Nash Equilbria for Quadratic Voting

Voters making a binary decision purchase votes from a centralized clearing house, paying the square of the number of votes purchased. The net payoff to an agent with utility $u$ who purchases $v$ votes is $\Psi (S_{n+1})u-v^{2}$, where $\Psi$ is a monotone function taking values between -1 and +1 and $S_{n+1}$ is the sum of all votes purchased by the $n+1$ voters participating in the election. The utilities of the voters are assumed to arise by random sampling from a probability distribution $F_{U}$ with compact support; each voter knows her own utility, but not those of the other voters, although she does know the sampling distribution $F_{U}$. Nash equilibria for this game are described. These results imply that the expected inefficiency of any Nash equilibrium decays like $1/n$.