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arXiv:2607.09477

Playing Bayesian games better with separable quantum states than with any classical correlation

Yaqing Xy Wang, Giannicola Scarpa, Andreas Winter

quant-phSubmitted Jul 10, 2026• Announced Jul 13, 2026• 0 citations
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Abstract

Bayesian games, also known as games of incomplete information, are a fruitful arena for exploring the impact of correlations on a set of independent agents (players) via the game equilibria to which they give rise. It was realised some time ago that quantum states shared between the players can lead to new and beneficial equilibria, compared to classical correlation. While until now examples of this effect required an entangled state, here we show that even separable states can create new, genuinely quantum equilibria in games, that are advantageous with respect to all classically correlated equilibria. This shows that non-classical correlations beyond entanglement are indeed a resource, even in otherwise entirely classical situations. Our result brings quantum advantage in games significantly closer to possible realisation.

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Bayesian games, also known as games of incomplete information, are a fruitful arena for exploring the impact of correlations on a set of independent agents (players) via the game equilibria to which they give rise. It was realised some time ago that quantum states shared between the players can lead to new and beneficial equilibria, compared to classical correlation. While until now examples of this effect required an entangled state, here we show that even separable states can create new, genuinely quantum equilibria in games, that are advantageous with respect to all classically correlated equilibria. This shows that non-classical correlations beyond entanglement are indeed a resource, even in otherwise entirely classical situations. Our result brings quantum advantage in games significantly closer to possible realisation.
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.
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.
We investigate what quantum advantages can be obtained in multipartite non-cooperative games by studying how different types of quantum resources can lead to new Nash equilibria and improve social welfare — a measure of the quality of an equilibrium. Two different quantum settings are analysed: a first, in which players are given direct access to an entangled quantum state, and a second, which we introduce here, in which they are only given classical advice obtained from quantum devices. For a given game G , these two settings give rise to different equilibria characterised by the sets of equilibrium correlations Qcorr(G) and Q(G) , respectively. We show that Q(G)⊆Qcorr(G) , and by exploiting the self-testing property of some correlations, that the inclusion is strict for some games G . We make use of SDP optimisation techniques to study how these quantum resources can improve social welfare, obtaining upper and lower bounds on the social welfare reachable in each setting. We investigate, for several games involving conflicting interests, how the social welfare depends on the bias of the game and improve upon a separation that was previously obtained using pseudo-telepathic solutions.
Game theory is a well established branch of mathematics whose formalism has a vast range of applications from the social sciences, biology, to economics. Motivated by quantum information science, there has been a leap in the formulation of novel game strategies that lead to new (quantum Nash) equilibrium points whereby players in some classical games are always outperformed if sharing and processing joint information ruled by the laws of quantum physics is allowed. We show that, for a bipartite non zero-sum game, input local quantum correlations, and separable states in particular, suffice to achieve an advantage over any strategy that uses classical resources, thus dispensing with quantum nonlocality, entanglement, or even discord between the players' input states. This highlights the remarkable key role played by pure quantum coherence at powering some protocols. Finally, we propose an experiment that uses separable states and basic photon interferometry to demonstrate the locally-correlated quantum advantage.
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.
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.