# Symmetric solution of the Bellman optimality equation for repeated harmony game

arXiv: https://arxiv.org/abs/2609.16289 · HTML: https://mlsanity.com/paper/2609.16289/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 4 of its 24 references that are in the map.
- First posted on arXiv 14 Sep 2026.

## Builds on (most cited first, 4 of 4)

- [Learning with Opponent-Learning Awareness](https://mlsanity.com/paper/1709.04326/) (2017)
- [Learning to Incentivize Other Learning Agents](https://mlsanity.com/paper/2006.06051/) (2020)
- [Symmetric equilibrium of multi-agent reinforcement learning in repeated prisoner's dilemma](https://mlsanity.com/paper/2101.11861/) (2021)
- [Strategical incoherence regulates cooperation in social dilemmas on multiplex networks](https://mlsanity.com/paper/1504.04735/) (2015)

## Abstract (arXiv)

In social dilemma games, additional rewards or punishments have been studied as means of promoting cooperation. Therefore, it is important to investigate the ideal situation, in which such an additional payoff would change the game. In this study, we investigated the symmetric solution of the Bellman optimality equation for a repeated harmony game. The calculations showed that three types of symmetric solutions exist. One of them corresponds to the trivial All-C strategy, and another to the Win-stay Lose-shift strategy of the prisoners dilemma game. The nontrivial behavior of the strategy corresponding to the last solution is also discussed in detail. In addition, we numerically investigated which strategy the agents actually learn by the reinforcement learning algorithm.