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Evolution & Economics No. 4. Evolutionary Stability in Repeated Games Played by Finite Automata. Automata. K. Binmore & L. Samuelson J.E.T. 1991. C. C. C,D. D. D. D. C. D. C. D. C. Grim. Tit For Tat (TFT). C. C. C. C. D. D. D. D. C. C. D. D. Tweedledum.

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Evolution & Economics No. 4

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Evolution economics no 4

Evolution & Economics

No. 4


Evolutionary stability in repeated games played by finite automata

Evolutionary Stability in Repeated Games Played by Finite Automata

Automata

K. Binmore & L. SamuelsonJ.E.T. 1991


Evolution economics no 4

C

C

C,D

D

D

D

C

D

C

D

C

Grim

Tit For Tat (TFT)

C

C

C

C

D

D

D

D

C

C

D

D

Tweedledum

Tat For Tit (TAFT)

Finite Automata playing the Prisoners’ Dilemma

transitions

states

(& actions)


Evolution economics no 4

C

C

C,D

C,D

D

D

C

D

C

D

Tweedledee

CA

C,D

C,D

C

D

C

D

Automata playing the Prisoners’ Dilemma


Evolution economics no 4

  • Two Automata playing together, eventually follow a cycle

  • (handshake)

  • The payoff is the limit of the means.

  • The cost of an automaton is the number of his states.

  • The cost enters the payoffs lexicographically.


Evolution economics no 4

The Structure of Nash Equilibrium in Repeated Games with Finite Automata

Dilip Abreu & Ariel Rubinstein

Econometrica,1988


Evolution economics no 4

(-1,3)

(2,2)

(0,0)

(3,-1)

The Structure of Nash Equilibrium in Repeated Games with Finite Automata

Dilip Abreu & Ariel Rubinstein

Econometrica,1988

N.E. of repeated Game

N.E in Repeated Games with Finite Automata

(Abreu Rubinstein)


Evolution economics no 4

Binmore Samuelson:


Evolution economics no 4

x

x

?

y

a

x

x

x

x

?

y

y

?

b

a

If

then:


Evolution economics no 4

x

x

?

y

a

x

x

x

x

?

y

?

y

b

a

If

then:


Evolution economics no 4

x

x

?

y

a

Q.E.D.


Evolution economics no 4

C

D

D

C

D

C

Tit For Tat (TFT)

C,D

C,D

C

D

C

D

Cis not an ESS, it can be invaded byD.

Dis not an ESS, it can be invaded byTit For Tat.


Evolution economics no 4

Q.E.D.


Evolution economics no 4

C

D

C

C,D

D

D

C

D

C

D

C

Grim

Tit For Tat (TFT)

Q.E.D.

In the P.D. Tit For Tat and Grim are not MESS

(they do not use one state against themselves)


Evolution economics no 4

For a general, possibly non symmetric game G.

Define the symmetrized version of G: G # #.

A player is player 1with probability 0.5 and player 2 with probability 0.5

  • The previous lemmas apply to (a1,a2)

  • An ESS has a single state │a1│=│a2│=1

  • If (a1,a2) is a MESS it uses all its states when playing against itself, i.e. a1,a2use all their states when playing against the other.


Evolution economics no 4

Q.E.D.


Evolution economics no 4

C

C,D

D

C

D

C

C,D

D

C

CA

C,D

D

AC

C,D

D

C

C

C

AA

D

C

C

C

D

C

D

D

D

C

D

C

D

Tat For Tit (TAFT)

D

C,D

CC

CD

It can be invaded by:


Evolution economics no 4

C

C,D

D

C

D

AC

C

C

D

D

C

D

Tat For Tit (TAFT)

It can be invaded by:


Evolution economics no 4

C

C,D

D

D

C

CA

C

C

D

D

C

D

Tat For Tit (TAFT)

It can be invaded by:


Evolution economics no 4

C

C

D

C

C

D

C

D

D

C

D

Tat For Tit (TAFT)

D

CC

It can be invaded by:


Evolution economics no 4

C

C

D

D

C

D

Tat For Tit (TAFT)

No other (longer and more sophisticated) automaton can invade.

Any exploitation of TAFT (playing D against his C) makes TAFT play D,

so the average of these two periods is (3+0)/2 = 1.5 < 2, the average of cooperating.


Evolution economics no 4

C

C,D

C

C,D

D

C

D

D

D

C

AC

CA

C

C

D

C

C

C

D

C

D

D

D

C

D

C

D

Tat For Tit (TAFT)

D

C,D

CC

CD

A population consisting of:

can be invaded only by:

If AC invaded, it does not do well against CD

D C D C …….

C D C D …….


Evolution economics no 4

C

C,D

C,D

D

D

C

CA

C,D

D

C

C

C

AA

D

C

C

C

D

C

D

D

D

C

D

C

D

Tat For Tit (TAFT)

D

C,D

CC

CD

A population consisting of:

can be invaded only by CA

If AA invaded, it does not do well against CC

D C C C C…….

C D D D D…….


Evolution economics no 4

C

C,D

D

D

C

CA

C

C

D

C

C

C

D

C

D

D

D

C

D

C

D

Tat For Tit (TAFT)

D

C,D

CC

CD

A population consisting of:

can be invaded only by CA

but if CA invaded then a sophisticated automaton S can invade and exploit CA .

S starts with C. if it saw C it continues with C forever (the opponent must be CD or CC ).

If it saw D, it plays D again, if the other then plays D it must be TAFT. S plays another D and then C forever.

If, however, after 2x D, the other played C, then it must be CA, and S should play D forever.


Evolution economics no 4

C

C,D

D

D

C

CA

C

C

D

C

C

C

D

C

D

D

D

C

D

C

D

Tat For Tit (TAFT)

D

C,D

CC

CD

A population consisting of:

can be invaded only by CA

When S invades, CA will vanish, and then S which is a complex automaton will die out.

Evolution - 5


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