# Approximate Nash Equilibria in Partially Observed Stochastic Games with Mean-Field Interactions

## TL;DR

This paper proves the existence of Nash equilibria in partially observed mean-field games and demonstrates that these equilibria serve as good approximations in large-agent settings.

## Contribution

It introduces a method to establish Nash equilibria in partially observed mean-field games using belief space transformation and dynamic programming.

## Key findings

- Existence of Nash equilibria under mild conditions.
- Mean-field equilibrium policies are approximate Nash equilibria in large-agent games.
- Applicable to infinite-horizon discounted cost scenarios.

## Abstract

Establishing the existence of Nash equilibria for partially observed stochastic dynamic games is known to be quite challenging, with the difficulties stemming from the noisy nature of the measurements available to individual players (agents) and the decentralized nature of this information. When the number of players is sufficiently large and the interactions among agents is of the mean-field type, one way to overcome this challenge is to investigate the infinite-population limit of the problem, which leads to a mean-field game. In this paper, we consider discrete-time partially observed mean-field games with infinite-horizon discounted cost criteria. Using the technique of converting the original partially observed stochastic control problem to a fully observed one on the belief space and the dynamic programming principle, we establish the existence of Nash equilibria for these game models under very mild technical conditions. Then, we show that the mean-field equilibrium policy, when adopted by each agent, forms an approximate Nash equilibrium for games with sufficiently many agents.