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problem set 6
from Osborne’sIntrod. To G.T.
p.210 Ex. 210.1p.234 Ex. 234.1
p.337 Ex. 26,27
from Binmore’sFun and Games
3
Minimax & Maximin Strategies
Minimax & Maximin Strategies
Given a game G( , ) and a strategy s of player 1:
min 1t
G s,t
is the worst that can happen to player 1 when he plays strategy s.
maxmin 1ts
G s,t
He can now choose a strategy s for which this ‘worst scenario’ is the best
4
A strategy s is called a maximin (security) strategy if
min maxmint ts
G s,t G s,t min maxmin .1 1t tσ
G s,t G σ,t
min
min
1t
1t
G s,t
G s',t
min 1t
G s,t
min 1t
G s',t
{{
s
s'max
s
5
A strategy s is called a maximin (security) strategy if
min maxmin1 1t ts
G s,t G s,t min maxmin .1 1t tσ
G s,t G σ,t
These can be defined for mixed strategies as well.
Similarly, one may define
minmax 1t s
G s,t
If the game is strictly competitive then this is the best of the ‘worst case scenarios’ of player 2.
max sup min inf= , =
6
where s,t are mixed strategies
Lemma:
minmax maxmint ts s
G s,t G s,t
Take the matrix to be the matrix of player 1’s payoffs of a game G,
i.e. G1
For any matrix G:
7
Lemma:For any matrix G:
minmax maxmint ts s
G s,t G s,t
Proof:For any two strategies s,t :
max minτσ
G σ,t G s,τ
max min τσ
G σ,t G s,t G s,τ
??
where s,t are mixed strategies
hence:
max mimi nn maxt τσ s
G σ,t G s,τ minmax maxmint ts s
G s,t G s,t
8
Theorem: (von Neumann) For any matrix G:
minmax maxmint ts s
G s,t G s,t
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:
The max & min is taken over mixed strategies
No proof is provided in the lecture
9
min
max
t
s
G s,t G s,t
G s,t
s
tProof:
max min ts
G s,t G s,t G s,t
=but
hence max mints
G s,t = G s,t G s,tmaxmin = minmax
10
max mints
G s,t = G s,t G s,t
t is a best response against s
s is a best response against t
( s , t ) is a Nash Equilibrium.
11
Mixed Strategies Equilibria in Infinite
GamesThe ‘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
12
is not an equilibriumx, x x <1. K
.
increasing the bid by increases payoff
from to
ε
K/2 - x K - x - ε
is not an equilibrium2. K,K
.
lowering the bid to increases payoff
from to
0
K/2 - K = -K/2 0
is not an equilibriumx, y x < y3.
.
lowering the bid from to increases payoff
from to
y y - ε
K - y K - y + ε
There are no equilibria in pure strategies
13
Equilibrium in mixed strategies
, .
A mixed strategy is a (cumulative) probability distribution
over with a density function F 0, K f x
at most is the probability that the player bids F x x.
assume that the support of is an interval F a,b 0,K
x
0
F x = f s ds
a b0 K
F1
iff f x > 0 x a,b
a b0 K
f
xF(x)
F a = 0, F b = 1
14
When player bids and player uses a mixed strategy
, then player 's payoff is :2 •
1
F
2x
1
2 2F x K - x + 1 - F x -x
2= KF x - x
Player 's mixed strategy is a best response to if
for all 1 2
1 1 2
F F
x a ,b KF x - x = C
1
and
for all 1 1 2y a ,b KF y - y C.
15
Player 's mixed strategy is a best response to if
for all 1 2
1 1 2
F F
x a ,b KF x - x = C
1
and
for all 1 1 2y a ,b KF y - y C.
2KF x - 1 = 0
2Kf x - 1 = 0 2
1f x
K
is uniform and s ince 2
2 2
F f 1/K
a ,b = 0,K
.Similarly is uniform over 1F 0,K
16
In equilibrium, the expected payoff of a given bid
(of each player) is :
1
KF(x) - x = K x - x 0K
1 2
xF (x) = F (x) = F(x) =
K
In equilibrium, the expected payoff of each player is . 0
17
Rosenthal’s Centipede Game
1 2
0 , 101, 0
1 2
0 , 103102 , 0
1 2
0 , 105104 , 0
0 , 0
D
A
‘Exploding’ payoffsdue to P. Reny
‘Centipede’due to
K.G.Binmore
18
Rosenthal’s Centipede Game
1 2
0 , 101, 0
1 2
0 , 103102 , 0
1 2
0 , 105104 , 0
0 , 0
D
A
Sub-game perfect equilibrium
19
Rosenthal’s Centipede Game
1 2
1 , 32, 0
1 2
3 , 54 , 2
1 2
5 , 76 , 4
8 , 6
D
A
Sub-game perfect equilibriumdifferent payoffs
1 2
0 , 101, 0
1 2
0 , 103102 , 0
1 2
0 , 105104 , 0
0 , 0
D
A
20
1
2, 2
Quietevening
A Variation of the Battle of the Sexes
Noisyevening
B X
B 3 , 1 0 , 0
X 0 ,0 1 , 3
Player 1 has 4 strategiesPlayer 2 has 2 strategies
21
1
2, 2
Quietevening
A Variation of the Battle of the Sexes
Noisyevening
B X
B 3 , 1 0 , 0
X 0 ,0 1 , 3
Nash Equilibria
B X
B 3 , 1 0 , 0
X 0 ,0 1 , 3
[ (N,B), B ]
B X
B 3 , 1 0 , 0
X 0 ,0 1 , 3[ (Q,X), X ]
B X
B 3 , 1 0 , 0
X 0 ,0 1 , 3
[ (Q,B), X ]
22
1
2, 2
Quietevening
A Variation of the Battle of the Sexes
Noisyevening
B X
B 3 , 1 0 , 0
X 0 ,0 1 , 3
Nash Equilibria
[ (N,B), B ]
[ (Q,X), X ]
[ (Q,B), X ]
not a sub-game perfect equilibrium !!!These S.P.E. guarantee player 1
a payoff of at least 27
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