Implementation by vote-buying mechanisms

A vote-buying mechanism is such that each agent buys a quantity of votes x to cast for an alternative of her choosing, at a cost c(x), and the outcome is determined by the total number of votes cast for each alternative. In the context of binary decisions, we prove that the choice rules that can be implemented by vote-buying mechanisms in large societies are parameterized by a positive parameter rho, which measures the importance of individual preference intensities on the social choice: The limit with rho= 0 is majority rule, rho = 1 is utilitarianism, and rho?8 is the Rawlsian maximin rule. We show that any vote-buying mechanism with limit cost elasticity (1 rho)/rho as x?0 implements the choice rule defined by rho. The utilitarian efficiency of quadratic voting (Lalley and Weyl, 2016) follows as a special case.