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Published January 2017 | Published
Book Section - Chapter Open

The duality gap for two-team zero-sum games


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, due to the von Neumann minimax theorem. 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. In our main result we show that the gap equals 2(1 – 2^(1−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. At a finer level, the cost to a team of individual players acting independently while the opposition employs joint randomness is 1 – 2^(1−k) for k = 2, and 1 – m^(−(1−o(1))k) for m > 2. This class of multiplayer games, apart from being a natural bridge between two-player zero-sum games and general multiplayer games, is motivated from Biology (the weak selection model of evolution) and Economics (players with shared utility but poor coordination).

Additional Information

© 2017 Leonard Schulman and Umesh Vazirani; licensed under Creative Commons License CC-BY. LJS was supported in part by NSF grants 1319745/1618795 and BSF grant 2012333; UVV was supported in part by NSF Grant CCF-1410022.

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