# Characterizing the Nash equilibria of a three-player Bayesian quantum game

## TL;DR

This paper analyzes the Nash equilibria in a three-player Bayesian quantum Prisoner's Dilemma, revealing complex structures influenced by entanglement and probabilistic types, offering insights into quantum advantages in game theory.

## Contribution

It characterizes the Nash equilibria in a three-player Bayesian quantum game, highlighting the impact of entanglement and probabilistic types on equilibrium structures.

## Key findings

- Rich Nash equilibrium structures with phase relationships.
- Quantum Bayesian game advantages over classical versions.
- Potential for referee-controlled equilibrium selection.

## Abstract

Quantum games with incomplete information can be studied within a Bayesian framework. We consider a version of prisoner's dilemma (PD) in this framework with three players and characterize the Nash equilibria. A variation of the standard PD game is set up with two types of the second prisoner and the first prisoner plays with them with probability p and 1-p respectively. The Bayesian nature of the game manifests in the uncertainty that the first prisoner faces about his opponent's type which is encoded either in a classical probability or in the amplitudes of a wave function. Here, we consider scenarios with asymmetric payoffs between the first and second prisoner for different values of the probability, p, and the entanglement. Our results indicate a class of Nash equilibria (NE) with rich structures, characterized by a phase relationship on the strategies of the players. The rich structure that can be exploited by the referee to set up rules of the game to push the players towards a specific class of NE. These results provide a deeper insight into the quantum advantages of Bayesian games over their classical counterpart.