TY  - JOUR
AU  - Wang, Yifei
AU  - Meng, Xinzhu
PY  - 2026
DA  - 2026/05/10
TI  - Strategy Dynamics of Three-Strategy Snowdrift Game Induced by Reward Strategy and Payoff Delay
JO  - Journal of Mathematics and Interdisciplinary Applications
T2  - Journal of Mathematics and Interdisciplinary Applications
JF  - Journal of Mathematics and Interdisciplinary Applications
VL  - 2
IS  - 2
SP  - 74
EP  - 90
DO  - 10.62762/JMIA.2026.346326
UR  - https://www.icck.org/article/abs/JMIA.2026.346326
KW  - evolutionary game theory
KW  - reward
KW  - three-strategy
KW  - time delay
KW  - hopf bifurcation
AB  - In the evolutionary game, the introduction of the third or more specific strategies to study cooperative evolution has attracted widespread attention recently. The main feature of this paper is to adopt a special reward strategy as the third choice to observe the evolution dynamics. This strategy can improve the utilization of resources and help cooperators and defectors get greater benefits. We investigate the evolutionary game dynamics of a three-strategy snowdrift game with special reward strategy and payoff delay. First, for the nondelay system, we discuss the dynamic properties of strategies, including the existence, stability and bistability of equilibrium states. Furthermore for the time-delay system, we explore the existence conditions of Hopf bifurcation with payoff delay as a parameter. Then we get the direction and stability of Hopf bifurcation through theoretical derivations and numerical simulations. The results show that (i) cooperators, defectors and rewarders can coexist stably; (ii) as the reward increases, the defective strategy gradually disappears, which helps to promote cooperation; (iii) the stable equilibrium becomes unstable when the payoff delay is large, and Hopf bifurcation and stable periodic oscillation appears; (iv) when the payoff delay is large enough, the defective strategy disappears. Our work may promote the prosperity of cooperation among biological populations.
SN  - 3070-393X
PB  - Institute of Central Computation and Knowledge
LA  - English
ER  - 

@article{Wang2026Strategy,
  author = {Yifei Wang and Xinzhu Meng},
  title = {Strategy Dynamics of Three-Strategy Snowdrift Game Induced by Reward Strategy and Payoff Delay},
  journal = {Journal of Mathematics and Interdisciplinary Applications},
  year = {2026},
  volume = {2},
  number = {2},
  pages = {74-90},
  doi = {10.62762/JMIA.2026.346326},
  url = {https://www.icck.org/article/abs/JMIA.2026.346326},
  abstract = {In the evolutionary game, the introduction of the third or more specific strategies to study cooperative evolution has attracted widespread attention recently. The main feature of this paper is to adopt a special reward strategy as the third choice to observe the evolution dynamics. This strategy can improve the utilization of resources and help cooperators and defectors get greater benefits. We investigate the evolutionary game dynamics of a three-strategy snowdrift game with special reward strategy and payoff delay. First, for the nondelay system, we discuss the dynamic properties of strategies, including the existence, stability and bistability of equilibrium states. Furthermore for the time-delay system, we explore the existence conditions of Hopf bifurcation with payoff delay as a parameter. Then we get the direction and stability of Hopf bifurcation through theoretical derivations and numerical simulations. The results show that (i) cooperators, defectors and rewarders can coexist stably; (ii) as the reward increases, the defective strategy gradually disappears, which helps to promote cooperation; (iii) the stable equilibrium becomes unstable when the payoff delay is large, and Hopf bifurcation and stable periodic oscillation appears; (iv) when the payoff delay is large enough, the defective strategy disappears. Our work may promote the prosperity of cooperation among biological populations.},
  keywords = {evolutionary game theory, reward, three-strategy, time delay, hopf bifurcation},
  issn = {3070-393X},
  publisher = {Institute of Central Computation and Knowledge}
}