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Nonlocal correlations: Fair and Unfair Strategies in Conflicting Bayesian Game

Arup Roy, Amit Mukherjee, Tamal Guha, Sibasish Ghosh A, Some Sankar Bhattacharya, Manik Banik

Published Jan 11, 2016• 0 citations
OpenAlex

Abstract

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.

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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.
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.
The correlations, established through quantum sources, have been demonstrated to outperform classical ones in the game theory. As shown in the game firstly proposed in [Phys. Rev. Lett. 114, 020401 (2015)], the considered conflicting Bayesian game can be converted into a common interest one and end in a better solution than all classical possible when the players share a maximally entangled state. In this paper, motivated by the possible constraints on the states in practical applications, we would like to discuss more general quantum advice and analyze the effects on the aforementioned Bayesian game.
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.
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.
We analyse different Bayesian games where payoffs of players depend on the types of players involved in a two-player game. The dependence is assumed to commensurate with the CHSH game setting. For this, we consider two different types of each player (Alice and Bob) in the game, thus resulting in four different games clubbed together as one Bayesian game. Considering different combinations of common interest, and conflicting interest coordination and anti-coordination games, we find that quantum strategies are always preferred over classical strategies if the shared resource is a pure non-maximally entangled state. However, when the shared resource is a class of mixed state, then quantum strategies are useful only for a given range of the state parameter. Surprisingly, when all conflicting interest games (Battle of the Sexes game and Chicken game) are merged into the Bayesian game picture, then the best strategy for Alice and Bob is to share a set of non-maximally entangled pure states. We demonstrate that this set not only gives higher payoff than any classical strategy, but also outperforms a maximally entangled pure Bell state, mixed Werner states, and Horodecki states. We further propose the representation of a special class of Bell inequality- tilted Bell inequality, as a common as well as conflicting interest Bayesian game. We thereafter, study the effect of sharing an arbitrary two-qubit pure state and a class of mixed state as quantum resource in those games; thus verifying that non-maximally entangled states with high randomness help attain maximum quantum benefit. Additionally, we propose a general framework of a two-player Bayesian game for d-dimensions Bell-CHSH inequality, with and without the tilt factor.