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Axioms

Research on Repeated Quantum Games with Public Goods under Strong Reciprocity

Simo Sun, Yadong Shu, Jinxiu Pi, Die Zhou

Published Nov 10, 2023• 1 citation
Publisher

Abstract

We construct a repeated quantum game with public goods by the quantum entanglement and strong reciprocity. By using the paradigm of quantum game analysis, we obtain that both the quantum entanglement and strong reciprocity are helpful to intensify cooperation and achieving Pareto optimality. In addition, under an example which is greenhouse green viable planting industry, we verify that the quantum entanglement and strong reciprocity have a positive role in cooperation.

DOI
10.3390/axioms12111044

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We construct a repeated quantum game with public goods by the quantum entanglement and strong reciprocity. By using the paradigm of quantum game analysis, we obtain that both the quantum entanglement and strong reciprocity are helpful to intensify cooperation and achieving Pareto optimality. In addition, under an example which is greenhouse green viable planting industry, we verify that the quantum entanglement and strong reciprocity have a positive role in cooperation.
Repeated quantum game theory addresses long term relations among players who choose quantum strategies. In the conventional quantum game theory, only single round quantum games or at most finitely repeated games have been studied, therefore nothing is known for infinitely repeated quantum game. Investigating infinitely repeated games is crucial since finitely repeated games does not much differ from single round games. In this work we establish the concept of general repeated quantum games and show the Quantum Folk Theorem, which claims that by iterating a game one can find an equilibrium strategy of the game and receive reward that is not obtained by a Nash equilibrium of the corresponding single round quantum game. A significant difference between repeated quantum prisoner's dilemma and repeated classical prisoner's dilemma is that the classical Pareto optimal solution is not always an equilibrium of the repeated quantum game when entanglement is sufficiently strong. In addition we present several concrete equilibrium strategies of the repeated quantum prisoner's dilemma.
Quantum generalizations of conventional games broaden the range of available strategies, which can help improve outcomes for the participants. With many players, such quantum games can involve entanglement among many states which is difficult to implement, especially if the states must be communicated over some distance. This paper describes a quantum mechanism for the economically significant nnn-player public goods game that requires only two-particle entanglement and is thus much easier to implement than more general quantum mechanisms. In spite of the large temptation to free ride on the efforts of others in this game, two-particle entanglement is sufficient to give near optimal expected payoff when players use a simple mixed strategy for which no player can benefit by making different choices. This mechanism can also address some heterogeneous preferences among the players.
Quantum entanglement has emerged as a new resource to enhance cooperation and remove dilemmas. This paper aims to explore conditions under which full cooperation is achievable even when the information of payoff is incomplete. Based on the quantum version of the extended classical cash in a hat game, we demonstrate that quantum entanglement may be used for achieving full cooperation or avoiding moral hazards with the reasonable profit distribution policies even when the profit is uncertain to a certain degree. This research further suggests that the fairness of profit distribution should play an important role in promoting full cooperation. It is hopeful that quantum entanglement and fairness will promote full cooperation among distant people from various interest groups when quantum networks and quantum entanglement are accessible to the public.
Quantum generalizations of conventional games exploit entangled states to improve performance. With many players, quantum games can require entangling many states. Such entanglement is difficult to implement, especially if the states must be communicated over some distance. To simplify possible implementations, we examine some quantum versions of social dilemma games and show their use of entanglement can be substantially reduced by randomly replacing some of the entangled states by unentangled ones. For the example of public goods games, we identify a unique Nash equilibrium invariant with respect to the amount of this replacement. We also show players obtain no advantage from adding more entanglement to states which they control. With many players, a fairly small number of entangled states can give nearly as good performance as using the full number of such states.
Purpose The current paper is a brief review of the emerging field of quantum-like modelling in game theory. This paper aims to explore several quantum games, which are superior compared to their classical counterparts, which means either they give rise to superior Nash equilibria or they make the game fairer. For example, quantum Prisoners Dilemma generates Pareto superior outcomes as compared to defection outcome in the famous classical case. Again, a quantum-like version of cards game can make the game fairer, increasing the chance of winning of players who are disadvantaged in the classical case. This paper explores all the virtues of simple quantum games, also highlighting some findings of the authors as regards Prisoners Dilemma game. Design/methodology/approach As this is a general review paper, the authors have not demonstrated any specific mathematical method, rather explored the well-known quantum probability framework, used for designing quantum games. They have a short appendix which explores basic structure of Hilbert space representation of human decision-making. Findings Along with the review of the extant literature, the authors have also highlighted some new findings for quantum Prisoners Dilemma game. Specifically, they have shown in the earlier studies (which are referred to here) that a pure quantum entanglement set up is not needed for designing better games, even a weaker condition, which is classical entanglement is sufficient for producing Pareto improved outcomes. Research limitations/implications Theoretical research, with findings and implications for future game designs, it has been argued that it is not always needed to have true quantum entanglement for superior Nash Equilibria. Originality/value The main purpose here is to raise awareness mainly in the social science community about the possible applications of quantum-like game theory paradigm. The findings related to Prisoners Dilemma game are, however, original.
A quantum Cournot game whose classical form game has multiple Nash equilibria is examined. Although the classical equilibria fail to be Pareto optimal, the quantum equilibrium exhibits the following two properties: (i) if the measurement of entanglement between strategic variables chosen by the competing firms is sufficiently large, the multiplicity of equilibria vanishes, and (ii) the more strongly the strategic variables are entangled, the more closely the unique equilibrium approaches to the optimal one.