Equilibrium transitions in stochastic evolutionary games

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1 Δεκ 2013 (πριν από 3 χρόνια και 7 μήνες)

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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

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

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

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

Random matching of players, Robson
-

Vega Redondo, 1996

p
t

# 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 response

with 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
μ
ε
Λ

which has the following property



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.