An Optimal Single-Winner Preferential Voting System Based on Game Theory∗

We present an optimal single-winner preferential voting system, called the “GT method” because of its interesting use of symmetric two-person zero-sum game theory to determine the winner. Game theory is not used to describe voting as a multi-player game between voters, but rather to define when one voting system is better than another one. The cast ballots determine the payoff matrix, and optimal play corresponds to picking winners optimally. The GT method is quite simple and elegant, and works as follows: • Voters cast ballots listing their rank-order preferences of the alternatives. • The margin matrix M is computed so that for each pair (x, y) of alternatives, the pairwise margin M(x, y) is the number of voters preferring x over y, minus the number of voters preferring y over x. • A symmetric two-person zero-sum game G is defined where the payoff matrix is M . Each player picks an alternative and wins M(x, y) points when he picks x and the opponent picks y. • An optimal mixed strategy p∗ is computed, where p∗(x) is the probability that a player picks x under optimal play. If the optimal mixed strategy is not unique, we use the “most balanced” one. This optimal strategy is the same for both players, since the game is symmetric. ∗Version April 8, 2010 • The election winner is determined by a randomized method, based for example on the use of ten-sided dice, that picks outcome x with probability p∗(x). If there is a Condorcet winner x, then p∗(x) = 1, and no dice are needed. We also briefly discuss a deterministic variant of GT, which we call GTD. The GT system, essentially by definition, is optimal : no other voting system can produce election outcomes that are liked better by the voters, on the average, than those of the GT system. We also look at whether the GT system has several standard properties, such as monotonicity, consistency, etc. The GT system is not only theoretically interesting and optimal, but simple to use in practice; it is probably easier to implement than, say, IRV. We feel that it can be recommended for practical use.