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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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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


C Automata

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)


C Automata

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



The Structure of Nash Equilibrium in Repeated Games with Finite Automata

Dilip Abreu & Ariel Rubinstein

Econometrica,1988


(-1,3) Finite Automata

(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)


Binmore Samuelson: Finite Automata


x Finite Automata

x

?

y

a

x

x

x

x

?

y

y

?

b

a

If

then:


x Finite Automata

x

?

y

a

x

x

x

x

?

y

?

y

b

a

If

then:


x Finite Automata

x

?

y

a

Q.E.D.


C Finite Automata

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.


Q.E.D. Finite Automata


C Finite Automata

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)


For a general, possibly non symmetric game Finite AutomataG.

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.


Q.E.D. Finite Automata


C Finite Automata

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:


C Finite Automata

C,D

D

C

D

AC

C

C

D

D

C

D

Tat For Tit (TAFT)

It can be invaded by:


C Finite Automata

C,D

D

D

C

CA

C

C

D

D

C

D

Tat For Tit (TAFT)

It can be invaded by:


C Finite Automata

C

D

C

C

D

C

D

D

C

D

Tat For Tit (TAFT)

D

CC

It can be invaded by:


C Finite Automata

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.


C Finite Automata

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 …….


C Finite Automata

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…….


C Finite Automata

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.


C Finite Automata

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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