# Pure Strategy Best Responses to Mixed Strategies in Repeated Games

## TL;DR

This paper demonstrates that in repeated games, particularly the iterated prisoner's dilemma, pure strategy best responses to mixed strategies can be computed efficiently, but such responses may not exist in multi-agent tournaments.

## Contribution

It introduces a method to compute pure strategy best responses in repeated games using SMT or MDP solvers and generalizes the results to k-memory strategies.

## Key findings

- Pure strategy best responses exist for one-memory strategies.
- Such responses can be computed efficiently with SMT or MDP solvers.
- Pure strategy best responses may not exist in multi-agent tournaments.

## Abstract

Repeated games are difficult to analyze, especially when agents play mixed strategies. We study one-memory strategies in iterated prisoner's dilemma, then generalize the result to k-memory strategies in repeated games. Our result shows that there always exists a pure strategy best response, which can be computed with SMT or MDP solvers. However, there may not exist such pure strategy best response in multi-agent tournaments. All source code is released for verification.