An Ordinal Minimax Theorem

TL;DR

This paper introduces a new theoretical result in zero-sum game theory, proving the uniqueness and interchangeability of weak saddles, which leads to a well-defined set-based value for these games.

Contribution

It establishes the equivalence and interchangeability of all weak saddles in zero-sum games, resulting in a unique set-based value.

Findings

All weak saddles are interchangeable and equivalent.

Zero-sum games have a unique set-based value.

The result generalizes the concept of game solutions.

Abstract

In the early 1950s Lloyd Shapley proposed an ordinal and set-valued solution concept for zero-sum games called \emph{weak saddle}. We show that all weak saddles of a given zero-sum game are interchangeable and equivalent. As a consequence, every such game possesses a unique set-based value.