# Time-aware uniformization of winning strategies

## TL;DR

This paper investigates how to simplify winning strategies in infinite-duration two-player games by ensuring strategies are uniform across equivalent histories with respect to their length, aiding implementation and strategic concealment.

## Contribution

It introduces a sufficient condition for uniformizing winning strategies in time-aware histories, preserving strategy finiteness and providing constructive proofs with relevant corollaries.

## Key findings

- Uniform strategies can be constructed under time-awareness conditions.
- The approach preserves finite memory in strategies.
- Counterexamples demonstrate the tightness of the conditions.

## Abstract

Two-player win/lose games of infinite duration are involved in several disciplines including computer science and logic. If such a game has deterministic winning strategies, one may ask how simple such strategies can get. The answer may help with actual implementation, or to win despite imperfect information, or to conceal sensitive information especially if the game is repeated. Given a concurrent two-player win/lose game of infinite duration, this article considers equivalence relations over histories of played actions. A classical restriction used here is that equivalent histories have equal length, hence \emph{time awareness}. A sufficient condition is given such that if a player has winning strategies, she has one that prescribes the same action at equivalent histories, hence \emph{uniformization}. The proof is fairly constructive and preserves finiteness of strategy memory, and counterexamples show relative tightness of the result. Several corollaries follow for games with states and colors.