# On Completely Mixed Stochastic Games

## TL;DR

This paper proves that certain undiscounted zero-sum stochastic games with specific properties are also completely mixed when considering discounted versions with discount factors close to 1, and explores related value properties.

## Contribution

It establishes conditions under which discounted stochastic games inherit the complete mixing property from their undiscounted counterparts, and provides counterexamples and value analysis.

## Key findings

- Discounted games with high discount factors are completely mixed under given conditions.
- Counterexample shows the converse of the main result does not hold.
- Non-zero value in an undiscounted game implies non-zero value in the discounted game for high discount factors.

## Abstract

In this paper, we consider a zero-sum undiscounted stochastic game which has finite state space and finitely many pure actions. Also, we assume the transition probability of the undiscounted stochastic game is controlled by one player and all the optimal strategies of the game are strictly positive. Under all the above assumptions, we show that the $\beta$-discounted stochastic games with the same payoff matrices and $\beta$ sufficiently close to 1 are also completely mixed. We give a counterexample to show that the converse of the above result in not true. We also show that, if we have non-zero value in some state for the undiscounted stochastic game then for $\beta$ sufficiently close to 1 the $\beta$-discounted stochastic game also possess nonzero value in the same state.