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arXiv:1609.03835 · Quantum Information Processing

Three-player conflicting interest games and nonlocality

Katarzyna Bolonek-Lasoń

Published Jun 19, 2017• 8 citations
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Abstract

We outline the general construction of three-player games with incomplete information which fulfil the following conditions: (i) symmetry with respect to the permutations of players; (ii) the existence of an upper bound for total payoff resulting from Bell inequalities; (iii) the existence of both fair and unfair Nash equilibria saturating this bound. Conditions (i)–(iii) imply that we are dealing with conflicting interest games. An explicit example of such a game is given. A quantum counterpart of this game is considered. It is obtained by keeping the same utilities but replacing classical advisor by a quantum one. It is shown that the quantum game possesses only fair equilibria with strictly higher payoffs than in the classical case. This implies that quantum nonlocality can be used to resolve the conflict between the players.

DOI
10.1007/s11128-017-1635-6

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We outline the general construction of three-player games with incomplete information which fulfil the following conditions: (i) symmetry with respect to the permutations of players; (ii) the existence of an upper bound for total payoff resulting from Bell inequalities; (iii) the existence of both fair and unfair Nash equilibria saturating this bound. Conditions (i)–(iii) imply that we are dealing with conflicting interest games. An explicit example of such a game is given. A quantum counterpart of this game is considered. It is obtained by keeping the same utilities but replacing classical advisor by a quantum one. It is shown that the quantum game possesses only fair equilibria with strictly higher payoffs than in the classical case. This implies that quantum nonlocality can be used to resolve the conflict between the players.
Nonlocality enables two parties to win specific games with probabilities strictly higher than allowed by any classical theory. Nevertheless, all known such examples consider games where the two parties have a common interest, since they jointly win or lose the game. The main question we ask here is whether the nonlocal feature of quantum mechanics can offer an advantage in a scenario where the two parties have conflicting interests. We answer this in the affirmative by presenting a simple conflicting interest game, where quantum strategies outperform classical ones. Moreover, we show that our game has a fair quantum equilibrium with higher payoffs for both players than in any fair classical equilibrium. Finally, we play the game using a commercial entangled photon source and demonstrate experimentally the quantum advantage.
Interesting connection has been established between two apparently unrelated concepts, namely, quantum nonlocality and Bayesian game theory. It has been shown that nonlocal correlations in the form of advice can outperform the best classical strategies in common interest Bayesian games and can also surpass the fair classical equilibrium payoffs in conflicting games. However a conflicting game may have unfair equilibrium strategies, which are the archetypal of such games and never occur in common interest games. In this work, making the connection between nonlocality and Bayesian game more striking, we show that quantum strategies can outperform even the unfair classical equilibrium strategies in conflicting games. For this purpose here we consider the conflicting game proposed in [Phys. Rev. Lett. 114, 020401 (2015)]. Moreover, we provide a complete characterization of the nonlocal correlations that are advantageous over the best fair and unfair classical strategies in the said game.
An interesting connection has been established between two apparently unrelated concepts, namely, quantum nonlocality and Bayesian game theory. It has been shown that nonlocal correlations in the form of advice can outperform classical equilibrium strategies in common-interest Bayesian games and also in conflicting-interest Bayesian games. Classical equilibrium strategies can be of two types, fair and unfair. Whereas in fair equilibrium payoffs of different players are equal, in the unfair case they differ. An advantage of nonlocal correlation has been demonstrated over fair strategies only. We show that quantum strategies can outperform even the unfair classical equilibrium strategies. For this purpose we consider a class of two-player Bayesian games. It becomes that such games can have only fair equilibria, both fair and unfair equilibria, or only unfair ones. We provide a simple analytic method to characterize the nonlocal correlations that are advantageous over the classical equilibrium strategies in these games. We also show that quantum advice provides a better social optimality solution (a relevant notion of equilibrium for the unfair case) over the classical one.
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.
Quantum entanglement has been recently demonstrated as a useful resource in conflicting-interest games of incomplete information between two players, Alice and Bob [Pappa et al., Phys. Rev. Lett. 114, 020401 (2015)]. The general setting for such games is that of correlated strategies where the correlation between competing players is established through a trusted common adviser; however, players need not reveal their input to the adviser. So far, the quantum advantage in such games has been revealed in a restricted sense. Given a quantum correlated equilibrium strategy, one of the players can still receive a higher than quantum average payoff with some classically correlated equilibrium strategy. In this work, by considering a class of asymmetric Bayesian games, we show the existence of games with quantum correlated equilibrium where the average payoff of both the players exceeds the respective individual maximum for each player over all classically correlated equilibriums.