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Physica Scripta

Quantum games with separated entanglements

Nengfa Wang, Zhe Yang

Published Oct 30, 2023• 2 citations
Publisher

Abstract

Abstract In this paper, we propose quantum games with continuous-variable strategies and separated entanglements, and analyze the outcomes of quantum games with respect to entanglement levels. We find that the entanglement level of own environment has a positive role in own quantum equilibrium supply quantities, and the properties of quantum equilibrium utilities are determined by the entanglement levels of two environments. Finally, we provide some numerical evidences to explain quantum games with separated entanglements.

DOI
10.1088/1402-4896/ad0819

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Abstract In this paper, we propose quantum games with continuous-variable strategies and separated entanglements, and analyze the outcomes of quantum games with respect to entanglement levels. We find that the entanglement level of own environment has a positive role in own quantum equilibrium supply quantities, and the properties of quantum equilibrium utilities are determined by the entanglement levels of two environments. Finally, we provide some numerical evidences to explain quantum games with separated entanglements.
Quantum games has becoming a new arisen branch of the quantum information. Du etc. utilize a two mode electromagnetic fileds in discussing the continuous-variable quantum games. From the model, they realized the entanglement in the game which might lead superior consults than the coresponding classical games. A multiplayer quantum game that the players can access to a continuous set of strategies is studied. By utilizing a kind of multi-mode electromagnetic fileds and let every two mode of the fields entangled pairwise. We discovered, base on this scheme, the conclusion above could be extended to arbitrary number of participators. With the increasing of entangle parameter, The sum of the profits will also increase monotonously. When each two-mode electromagnetic fileds entangle to maximum, i.e. EPR state, they will also attain highest profits.
Coopetition is defined as the existence of simultaneous competition and cooperation between the same set of players, leading to entanglement of payoffs and actions of the players. This paper provides insights into the game theoretical application of quantum games to model simultaneity and entanglement that occur in coopetition. Modeling as a quantum game also allows for a larger action space and hence new equilibrium that may not have existed earlier. The impact of the level of entanglement on the equilibrium of the game can also be studied. We demonstrate the same through an example of two players who currently compete in the domestic market and are considering cooperating simultaneously in the international market. They need to determine the equilibrium strategy to adopt under coopetition that maximizes their payoffs. We also arrive at how to ensure that the quantum strategy is the equilibrium strategy for both players, namely, how to design the quantum strategy and how to define the unitary operator.
This work, based on the author's MA thesis, concentrates on simultaneous move quantum games of two players. A numerical algorithm based on the method of best response functions, designed to search for pure strategy Nash equilibrium in quantum games is constructed. It is then used show that there is a critical degree of entanglement above which a pure strategy Nash Equilibrium does not exist. The behavior of the two payoffs approach the cooperative classical ones. Bayesian quantum games are defined, and it is shown that under certain conditions, there {\it is} a pure strategy Nash equilibrium in such games even when entanglement is maximal. The basic ingredients of a quantum game based on a two-players {\it three} decisions classical games are presented. This requires the definition of trits (instead of bits) and quantum trits (instead of quantum bits). It is shown 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. An entanglement operator transforming a pure two qutrit state into a maximally entangled one is explicitly constructed.
Game theory is the mathematical framework for analyzing strategic interactions in conflict and competition situations.In recent years quantum game theory has earned the attention of physicists, and has emerged as a branch of quantum information theory [1].With the aid of entanglement and linear superposition of strategies, quantum games are shown to yield significant advantage over their classical counterparts.In this paper we explore optimal and equilibrium solutions to quantum minority games.Initial states with different level of entanglement are investigated.Focus will be on 4 and 6 player games with some N -player generalizations.
Game theory is the mathematical framework for analyzing strategic interactions in conflict and competition situations. In recent years quantum game theory has earned the attention of physicists, and has emerged as a branch of quantum information theory [1]. With the aid of entanglement and linear superposition of strategies, quantum games are shown to yield signifcant advantage over their classical counterparts. In this paper we explore optimal and equilibrium solutions to quantum minority games. Initial states with different level of entanglement are investigated. Focus will be on 4 and 6 player games with some N-player generalizations.