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arXiv:1802.00260

When quantum games can be played classically: in support of van Enk-Pike's assertion

Berry Groisman

Published Feb 1, 2018• 1 citation
arXivHTMLPDF

Abstract

N. Vyas and C. Benjamin (arXiv:1701.08573[quant-ph]) propose a new mixed strategy for the (quantum) Hawk-Dove and Prisoners' Dilemma games and argue that this strategy yields payoffs, which cannot be obtained in the corresponding classical games. They conclude that this refutes the earlier assertion by S.J. van Enk and R. Pike that the quantum equilibrium solution is present in a corresponding extended classical game. This paper argues that the scheme suggested by N. Vyas and C. Benjamin changes the rules of the original game, and hence it does not refute the argument put forward by van Enk and Pike.

DOI
10.48550/arxiv.1802.00260

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N. Vyas and C. Benjamin (arXiv:1701.08573[quant-ph]) propose a new mixed strategy for the (quantum) Hawk-Dove and Prisoners' Dilemma games and argue that this strategy yields payoffs, which cannot be obtained in the corresponding classical games. They conclude that this refutes the earlier assertion by S.J. van Enk and R. Pike that the quantum equilibrium solution is present in a corresponding extended classical game. This paper argues that the scheme suggested by N. Vyas and C. Benjamin changes the rules of the original game, and hence it does not refute the argument put forward by van Enk and Pike.
S. J. van Enk and R. Pike in PRA 66, 024306 (2002), argue that the equilibrium solution to a quantum game isn't unique but is already present in the classical game itself. In this work, we debunk this assertion by showing that a random strategy in a particular quantum (Hawk-Dove) game is unique to the quantum game. In other words the equilibrium solution of the quantum Hawk-Dove game can not be obtained in the classical Hawk-Dove game. Moreover, we provide an analytical solution to the quantum 2×22\times22×2 strategic form Hawk-Dove game using random mixed strategies. The random strategies which we describe are evolutionary stable implying both Pareto optimality and Nash equilibrium with their payoff's classically unobtainable.
A new approach to play games quantum mechanically is proposed. We consider two players who perform measurements in an EPR-type setting. The payoff relations are defined as functions of correlations, i.e. without reference to classical or quantum mechanics. Classical bi-matrix games are reproduced if the input states are classical and perfectly anti-correlated, that is, for a classical correlation game. However, for a quantum correlation game, with an entangled singlet state as input, qualitatively different solutions are obtained. For example, the Prisoners' Dilemma acquires a Nash equilibrium if both players apply a mixed strategy. It appears to be conceptually impossible to reproduce the properties of quantum correlation games within the framework of classical games.
A new approach to play games quantum mechanically is proposed. We consider two players who perform measurements in an EPR-type setting. The payoff relations are defined as functions of correlations , i.e. without reference to classical or quantum mechanics. Classical bi-matrix games are reproduced if the input states are classical and perfectly anti-correlated, that is, for a classical correlation game. However, for a quantum correlation game, with an entangled singlet state as input, qualitatively different solutions are obtained. For example, the Prisoners' Dilemma acquires a Nash equilibrium if both players apply a mixed strategy. It appears to be conceptually impossible to reproduce the properties of quantum correlation games within the framework of classical games.
A quantized version of the prisoner’s dilemma game based on the Marinatto–Weber quantization scheme is introduced. A four-qubit-basis initial entangled quantum state is assumed. Two domains for Nash equilibria are observed in terms of the degree of entanglement of the game. Each Nash equilibrium point is found to be Pareto optimal in its corresponding domain. Therefore, there is a unique solution to the game dilemma in each domain of the entanglement parameter. The maximally entangled limit is studied, and the players’ payoffs are found to equal the average of the payoff matrix independently of their choices. Consequently, any combination of players’ choices is a solution to the dilemma of the game. The nonentangled limits are also studied and found to coincide with the classical mixed-strategy game. Our results are equivalent to those of the noisy schemes of the quantum prisoner’s dilemma game. In our scheme, entanglement can be interpreted as a noise.