B Oligopoly Theory

8/13/2019 B Oligopoly Theory

1/47

(B) Oligopoly TheoryThe main solution concept for this course is game

theory; We will apply it a lot.

Plan

1. Cournot

2. Bertrand

3. A Simple Auction Game

4. Multi-Stage (Period) Games

5. Stackelberg

6. Collusion, innitely repeated games

7. Sequential Bargaining


2/47

Duopoly Models

Assumptions

1. Two rms with constant marginal cost c

2. Identical products

3. Inverse market demand

P(Q) = a bQQ = q1+ q2

Consider two strategy spaces: quantity and price

(i) Quantity: Cournot Equilibrium

(ii) Price: Bertrand Equilibrium


3/47

1. Cournot Equilibrium: (Cournot NE)

1. (a) Pure strategy: q1 for rm 1, q2 for rm 2.

(b) Payos:

i= [a b(q1+q2) c] qi for i= 1; 2

How to nd a NE?

FOC@1

@q1

=a 2bq1 bq2 c= 0

or expressed as a reaction function

qR1(q2) =a c bq2

2bSimilarly for rm 2

qR

2(q1) =

a c bq12b

Two equations in two unknowns q1 =q2= ac

3b


4/47

Result: Cournot NE

q1=q2=a c

3b

Illustration using reaction functions

(a-c)/2b

q2

C-N

q2

(a-c)/3b

(a-c)/3b (a-c)/2b

q1R

q2R

M


5/47

Nrm Cournot ModelPayo

i = [a b(q1+: : :+qN) c] qi

for i = 1; 2; : : : ; N

How to solve for the reaction functions?

FOC

@1@q1

=a b(q1+: : :+qN) bq1 c= 0

reaction function

qRi =a c b

Pj6=i qj

2b


6/47

Suppose a symmetric solution exists:

q1 =q2=: : :=qN

Result: Cournot Equilibrium

qi = a c

b(N+ 1)

Comments

1. SOC satised

2. Q= acb

NN+1 and P =

a+N cN+1

3. As N! 1; P !c

converge to competitive equilibrium


7/47

3. Bertrand equilibrium

an alternative strategy space

(a) Pure strategy: p1 0 for rm 1, p2 0 for rm

2

(b) Payos: Consider the demand curve

Pi

P

Q

D

Pj

IfPj < Pi, then rm j gets all demandIfPj =Pi, then each rm gets half the customers. (a

convention)


8/47

Notice: Payos are discontinuous

because rm specic demand is discontinuousImplication: Cannot use the rst order condition

NE?

Result: Unique Bertrand EquilibriumP1=P2=c

How to show this?

1. At P1

=P2

=cconsider a deviation, no rm benets from de-viating

2. Pi > c; Pj =c cannot be a NEFirm j could increase it's price and make pos-itive prot.

3. P1; P2 > c cannot be a NEIfPi Pj, then rm i can protably "under-cut".


9/47

Comments

1. P = c is the unique symmetric NE for any

N2

2. P =c seems extreme

Note: Most products are dierentiated

3. Constant marginal cost is extreme

with increasing marginal costs, Bertrand out-

come is closer to Cournot

Indeed: With capacity constraints the two out-

comes may coincide (Kreps and Scheinkman)


10/47


11/47


12/47

NE?

Pure strategies

Claim: There is no pure strategy equilibrium.

How to show this?

Suppose uninformed uses a pure strategy bU.

How would informed rm respond?

1. Ifv > bU?Then bI=bU+" with " small.

2. Ifv bU?

Then submit a low bid (bI < bU)

Conclusion: Uninformed would make a loss

Uninformed bidder has to "disguise" his strategy

by using a mixed strategy


13/47

Result: NE: bI = v2; bU uniformly distributed on

[0;12]:

How to show this?

Verify

Uninformed bidder

U wins if bU > bI

or ifv < 2bU

Expected value of the object conditional on bidder

U winning?

E[vjUwins] =bU

Why? becausevis distributed uniformly on [0; 1]:


14/47

Expected payo for U?

1. For bU > 1

2

, make a loss U bI)| {z }Payo if win Winning probability

= [bU bU] Pr(bU > bI)

= 0 Pr(bU > bI)

= 0

Hence, optimal (constant payos)


15/47

Informed bidder

1. What is the winning probability?

Pr(bI > bU) =

8>>>:

0 ifbI 12

Since bUuniformly distributed on [0;12]:

2. Expected payo?

I = [v bI]| {z } Pr(bI > bU)| {z }Payo if win Winning probability

3. FOC for 0bI 12

(v bI)2 2bI = 0

=) bI=v

2

4. SOC satised


16/47

Properties of Equilibrium

1. Pure strategy by informed rmMixed strategy by uninformed rm (to disguise)

Interpretation of mixed strategies:

Non-payo relevant types (aggressive or not)

2. Expected Prots:

U = 0

I =

v

2

v >0


17/47


18/47


19/47

Oligopoly Theory (continued)

So far: Static Oligopoly Games

Now: Multi Stage Games

Plan

1. Review: Subgame Perfect Equilibrium

2. Stackelberg Game

3. Review: N-player Multi Period Games

4. Repeated Oligopoly Games

5. Rubinstein's Bargaining Model


20/47

1. Review: Subgame Perfect Equilibrium

Games in extensive form (game tree)

Example: A game with two stages (subgames

0

2

Player 1

Player 2

D

-1

-1

1

1


21/47

here there are two subgames

Payos are given at the end nodes: x

y

!where x is the payo to player 1 and y is

the payo to player 2

Recall: A Subgame perfect Nash equilibrium (SPNE)

is a NE in every subgame

NE?


22/47

0

2

Player 1

Player 2

D

-1-1

11

NE: (U; L); (D; R)

but (U; L) is not subgame perfect as it is not a NE inthe subgame (not credible!)

SPNE?

(D; R)

How to solve? Backwards Induction (start in the

last subgame and work backwards)


23/47

2. Stackelberg Game

Assumptions

1. Two rms constant marginal cost c

2. Firms choose quantities q1; q20

3. Firm 1 moves rst and then, after observing

output q1 rm 2 moves

4. Inverse demand curve

P =a b (q1+q2)

with a; b >0.

5. Prots

i= [a b (q1+q2) c]qi for i= 1; 2


24/47

How to solve a multi-stage game?

Backwards

1. Solve stage 2: for all possible quantities q1

2. Solve stage 1: anticipating rm 2's reaction

qR2(q1)

Stage 2: Firm 2's problem

maxq20

2= [a b (q1+q2) c]q2

FOC

a b (q1+ q2) c bq2 = 0

SOC is satised as 2b


25/47

Stage 1: Firm 1 anticipates 2's reaction

What is rm 1's problem?

maxq10

1= ha b

q1+q

R2(q1)

c

i q1

Substituting qR2(q1) = acbq1

2b yields

1=

a c

2 b

q1

2

q1

FOC

a c

2 bq1 = 0

q1 = a c

2b

SOC is satised as b


26/47

Result: The SPNE is given by

q1 = a c

2b

q2(q1) = a c bq1

2b

Note: In equilibrium q2 = ac

4b


27/47

Comments

1. Dierent outcome as under Cournot (order of

moves matters)

(a) Quantities: q1 = 2q2

(b) Industry output: QS = 34 acb > 23 acb =QCN

(c) Price:PS = ac4 +c < ac

3 +c =PCN

2. First Mover Advantage

(a) S1 = (ac)2

8b > (ac)2

9b = CN1

(b) S2 = (ac)2

16b < (ac)2

9b = CN2

(c) Intuition: depends on the slope of the reac-

tion function

downward sloping: rst mover advantage

upward sloping: rst mover disadvantage


28/47

Graphical Illustration of Stackelberg Equilib-rium

(a-c)/2b

q2

S

C-N

q2

(a-c)/4b

(a-c)/4b (a-c)/2b

q1R

q2R

M


29/47

3. N Player Multi-Period Game

Notation

1. N Players: indexed by i

2. T periods: indexed by t= 1; 2; : : : ; T

(where T is nite or innite)

3. Actions: a

t= a1t ; : : : ; aNt 2A;ait2A

i (action set) for i= 1; : : : ; N

A=A1 : : : AN

4. History: ht = (a1; : : : ; at1)2Ht(history set), (ht is a node in the tree)

H1=;


30/47

5. Time t payo:

i :A Ht! 0


33/47

4. DemandD pmint wherepmint = min(p1t ; : : : ; pNt )Sales of rm i:

qit(pt) =

8>>>>>>>:

0 ifpit > pmint

Dpmint

k

ifpitpmint and

k= #fijpit=pmint g

(equal rationing rule)

5. Discount factor = 11+r

where r is the interest rate

6. Period payo

i (pt) =pit c

qit(pt)

7. Game payo

TXt=1

t1i (pt)

8. History: ht = (p1; : : : ; pt1)2Ht


34/47

One shot: T = 1

N2: Bertrand game:

pt=c

is the unique symmetric equilibrium

Monopolist:

max(p c) D(p)

=) m = (pm c) D(pm)

Repeated game: T >1

Strategy: pit:Ht!


35/47

Finite T

What is the SPNE?

Result: The SPNE is given by

pit(:) =c for any t; i

How to show this?

Backwards induction

1. At T: for all histories hT rm ichooses piT to

maximize i (pT) given pjT(hT).

The only symmetric NE is

piT(hT) =c for all hT:


36/47

2. At T 1: for all histories hT1 rm i chooses

piT1to maximize ipT1

+i (pT) given

pjT1(hT1):

Since, piT(hT) =c for all hT, implies

i (pT) = 0;

we can rewrite the objective function as:

maximize ipT1

given pjT1(hT1);

Again: The only symmetric NE is

piT1(hT1) =c for all hT1:

(...)

3. Induction

Att= 1 : same argument applies

Unique symmetric SPNE is pit =c for all t; i, the

Bertrand outcome in every period.


37/47

Innite T. What are SPNE?

1. pit(h

t) =c for all t; i is a SPNE

2. Trigger strategies

(Aumann, Shubik, Telser, Friedman)

p

i

t(ht) =

8>>>>>>>:

pmift= 1; or

ifht = (pm; : : : ; pm)| {z };N (t-1) times

c otherwise.

(Intuition: a deviation triggers marginal cost

pricing.)


38/47

Result: If (N), then trigger strategy is a

SPNE.

Proof:

Follow strategy: get discounted present value

V =1

Xt=0 t

m

N

= 1

N

m

1

(why? because it is a geometric sum).


39/47

Now, suppose rm i cheats in period t: Firm i

gets at most m in period t, and 0 thereafter:

Vd =t1X=0

m

N

| {z }+ tm

| {z }+

1X=t+1

0

| {z }collude defect punishment=

t1X=0

m

N+tm

Defect in period t ifVd > V :

m

>

m

N +

1

X=1

m

N

= 1

N

m

1

equivalently, defect if:


40/47

Result: If (N), then any price r2[c; pm]

is sustainable in a SPNE.

Proof?

use the trigger strategy, same argument as before.

Folk Theorems

All symmetric period payo divisions from 0 tom

N are sustainable in a SPNE


41/47

Illustration

1

2

m

m

o

All payoffs are sustainable for

sufficientl lar e discount factors

Conclusion: Many equilibria

(a negative result)


42/47

5. Rubinstein's Bargaining Model

Story: A pie of unit size.

Two players must agree on how to share the pie.

An alternating oer game.

Period 0; 2; 4; : : ::

1. Player 1 oers a sharing rule (x; 1 x) wherex is the share for player 1.

2. Player 2 accepts or rejects the oer

acceptance =) the game ends

reject =) the game continuous


43/47


44/47

NE?

1. Player 1 always demands x = 1, and rejects

all smaller shares.

Player 2 always oers y = 1 and accepts any

oer.

2. The same strategies as above, but with iden-

tities reversed.

Above NE are not SPNE: Why?

Because if 2 rejects 1's rst oer, and oers a

share 1 > x > , then 1 should accept. Why? If

1 rejects, then at best he receives the entire pie

tomorrow which is worth only.


45/47

Result: The SPNE is given by both players adopt-

ing the following strategy:demand x = 11+ and accept any share

1+ .

How to show this?

1. Proposer: Oer x

is the highest share for i

that will be accepted.

Suppose x > x; this will be rejected;

=) can get at most

1 1

1 +

=

2

1 +

< 11 +

2. Acceptance Rule: Suppose oerx is rejected.

Then next period receive:

1

1 + 1

1

1 +

=

1 +


46/47

Can show that SPNE is unique (omitted here)

Features of SPNE

1. unique

2. agreement is reached immediately

3. First oer advantage

4. As ! 1, then rst oer advantage disap-

pears. Players split the pie equally.


47/47

What happens if the game ends after period 0?

That is:

1. Player 1 oers a sharing rule (x; 1 x) where

x is the share for player 1.

2. Player 2 accepts or rejects the oer, and the

game ends.

NE?

1. (x; 1 x) = 12;122. any share x2(0; 1)

SPNE?

solve backwards2nd stage: accept any oer such that 1 x0

1st stage: oer (1; 0)

Documents

B Oligopoly Theory