1. problem set 6 from osborne’s introd. to g.t. p.210 ex. 210.1 p.234 ex. 234.1 p.337 ex. 26,27...
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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
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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
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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
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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= , =
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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:
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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
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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
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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
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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.
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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
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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
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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
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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.
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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
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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
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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
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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
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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
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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
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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 ]
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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