Abstract
This study considers Nash implementation in the problem of selecting an alternative when an agent has single-peaked preferences. We investigate condition μ (Moore and Repullo, 1990) and condition μ2 (Moore and Repullo, 1990; Dutta and Sen, 1991), which are necessary and sufficient conditions for implementation. We establish that a rule satisfies condition μ if and only if it is a minimax rule as introduced by Moulin (1980). This leads to the characterization of Nash implementable rules in the case of three or more agents. In addition, in the two-agent case, we identify the set of rules that satisfy condition μ2 or, equivalently, that are Nash implementable. The set of such rules is strictly included in the set of minimax rules, and hence, some minimax rules cannot be Nash implementable in the two-agent case. Since our analysis covers the fair division problem (Sprumont, 1991) with two agents, our theorem in the two-agent case also identifies the set of Nash implementable rules in the problem.
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