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arXiv:2311.08495 · Physical Review Research

Photonic implementation of the quantum Morra game

Andrés Ulibarrena, Alejandro Sopena, Russell Brooks, Daniel Centeno, Joseph Ho, Germán Sierra, Alessandro Fedrizzi

Published Jun 5, 2024• 3 citations
arXivHTMLPDF

Abstract

The Morra game, an age-old noncooperative game, traditionally played on one's hand, has proved to be a rich setting to study game-theoretic strategies, both classically and within the quantum realm. In this paper, we study a faithful translation of a two-player quantum Morra game, which builds on previous study by including the classical game as a special case. We propose a natural deformation of the game in the quantum regime in which Alice has a winning advantage, breaking the balance of the classical game. A Nash equilibrium can be found in some cases by employing a pure strategy, which is impossible in the classical game where a mixed strategy is always required. We prepared our states using photonic qubits on a linear optics setup, with an average deviation ≤2% with respect to the measured outcome probabilities. Finally, we discuss potential applications of the quantum Morra game to the study of quantum information and communication. Published by the American Physical Society 2024

DOI
10.1103/physrevresearch.6.023248

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The Morra game, an age-old noncooperative game, traditionally played on one's hand, has proved to be a rich setting to study game-theoretic strategies, both classically and within the quantum realm. In this paper, we study a faithful translation of a two-player quantum Morra game, which builds on previous study by including the classical game as a special case. We propose a natural deformation of the game in the quantum regime in which Alice has a winning advantage, breaking the balance of the classical game. A Nash equilibrium can be found in some cases by employing a pure strategy, which is impossible in the classical game where a mixed strategy is always required. We prepared our states using photonic qubits on a linear optics setup, with an average deviation ≤2% with respect to the measured outcome probabilities. Finally, we discuss potential applications of the quantum Morra game to the study of quantum information and communication. Published by the American Physical Society 2024
This work concentrates on simultaneous move non-cooperating quantum games. Part of it is evidently not new, but it is included for the sake self consistence, as it is devoted to introduction of the mathematical and physical grounds of the pertinent topics, and the way in which a simple classical game is modified to become a quantum game (a procedure referred to as a quantization of a classical game). The connection between game theory and information science is briefly stressed, and the role of quantum entanglement (that plays a central role in the theory of quantum games), is exposed. Armed with these tools, we investigate some basic concepts like the existence (or absence) of a pure strategy and mixed strategy Nash equilibrium and its relation with the degree of entanglement. The main results of this work are as follows: 1) Construction of a numerical algorithm based on the method of best response functions, designed to search for pure strategy Nash equilibrium in quantum games. The formalism is based on the discretization of a continuous variable into a mesh of points, and can be applied to quantum games that are built upon two-players two-strategies classical games, based on the method of best response functions. 2) Application of this algorithm to study the question of how the existence of pure strategy Nash equilibrium is related to the degree of entanglement (specified by a continuous parameter γ ). It is shown that when the classical game GC has a pure strategy Nash equilibrium that is not Pareto efficient, then the quantum game GQ with maximal entanglement (γ = π/2) has no pure strategy Nash equilibrium. By studying a non-symmetric prisoner dilemma game, it is found that there is a critical value 0γc such that for γγc there is a pure strategy Nash equilibrium and for γ≥γc there is no pure strategy Nash equilibrium. The behavior of the two payoffs as function of γ starts at that of the classical ones at (D, D) and approaches the cooperative classical ones at (C, C) (C = confess, D = don’t confess). 3) We then study Bayesian quantum games and show that under certain conditions, there is a pure strategy Nash equilibrium in such games even when entanglement is maximal. 4) We define the basic ingredients of a quantum game based on a two-player three strategies classical game. This requires the introduction of trits (instead of bits) and quantum trits (instead of quantum bits). It is proved that in this quantum game, there is no classical commensurability in the sense that the classical strategies are not obtained as a special case of the quantum strategies.
We develop a rigorous mathematical framework for quantum game theory applied to static 2x2 games, extending classical concepts to the quantum setting where players may employ arbitrary unitary operations (pure strategies) or probability measures over the continuous group SU(2) (mixed strategies). The Eisert-Wilkens-Lewenstein protocol is introduced as the standard implementation of quantum 2x2 games. We prove the existence of Nash equilibria for continuous quantum mixed strategies via a fixed-point argument, generalising the classical Nash theorem to the quantum case.
Quantum game theory is a multidisciplinary field which combines quantum mechanics with game theory by introducing non-classical resources such as entanglement, quantum operations and quantum measurement. By transferring two-player two-strategy (2 × 2) dilemma containing classical games into the quantum realm, dilemmas can be resolved in quantum pure strategies if entanglement is distributed between the players who use quantum operations. Moreover, players receive the highest sum of payoffs available in the game, which are otherwise impossible in classical pure strategies. Encouraged by the observation of rich dynamics of physical systems with many interacting parties and the power of entanglement in quantum versions of 2 × 2 games, it became generally accepted that quantum versions can be easily extended to N -player situations by simply allowing N -partite entangled states. In this article, however, we show that this is not generally true because the reproducibility of classical tasks in the quantum domain imposes limitations on the type of entanglement and quantum operators. We propose a benchmark for the evaluation of quantum and classical versions of games, and derive the necessary and sufficient conditions for a physical realization. We give examples of entangled states that can and cannot be used, and the characteristics of quantum operators used as strategies.
Game theory is a well established field with applications in many areas including the social sciences, biology, and computer science [6]. Recently, however, researchers in quantum computation have begun to study the implications of games set in the quantum domain (i.e. game theoretical situations where either one or more players have quantum computational power or where the implementation of strategies uses some form of quantum information). Such a setting allows for entanglement and superpositions of strategies and evaluation of payoffs using quantum measurement. This approach to game theory, according to some scholars in the field, could help us understand certain physical systems, improve economic markets, and develop quantum algorithms. I have conducted an investigation into quantum game theory. I have constructed simulation tools using Maple that, along with research into related work, have helped gain some insight into this new field. The important solutions in classical game theory are usually Nash equilibria. What do they look like in the quantum realm? In what
Research in quantum games has flourished during recent years. However, it seems that opinion remains divided about their true quantum character and content. For example, one argument says that quantum games are nothing but 'disguised' classical games and that to quantize a game is equivalent to replacing the original game by a different classical game. The present thesis contributes towards the ongoing debate about quantum nature of quantum games by developing two approaches addressing the related issues. Both approaches take Einstein-Podolsky-Rosen (EPR)-type experiments as the underlying physical set-ups to play two-player quantum games. In the first approach, the players' strategies are unit vectors in their respective planes, with the knowledge of coordinate axes being shared between them. Players perform measurements in an EPR-type setting and their payoffs are defined as functions of the correlations, i.e. without reference to classical or quantum mechanics. Classical bimatrix games are reproduced if the input states are classical and perfectly anti-correlated, as for a classical correlation game. However, for a quantum correlation game, with an entangled singlet state as input, qualitatively different solutions are obtained. The second approach uses the result that when the predictions of a Local Hidden Variable (LHV) model are made to violate the Bell inequalities the result is that some probability measures assume negative values. With the requirement that classical games result when the predictions of a LHV model do not violate the Bell inequalities, our analysis looks at the impact which the emergence of negative probabilities has on the solutions of two-player games which are physically implemented using the EPR-type experiments.
Game theory is central to the understanding of competitive interactions arising in many fields, from the social and physical sciences to economics. Recently, as the definition of information is generalized to include entangled quantum systems, quantum game theory has emerged as a framework for understanding the competitive flow of quantum information. Up till now, only two- and three-player quantum games have been demonstrated with restricted strategy sets. Here, we report the first experiment that implements a four-player quantum minority game over tunable four-partite entangled states encoded in the polarization of single photons. Experimental application of appropriate player strategies gives equilibrium payoff values well above those achievable in the classical game. These results are in excellent quantitative agreement with our theoretical analysis of the symmetric Pareto optimal strategies. Our results demonstrate for the first time how nontrivial equilibria can arise in a competitive situation involving quantum agents.