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

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 (1953), 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 and all entries of the constraint matrix and right hand side vector are equal to one. 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. We end by discussing evidence that von Neumann possessed an understanding of the notion of polynomial-time solvability.