1
problem set 6
from Osborne’s
Introd. To G.T.
p.210 Ex. 210.1
p.234 Ex. 234.1
from Binmore’s
Fun and Games
p.337 Ex. 26,27
Minimax & Maximin
Strategies
Given a game G( , ) and a strategy s of player 1:
min G1  s,t 
t
is the worst that can happen to player 1 when he
plays strategy s.
He can now choose a strategy s for which this
‘worst scenario’ is the best
max min G1  s,t 
s
t
3
A strategy s is called a maximin (security) strategy if
min
minGG
s,t  max
maxmin
minG
G1s,t
σ,t .
1 s,t
σs
t t
max
s



min G1  s,t  s 
t

min G1  s',t  s'
t



{
{
tt









min G1  s,t 
t
min G1  s',t 
t
4
A strategy s is called a maximin (security) strategy if
minG
G11 s,t
s,t  max
maxmin
minG
G11σ,t
s,t .
min
t
t
s
σ
t
t
These can be defined for mixed strategies as well.
max = sup , min = inf
Similarly, one may define
min max G1  s,t 
t
s
If the game is strictly competitive then this is the
best of the ‘worst case scenarios’ of player 2.
5
Lemma:
For any matrix G:
min max G  s,t   max min G  s,t 
t
s
s
t
where s,t are mixed strategies
Take the matrix to be
the matrix of player 1’s
payoffs of a game G,
i.e. G1
6
Lemma:
For any matrix G:
min max G  s,t   max min G  s,t 
t
s
t
s
where s,t are mixed strategies
Proof:
For any two strategies s,t :
??
max G  σ,t   min G  s,τ 
σ
τ
σ,t   G  s,t   min G  s,τ 

τ
σ
mi
n max
G
σ,t
max
mi
n GG s,τ



min
max
G
s,t
max
min
s,t




tt
τ t
σs
ss
hence:
max G
7
Theorem: (von Neumann) For any matrix G:
min max G  s,t   max min G  s,t 
t
s
s
t
The max & min is taken over mixed strategies
No proof is provided in the lecture
Lemma:
If s is a maximin strategy and t is a minimax strategy
of a strictly competitive game, then (s,t) is a Nash
equilibrium.
Proof:
8
t
Proof:









s
min G  s,t 
t
G  s,t 
max G  s,t 
s









max G  s,t   G  s,t   min G  s,t 
t
s
=
but
hence
maxmin = minmax
max G  s,t  = G  s,t   min G  s,t 
s
t
9
max G  s,t  = G  s,t   min G  s,t 
t
s
t is a best response against s
s is a best response against t
( s , t ) is a Nash Equilibrium.
10
Mixed Strategies Equilibria in
Infinite Games
The ‘All Pay’ Auction




Two players bid simultaneously for a good of value K
the bids are in [0,K].
Each pays his bid.
The player with the higher bid gets the object.
If the bids are equal, they share the object.
There are no equilibria in pure strategies
11
There are no equilibria in pure strategies
1.
 x, x 
x < K is not an equilibrium
increasing the bid by ε increases payoff
from K/2 - x to K - x - ε.
2.
 K, K  is not an equilibrium
lowering the bid to 0 increases payoff
from K/2 - K = -K/2 to 0.
3.
 x, y 
x < y is not an equilibrium
lowering the bid from y to y - ε increases payoff
from K - y to K - y + ε.
12
Equilibrium in mixed strategies
A mixed strategy is a (cumulative) probability distribution
F over 0, K  , with a density function f  x  .
x
F  x  =  f  s  ds
F  x  is the probability that the player bids at most
x.
0
assume that the support of F is an interval a,b  0, K 
F F  a  = 0, F  b  = 1
f  x  > 0 iff x a,b
f
1
F(x)
0
a
b K
0
a
x
13
b K
When player 1 bids x and player 2 uses a mixed strategy
F2 •  , then player 1's payoff is :
F2  x   K - x  + 1 - F2  x    -x 
= KF2  x  - x
Player 1's mixed strategy F1 is a best response to F2 if
for all x  a1 ,b1 
KF2  x  - x = C
and
for all y  a1 ,b1 
KF2  y  - y  C.
14
KF2  x  - 1 = 0
Player 1's mixed strategy F11 is a best response to F2 if
Kf 2  x  - 1 = 0
f2  x  
for all x  a1 ,b1 
KF2 K
 x - x = C
F2 and
is uniform and since f  1/K
a2 ,by2=a10,
forall
,b1K  KF2  y  - y  C.
Similarly F1 is uniform over 0, K .
15
x
F1 (x) = F2 (x) = F(x) =
K
In equilibrium, the expected payoff of a given bid
(of each player) is :
1
KF(x) - x = K x - x  0
K
In equilibrium, the expected payoff of each player is 0 .
16
Rosenthal’s Centipede Game
1
A
2
D
1, 0
0 , 10
1
1
2
‘Centipede’
due to
K.G.Binmore
102 , 0 0 , 103 104 , 0
‘Exploding’ payoffs
due to
P. Reny
2
0,0
0 , 105
17
Rosenthal’s Centipede Game
A
1
2
1
2
1
2
0 , 10
102 , 0
0 , 103
104 , 0
0 , 105
0,0
D
1, 0
Sub-game perfect equilibrium
18
Rosenthal’s Centipede Game
A
1
2
1
2
1
2
1,3
4,2
3,5
6,4
5,7
8,6
D
2, 0
Sub-game perfect equilibrium
different payoffs
1
A
2
1
2
1
2
0 , 10
102 , 0
0 , 103
104 , 0
0 , 105
0,0
D
1, 0
19
A Variation of the
Battle of the Sexes
1
Quiet
evening
2, 2
Noisy
evening
B
X
B 3,1 0,0
X 0 ,0
1,3
Player 1 has 4 strategies
Player 2 has 2 strategies
20
A Variation of the
Battle of the Sexes
Nash Equilibria
1
Quiet
evening
2, 2
[ (N,B), B ]
[ (Q,X), X ]
Noisy
evening
BB
XX
BB 33, ,11 00, ,00
XX 00,0,0 11, ,33
[ (Q,B), X ]
21
A Variation of the
Battle of the Sexes
Nash Equilibria
1
Quiet
evening
2, 2
[ (N,B), B ]
[ (Q,X), X ]
Noisy
evening
B
X
B 3,1 0,0
X 0 ,0
1,3
[ (Q,B), X ]
These S.P.E. guarantee player 1
not a sub-game
equilibrium
!!!
a payoff perfect
of at least
2
7
22