1. Equilibrium transitions in stochastic evolutionary games Dresden, ECCS’07 Jacek Miękisz Institute of Applied Mathematics University of Warsaw

2. Population dynamics time A and B are two possible behaviors, fenotypes or strategies of each individual

3. Matching of individuals everybody interacts with everybody random pairing of individuals space –structured populations

4. Simple model of evolution Selection individuals interact in pairs – play games receive payoffs = # of offspring Fenotypes are inherited Offspring may mutate

5. Main Goals Equilibrium selection in case of multiple Nash equilibria Dependence of the long-run behavior of population on --- its size --- mutation level

6. Stochastic dynamics of finite unstructured populations n - # of individuals z t - # of individuals playing A at time t Ω = {0,…,n} - state space selection z t+1 > z t if „average payoff” of A > „average payoff” of B mutation each individual may mutate and switch to the other strategy with a probability ε

7. Markov chain with n+1 states and a unique stationary state μ ε n

8. Previous results Playing against the field, Kandori-Mailath-Rob 1993 (A,A) and (B,B) are Nash equilibria A is an efficient strategy B is a risk-dominant strategy A B A a b B c d a>c, d>b, a>d, a+b<c+d

9. Random matching of players, Robson - Vega Redondo, 1996 p t # of crosspairings

10. Our results, JM J. Theor. Biol, 2005 Theorem (random matching model)

11. Spatial games with local interactions deterministic dynamics of the best-response rule i

12. Stochastic dynamics a) perturbed best response with the probability 1-ε, a player chooses the best response with the probability ε a player makes a mistake b) log-linear rule or Boltzmann updating

13. Example without A B is stochastically stable A is a dominated strategy with A C is ensemble stable at intermediate noise levels in log-linear dynamics

14. ε α C B A B C A 0 0.1 1 B 0.1 2+α 1.1 C 1.1 1.1 2 where α > 0

15. Open Problem Construct a spatial game with a unique stationary state μ ε Λ which has the following property

16. Real Open Problem Construct a one-dimensional cellular automaton model with a unique stationary state μ ε Λ such that when you take the infinite lattice limit you get two measures.