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, 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.

💡 Research Summary

The paper revisits the classic reduction introduced by John von Neumann that maps the assignment problem onto a zero‑sum “hide‑and‑seek” game. While von Neumann’s construction works only for a highly specialized linear program—positive objective coefficients, a unit right‑hand side, and a 0/1 incidence matrix—the authors broaden the scope in two orthogonal directions.

First, they keep the right‑hand side vector b and the objective vector c strictly positive, but allow the constraint matrix A to be an arbitrary real matrix. By scaling each row of A by 1/b_i and each column by 1/c_j, they obtain a game matrix M = diag(1/b)·A·diag(1/c). In the resulting zero‑sum game the row player’s maximin strategy coincides with an optimal primal solution x* of the LP, while the column player’s minimax strategy coincides with an optimal dual solution y*. Strong duality guarantees cᵀx* = bᵀy*, and the game’s value is exactly the reciprocal of this common optimal value: v = 1/(cᵀx*) = 1/(bᵀy*).

Second, they relax the positivity requirement on b and c, assuming instead that the constraint matrix A is non‑negative (A ≥ 0). Here the scaling is performed with absolute values, e.g., M = A·diag(1/|c|) or M = diag(1/|b|)·A, so that negative entries in b or c are handled naturally. The same correspondence holds: the row player solves the primal LP, the column player solves the dual LP, and the game value again equals the reciprocal (up to sign) of the LP optimum.

These two extensions encompass a wide variety of economic and operations‑research models. When both b and c are positive, the reduction models classic production‑planning problems: b represents resource capacities, c represents product profits, and A encodes input‑output coefficients. The row player can be interpreted as a “producer” choosing a resource allocation, while the column player acts as a “price‑setter” choosing dual prices; their equilibrium strategies yield the optimal production plan and shadow prices simultaneously. When A is non‑negative but b or c may be negative, the framework captures transportation‑matching, b‑matching, and other network flow problems where costs can be subtracted or revenues added.

The authors also address the “difficult direction” of the von Neumann–LP duality equivalence, which historically relied on a symmetric zero‑sum game that mixes primal and dual variables in each strategy. That construction yields strategies of a composite form that are hard to interpret as pure primal or dual solutions. By contrast, the new reductions keep the two players’ roles separate, giving each player a clean LP to solve. This separation restores the intuitive game‑theoretic meaning that von Neumann originally sought.

Moreover, the paper shows that the reduction remains meaningful even when the LP is infeasible or unbounded. If the primal is infeasible, the associated game has a negative infinite value; if the primal is feasible but unbounded, the game’s value collapses to zero. Thus the zero‑sum game provides a unified certificate of feasibility, infeasibility, or unboundedness—what the authors term a “degree of (in)feasibility.”

In summary, the work delivers a comprehensive generalization of von Neumann’s hide‑and‑seek reduction, extending it from the narrow assignment‑LP setting to two broad classes of linear programs. It clarifies the distinct strategic roles of the two players, links game value directly to the LP optimum (or its reciprocal), and offers new game‑theoretic insights into feasibility and unboundedness. The results bridge classic game theory, linear programming duality, and practical optimization models in economics and operations research.