Equilibrium transitions in stochastic evolutionary games

Dresden, ECCS’07

Jacek Miękisz

Institute of Applied Mathematics

University of Warsaw

Population dynamics

time

A and B are two possible behaviors, fenotypes or strategies of each individual

Matching of individuals

everybody interacts with everybody

random pairing of individuals

space –structured populations

Simple model of evolution

Selection individuals interact in pairs – play games receive payoffs = # of offspring

Fenotypes are inherited

Offspring may mutate

Main Goals

Equilibrium selection in case of multiple Nash equilibria

Dependence of the long-run behavior of population on

--- its size

--- mutation level

Stochastic dynamics of finite unstructured populations

n - # of individuals

zt - # of individuals playing A at time t

Ω = {0,…,n} - state space

selection

zt+1 > zt if „average payoff” of A > „average payoff” of B

mutationeach individual may mutate and switch to the other strategy with a probability ε

Markov chain with n+1 states and a unique stationary state με

n

Previous results

Playing against the field, Kandori-Mailath-Rob 1993

(A,A) and (B,B) are Nash equilibria

A is an efficient strategyB is a risk-dominant strategy

A B A a b B c d

a>c, d>b, a>d, a+b<c+d

Random matching of players, Robson - Vega Redondo, 1996

pt # of crosspairings

Our results, JM J. Theor. Biol, 2005

Theorem (random matching model)

Spatial games with local interactions

deterministic dynamics of the best-response rule

i

Stochastic dynamics

a) perturbed best response

with the probability 1-ε, a player chooses the best responsewith the probability ε a player makes a mistake

b) log-linear rule or Boltzmann updating

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

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

Open Problem

Construct a spatial game with a unique stationary state με

Λ

which has the following property

Real Open Problem

Construct a one-dimensional cellular automaton model with a unique stationary state με

Λ

such that when you take the infinite lattice limit

you get two measures.