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(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
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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
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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
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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
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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
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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
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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)
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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".
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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)
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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
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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]:
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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)
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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
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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
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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
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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
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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?
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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)
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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
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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
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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
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Result: The SPNE is given by
q1 = a c
2b
q2(q1) = a c bq1
2b
Note: In equilibrium q2 = ac
4b
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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
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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
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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=;
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5. Time t payo:
i :A Ht! 0
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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
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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!
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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:
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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.
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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.)
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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).
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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:
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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
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Illustration
1
2
m
m
o
All payoffs are sustainable for
sufficientl lar e discount factors
Conclusion: Many equilibria
(a negative result)
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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
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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.
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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 +
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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.
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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)