# Solutions for Zero-Sum Two-Player Games with Noncompact Decision Sets and Unbounded Payoffs

## TL;DR

This paper establishes conditions under which solutions and values exist in two-player zero-sum games with possibly unbounded payoffs and noncompact decision sets, extending classic game theory results.

## Contribution

It introduces new sufficient conditions for the existence of solutions and values in complex zero-sum games without assuming convexity or compactness.

## Key findings

- Existence of solutions under inf/sup-compact payoffs
- Conditions for the existence of game value
- Application to the number guessing game

## Abstract

This paper provides sufficient conditions for the existence of solutions for two-person zero-sum games with inf/sup-compact payoff functions and with possibly noncompact decision sets for both players. Payoff functions may be unbounded, and we do not assume any convexity/concavity-type conditions. For such games expected payoff may not exist for some pairs of strategies. The results of this paper imply several classic facts. The paper also provides sufficient conditions for the existence of a value and solutions for each player. The results of this paper are illustrated with the number guessing game.