# Submixing and Shift-invariant Stochastic Games

arXiv: https://arxiv.org/abs/1401.6575 · HTML: https://mlsanity.com/paper/1401.6575/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 25 Jan 2014.

## Built on by (most cited first, 2 of 2)

- [Mixing Any Cocktail with Limited Ingredients: On the Structure of Payoff Sets in Multi-Objective POMDPs and its Impact on Randomised Strategies](https://mlsanity.com/paper/2502.18296/) (2025)
- [Simplicity Lies in the Eye of the Beholder: A Strategic Perspective on Controllers in Reactive Synthesis](https://mlsanity.com/paper/2509.04129/) (2025)

## Abstract (arXiv)

We consider zero-sum stochastic games with perfect information and finitely many states and actions. The payoff is computed by a function which associates to each infinite sequence of states and actions a real number. We prove that if the payoff function is both shift-invariant and submixing, then the game is half-positional, i.e. the first player has an optimal strategy which is both deterministic and stationary. This result relies on the existence of epsilon-subgame-perfect strategies in shift-invariant games, a second contribution of the paper. The techniques can be used to establish a third result: for shift-invariant and submixing payoff functions, the existence of finite-memory strategies for player 2 in one-player games implies the same property for two-player games as well.