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International Journal of Parallel Emergent and Distributed Systems

A quantum prisoner’s dilemma cellular automaton with probabilistic updating

Ramón Alonso‐Sanz

Published Oct 28, 2015• 2 citations
Publisher

Abstract

The effect of variable entangling on the dynamics of a spatial quantum formulation of the iterated prisoner’s dilemma game is studied in this work. The game is played in the cellular automata manner, i.e. with local and synchronous interaction. The effect of probabilistic updating is assessed in fair and unfair contests, both in the two and three parameter strategy spaces.

DOI
10.1080/17445760.2015.1096360

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The effect of variable entangling on the dynamics of a spatial quantum formulation of the iterated prisoner’s dilemma game is studied in this work. The game is played in the cellular automata manner, i.e. with local and synchronous interaction. The effect of probabilistic updating is assessed in fair and unfair contests, both in the two and three parameter strategy spaces.
The dynamics of a spatial quantum formulation of the iterated prisoner's dilemma game is studied in this work. The game is played in the cellular automata manner, i.e. with local and synchronous interaction. The evolution is driven by the imitation of the best neighbour. It is shown how spatial structure enables in fully quantum contests a dramatically rapid emergence of mutual cooperation. In unfair contests, such as quantum versus classic players, the velocity in which the quantum players take advantage of its privileged status is also very notable.
Decision-making by the two negotiating parties is simulated by a prisoner's dilemma game.The game is formulated in a quantum manner, where players strategies are unitary transformations of qubits built over the basis of opposite decision options.Quantum strategies are correlated through the mechanism of quantum entanglement and the result of the game is obtained by the collapse of the resulting transformed state.The range of strategies allowed for quantum players is richer than in case of a classical game and therefore the result of the game can be better optimized.On the other hand, the quantum game is save against eavesdropping and the players can be assured that this type of quantum arbitration is fair.We show that quantum prisoner's dilemma has more favorable Nash equilibria than its classical analog and they are close to the Pareto optimal solutions.Some economical examples of utilizing quantum game Nash equilibria are proposed.
Classic game theory is an important field with a long tradition of useful results. Recently, the quantum versions of classical games, such as the Prisoner's Dilemma (PD), have attracted a lot of attention. Similarly, state machines and specifically finite automata have also been under constant and thorough study for plenty of reasons. The quantum analogues of these abstract machines, like the quantum finite automata, have been studied extensively. In this work, we examine some well-known game conditional strategies that have been studied within the framework of the repeated PD game. Then, we try to associate these strategies to proper quantum finite automata that receive them as inputs and recognize them with probability 1, achieving some interesting results. We also study the quantum version of PD under the Eisert-Wilkens-Lewenstein scheme, proposing a novel conditional strategy for the repeated version of this game.
In this paper, we extend the quantum game theory of Prisoner’s Dilemma to the N-player case. The final state of quantum game theory of N-player Prisoner’s Dilemma is derived, which can be used to investigate the payoff of each player. As demonstration, two cases (2-player and 3-player) are studied to illustrate the superiority of quantum strategy in the game theory. Specifically, the non-unique entanglement parameter is found to maximize the total payoff, which oscillates periodically. Finally, the optimal strategic set is proved to depend on the selection of initial states.