Computing the Optimal Game

In many multiagent environments, a designer has some, but limited control over the game being played. In this paper, we formalize this by considering incompletely specified games, in which some entries of the payoff matrices can be chosen from a specified set. We show that it is NP-hard for the designer to decide whether she can make her choices so that no action in a given set gets played in equilibrium. Hardness holds even in zero-sum games and even in weak tournament games (which are symmetric zero-sum games whose entries are all −1, 0, or 1). The latter case is closely related to the necessary winner problem for a social-choice-theoretic solution concept. We then give a mixed-integer linear programming formulation for weak tournament games and evaluate it experimentally.

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