Evolution

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Evolution Economics No. 5
p 0.53
p 0.58
Forest fire
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Best Response Dynamics

• A finite population playing a (symmetric) game
  G.
• Each individual is randomly matched and plays a
  pure strategy of G.
• At each point in (discrete) time each
  individual (or alternatively each with
  probability) chooses the best response to the
  mixed strategy played by the population.

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A B
A 3 , 3 0 , x
B x , 0 x , x
A B
A 3 , 3 0 , x
B x , 0 x , x
Payoff Dominant Equilibrium
3 gt x gt 0
Nash Equilibria
Risk Dominant Equilibrium
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A B
A 3 , 3 0 , x
B x , 0 x , x
Payoff Dominant Equilibrium
Risk Dominant Equilibrium
A Population of n individuals playing either A or
B.
1
0
k/n
All playing A
All playing B
k playing B
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A B
A 3 , 3 0 , x
B x , 0 x , x
A B
A 3 , 3 0 , 2
B 2 , 0 2 , 2
Payoff Dominant Equilibrium
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2
2
2
Risk Dominant Equilibrium
1-a a
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A B
A 3 , 3 0 , x
B x , 0 x , x
A B
A 3 , 3 0 , 2
B 2 , 0 2 , 2
Payoff Dominant Equilibrium
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2
2
2
Risk Dominant Equilibrium
In the Best Response Dynamics The Basin of
Attraction of the Risk Dominant Equilibrium is
larger than that of the Pareto Dominant
Equilibrium
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A
B
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A
B
A deterministic dynamics can be described as a
transition function
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The replicator Dynamics (in which all learn
revise their strategy) is
If each individual has probability ? of learning
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A state of the population becomes a distribution
of states
If each individual has probability ? of learning
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If each individual (when learning) has a small
probability e of learning the wrong strategy
B (B)
B (A)
The prob. of learning may be dropped.
Each state has a positive probability of becoming
any other state.
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A B
A 3 , 3 0 , 2
B 2 , 0 2 , 2
e-small, mistakes are rare
The process converges to B (A) and will stay
there a long time.
In the long-run mistakes occur (with prob. 1) and
the process will be swept away from B.
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If few individuals make mistakes, the process
will remain in the basin of attraction of B, and
will return to B.
If more individuals made mistakes, the process
moves to the basin of attraction of A, from there
to A, and will remain there a long time.
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The process remains a long time in A and in B.
What proportion of the time is spent in A ??
To leave the basin of attraction of A, at least
1/3 of the population has to err. This happens
with prob.
To leave the basin of attraction of B, at least
2/3 of the population has to err. This happens
with prob.
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It is a rare move from B to the basin of
attraction of A. The reverse is much more
frequent !!!
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In the very long run the process will spend
infinitely more time in B than in A.
In the very long run the process will spend 0
time in A.
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A B
A 3 , 3 0 , 2
B 2 , 0 2 , 2
A B
A 3 , 3 0 , 2
B 2 , 0 2 , 2
The Best Response Dynamics with vanishing
noise, chooses the Risk Dominant Equilibrium. The
equilibrium with a larger Basin of attraction.
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Kandori, Michihiro Mailath, George J Rob,
Rafael, 1993. "Learning, Mutation, and Long Run
Equilibria in Games," Econometrica, Econometric
Society, vol. 61(1), pages 29-56. H Peyton,
1993. "The Evolution of Conventions,"
Econometrica, EconometricSociety, vol. 61(1),
pages 57-84
Based on
M.I. Freidlin and A.D. Wentzell, Random
Perturbations of Dynamical Systems , Springer New
York, 1984