A Generalization of von Neumann's Reduction from the Assignment Problem to Zero-Sum Games

TL;DR

This paper generalizes von Neumann's reduction from the assignment problem to zero-sum games, explicitly assigning roles to players and revealing LP values, thus broadening the theoretical connection between game theory and optimization.

Contribution

It extends von Neumann's reduction to include LPs with negative entries, clarifies player roles, and links game value to LP value, covering more economic scenarios.

Findings

Reduces LPs with negative entries to zero-sum games

Explicitly assigns roles to game players as LP solutions

Game value corresponds to LP value

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.