# Characterizing the Nash equilibria of three-player Bayesian quantum games

## TL;DR

This paper investigates the Nash equilibria of three-player Bayesian quantum games within the Eisert-Wilkens-Lewenstein framework, revealing how entanglement and uncertainty influence strategic outcomes and potential advantages over classical games.

## Contribution

It provides a comprehensive analysis of Bayesian quantum game equilibria, including phase diagram structures and the impact of entanglement, extending prior work on two-player quantum games.

## Key findings

- Equilibria depend on uncertainty and entanglement levels.
- Quantum games can outperform classical counterparts when Pareto-optimal solutions are not Nash.
- Some games exhibit a continuum of strategies bounded by simpler restricted solutions.

## Abstract

Quantum games with incomplete information can be studied within a Bayesian framework. We analyze games quantized within the EWL framework [Eisert, Wilkens, and Lewenstein, Phys Rev. Lett. 83, 3077 (1999)]. We solve for the Nash equilibria of a variety of two-player quantum games and compare the results to the solutions of the corresponding classical games. We then analyze Bayesian games where there is uncertainty about the player types in two-player conflicting interest games. The solutions to the Bayesian games are found to have a phase diagram-like structure where different equilibria exist in different parameter regions, depending both on the amount of uncertainty and the degree of entanglement. We find that in games where a Pareto-optimal solution is not a Nash equilibrium, it is possible for the quantized game to have an advantage over the classical version. In addition, we analyze the behavior of the solutions as the strategy choices approach an unrestricted operation. We find that some games have a continuum of solutions, bounded by the solutions of a simpler restricted game. A deeper understanding of Bayesian quantum game theory could lead to novel quantum applications in a multi-agent setting.