In a recent paper, Eisert et al. presented a quantum mechanical generalization of Prisoner's Dilemma. They asserted that the maximally entangled game exhibits a unique Nash equilibrium which yields a pay-off equivalent to cooperative behaviour. In this comment we show that their observation is incorrect: there is no Nash equilibrium in the space of deterministic quantum strategies.
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 s…
We investigate the quantization of non-zero sum games. For the particular case of the Prisoners' Dilemma we show that this game ceases to pose a dilemma if quantum strategies are allowed for. We also construct a particular quantum strategy which always gives reward if played against any classical strategy.
We propose a scheme for a quantum game based on performing an EPR type experiment and in which each player's spatial directional choices are considered as their strategies. A classical mixed-strategy game is recovered by restricting the players' choices to specific spatial trajectories. We show that for players' directional choices for which the Bell-CHSH inequality is violated, the players' payoffs in the quantum game have no mapping within the classical mixed-strategy game. The scheme provides a more direct link between classical and quantum games.
We point out a flaw in the unfair case of the quantum Prisoner's Dilemma as introduced in the pioneering Letter "Quantum Games and Quantum Strategies" of Eisert, Wilkens and Lewenstein. It is not true that the so-called miracle move therein always gives quantum Alice a large reward against classical Bob and outperforms tit-for-tat in an iterated game. Indeed, we introduce a new classical strategy that becomes Bob's dominant strategy, should Alice play the miracle move. Finally, we briefly survey the subsequent literature and turn to the 3-parameter strategic space instead of the 2-parameter on…
The explicit construction is presented of two-player game satisfying: (i) symmetry with respect to the permutation of the players; (ii) the existence of upper bound on total payoff following from Bell inequality; (iii) the existence of unfair equilibrium with total payoff saturating the above bound. The quantum counterpart of the game is considered which possesses only fair equilibria and strategies outperforming the classical ones.
The CHSH no-signalling game studies Bell nonlocality by showcasing a gap between the win rates of classical strategies, quantum-entangled strategies, and no-signalling strategies. Similarly, the CHSH* single-system game explores the advantage of irreversible processes by showcasing a gap between the win rates of classical reversible strategies, quantum reversible strategies, and irreversible strategies. The irreversible process of erasure rules supreme for the CHSH* single-system game, but this ``erasure advantage'' does not necessarily extend to every single-system game: We introduce the 32-G…
We unify and consolidate various results about non-signall-ing games, a subclass of non-local two-player one-round games, by introducing and studying several new families of games and establishing general theorems about them, which extend a number of known facts in a variety of special cases. Among these families are {\it reflexive games,} which are characterised as the hardest non-signalling games that can be won using a given set of strategies. We introduce {\it imitation games,} in which the players display linked behaviour, and which contains as subclasses the classes of variable assignmen…
We study the advantages of quantum strategies in evolutionary social dilemmas on evolving random networks. We focus our study on the two-player games: prisoner's dilemma, snowdrift and stag-hunt games. The obtained result show the benefits of quantum strategies for the prisoner's dilemma game. For the other two games, we obtain regions of parameters where the quantum strategies dominate, as well as regions where the classical strategies coexist.
We consider quantum XOR games, defined in [11], from the perspective of unitary correlations defined in [7]. We show that Connes' embedding problem has a positive answer if and only if every quantum XOR game has entanglement bias equal to the commuting bias. In particular, the embedding problem is equivalent to determining whether every quantum XOR game $G$ with a winning strategy in the commuting model also has a winning strategy in the approximate finite-dimensional model.
Linear system games are a generalization of Mermin's magic square game introduced by Cleve and Mittal. They show that perfect strategies for linear system games in the tensor-product model of entanglement correspond to finite-dimensional operator solutions of a certain set of non-commutative equations. We investigate linear system games in the commuting-operator model of entanglement, where Alice and Bob's measurement operators act on a joint Hilbert space, and Alice's operators must commute with Bob's operators. We show that perfect strategies in this model correspond to possibly-infinite-dim…
