# A Stay-in-a-Set Game without a Stationary Equilibrium

## TL;DR

This paper presents a finite-state two-player turn-based stochastic game with safety objectives that lacks a stationary Nash equilibrium, addressing an open question in game theory.

## Contribution

It provides the first example of such a game, demonstrating the non-existence of stationary Nash equilibria under these conditions.

## Key findings

- No stationary Nash equilibrium exists for the constructed game.
- The example answers an open question in the theory of stochastic games.
- Highlights limitations of stationary strategies in certain game settings.

## Abstract

We give an example of a finite-state two-player turn-based stochastic game with safety objectives for both players which has no stationary Nash equilibrium. This answers an open question of Secchi and Sudderth.