Nonlocal correlations: Fair and unfair strategies in Bayesian games

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 classical equilibrium strategies in common interest Bayesian games and also in conflicting interest games. However, classical equilibrium strategies can be of two types, fair and unfair. Whereas in fair equilibrium payoffs of different players are same, in unfair case they differ. Advantage of nonlocal correlation has been demonstrated over fair strategies. In this work we show that quantum strategies can outperform even the unfair classical equilibrium strategies. For this purpose we consider a class of two players games which as a special case includes the conflicting game proposed in [Phys. Rev. Lett. 114, 020401 (2015)]. These games can have both fair and unfair classical equilibria and also can have only unfair ones. We provide a simple analytic characterization of the nonlocal correlations that are advantageous over the classical equilibrium strategies in these games.Undoubtedly one of the most fundamental contradictions of Quantum mechanics (QM) with classical physics gets manifested in its nonlocal behavior. This bizarre feature of QM was first established in the seminal work of J. S. Bell [1], where he has shown that QM is incompatible with the local-realistic world view of classical physics. More precisely, Bell showed that measurement statistics of multipartite entangled quantum systems can violate an empirically testable local realistic inequality (in general called Bell type inequalities) which establishes the denial of local realism underlying QM. Since Bell's work, nonlocality remains at the center of quantum foundational research (see [2] and references therein) and it has been verified in numerous successful experiments, starting from the famous Aspect's experiment [3] to very recent loop-hole free tests [4]. Apart from foundational interest quantum nonlocal correlations have been proved to be the key resource in various device-independent protocols [5]. Very recently Brunner and Linden have established that Bell nonlocality has interesting connection with a seemingly different area of research, namely, theory of Bayesian game [6]. A Bayesian game can be played under classical equilibrium strategies which are of two types, fair equilibria and unfair equilibria. Whereas in fair equilibria payoffs of different parties are same, in unfair equilibria they differ. It has been shown that QM can provide advantageous strategies over the best classical strategies in common interest Bayesian games [6] and can also outperform the fair classical equilibrium strategies in conflicting games [7]. The aim of this present paper is to study whether nonlocal correlation can be advantageous over the classical unfair equilibrium strategies in such a games.Operationally Bell type inequalities can be best understood in terms of games involving several number of spatially separated parties. Each part...

Strong quantum solutions in conflicting-interest Bayesian games

“…However, we will see in the following section that these bounds are sufficient to reveal a feature of strategies which use PR-box correlations in which they outperform all classically correlated strategies in the parameter range 0 ≤ ǫ ≤ 5/8. Incidentally, for a different class of conflicting interest games, such a feature of strategies which use PR-box correlation was also revealed in a recent work [11].…”

“…Theorem (2) shows that in the class of conflictinginterest Bayesian games that we consider, there are several examples of games where quantum entanglement gives a much stronger advantage than any such game proposed so far [9,11]. For example, G(2/5), G(3/8), etc.…”

“…An interesting connection between Bell nonlocality [5] (which can be realized in quantum mechanics through entangled states) and Bayesian games introduced by Harsanyi [14] was established in [13]. Soon after, an explicit example of a two-party game with conflicting interests where an entangled state leads to a better solution was provided in [9] and has inspired a number of interesting works along this direction [10,11,[15][16][17][18][19]. There had also been previous attempts to devise quantum strategies for nonBayesian games, providing an advantage under certain specific restrictions [20]; however, the physical applicability of such results had been debated [21][22][23].…”

“…Next, we analyze classically correlated Nash equilibriums for our games and find that the sum of average payoff of two players in these games is equivalent to the Bell-ClauserHorne-Shimony-Holt (CHSH) expression up to scaling. Then we show that with Popescu-Rohrlich-box (PRbox) correlations the achieved Nash equilibrium has a strong feature (also noted in [11]). Finally, we discuss a quantum correlated strategy and show that in a certain parameter range there are quantum Nash equilibriums in which individual rewards to each player exceed their respective average payoff over the set of all possible classically correlated Nash equilibriums.…”

“…Several applications of nonlocal correlations exists, for example, in quantum cryptography [7] and nonlocal games of full cooperation [8]. However, applications of quantum nonlocality to achieve the best possible solution in Bayesian games of conflicting interests is a very recent development [9][10][11].…”