Preference manipulations lead to the uniform rule

For the problem of fully allocating a social endowment of a commodity among a group of agents with single-peaked preferences, we study the consequences of manipulation for several families of rules that are not strategy-proof. Given a rule and a true preference profile, we consider the induced direct revelation game, and characterize its equilibrium allocations in terms of the profile. Our results are unequivocal: for any rule we consider, and for each true preference profile, there is a unique Nash equilibrium allocation. For the profile, it is the allocation of the uniform rule (Sprumont, 1991), the unique strategy-proof, efficient, and symmetric rule in this literature. These conclusions are drawn from two distinct sets of assumptions on the rules.