The duality gap for two-team zero-sum games

Abstract

We consider multiplayer games in which the players fall in two teams of size k, with payoffs equal within, and of opposite sign across, the two teams. In the classical case of k=1, such zero-sum games possess a unique value, independent of order of play. However, this fails for all k>1; we can measure this failure by a duality gap, which quantifies the benefit of being the team to commit last to its strategy. We show that the gap equals 2(1−21−k) for m=2 and 2(1−m−(1−o(1))k) for m>2, with m being the size of the action space of each player. Extensions hold also for different-size teams and players with various-size action spaces.We further study the effect of exchanging order of commitment among individual players (not only among the entire teams).The class of two-team zero-sum games is motivated from the weak selection model of evolution, and from considering teams such as firms in which independent players (ideally) have shared utility.