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Evolution & Economics No. 4

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Evolution & Economics

No. 4

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

It can be invaded by:

C

C,D

D

C

D

AC

C

C

D

D

C

D

Tat For Tit (TAFT)

It can be invaded by:

C

C,D

D

D

C

CA

C

C

D

D

C

D

Tat For Tit (TAFT)

It can be invaded by:

C

C

D

C

C

D

C

D

D

C

D

Tat For Tit (TAFT)

D

CC

It can be invaded by:

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:

can be invaded only by:

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