How to Win a Large Election

Abstract

We consider the optimization problem of a campaign trying to win an election when facing aggregate uncertainty, where agents' voting probabilities are uncertain. Even a small amount of uncertainty will in a large electorate eliminate many of counterintuitive results that arise when voting probabilities are known. In particular, a campaign that can affect the voting probabilities of a fraction of the electorate should maximize the expected difference between its candidate's and the opposing candidate's share of the fraction's potential vote. When a campaign can target only finitely many voters, maximization of the same objective function remains optimal if a convergence condition is satisfied. When voting probabilities are certain, this convergence condition obtains only at knife-edge combinations of parameters, but when voting probabilities are uncertain the condition is necessarily satisfied.