Soft Matter and Complex Systems Seminar
sala 1.40, ul. Pasteura 5
2026-04-10 (09:30) 
Jacek Miękisz (MIM UW)
Phase transitions in stochastic evolutionary games on graphs
Short abstract, general discussion
We discuss similarities and differences between systems of interacting spins in the ferromagnetic Ising model and systems of interacting players in evolutionary games. We compare Nash equilibria to ground states and phase transitions in both models.
Long abstract, a case study
We examine the impact of the maintenance cost of social links on cooperative behavior in the Prisoner’s Dilemma game on the Barabási-Albert scale-free network with a pairwise stochastic imitation. We show by means of Monte Carlo simulations and pair approximation that the cooperation frequency changes abruptly from an almost full cooperation to a much smaller value when we increase the cost of maintaining links. In the critical region, the stationary distribution is bimodal and the system oscillates between two states: the state with almost full cooperation and one with coexisting strategies. We show that the critical region shrinks with the increasing size of the population. However, the expected time the system spends in a metastable state before switching to the other one does not change as a function of the system’s size, which precludes the existence of two stationary states in the thermodynamic limit of the infinite population.
Bibliography
J. Miękisz and J. Mohamadichamgavi, Phase transitions in the prisoner’sdilemma game on the Barabási-Albert graph with participation cost, Phys.Rev. E 112: L032302 (2025).
https://www.mimuw.edu.pl/~miekisz/phtrpre.pdf
J. Miękisz, J. Mohamadichamgavi and J. Łacki, Phase transitions in thePrisoner’s Dilemma game on scale-free networks, BioPhysMath 1: 9 pages(2024).
https://www.mimuw.edu.pl/~miekisz/phasetrpdbpm.pdf
We discuss similarities and differences between systems of interacting spins in the ferromagnetic Ising model and systems of interacting players in evolutionary games. We compare Nash equilibria to ground states and phase transitions in both models.
Long abstract, a case study
We examine the impact of the maintenance cost of social links on cooperative behavior in the Prisoner’s Dilemma game on the Barabási-Albert scale-free network with a pairwise stochastic imitation. We show by means of Monte Carlo simulations and pair approximation that the cooperation frequency changes abruptly from an almost full cooperation to a much smaller value when we increase the cost of maintaining links. In the critical region, the stationary distribution is bimodal and the system oscillates between two states: the state with almost full cooperation and one with coexisting strategies. We show that the critical region shrinks with the increasing size of the population. However, the expected time the system spends in a metastable state before switching to the other one does not change as a function of the system’s size, which precludes the existence of two stationary states in the thermodynamic limit of the infinite population.
Bibliography
J. Miękisz and J. Mohamadichamgavi, Phase transitions in the prisoner’sdilemma game on the Barabási-Albert graph with participation cost, Phys.Rev. E 112: L032302 (2025).
https://www.mimuw.edu.pl/~miekisz/phtrpre.pdf
J. Miękisz, J. Mohamadichamgavi and J. Łacki, Phase transitions in thePrisoner’s Dilemma game on scale-free networks, BioPhysMath 1: 9 pages(2024).
https://www.mimuw.edu.pl/~miekisz/phasetrpdbpm.pdf