL(gp(gt, l~,a,R),R*) and C2(a,R)~C*(a,R;) _ L (4) (~,/~, a, R), R2*), so (9 (fi, 1~, a, R) e F(R*). Therefore (iv) is satisfied for C* (a, R~). Hence, F satisfies condition M2. Conversely, suppose F satisfies condition M2: By Lemma 3, condition p holds for B =-B* and Ci (a, R) = C* (a, R~). Condition (iv) holds by definition. Therefore, F satisfies condition/~ 2. []