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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

### Evolutionary Stability in Repeated Games Played by Finite Automata

Automata

K. Binmore & L. SamuelsonJ.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

Tat For Tit (TAFT)

Finite Automata playing the Prisoners’ Dilemma

transitions

states

(& actions)

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

• 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.

The Structure of Nash Equilibrium in Repeated Games with Finite Automata

Dilip Abreu & Ariel Rubinstein

Econometrica,1988

(-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)

Binmore Samuelson:

x

x

?

y

a

x

x

x

x

?

y

y

?

b

a

If

then:

x

x

?

y

a

x

x

x

x

?

y

?

y

b

a

If

then:

x

x

?

y

a

Q.E.D.

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.

Q.E.D.

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)

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.

Q.E.D.

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

C

C,D

D

C

D

AC

C

C

D

D

C

D

Tat For Tit (TAFT)

C

C,D

D

D

C

CA

C

C

D

D

C

D

Tat For Tit (TAFT)

C

C

D

C

C

D

C

D

D

C

D

Tat For Tit (TAFT)

D

CC

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.

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:

If AC invaded, it does not do well against CD

D C D C …….

C D C D …….

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…….

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.

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