Dominance of truthtelling and the lattice structure of Nash equilibria

Truthtelling is often viewed as focal in the direct mechanisms associated with strategy-proof decision rules. Yet many direct mechanisms also admit Nash equilibria whose outcomes differ from the one under truthtelling. We study a model that has been widely discussed in the mechanism design literature (Sprumont, 1991) and whose strategy-proof and efficient rules typically suffer from the aforementioned deficit. We show that when a rule in this class satisfies the mild additional requirement of replacement monotonicity, the set of Nash equilibrium allocations of its preference revelation game is a complete lattice with respect to the order of Pareto dominance. Furthermore, the supremum of the lattice is the one obtained under truthtelling. In other words, truthtelling Pareto dominates all other Nash equilibria. For the rich subclass of weighted uniform rules, the Nash equilibrium allocations are, in addition, strictly Pareto ranked. We discuss the tightness of the result and some possible extensions.