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Quantum Games and Quantum Strategies. (arXiv:quant-ph/9806088v4 UPDATED)

Jens Eisert, Martin Wilkens, Maciej Lewenstein

Published Mar 15, 2020• 0 citations
OpenAlex

Abstract

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.

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We present a perspective on quantum games that focuses on the physical aspects of the quantities that are used to implement a game. If a game is to be played, it has to be played with objects and actions that have some physical existence. We call such games playable. By focusing on the notion of playability for games we can more clearly see the distinction between classical and quantum games and tackle the thorny issue of what it means to quantize a game. The approach we take can more properly be thought of as gaming the quantum rather than quantizing a game and we find that in this perspective we can think of a complete quantum game, for a given set of preferences, as representing a single family of quantum games with many different playable versions. The versions of Quantum Prisoners Dilemma presented in the literature can therefore be thought of specific instances of the single family of Quantum Prisoner's Dilemma with respect to a particular measurement. The conditions for equilibrium are given for playable quantum games both in terms of expected outcomes and a geometric approach. We discuss how any quantum game can be simulated with a classical game played with classical coins as far as the strategy selections and expected outcomes are concerned.
This paper investigates Nash equilibria in pure strategies for quantum approach to the Prisoner's Dilemma. The quantization process involves extending the classical game by introducing two additional unitary strategies. We consider five classes of such quantum games, which remain invariant under isomorphic transformations of the classical game. For each class, we identify and analyse all possible Nash equilibria. Our results reveal the complexity and diversity of strategic behaviour in the quantum setting, providing new insights into the dynamics of classical decision-making dilemmas. In the case of the standard Prisoner's Dilemma, the resulting Nash equilibria of quantum extensions are found to be closer to Pareto optimal solutions than those of the classical equilibrium.
In a normal form, prisoners' dilemma (PD) is represented by a payoff matrix showing players' strategies and payoffs. To obtain the distinguishing trait and strategic form of PD, certain constraints are imposed on the elements of its payoff matrix. We quantize PD by a generalized quantization scheme to analyze its strategic behavior in the quantum domain. The game starts with a general entangled state of the form and the measurement for payoffs is performed in entangled and product bases. We show that for both measurements, there exist respective cutoff values of entanglement of the initial quantum state up to which the strategic form of the game remains intact. Beyond these cutoffs the quantized PD behaves like the chicken game (CG) up to another cutoff value. For the measurement in the entangled basis the dilemma is resolved for sinξ > 1/7 with Q ⊗ Q as a Nash Equilibrium (NE). However, the quantized game behaves like PD when sinξ > 1/3; whereas in the range 1/7 < sinξ < 1/3 it behaves like CG with Q ⊗ Q as an NE. For the measurement in the product basis the quantized PD behaves like classical PD for with D ⊗ D as an NE. In region , the quantized PD behaves like classical CG with C ⊗ D and D ⊗ C as NEs.
The concept of Nash equilibria in pure strategies for quantum extensions of the general form of the Prisoner's Dilemma game is investigated. The process of quantization involves incorporating two additional unitary strategies, which effectively expand the classical game. We consider five classes of such quantum games, which remain invariant under isomorphic transformations of the classical game. The resulting Nash equilibria are found to be more closely aligned with Pareto-optimal solutions than those of the conventional Nash equilibrium outcome of the classical game. Our results demonstrate the complexity and diversity of strategic behavior in the quantum setting, providing new insights into the dynamics of classical decision-making dilemmas. In particular, we provide a detailed characterization of strategy profiles and their corresponding Nash equilibria, thereby extending the understanding of quantum strategies' impact on traditional game-theoretical problems.