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Game TheoryLecture 2

Game TheoryLecture 2

Spieltheorie-Übungen

P. Kircher: Dienstag – 09:15 - 10.45 HS M

S. Ludwig: Donnerstag - 9.30-11.00 Uhr HS L

T. Troeger: Mittwoch – 8.30-10.00 Uhr HS N

A. Shaked: Freitag - 14.15-15.45 Uhr - HS N

33

Game Theory WS 2003

Problem Set 2 From Binmore's Fun and Games p. 62 Exercises

21, 22, 23, 24, (25).

Theorem (Zermelo, 1913): Either player 1 can force an outcome in T or player 2 can force an outcome in T’

Let G be a finite, two player game of perfect information without chance moves.

A reminder

A reminder

Zermelo’s proof uses Backwards Induction

A reminder

A game G is strictly Competitive if for any two

terminal nodes a,b

a1b b2a

An application of Zermelo’s theorem to Strictly Competitive Games

Let a1,a2,….an be the terminal nodes of a strictly

competitive game (with no chance moves and with perfect

information) and let:

an 1 an-1 1 …. 1 a2 1 a1

(i.e. an 2 an-1 2 …. 2 a2 2 a1 ).

?

Then there exists k, n k 1 s.t. player 1 can force an outcome in

an , an-1… … akAnd player 2 can force an outcome in

ak , ak-1… … a1

?an 1 an-1 1.. 1 ak 1 .. 1 a2 1 a1

G(s,t)= ak

Player 1 has a strategy s which forces an outcome

better or equal to ak (1)

Player 2 has a strategy t which forces an outcome

better or equal to ak (2)

Proof :

Let wj =an , an-1… … ,aj , j =1,…,n wn+1 =

an , an-1… aj …, a2, a1w1

w2

wjwn+1

wn

Proof :

Player 1 can force an outcome in

W1 =an , an-1…,a1 ,

and cannot force an outcome in wn+1 =.

Let wj =an , an-1… … ,aj , j =1,…,n wn+1 =

w1 , w2 , ….wn ,wn+1 w1 , w2 , ….wn ,wn+1

can force cannot forcecan force ??

Let k be the maximal integer s.t. player 1 can force an outcome in Wk

Proof :

w1 , w2 , … wk , wk+1...,wn+1 w1 , w2 , … wk , wk+1...,wn+1

Player 1 can force Player 1 cannot force

Let k be the maximal integer s.t. player 1 can force an outcome in Wk

an , an-1… ak+1 , ak …, a2, a1

w1

wk+1

wk

Player 2 can force an outcome in T -wk+1

by Zermelo’s theorem

!!!!!an 1 an-1 1.. 1 ak 1 .. 1 a2 1 a1

G(s,t)= ak

Player 1 has a strategy s which forces an outcome

better or equal to ak (1)

Player 2 has a strategy t which forces an outcome

better or equal to ak (2)

Now consider the implications of this result for the strategic form game

s

t

ak

player 1’s strategy s guarantees at least ak

player 2’s strategy t guarantees him at least ak -

--

-

--

+++ ++

i.e. at most ak for player 1

s

t

ak

---

-

--

+++ ++

The point (s,t) is a Saddle point

s

t

ak

---

-

--

+++ ++

Given that player 2 plays t,Player 1 has

no better strategy than s

strategy s is player 1’s best response

to player 2’s strategy t

Similarly, strategy t is player 2’s best response

to player 1’s strategy s

A pair of strategies (s,t) such that each is a best response to the other is

a Nash Equilibrium

Awarding the Nobel Prize in Economics - 1994

John F. Nash Jr.

This definition holds for any game, not only for strict competitive ones

1

22

2

1

L W

W

W L

WW

rl

RM

L

1

2

1

2

Example

3

R

L

R

r

L

l

backwards Induction(Zermelo)

r

( l , r ) ( R, , )

1

22

2

1

L W

W

W L

WW

rl

RM

L

1

2

1

2

Example

3

R

L

R

r

L

l r

( l , r ) ( R, , )All those

strategy pairs are Nash equilibria

But there are other Nash equilibria …….

( l , r ) ( L, , )( l , r ) ( L, , )( l , r ) ( L, , )

1

22

2

1

L W

W

W L

WW

rl

RM

L

1

2

1

2

Example

3

R

L

R

r

L

l r

( l , r ) ( R, , )

The strategies obtained by backwards induction

Are Sub-Game Perfect equilibria

in each sub-game they prescribe a Nash equilibrium

1

22

2

1

L W

W

W L

WW

rl

RM

L

1

2

1

2

Example

3

R

L

R

r

L

l r

( l , r ) ( R, , )

Whereas, the non Sub-Game Perfect

Nash equilibrium

prescribes a non equilibrium behavior in some sub-games

( l , r ) ( L, , )

A Sub-Game Perfect equilibriaprescribes a Nash equilibrium

in each sub-game

Awarding the Nobel Prize in Economics - 1994

R. Selten

22

Chance Moves

Nature (player 0), chooses randomly, with known probabilities, among some actions.

0

+ + + = 1

Russian Roulette0

1/6

1 1 1 1 1 1

1/6

123456

information setN.S.

S.N.S.

S.N.S.

S.N.S.

S.N.S.

S.N.S.

S.

Payoffs: W (when the other dies, or when the other chose

not shoot in his turn)D (when not shooting)L (when dead)

Russian Roulette0

1/6

1 1 1 1 1 1

1/6

123456

N.S.S.

N.S.S.

N.S.S.

N.S.S.

N.S.S.

N.S.S.

Payoffs: W (when the other dies, or when the other did

not shoot in his turn)D (when not shooting)L (when dead)

W D L

Russian Roulette0

1/6

1 1 1 1 1 1

1/6

123456

N.S.S.

N.S.S.

N.S.S.

N.S.S.

N.S.S.

N.S.S.

DDDDDD

L

22222

Russian Roulette0

1/6

1 1 1 1 1 1

1/6

123456

N.S.S.

N.S.S.

N.S.S.

N.S.S.

N.S.S.

N.S.S.

DDDDDD

L

22222

N.S.S.

DL

N.S.S.

D

N.S.S.

D

N.S.S.

D

N.S.S.

D