A Note on Gale, Kuhn, and Tucker's Reductions of Zero-Sum Games

TL;DR

This paper examines Gale, Kuhn, and Tucker's methods for reducing zero-sum games, clarifies the conditions for desirable reductions based on optimal strategies, and corrects a previous example to refine their theoretical framework.

Contribution

It demonstrates that desirable reductions depend on original game strategies and corrects an earlier example, refining the theoretical understanding of game reduction methods.

Findings

Desirable reductions rely on optimal strategies in the original game.

A correction is provided for an improper example from the original work.

The reverse of a key theorem does not hold, clarifying theoretical limitations.

Abstract

Gale, Kuhn and Tucker (1950) introduced two ways to reduce a zero-sum game by packaging some strategies with respect to a probability distribution on them. In terms of value, they gave conditions for a desirable reduction. We show that a probability distribution for a desirable reduction relies on optimal strategies in the original game. Also, we correct an improper example given by them to show that the reverse of a theorem does not hold.