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arXiv:2310.18067 · Journal of Physics A Mathematical and Theoretical

Quantization of two- and three-player cooperative games based on QRA

Ivan Eryganov, Jaroslav Hrdina, Aleš Návrat

Published Sep 18, 2024• 2 citations
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Abstract

Abstract In this paper, a novel quantization scheme for cooperative games is proposed. The circuit is inspired by the Eisert–Wilkens–Lewenstein protocol, which was modified to represent cooperation between players and extended to 3–qubit states. The framework of Clifford algebra is used to perform necessary computations. In particular, we use a direct analogy between Dirac formalism and Quantum Register Algebra (QRA) to represent circuits. This analogy enables us to perform automated proofs of the circuit equivalence in a simple fashion. The expected value of the Shapley value concerning quantum probabilities is employed to distribute players’ payoffs after the measurement. We study how entanglement, representing the level of pre-agreement between players, affects the final utility distribution. The paper also demonstrates how the QRA and GAALOP software can automate all necessary calculations.

DOI
10.1088/1751-8121/ad7c9c

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Abstract In this paper, a novel quantization scheme for cooperative games is proposed. The circuit is inspired by the Eisert–Wilkens–Lewenstein protocol, which was modified to represent cooperation between players and extended to 3–qubit states. The framework of Clifford algebra is used to perform necessary computations. In particular, we use a direct analogy between Dirac formalism and Quantum Register Algebra (QRA) to represent circuits. This analogy enables us to perform automated proofs of the circuit equivalence in a simple fashion. The expected value of the Shapley value concerning quantum probabilities is employed to distribute players’ payoffs after the measurement. We study how entanglement, representing the level of pre-agreement between players, affects the final utility distribution. The paper also demonstrates how the QRA and GAALOP software can automate all necessary calculations.
This work explores the application of quantum entanglement to the analysis of cooperative three-player games. The interactions between players influence their behaviour and the overall outcomes of the transferable utility (TU) cooperative game. The mathematical framework employed is the quantum register algebra (QRA), which algebraizes the Dirac formalism. By using QRA, we can simulate the game and explore how different levels of entanglement impact players. This article is part of the theme issue 'Modern applications of geometric algebra'.
This paper introduces an application of complex Clifford algebra in a representation of the quantum prisoner's dilemma. The authors propose a novel modification of the Eisert–Lewenstein–Wilkens protocol to represent a repeated version of the quantum game. This repeated modification allows to embed entanglement into players' strategy sets and to see how players will operate with it. The apparatus of complex Clifford algebra enables an intuitive representation of the suggested protocol and efficient computation of the resulting payoff functions. The presented findings provide a new point of view on the interpretation of entanglement as a measure of information transition between rounds of the game.
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.
The N-player quantum games are analyzed that use an Einstein-Podolsky-Rosen (EPR) experiment, as the underlying physical setup. In this setup, a player's strategies are not unitary transformations as in alternate quantum game-theoretic frameworks, but a classical choice between two directions along which spin or polarization measurements are made. The players' strategies thus remain identical to their strategies in the mixed-strategy version of the classical game. In the EPR setting the quantum game reduces itself to the corresponding classical game when the shared quantum state reaches zero entanglement. We find the relations for the probability distribution for N-qubit GHZ and W-type states, subject to general measurement directions, from which the expressions for the players' payoffs and mixed Nash equilibrium are determined. Players' N x N payoff matrices are then defined using linear functions so that common two-player games can be easily extended to the N-player case and permit analytic expressions for the Nash equilibrium. As a specific example, we solve the Prisoners' Dilemma game for general N ≥ 2. We find a new property for the game that for an even number of players the payoffs at the Nash equilibrium are equal, whereas for an odd number of players the cooperating players receive higher payoffs. By dispensing with the standard unitary transformations on state vectors in Hilbert space and using instead rotors and multivectors, based on Clifford's geometric algebra (GA), it is shown how the N-player case becomes tractable. The new mathematical approach presented here has wide implications in the areas of quantum information and quantum complexity, as it opens up a powerful way to tractably analyze N-partite qubit interactions.
Two-player |NNN|-strategy games quantized according to the Eisert–Lewenstein–Wilkens scheme [Phys. Rev. Lett. 83, 3077 (1999)] are considered. Group-theoretical methods are applied to the problem of finding a general form of gate operators (entanglers) under the assumption that the set of classical pure strategies is contained in the set of pure quantum ones. The role of the stability group of the initial state of the game is stressed. As an example, it is shown that maximally entangled games do not admit nontrivial pure Nash strategies. The general arguments are supported by explicit computations performed in the three-strategy case.
We present a real geometric algebra framework designed for the direct translation of the Dirac formalism into geometric algebra representations. Unlike previous approaches based on positive-definite signatures, QCA employs a split-signature construction that enables a natural realization of quantum states and operators while simplifying computational implementation. We further present an implementation of QCA using the \textit{GAALOP} software and show how quantum gates and multi-qubit systems can be efficiently represented and generated computationally. As an application, we demonstrate the use of QCA in quantum game theory, where the real-algebraic formulation provides computational advantages for modeling entangled strategies and quantum interactions. The proposed framework establishes a practical bridge between the abstract formalism of quantum computation and efficient geometric algebra implementations.