A solution to the two-person implementation problem

We propose a solution to the classical problem of Hurwicz and Schmeidler [1978] and Maskin [1999] according to which, in two-person societies, no Pareto efficient rule is Nash-implementable. To this end, we consider implementation through mechanisms that are deterministic-in-equilibrium while lotteries are allowed off-equilibrium. For strict preferences over alternatives and under a very weak condition for extending preferences over lotteries, we build simple veto mechanisms that Nash implement a class of Pareto efficient social choice rules called Pareto-and-veto rules. Moreover, under mild richness conditions on the domain of preferences over lotteries, any Pareto efficient Nashimplementable rule is a Pareto-and-veto rule and hence is implementable through one of our simple veto mechanisms