Game Theory
Lecture 2
1
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
Game Theory
WS 2003
Problem Set 2
From Binmore's Fun and Games
p. 62 Exercises
21, 22, 23, 24, (25).
3
A reminder
Let G be a finite, two player game of perfect
information without chance moves.
Theorem (Zermelo, 1913): Either
player 1 can force an outcome in T or
player 2 can force an outcome in T’
A reminder
Zermelo’s proof uses
Backwards Induction
A reminder
A game G is strictly Competitive if for any two
terminal nodes a,b
a1b
 b2a
?
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… … ak
And player 2 can force an outcome in
ak , ak-1… … a1
?
an 1 an-1 1.. 1 ak 1 .. 1 a2 1 a1
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)
G(s,t)= ak
Proof :
Let wj =an , an-1… … ,aj , j =1,…,n
wn+1 =
wn
w2
an , an-1… aj …, a2, a1
wn+1
wj
w1
Proof :
Let wj =an , an-1… … ,aj , j =1,…,n
wn+1 =
Player 1 can force an outcome in
W1 =an , an-1…,a1 ,
and cannot force an outcome in wn+1 =.
Let k be the maximal integer s.t. player 1
can force an outcome in Wk
w1 , w2 , ….wn ,wn+1
can force
can force ??
cannot force
Proof :
Let k be the maximal integer s.t. player 1
can force an outcome in Wk
w1 
, w2 , … wk , wk+1...,wn+1
Player 1 can force
Player 1 cannot force
wk+1
an , an-1… ak+1 , ak …, a2, a1
by Zermelo’s theorem
wk
Player 2 can force an outcome in T
w1
-wk+1
!!!!!
an 1 an-1 1.. 1 ak 1 .. 1 a2 1 a1
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)
G(s,t)= ak
Now consider the implications of this result for the
strategic form game
t
player 1’s strategy s
guarantees at least ak
s +
player 2’s strategy t
guarantees him at least ak
i.e. at most ak
for player 1
+
ak
-
+
+
+
t
-
The point (s,t) is
a Saddle point
s +
+
ak
-
+
+
+
t
-
Given that player 2 plays t,
Player 1 has
no better strategy
than s
s +
strategy s is player 1’s
best response
to player 2’s strategy t
+
ak
+ +
+
-
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
John F. Nash Jr.
This definition holds for any game,
not only for strict competitive ones
Awarding the Nobel Prize in Economics - 1994
Example
1 1
r
l
2
L
3
2
2
M
W
2 1
R
L
R
W
L
W
L
R
backwards Induction
(Zermelo)
2 1
W
( l , r ) ( R,  ,  )
l
L
r
r
W
Example
r
l
2
L
3
2
R
2 1
W
l
L
2
M
W
L
r
( l , r ) ( R,  ,  )
1 1
2 1
R
W
All those
strategy pairs are
Nash equilibria
L
R
W
L
But there are other
Nash equilibria …….
( l , r ) ( L,  ,  )
r
W
Example
r
l
2
L
3
2
R
2 1
W
L
r
2
M
W
L
l
( l , r ) ( R,  ,  )
1 1
r
W
2 1
R
L
R
W
L
W
The strategies obtained by
backwards induction
Are Sub-Game Perfect equilibria
in each sub-game they prescribe
a Nash equilibrium
Example
r
l
2
L
3
2
R
2 1
W
L
r
2
M
W
L
l
( l , r ) ( R,  ,  )
1 1
r
W
2 1
R
W
L
R
W
L
Whereas, the
non Sub-Game Perfect
Nash equilibrium
( l , r ) ( L,  ,  )
prescribes a non equilibrium
behavior in some sub-games
A Sub-Game Perfect equilibria
prescribes a Nash equilibrium
in each sub-game
R. Selten
Awarding the Nobel Prize in Economics - 1994
Chance Moves
Nature (player 0), chooses randomly, with
known probabilities, among some actions.
0




+++=1
22
Russian Roulette
0
1/6
1/6
6
1
S.
N.S.
5
1
S.
N.S.
4
3
1
S.
N.S.
information set
1
2
1
S.
N.S.
1
S.
N.S.
1
S.
Payoffs:
W
(when the other dies, or when the other chose
not shoot in his turn)
D
(when not shooting)
L
(when dead)
N.S.
Russian Roulette
0
1/6
1/6
6
1
S.
N.S.
5
1
S.
N.S.
4
3
1
S.
N.S.
1
2
1
S.
N.S.
1
S.
N.S.
Payoffs:
W
(when the other dies, or when the other did
not shoot
hisD
turn)
Win
L
D
(when not shooting)
L
(when dead)
1
S.
N.S.
Russian Roulette
0
1/6
1/6
6
1
S.
5
4
1
N.S.
D
1
N.S.
S.
D
1
2
1
N.S.
S.
3
S.
D
1
N.S.
S.
D
1
N.S.
S.
D
D
L
2
2
2
2
2
N.S.
Russian Roulette
0
1/6
1/6
6
5
1
S.
1
N.S.
S.
2
D
S.
S.
D
S.
D
2
N.S.
S.
1
2
1
N.S.
D
2
N.S.
3
1
N.S.
D
S.
4
1
N.S.
S.
D
D
S.
N.S.
D
S.
L
N.S.
D
N.S.
D
L
2
N.S.
S.
D
2
N.S.
1