Equilibrium transitions in stochastic evolutionary games

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Equilibrium transitions in stochastic
evolutionary games
Dresden, ECCS07
Jacek Miekisz Institute
of Applied Mathematics University of
Warsaw
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Population dynamics
time
A and B are two possible behaviors, fenotypes or
strategies of each individual
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Matching of individuals
everybody interacts with everybody
random pairing of individuals
space structured populations
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Simple model of evolution
Selection individuals interact in pairs
play games receive payoffs of
offspring
Fenotypes are inherited
Offspring may mutate
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Main Goals
Equilibrium selection in case of multiple Nash
equilibria
Dependence of the long-run behavior of population
on --- its size --- mutation level
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Stochastic dynamics of finite unstructured
populations
n - of individuals
zt - of individuals playing A at time t
O 0,,n - state space
selection
zt1 gt zt if average payoff of A gt
average payoff of B
mutation
each individual may mutate and switch to the
other strategy with a probability e
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Markov chain with n1 states and a unique
stationary state µen
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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 agtc, dgtb, agtd, abltcd
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Random matching of players, Robson - Vega
Redondo, 1996
pt of crosspairings
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Our results, JM J. Theor. Biol, 2005
Theorem (random matching model)
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Spatial games with local interactions
deterministic dynamics of the best-response rule
i
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Stochastic dynamics
a) perturbed best response
with the probability 1-e, a player chooses the
best response with the probability e a
player makes a mistake
b) log-linear rule or Boltzmann updating
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Example
A is a dominated strategy
without A B is stochastically stable
with A C is ensemble stable at
intermediate noise levels in
log-linear dynamics
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A B C
A 0 0.1 1 B
0.1 2a 1.1 C 1.1
1.1 2
where a gt 0
e
C
B
a
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Open Problem
Construct a spatial game with a unique
stationary state µe? which has the following
property
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Real Open Problem
Construct a one-dimensional cellular automaton
model with a unique stationary state µe?
such that when you take the infinite lattice
limit you get two measures.