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Evolution & EconomicsEvolution & EconomicsNo. 5No. 5

Forest fire

p = 0.53 p = 0.58

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Best Response DynamicsBest 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

Payoff Dominant EquilibriumA B

A 3 , 3 0 , x

B x , 0 x , x

3 > x > 0

Nash EquilibriaNash Equilibria

When then: 3+ 0x >

2

Risk Dominant Equilibrium

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Payoff Dominant Equilibrium

Risk Dominant Equilibrium

A B

A 3 , 3 0 , x

B x , 0 x , x

A Population of n individuals playing either A or B.

A B

All playing A All playing B

0 1k/n

k playing B

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Payoff Dominant Equilibrium

Risk Dominant Equilibrium

A B

A 3 , 3 0 , x

B x , 0 x , x

A B

A 3 , 3 0 , 2

B 2 , 0 2 , 2

2

222

1-α α Best Response to is:

when when

1- α,α

1 1A > α, B < α.3 3

A B13

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Payoff Dominant Equilibrium

Risk Dominant Equilibrium

A B

A 3 , 3 0 , x

B x , 0 x , x

A B

A 3 , 3 0 , 2

B 2 , 0 2 , 2

2

222

A B13

In the Best Response Dynamics:In the Best Response Dynamics:The The Basin of AttractionBasin of Attraction of the Risk Dominant Equilibrium of the Risk Dominant Equilibrium

is is largerlarger than that of the Pareto Dominant Equilibrium than that of the Pareto Dominant Equilibrium

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Let a state of the population be a complete description

of the population:

ns Rwho is playing what.

011s =.........

A

B

There are states of the population.n2

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Let a state of the population be a complete description

of the population:

ns Rwho is playing what.

011s =.........

A

B

There are states of the population.n2

A deterministic dynamics can be described as a transition functionA deterministic dynamics can be described as a transition function

.i js s

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The replicator Dynamics The replicator Dynamics (in which all learn – revise their strategy) is:(in which all learn – revise their strategy) is:

is 0 or 1

If each individual has probabilityIf each individual has probability λ of learning: of learning:

A B13

10If each individual has probabilityIf each individual has probability λ of learning: of learning:

A B13

A state of the population becomes a distribution of statesA state of the population becomes a distribution of states

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The prob. of learning may be dropped.The prob. of learning may be dropped.

A B13

If each individual (when learning) has a small probabilityIf each individual (when learning) has a small probability εof learning the ‘wrong’ strategy:of learning the ‘wrong’ strategy:

B (B)

B (A)

Each state has a positive probability of becoming any other state.Each state has a positive probability of becoming any other state.

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Let be a distribution over the states of the population.n p 2

Let be the probability that state becomes

jij

ji i j(ε) > 0

t = 1.

t s s

,Let be the transition matrix.

describes a

ji

tt+1

ε

T = t

p = Tp Markov process.

lim

is independent of

the columns of are identical.

t 00

t

tt 0

p p

T

p = T

T

p

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lim..

n n n

1 1 1

2 2 2tt

2 2 2

p p .. pp p p

T = T = .. .. ..

p p .. p

lim t 0 0

t p = T p = pT lim t

tε ε0 0ε ε p = T p = pT

lim

What is the limit distribution when

ε 0ε

ε 0 ????

p = ???

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A B13

A B

A 3 , 3 0 , 2

B 2 , 0 2 , 2

ε-small, mistakes are rare

The process converges to B (A) and will stay there a long time.

ε 0

In the long-run mistakes occur (with prob. 1) and the process will be swept away from B.

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A B13

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.

ε 0

If few individuals make mistakes, the process will remain in the basin of attraction of B, and will return to B.

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A B13

What proportion of the time is spent in A ??

ε 0

The process remains a long time in A and in B.

To leave the basin of attraction of B, at least 2/3 of the population has to err. This happens with prob.:

n

k

2k= n3

2n3n

εk

ε

To leave the basin of attraction of A, at least 1/3 of the population has to err. This happens with prob. :

n

k

1k= n3

1n3n

εk

ε

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A B13

ε 0

1n3ε

22 1n n3 3=ε ε

It is a rare move from B to the basin of attraction of A.The reverse is much more frequent !!!

1n

13

3

3

n2n

leaving A

leaving =

B= εε

ε0

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A B13

ε 0

1n3ε

22 1n n3 3=ε ε

In the very long run the process will spend infinitely more time in B than in A.

1n

13

3

3

n2n

leaving A

leaving =

B= εε

ε0

In the very long run the process will spend 0 time in A.

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The Best Response Dynamics – with vanishing noise, chooses the Risk Dominant Equilibrium. The equilibrium with a larger Basin of attraction.

A B

A 3 , 3 0 , 2

B 2 , 0 2 , 2

A B

A 3 , 3 0 , 2

B 2 , 0 2 , 2

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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

M.I. Freidlin and A.D. Wentzell, Random Perturbations of Dynamical Systems , Springer New York, 1984

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