Bifurcation analysis of an asymmetric three-player three-strategy game with logit dynamics and reverse-cyclic mutations

Jie Liu; Wenjun Hu; Jie Liu; Wenjun Hu

doi:10.3934/math.2025988

AIMS Mathematics

2025, Volume 10, Issue 9: 22206-22220. doi: 10.3934/math.2025988

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Research article

Bifurcation analysis of an asymmetric three-player three-strategy game with logit dynamics and reverse-cyclic mutations

Jie Liu ¹,
Wenjun Hu ^{1,2
,
,}

1.
School of Mathematics, North University of China, Taiyuan 030051, China
2.
Department of Mathematics, Lyuliang University, Lvliang 033001, China

Received: 22 July 2025 Revised: 11 September 2025 Accepted: 15 September 2025 Published: 25 September 2025
MSC : 37G15, 91A22

Rock-Paper-Scissors (RPS) games have been widely studied, while research on asymmetric multiplayer cases has remained relatively limited. In this paper, we constructed a three-player asymmetric RPS model by introducing a novel payoff matrix, which then serves as the basis for developing a reverse-cyclic mutation logit system. Analysis of the interior equilibrium showed that both mutation rate and selection intensity can destabilize the equilibrium and generate oscillatory dynamics through Hopf bifurcations. When the first Lyapunov coefficient vanished, the second Lyapunov coefficient was derived to classify the bifurcation type, enabling us to distinguish supercritical and subcritical cases. Numerical simulations confirmed the theoretical predictions. These results revealed stability conditions and bifurcation mechanisms in asymmetric multiplayer interactions and demonstrated how higher-order Lyapunov coefficients enhance the analysis of complex dynamics.
- three-player asymmetric RPS game,
- reverse cyclic mutation,
- logit system,
- degenerate Hopf bifurcation
Citation: Jie Liu, Wenjun Hu. Bifurcation analysis of an asymmetric three-player three-strategy game with logit dynamics and reverse-cyclic mutations[J]. AIMS Mathematics, 2025, 10(9): 22206-22220. doi: 10.3934/math.2025988

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Abstract

Rock-Paper-Scissors (RPS) games have been widely studied, while research on asymmetric multiplayer cases has remained relatively limited. In this paper, we constructed a three-player asymmetric RPS model by introducing a novel payoff matrix, which then serves as the basis for developing a reverse-cyclic mutation logit system. Analysis of the interior equilibrium showed that both mutation rate and selection intensity can destabilize the equilibrium and generate oscillatory dynamics through Hopf bifurcations. When the first Lyapunov coefficient vanished, the second Lyapunov coefficient was derived to classify the bifurcation type, enabling us to distinguish supercritical and subcritical cases. Numerical simulations confirmed the theoretical predictions. These results revealed stability conditions and bifurcation mechanisms in asymmetric multiplayer interactions and demonstrated how higher-order Lyapunov coefficients enhance the analysis of complex dynamics.

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