game theory and applications following h. varian chapters 28 & 29
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Game Theory and Applications
following H. Varian Chapters 28 & 29
A
P L A Y E R B
le ft rig ht
To p
Bo tto m
1,2 0,1
2,1 1,0
A p a yo ff m a trix o f a g a m e
A
P L A Y E R B
le ft rig ht
To p
Bo tto m
2,1
0,0
0,0
1,2
A Na sh Eq uilib irum
In the p a irs o f num b e rs, re a d the le ft d ig it a s Pla ye r A 's p a yo ff a nd the rig htd ig it a s Pla ye r B's.
Defining a Nash Equilibrium
The situation when Player A's choice is optimalfor him given Player B's choice; and Player B'schoice at the same time is optimal for him given Player A's choice.
Or: A pair of strategies (r*,c*) such that c*=bc(r*) and r*=br(c*),
where c* is Player B's best response, and r* is Player A's best response, and b is a functionthat chooses the best response for player i=r,c.
Mixed Strategies:
A
P L A Y E R B
le ft rig ht
To p
Bo tto m
2.1 0,0
0,0 1,2
A Sim p le G a m e
Let r be the probability that "Row" plays Topand let c be the probability that "Column" plays Left.
Combination Probability PayofftoRow
Top, Left rc 2
Bottom, Left (1-r)c 0
Top, Right r(1-c) 0
Bottom,Right (1-r)(1-c) 1
Row's payoff= 2rc + 1 -r - c +rc; In changes: RowPay=(3c-1)rCol's payoff=cr +2(1-c)(1-r); In changes: ColPay=(3r-2)c
Ro w's Be stRe sp o nse
C o lum n's Be st Re sp o nse
2/3
1/3
c 1
0 1r
Best Resp onse C urves
A
P L A Y E R B
le ft rig ht
To p
Bo tto m
2,1 0,0
0,0 1,2
The Ba ttle o f the Se xe s
A
P L A Y E R B
le ft rig ht
To p
Bo tto m
The Priso ne r's Dile m m a
-3,-3 0,-6
-6,0 -1,-1
He re re a d "To p " a nd "le ft" a s "c o n fe ss"
A
P L A Y E R B
le ft rig ht
To p
Bo tto m
4,4 1,3
3,1 2,2
An a rm s ra c e
A
P L A Y E R B
le ft rig ht
To p
Bo tto m
0,0 -1,1
1,-1 -2,-2
C hic ke n
A
P L A Y E R B
le ft rig ht
To p
Bo tto m
50,-50 80,-80
90,-90 20,-20
Pe na lty Po int in So c c e r
A
P L A Y E R B
le ft rig ht
To p
Bo tto m
-2,-2 4,0
0,4 2,2
Ha wk-d o ve G a m e
Pa yo ff
4
2
0.5 0.66 1.0
Pa yo ff in the Ha wk-Do ve G a m e
.
.
.
Sc o rp io nC ho o se s
0,0
5,3
-10,5
Re fuse
To p
Pla ye r A p a yo ff is le ft d ig it in e a c h p a ir.
Sting
Re fra in
The Fro g a nd the Sc o rp io n
.
.
.
Sc o rp io nC ho o se s
0,0
5,3
-10,2
Re fuse
To p
Pla ye r A p a yo ff is le ft d ig it in e a c h p a ir.
Sting
Re fra in
The Fro g a nd the Sc o rp io n Pa rt II
.
.
.
Ho sta g eC ho o se s
-3 ,-10
5,3
-5 ,5
Kill
Re le a se
Kid na p p e r p a yo ff is le ft d ig it in e a c h p a ir.
Id e ntify
Re fra in
The Kid na p G a m e
A
P L A Y E R B
le ft rig ht
To p
Bo tto m
0,0 4,1
0,5 2,3
Pig s Pre ssing Le ve rs
P L A Y E R Young
le ft rig ht
To p
Bo tto m
3,-1 1,1
2,-1 -2,-2
Inte rg e nera tio na lc o nflic t o ve r sa ving s
OLD
.
.
.
Yo ung C ho o se
1,1
-2,-2
2,-1
Sa ve
Sq ua nd e r
Kid na p p e r p a yo ff is le ft d ig it in e a c h p a ir.
Sup p o rt
Re fra in
The Sa ving s G a m e, Exte nd e d Fo rm
Yo ungC ho o se
Re fra in
Sup p o rt 3,-1
.
.
.
C lie nt C ho o se
0,1300
0,-100
1300,0
C ha rg eAc tua l C o st
Exto rt
Kid na p p e r p a yo ff is le ft d ig it in e a c h p a ir.
G ive in
Find a Pa inte r
The Ho ld -up Pro b le m
s