Lecture 4
Dominance
14.12 Game Theory
Muhamet Yildiz
1
Road Map
1. Dominance & Rationality
2. Dominant-Strategy Equilibrium
3. 2nd price auction
2
Prisoners' Dilemma
2
Cooperate
Defect
Cooperate
(5,5)
(0,6)
Defect
(6,0)
(1,1 )
1
3
Dominance
Definition: A pure strategy Sj * strictly
dominates Sj if and only if
u(s*. ,s .»u(s,s .)
I
I
- I
A mixed strategy
O)SiJ )UJSil' S _i)
I
(Jj
I
- [
\/s.
- I
strictly dominates
Sj
+ ... + (Ji (Sik )ui (Sik' S _i ) > Ui (Si' S _i )
iff
'lis .
- /
A rational player never plays a strictly
dominated strategy.
4
Prisoners' Dilemma
2
Cooperate
Defect
Cooperate
(5,5)
(0,6)
Defect
(6,0)
(1,1 )
1
5
vM = 0
V T = 2p-(I-p) = 3p-1
A Game
VB = -p+2 (I-p) = 2-3p
V
L
R
T
(2,0)
(-1,1)
M
(0,10) (0,0)
B
(-1,-6)
(2,0)
p
I-p
2
-I ~_ _ _ _ _ _ _-------'
o
P
1
6
Weak Dominance
Definition: A pure strategy Sj * weakly dominates
if and only if
u(s*,s .) > u .(s,s .)
I
I
- I
I
l
- l
Sj
\is.
- I
and at least one of the inequalities is strict. A mixed
strategy a/ weakly dominates Sj iff
and at least one of the inequalities is strict.
If a player is rational and cautious (i.e., he assigns
positive probability to each of his opponents'
strategies), then he will not playa weakly
dominated strategy.
7
Dominant-strategy equilibrium
Definition: A strategy Sj * is a dominant
strategy iff Sj * weakly dominates every
other strategy Sj.
Definition: A strategy profile s* is a
dominant-strategy equilibrium iff Sj* is a
dominant strategy for each player i.
8
Prisoners' Dilemma
2
1
Cooperate
Cooperate
Defect
(5,5)
(0,6)
. J. >
Defect
(6,0)
~7
r(c l,l ~
"-
./
9
Second-price auction
• N = {1,2} buyers;
• The value of the house
for buyer i is Vi;
• Each buyer i
simultaneously bids b i;
• i* with b i> = max b i gets
the house and pays the
second highest bid
Courtesy of Machovka on OpenClipart.org.
p = maxj*ibjo
10
2nd price Auction
• Strategies:
bi
E
[0,(0)
• Payoffs:
U i (bi,b) =
=
=
Vi - bj
ifb·I > bJ·
(Vi - b)12
ifb·I = b·J
°
otherwise.
11
bi =
Vi
is a dominant strategy
12
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14.12 Economic Applications of Game Theory
Fall 2012
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