Quantum Advantage in Bayesian Games

Quantum advantage in Bayesian games, or games with incomplete information, refers to the larger set of correlated equilibrium outcomes that can be obtained by using quantum rather than classical mechanisms. A mechanism (a stoplight, jointly controlled lottery, sunspots, etc.) is used as a correlation device. It sends to the players (either identical or different) random signals which they can use to coordinate their choice of actions. In a quantum mechanism, coordination is achieved through the (anti-)correlated measurements of entangled particles. It is a consequence of the laws of quantum mechanics that the class of correlations thus afforded is a superset of the classical one. (In particular, Bell inequalities do not have to hold.)

Previous examples of the use of quantum mechanisms in Bayesian games go under the title quantum pseudo-telepathy. By using measurements of entangled particles (spin of electron-positron pairs, polarization of photons, etc.), the players in the Mermin–Peres magic square game or the Greenberger–Horne–Zeilinger game can obtain a common payoff that is higher than that afforded by any classical mechanism. “Pseudo-telepathy” refers to the fact that improved coordination is achieved in spite of the fact that the measurements cannot be used by the players for communicating their intentions; the laws of physics categorically forbid that.

Astonishing as they are, these examples only scratch the surface of using quantum mechanisms as part of correlated equilibria in Bayesian games. For one thing, there is a sense in which they are not actually about games. What they truly show is that by using a quantum mechanism it is possible to get a joint distribution of type and action profiles that is not otherwise achievable. In other words, these examples really concern just the game forms. The common payoff function only helps presenting this fact more vividly. Mathematically, it indicates a separating hyperplane: classical distributions on one side, the quantum-mechanical one on the other.

A second, and more profound, sense in which these previous examples are limited is that there is a fundamental difference between the special case of games with identical payoff functions and general Bayesian games, where payoffs may differ not only across players but also across player types. The difference is that the implementation of specific correlated equilibrium outcomes generally requires a mechanism that is capable, not only of signaling the players, but also of doing so selectively, with different types of the same player getting different signals. Classical mechanisms can do that only if they know – or are told – the players’ types. But for quantum mechanisms, this is not so. As this paper shows, it is possible in this case to rely on the players themselves “reading” in their signal (one of two entangled particles) only the information meant for their type. This possibility relies on the inherently destructive nature of measurements in quantum mechanics, which forces the players to choose what part of the “encoded” information to read. In equilibrium, they choose the part meant for them.