A game-theoretic setting provides a mathematical basis for analysis of strategic interaction among competing agents and provides insights into both classical and quantum decision theory and questions of strategic choice. An outstanding mathematical question is to understand the conditions under which a classical game-theoretic setting can be transformed to a quantum game, and under which conditions there is an equivalence. In this paper, we consider quantum games as those that allow non-factorizable probabilities. We discuss two approaches for obtaining a non-factorizable game and study the outcome of such games. We demonstrate how the standard version of a quantum game can be analysed as a non-factorizable game and determine the limitations of our approach.
A game-theoretic setting provides a mathematical basis for analysis of strategic interaction among competing agents and provides insights into both classical and quantum decision theory and questions of strategic choice. An outstanding mathematical question is to understand the conditions under which a classical game-theoretic setting can be transformed to a quantum game, and under which conditions there is an equivalence. In this paper, we consider quantum games as those that allow non-factorizable probabilities. We discuss two approaches for obtaining a non-factorizable game and study the outcome of such games. We demonstrate how the standard version of a quantum game can be analysed as a non-factorizable game and determine the limitations of our approach.
A quantum game-theoretic setting enables the analysis of strategic interaction among agents with access to quantum resources. One of the original motivations for re-expressing quantum algorithms, and quantum communication protocols, in terms of competing quantum agents was to provide new insights into their workings. Looking closely, one finds that quantum strategic interactions utilize quantum probabilities and such probabilities are realizable in quantum mechanical experiments that allow implementation of unitary transformations, quantum measurements etc. This paper presents a perspective on a quantum game-theoretic framework involving quantum probabilities, which are not subject to the strong factorizability constraints that classical probabilities are found to possess. We suggest that a quantum version of a classical game can be obtained more directly if we replace the factorizability relations for probabilities in a classical game with two weaker constraints that are also obeyed by the quantum probabilities. The solutions of symmetric bimatrix games with such `non-factorizable' probabilities are then analyzed and compared to the solutions that appear in the standard version of a quantum 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.
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.
Quantum game theory (QGT) extends traditional game theory by using quantum mechanics to create new models for strategic decision-making. Traditional game theory provides effective tools for analyzing rational human interaction; yet it suffers from three main restrictions because it cannot properly handle uncertain situations, contradictory behavior, and complex relationships among agents. QGT addresses these challenges through its implementation of quantum phenomena, which enable strategic options that go beyond traditional mixed strategies. The extended system introduces fresh equilibrium solutions that allow easier resolution of standard problems. The chapter presents an organized structure to display QGT, which shows its value for strategic decision-making systems. The chapter shows fundamental concepts that link classical theories with quantum theories through its demonstration of how quantum strategies have developed from their original state to their present form, which expands traditional game models. The research investigates optimization methods, cybersecurity measures, economic systems, and artificial intelligence applications. The study identifies implementation obstacles and future research needs. The advanced QGT decision-making systems provide outstanding solutions that allow intelligent systems to interact with their environments and solve complex environmental problems.
Game theory has been studied extensively in recent centuries as a set of formal mathematical strategies for optimal decision making. This discipline improved its efficiency with the arrival, in the 20th century, of digital computer science. However, the computational limitations related to exponential time type problems in digital processors, triggered the search for more efficient alternatives. One of these choices is quantum computing. Certainly, quantum processors seem to be able to solve some of these complex problems, at least in theory. For this reason, in recent times, many research works have emerged related to the field of quantum game theory. In this paper we review the main studies about the subject, including operational requirements and implementation details. In addition, we describe various quantum games, their design strategy, and the used supporting tools. We also present the still open debate linked to the interpretation of the transformations of classical algorithms in fundamental game theory to their quantum version, with special attention to the Nash equilibrium.
Quantum games extend classical game theory into the quantum realm by incorporating quantum players and quantum strategies. Two-player quantum games have been widely studied and applied, among other areas, in optimization and decision-making. In this work, we introduce, formalize and analyze a novel class of three-player quantum games, referred to as Proxy Quantum Games (PQGs). In this setting, one player, the Controller, influences or governs the strategic operations executed by a second player, the Proxy, who, in turn, competes directly against a third player, the Opponent. The strategic influence is implemented via controlled unitary operations, whereby the Proxy’s available strategies are conditioned on the Controller’s quantum state. We further generalize the PQG framework by incorporating multiple Controllers acting on a common Proxy, thereby enabling the study of higher-order strategic interactions and multi-agent influence structures. The objective under consideration is the maximization of the Controller’s expected payoff, under various configurations of strategic control and entanglement. Comparative analyses are conducted between the PQG formulation and conventional multiplayer quantum games, highlighting key differences in strategic behavior and payoff distributions.