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Foundations of Physics

Quantum-Like Model for Decision Making Process in Two Players Game

Masanari Asano, Masanori Ohya, Andrei Yu. Khrennikov

Published Apr 12, 2010• 66 citations
Publisher
DOI
10.1007/s10701-010-9454-y

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The reachability of quantum games refers to the ability of two players to reach a certain subspace of Hilbert spaces in a quantum game modeled by a newly proposed quantum game process (QGP). We refer to this as quantum reachability games (QRGs). Reachability analysis is crucial in studying the strategic behavior and decision-making processes of players in quantum games, as it aids in devising optimal strategies to achieve desired objectives. In a QRG, players make decisions based on their quantum strategies related to decision epochs, which are represented by quantum operations or quantum measurements. We present an algorithm for evaluating two-player strategies, and investigate the optimal QRG for two players, where the goal of player 1 is to maximize his/her probability of reaching a given target subspace, while player 2 aims to minimize it. We show that any optimal QRG are determined, i.e., both of the two players have the same probability of winning in non-cooperative reachability games. Furthermore, our results also demonstrate that reachability games in the quantum world can be undecidable in certain cases. These findings are significantly distinct from those in the probabilistic or classical scenarios. Finally, we present three case studies, penny flipover, ambushing quantum walk and peer-to-peer scheme game on quantum network, to illustrate the potential applications and implications of QRGs.
In classical game theory, the rational behavior for the player is to make the decision which is approaching Nash equilibrium. The Prisoner's Dilemma, which is a canonical game, is often used to present the rationality. In real experiment in cognitive psychology which were performed by Tversky and Shafir, the statistical data show the existence of the irrational behaviors in reality. The phenomenon is called disjunction effect. To explain why it probably happens, we review the Asano-Khrennikov-Ohya model which is the mathematical modeling of the process of decision making in the game of Prisoner's Dilemma. It applies only the mathematical apparatus of quantum mechanics to the decision making process rather than the quantum physical model. In this paper, we present several numerical simulations for the Asano-Khrennikov-Ohya model together with the graphs of the von Neumann entropy for the solutions. By analyzing the simulation results, we explicitly and numerically present the existence of the irrational behavior for the player which is generated by the Asano-Khrennikov-Ohya model.
In quantum game theory, one of the most intriguing and important questions is, "Is it possible to get quantum advantages without any modification of the classical game?" The answer to this question so far has largely been negative. So far, it has usually been thought that a change of the classical game setting appears to be unavoidable for getting the quantum advantages. However, we give an affirmative answer here, focusing on the decision-making process (we call 'reasoning') to generate the best strategy, which may occur internally, e.g., in the player's brain. To show this, we consider a classical guessing game. We then define a one-player reasoning problem in the context of the decision-making theory, where the machinery processes are designed to simulate classical and quantum reasoning. In such settings, we present a scenario where a rational player is able to make better use of his/her weak preferences due to quantum reasoning, without any altering or resetting of the classically defined game. We also argue in further analysis that the quantum reasoning may make the player fail, and even make the situation worse, due to any inappropriate preferences.