Quadratic voting with multiple alternatives

Consider the following collective choice problem: a society of budget-constrained agents faces multiple alternatives and wants to reach an e¢ cient decision (i.e. to Nash implement the utilitarian maximum). In this paper, we propose a budget-balanced vote-buying mechanism for this setting: for each alternative, every voter can cast any number of votes, x, in support or against it, by transferring an amount x2 to the rest of the voters; and the outcome is determined by the net vote totals. We prove that as the society grows large, in every equilibrium of the mechanism, each agent's transfer converges to zero, and the probability that the mechanism chooses the socially efficient outcome converges to one.