On quantum game approach to correlated equilibrium

Alan Deckelbaum

We study the scenario where the players of a classical complete information game initially share an entangled pure quantum state. Each player may perform arbitrary local operations on his own qubits, but no direct communication is allowed. In this framework, we define the concept of quantum correlated equilibrium (QCE) for both normal and extensive form games of complete information. We show that in a normal form game, any outcome distribution implementable by a QCE can also be implemented by a classical correlated equilibrium (CE). We prove that the converse is surprisingly false: we give an example of an outcome distribution of a normal form game which is implementably by a CE, yet we prove that in any attempted quantum protocol beginning with a partition of a pure quantum state, at least one of the players will have incentive to deviate. We extend our analysis to extensive form games, and find that the relation between classical and quantum correlated equilibria becomes less clear. We compare the outcome distributions implementable in our quantum model to those implementable by a classical extensive form correlated equilibrium (EFCE). For example, we show that there exists an extensive form complete information game and a distribution of outcomes which can be implemented by a QCE but not by any EFCE, in contrast to the result for normal form games. We also consider the concept of an immediate-revelation extensive form correlated equilibrium (IR-EFCE) and compare the power of IR-EFCE to EFCE and to QCE.

Indranil Ghosh

Alan Deckelbaum

We ask whether players of a classical game can partition a pure quantum state to implement classical correlated equilibrium distributions. The main contribution of this work is an impossibility result: we provide an example of a classical correlated equilibrium that cannot be securely implemented without useful information leaking outside the system. We study the model where players of a classical complete information game initially share an entangled pure quantum state. Players may perform arbitrary local operations on their subsystems, but no direct communication (either quantum or classical) is allowed. We explain why, for the purpose of implementing classical correlated equilibria, it is desirable to restrict the initial state to be pure and to restrict communication. In this framework, we define the concept of pure quantum correlated equilibrium (PQCE) and show that in a normal form game, any outcome distribution implementable by a PQCE can also be implemented by a classical correlated equilibrium (CE), but that the converse is false. We extend our analysis to extensive form games, and compare the power of PQCE to extensive form classical correlated equilibria (EFCE) and immediate-revelation extensive form correlated equilibria (IR-EFCE).

Indranil Ghosh

Piotr Frąckiewicz

Players’ choices in quantum game schemes are often correlated by a quantum state. This enables players to obtain payoffs that may not be achievable when classical pure or mixed strategies are used. On the other hand, players’ choices can be correlated due to a classical probability distribution, and if no player benefits by a unilateral deviation from the vector of recommended strategies, the probability distribution is a correlated equilibrium. The aim of this paper is to investigate relation between correlated equilibria and Nash equilibria in the MW-type schemes for quantum games.

Indranil Ghosh