On the equivalence between the minimax theorem and strong duality of conic linear programming

TL;DR

This work addresses the problem of connecting the minimax theorem with strong duality for conic linear programming in reflexive Banach spaces, effectively extending the von Neumann–Dantzig linkage from finite-dimensional Euclidean spaces to infinite strategy spaces. It develops a general conic programming framework for two-player zero-sum games with bilinear payoffs $u(x,y)=\langle y,Ax\rangle$ and strategy sets that are cone-leveled or bases of convex cones, proving minimax via strong duality (Theorems 3.1 and 3.3) and showing that the converse holds almost always (Theorem 4.1), with a documented pathology for zero-value games. The paper further extends these results to finite intersections and unions of cone-leveled sets, and demonstrates broad applicability across semi-infinite, semidefinite, quantum, time-continuous, polynomial, and homogeneous separable games, mapping equilibria to primal–dual conic programs (and sometimes to SDP/SIP/GCAP formulations). Practically, the results provide a unified, computationally viable route to obtaining game values and Nash equilibria through conic programming, while offering criteria to certify strict feasibility and exposing the precise limits of the equivalence via a sharp pathological example. Overall, the work lays a foundational bridge between game theory and conic duality that encompasses a wide spectrum of infinite games and yields both theoretical and algorithmic insights.

Abstract

We prove the almost equivalence between two-player zero-sum games and conic linear programming problems in reflexive Banach spaces. The previous fundamental results of von Neumann, Dantzig, Adler, and von Stengel on the equivalence between linear programming and finite games with strategy sets defined over $\mathbb{R}^n$, are therefore extended to more general strategy spaces. More specifically, we show that for every two-player zero-sum game with a bilinear payoff function of the form $u(x,y)=\langle y,Ax\rangle$, for some linear operator $A$, and strategy sets that represent bases of convex cones, the minimax theorem holds, and its game value and Nash equilibria can be computed by solving a primal-dual pair of conic linear problems. Conversely, the minimax theorem for the same class of games "almost always" implies strong duality of conic linear programming. The main results are applied to a number of infinite zero-sum games, whose classes include those of semi-infinite, semidefinite, time-continuous, quantum, polynomial, and homogeneous separable games.

Theorems & Definitions (26)

• Proposition 2.1
• Definition 2.2
• Theorem 3.1: Minimax theorem for cone-leveled sets
• Lemma 3.2
• proof
• proof : Proof of Theorem 3.1.
• Theorem 3.3: Minimax theorem for bases
• Lemma 3.4
• Lemma 3.5
• proof