# Symmetric equilibrium of multi-agent reinforcement learning in repeated prisoner's dilemma

arXiv: https://arxiv.org/abs/2101.11861 · HTML: https://mlsanity.com/paper/2101.11861/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 28 Jan 2021.

## Built on by (most cited first, 4 of 4)

- [Intrinsic fluctuations of reinforcement learning promote cooperation](https://mlsanity.com/paper/2209.01013/) (2022)
- [A Cooperative Reinforcement Learning Environment for Detecting and Penalizing Betrayal](https://mlsanity.com/paper/2210.12841/) (2022)
- [Cooperation Dynamics in Multi-Agent Systems: Exploring Game-Theoretic Scenarios with Mean-Field Equilibria](https://mlsanity.com/paper/2309.16263/) (2023)
- [Symmetric solution of the Bellman optimality equation for repeated harmony game](https://mlsanity.com/paper/2609.16289/) (2026)

## Abstract (arXiv)

We investigate the repeated prisoner's dilemma game where both players alternately use reinforcement learning to obtain their optimal memory-one strategies. We theoretically solve the simultaneous Bellman optimality equations of reinforcement learning. We find that the Win-stay Lose-shift strategy, the Grim strategy, and the strategy which always defects can form symmetric equilibrium of the mutual reinforcement learning process amongst all deterministic memory-one strategies.