A Generalization of von Neumann's Reduction from the Assignment Problem to Zero-Sum Games
Ilan Adler, Martin Bullinger, Vijay V. Vazirani
TL;DR
The paper broadens von Neumann’s reduction from the assignment problem to a wider class of linear programs by a scaling construction that yields a zero-sum game whose value informs the original LP’s feasibility and optimal value. It presents two main extensions: (i) positive objective/constraint vectors via a scaled game matrix $M= B A C$ and a reciprocal relationship between LP value and reduced-game value, and (ii) a preprocessing-based approach that handles nonnegative constraint matrices $A\ge 0$ to reduce to the positive-case framework. The reductions explicitly separate the two players’ roles, linking the row player’s maximin to the primal and the column player’s maximin to the dual, and provide interpretations via the assignment problem, production planning, and broader “playing linear programs” viewpoints; the framework also clarifies certificates of infeasibility and unboundedness. Computationally, the work argues that these reductions are strongly polynomial and, in many settings, operate in log-space and apply to real numbers, suggesting a closer-than-before equivalence between LPs and zero-sum games with practical implications for algorithmic design and complexity.
Abstract
The equivalence between von Neumann's Minimax Theorem for zero-sum games and the LP Duality Theorem connects cornerstone problems of the two fields of game theory and optimization, respectively, and has been the subject of intense scrutiny for seven decades. Yet, as observed in this paper, the proof of the difficult direction of this equivalence is unsatisfactory: It does not assign distinct roles to the two players of the game, as is natural from the definition of a zero-sum game. In retrospect, a partial resolution to this predicament was provided in another brilliant paper of von Neumann, which reduced the assignment problem to zero-sum games. However, the underlying LP is highly specialized; all entries of its objective function vector are strictly positive, the constraint vector is all ones, and the constraint matrix is 0/1. We generalize von Neumann's result along two directions, each allowing negative entries in certain parts of the LP. Our reductions make explicit the roles of the two players of the reduced game, namely their maximin strategies are to play optimal solutions to the primal and dual LPs. Furthermore, unlike previous reductions, the value of the reduced game reveals the value of the given LP. Our generalizations encompass several basic economic scenarios.