# A set optimization approach to zero-sum matrix games with multi-dimensional payoffs

## TL;DR

This paper introduces a set optimization approach for two-player zero-sum matrix games with multi-dimensional payoffs, providing new equilibrium concepts, existence theorems, and an algorithm for strategy computation.

## Contribution

It develops a novel set relation-based solution concept for multi-dimensional payoff games, extending classical minimax and maximin strategies and establishing their relation to existing equilibrium notions.

## Key findings

- New set-based equilibrium concepts with existence proofs
- Algorithms for computing optimal strategies in multi-dimensional payoff games
- Relationships clarified between new and existing solution concepts

## Abstract

A new solution concept for two-player zero-sum matrix games with multi-dimensional payoff is introduced. It is based on extensions of vector orders in K-dimensional spaces to order relations in their power sets, so-called set relations, and strictly motivated by the interpretation of the payoff as multi-dimensional loss for one and gain for the other player. The new concept provides coherent worst case estimates, i.e. minimax and maximin strategies, for games with multi-dimensional payoffs. It is shown that-in contrast to games with one-dimensional payoff-minimax and maximin strategies are independent from Shapley's notion of equilibrium strategies for such games. Therefore, the two concepts are combined into new equilibrium notions for which existence theorems are given. By means of examples, relationships of the new concepts to existing ones such as Shapley and vector equilibria, vector minimax and maximin solutions as well as Pareto optimal security strategies are clarified. An algorithm for computing optimal strategies is presented.