Spieltheorie-Übungen
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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
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 :
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
---
-
--
+++ ++
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