On the Equivalence of Zero-Sum Games and Conic Programs

TL;DR

This paper establishes a comprehensive framework linking zero-sum games with conic linear programs, extending classical results to Banach spaces and various game classes, and demonstrating their near-equivalence.

Contribution

It generalizes the equivalence between zero-sum games and conic programs to Banach spaces and broad classes of games, unifying game theory and conic optimization.

Findings

Minimax equality holds for games with strategy sets as bases of convex cones.

Game value and Nash equilibria can be computed via primal-dual conic programs.

Minimax theorem implies strong duality of conic linear programming in this framework.

Abstract

We prove the almost equivalence of the minimax theorem and the strong duality theorem for a large class of games and conic programs. The previous fundamental results on the equivalence of linear programming and two-player zero-sum games with simplex-strategy sets are extended to Banach spaces, and a comprehensive framework unifying two-player zero-sum games and conic linear programs is established. Specifically, we show that for every zero-sum game with a bilinear payoff function and strategy sets that represent bases of convex cones, the minimax equality holds and its game value and Nash equilibria can be found by solving a primal-dual pair of conic programs. Conversely, the minimax theorem for the same class of games "almost always" implies strong duality of conic linear programming. In fact, we give a game-dependent characterization of strict feasibility, and show that minimax is equivalent to a generalized version of Ville's theorem of the alternative. Several well-established game classes are embedded in the introduced model, including (i) semi-infinite, (ii) semidefinite, (iii) quantum, (iv) time-dependent, and (v) polynomial games, as well as (vi) the mixed extension of any continuous game with compact strategy sets.