https://doi.org/10.1140/epjp/s13360-026-07650-4
Regular Article
The possibility of fairly mitigating penalties in the quantum Prisoner’s Dilemma via classical noise channels
1
Department of Mathematics, Faculty of Women for Art, Science and Education, Ain Shams University, Cairo, Egypt
2
School of Physics, University of Chinese Academy of Sciences, Yuquan Road 19A, 100049, Beijing, China
3
Mathematics Department, Faculty of Science, Al-Azhar University, 11884, Nassr City, Cairo, Egypt
4
Mathematics Department, College of Science, University of Bahrain, PO Box 320038, Sakhir, Bahrain
5
Department of Mathematics, Aswan University, 81528, Sahari, Aswan, Egypt
a
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Received:
3
January
2026
Accepted:
1
April
2026
Published online:
27
April
2026
Abstract
The effectiveness of different types of classical noise in maximizing or minimizing the expected payoffs for two players attempting to solve the Prisoner’s Dilemma is discussed. It is shown that the behavior of the payoffs depends on the initial states of the players, specifically their symmetric or polarized nature, the classical/quantum defection strategies, and the degree of Markovianity. The results show that when both players start in identical states and use the quantum defection strategy, they receive equal payoffs that do not exceed the classical bounds of cooperation. However, when the initial states differ, the dynamics change: if one player attains the maximum payoff, the other incurs the minimum payoff, and their payoffs oscillate within their respective upper and lower bounds, with each player’s gain occurring at the expense of the other. The oscillatory behavior is dominant, originating from the continuous time-dependent unitary evolution and, in non-Markovian regimes, the backflow of information from the environment. Extremely fast oscillations are predicted when the channel strength is exponentially time-dependent due to the rapid exponential accumulation of the entangling phase. Maximizing/minimizing the expected payoffs depends on the Markovianity of the used channel. In the presence of the Modified Ornstein–Uhlenbeck (MOU) noisy channel, the upper bounds of all expected payoffs become nearly identical, regardless of whether the players adopt cooperative strategies or employ the quantum defection strategy. In the presence of Random Telegraph Noise (RTN) and MOU noisy channels, the lower bound of the payoffs increases as the game duration increases, while they are similar if the game is implemented in the presence of the Power-Law Noise (PLN) channel.
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© The Author(s), under exclusive licence to Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2026
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
