Quantum Advantage in Bayesian Games

Quantum advantage in Bayesian games, or games with incomplete information, refers to the larger set of correlated and communication equilibrium outcomes that can be obtained by using quantum mechanisms rather than classical ones. Earlier examples of such an advantage go under the title of quantum pseudo-telepathy. By using measurements of entangled particles, the players in the Mermin–Peres magic square game and similar games can obtain a common payoff that is higher than that afforded by any classical mechanism. However, these games are very special, in that the payoff functions are purposely chosen to favor a specific physical outcome. In general games, where payoff functions are not artificially constructed but reflect a strategic interaction of interest, obtaining a specific correlated equilibrium outcomes may require limiting the information that different player types get from the signals or messages they receive from a correlation device or mechanism. Because of the inherently destructive nature of measurements in quantum mechanics, it is well suited for this task. In a quantum correlated equilibrium, players choose what part of the information “encoded” in the quantum state to read, and choosing the part meant for their actual type is required to be incentive compatible. This requirement makes the choice of measurement analogous to the choice of report to the mediator in a communication equilibrium, with the measurement value analogous to the massage sent back from the mediator.
The paper systematically explores the advantage quantum mechanisms has over comparable classical mechanisms in correlated and communication equilibria. It identifies the specific properties of quantum mechanisms responsible for these advantages. It presents a classification of the equilibrium outcomes (both type-action distributions and equilibrium payoffs) in correlated and communication equilibria according to the kind of (classical or quantum) mechanism employed.