Probabilistic Assignment : Implementation in Stochastic Dominance Nash Equilibria

We study the assignment of objects to people via lotteries. We consider the implementation of solutions that are based only on ordinal preferences over the objects. There are three natural ways of comparing lotteries, each of which corresponds to a different notion of Nash equilibrium. For each person, the first says that the equilibrium outcome is not stochastically dominated by what he can get by deviating. The second says that the equilibrium outcome stochastically dominates any other lottery that he can get by deviating. Finally, the third says that the he finds the equilibrium outcome to be at least as desirable under a particular von Neumann-Morgenstern index as what he can get by deviating. We provide a necessary and sufficient condition for implementability for each notion of Nash equilibrium. Our main result is to show the equivalence of implementability in these three notions of equilibrium. As an application of our result, we show that a Pareto efficient and envy-free solution is Nash implementable. For three agent economies, there is only one such solution: the Bogomolnaia-Moulin rule