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