Game Theoretic Analysis of Oligopoly

.

5-20

-5-5

y n

Y N

00

Y N

-205

1

2 2

The unique dominant strategy Nash Equilibrium is (y,Y)

A game of imperfect Information

The Prisoners’ DilemmaY y stand for compete

N n stand for collude

5-20

-5-5

y n

Y N

00

Y N

-205

1

2 2

The Prisoners’ DilemmaA game of Perfect Information

The only play at a Nash Equilibriumis (y, Y)

103

010

-3-4

TM

B

L RC

-211

-52

L RC

12-2

-3-4

L RC

10-1

34

1

2

2

2

A:1 plays T 2 plays R if T, R if M, R if BB: 1 plays B 2 plays L if T, R if M, C if B

C:1 plays M 2 plays R if T, L if M, C if B

Only C is a (Subgame) Perfect or ‘Credible’ Nash Equilibrium

1

2

Enter Stay Out

ToughSoft

3m3m

-1m 2m

07m

1- Entrant

2- Incumbent

1: Stay Out2: Tough if Enter

1: Enter2: Soft if Enter

The two Nash Equilibria are

Credible ThreatEquilibrium

Finitely Repeated Games

Prisoners’ Dilemma

5-20

-5-5

y n

Y N

00

Y N

-205

1

2 2

The Prisoners’ DilemmaA game of Perfect Information

Player 1 plays y and player 2 playsY if y and Y if n at the only Nash Equilibrium

Y y stand for compete

N n stand for collude

Game 2

5-20

-5-5

y n

Y N

00

Y N

-205

1

2 2

The Prisoners’ DilemmaA game of Perfect Information

Y y stand for compete

N n stand for collude

Game 200

Player 1 plays y and player 2 playsY if y and Y if n at the only Nash Equilibrium

Finite Sequence of Entry Games

1

2

Enter Stay Out

ToughSoft

3m3m

-1m 2m

07m

1- Entrant

2- Incumbent

1: Stay Out2: Tough if Enter

1: Enter2: Soft if Enter

The two Nash Equilibria are

Game withtwo sequentialentries

1

2

Enter Stay Out

ToughSoft

3m3m

-1m 2m

07m

1- Entrant

2- Incumbent

1: Stay Out2: Tough if Enter

1: Enter2: Soft if Enter

The two Nash Equilibria are

Game withtwo hundredsequentialentries

Collusive Behaviour

Reputation Building And Predatory Behaviour

Both play the Tit-for-Tat Strategy

Start with n or N (Collude)

Stick with n or N (Collude) until the other player deviates and plays Y

Play y (or Y) forever once the other player has played Y (or y)

Analysis of the Infinitely Repeated Game

Prisoners’ Dilemma

Either player payoff structure is as follows

Get 0 always if stick with n (or N)

Get 5 one-off with play y (or Y) and then (-5)forever

= 5-5/r

PDVY = 5 - 5/(1+r) -5/(1+r)2 - 5/(1+r)3 –

….. = 5 – (5/(1+r) +5/(1+r)2 + 5/(1+r)3 - …..)= 5 – 5/(1+r) *[1/1-{1/(1+r)}]

Present Discounted Value of playing collude forever (PDVN) is 0

Present Discounted Value of playing Compete now (PDVY) is

All Entrants : Play Stay out if the incumbent has no history of playing soft. Otherwise enter

Analysis of the case of anInfinite Chain of Sequential entry

Entry Games

• Incumbent: always play tough if enter

Payoff structure for incumbent:

Get 7m forever

Payoff structure for each entrant:

Get 0 forever

After any entry:Get 2m one-off with play tough and then 7mforever

Is the threat ‘credible’?

= 2 +7/r

PDVT = 2 + 7/(1+r) +7/(1+r)2 +7 /(1+r)3 –

….. = 2 +7 /(1+r) *[1/1-{1/(1+r)}]

Present Discounted Value of playing Threat strategy (PDVT) is

= 3(1+r)/r

PDVT = 3 + 3/(1+r) +3/(1+r)2 +3 /(1+r)3 –

….. = 3 *[1/1-{1/(1+r)}]

Present Discounted Value of playing Soft strategy (PDVS) is

2+ 7/r > 3(1+r)/rIf and only if r < 4

2

1

1

2

1

2

qA

qB

qB

(3, 1)

(2, 2)

(4, 1)

(2, 0) 

                   

A Duopoly Game involving two firmsA and B

Show that Cournot (Stackelberg) ideas are similar to Nash (Subgame Perfect Nash)

……………

S SSSS

SSS

S

S

G

S S

GGG

GGGG

GG

1

11

1

1

1

2

2

2

22

00

-10 1

1-10

-9 2

2 -9

-8 3

90101

101 90

91102

103 92

92103

102 91

2

Rosenthal’s Centipede Game

1

2 -1+1

-2-1

0 0

Y N

y n

Top Number is1’s Payoff

A Game of Loss Infliction

Y – Player 1 givesin to threat

y – Player 2 executes threat

Perfect Nash Equilibrium1 plays N2 plays n if N

But is 1 plays Y2 plays y if Nnon-credible?