Chapter 9: Information and Strategic
Behavior
• Asymmetric information.
• Firms may have better (private) information on
– their own costs,
– the state of the demand...
• Static game
– firm’s information can be partially revealed by its
action,
– myopic behavior.
• Dynamic game (repeated interaction)
– firm’s information can be partially revealed,
– can be exploited by rivals later,
– and thus manipulation of information.
• Accommodation
• entry deterrence (Limit Pricing model, MilgromRoberts (1982))
1
1 Static competition under Asymmetric Information
• 2 period model
• 2 risk-neutral firms: firm 1 (incumbent), firm 2
(potential entrant)
Timing:
Period 1. – Firm 1 takes a decision (price, advertising,
quantity...).
– Firm 2 observes firm 1’s decision, and takes an action
(entry, no entry...).
Period 2. If duopoly, firms choose they price simultaneously (Bertrand competition).
Period 2, if entry.
• Differentiated products.
• Demand curves are symmetric and linear
Di(pi, pj ) = a − bpi + dpj
for i, j = 1, 2 and i 6= j where 0 < d < b.
2
i
• The two goods are substitutes ( dD
dpj = d > 0) and
2
i
strategic complements ( dpd iΠ
dpj > 0).
• Marginal cost of firm 2 is c2, and common knowledge.
L
• Marginal cost of firm 1 can take 2 values c1 ∈ {cH
1 , c1 }
and is private information.
• Firm 2 has only prior beliefs concerning the cost of its
rival, x. Thus(
cL1 with probability x
c1 =
cH
1 with probability (1 − x)
• Firm 1’s expected MC from the point of view of 2 is
ce1 = xcL1 + (1 − x)cH
1
• Ex post profit is
Πi(pi, pj ) = (pi − ci)(a − bpi + dpj )
• Firm 1’s program is
– if c1 = cL1
Max(p1 − cL1 )(a − bp1 + dp∗2)
p1
3
– If c1 = cH
1
∗
Max(p1 − cH
1 )(a − bp1 + dp2 )
p1
• Firm 2’s program
Max{x[(p2 − c2)(a − bp2 + dpL1 )]
p2
+(1 − x)[(p2 − c2)(a − bp2 + dpH
1 )]}
which is equivalent to
Max{(p2 − c2)(a − bp2) + (p2 − c2)pe1}
where
p2
pe1 = xpL1 + (1 − x)pH
1
• Best response functions are
a + bcL1 + dp2
L
p1 =
= R1L(p2)
2b
pH
1
• Graph
a + bcH
1 + dp2
=
= R1H (p2)
2b
a + bc2 + dpe1
= R2(pe1)
p2 =
2b
4
• Solution of the system of 3 equations gives
p∗2
• where
∂p∗2
∂ce1
2ab + ad + 2b2c2 + dbce1
=
4b2 − d2
> 0 and
∂p∗2
∂(1−x)
>0
• Then you plug p∗2 in R1L(p2) and R1H (p2) to find the
solution pL1 and pH
1 .
• Under asymmetric information, everything is “as if”
firm 1 has an average reaction curve
R1e (p2) = xR1L(p2) + (1 − x)R1H (p2)
a + bce1 + dp2
=
2b
• Firm 1 has an incentive to prove that it has a high cost
before engaging in price competition.
5
2 Dynamic Game
• Assume that direct disclosure is impossible.
Timing:
Period 1. Price competition
Period 2. Price competition
• If entry is not an issue (accommodate), firms want to
appear inoffensive so as to induce its rival to raise its
price.
• Thus, in first period: high price to signal high cost.
• Thus, accommodation calls for puppy dog strategy
(be small to look inoffensive).
• If deterrence is at stake, more aggressive behavior: the
firm wants to signal a low cost.
• Thus, in first period, low price to induce its rival to
doubt about the viability of the market (limit pricing
model).
• Thus, deterrence calls for top dog strategy.
6
3 Accommodation
• A firm may rise its price to signal high cost and soften
the behavior of its rival.
• Riordan (1985)’s model
• 2 firms
Timing:
Period A. Price competition
Period B. Price competition
• Marginal cost is 0.
• Firm i’s demand is
qi = a − pi + pj
• The demand intercept is unknown to both firms, and
has a mean ae.
• In a one-period version of the game, program of firm i
Max{E(a − pi + pj )pi = (ae − pi + pj )pi}
pi
• thus
ae + pj
pi =
,
2
7
• and by symmetry, the Static Bertrand equilibrium is
p1 = p2 = ae.
• 2 period version with same a for each period, and each
firm observes the realization of its own demand.
• In the symmetric equilibrium,
– each firm sets
A
pA
=
p
1
2 =α
in the first period.
– Thus, each firm learns perfectly a as
DiA = a − α + α = a
– and the second-period is of complete information,
and the program of firm i
B
B B
Max
(a
−
p
+
p
i
j )pi
B
pi
• thus
pB
i
a + pB
j
=
,
2
8
• and the symmetric equilibrium of second period is
B
pB
1 = p2 = a.
• Consider a strategic behavior in period A: firm i
deviates and chooses
pA
i 6= α
• Firm j observes a demand of
DjA = a − α + pA
i
• Firm j has a wrong perception of a, and has a
perception e
a,
a − α + pA
a−α+α =e
a
i =e
and thus
A
e
a(pA
)
=
a
−
α
+
p
i
i
• In the second period, j believes it is playing a game of
perfect information, with intercept e
a(pA
i ), so it charges
A
pB
a(pA
j = e
i ) = a − α + pi
9
and thus
∂pB
j
=1
∂pA
i
• A unit increase in the first period triggers a unit increase
in the rival’s second period price.
• However i knows the intercept is not the right one, and
the program of i in the second period is
B
B
A
B
Max
{Π
=
(a
−
p
+
e
a
(p
))p
i
i
i
i }
B
pi
• Thus
a+e
a(pA
pA
−α
i )
=
=a+ i
2
2
• The derivative of the second period profit with respect
to pA
i is
B
∂ΠB
∂ΠB
dΠB
i
i ∂pi
i
=
+
A
dpA
∂pB
∂pA
i
i ∂pi
i
pB
i
A
∂e
a
(p
i )
B
= pi
∂pA
i
= pB
i
10
• Firm i maximizes its expected present discounted
profit, thus the FOC is
B
dΠA
dΠ
E Ai + δE Ai = 0
dpi
dpi
• where δ is the discount factor.
• Thus, it is equivalent to
e
a −
2pA
i
pA
−α
+ α + δ(a + i
) = 0
2
e
• In equilibrium pA
i = α, thus
α = ae(1 + δ) > ae
• In a dynamic model, a firm may induce its rival to raise
its price.
11
4 The Milgrom-Roberts (1982)
Model of Limit Pricing
• Asymmetric information drives firms to cut their price
in first period.
• 2 risk-neutral firms: firm 1 (incumbent), firm 2
(potential entrant)
• Asymmetric information on firm 1’s costs. Firm 2 has
only prior beliefs concerning the cost of its rival, x.
Thus
(
cL1 with probability x
c1 =
cH
1 with probability (1 − x)
Timing:
Period 1.
• Firm 1 chooses a first period price p1.
– Firm 2 observes p1 and decides whether to enter
{e, ne}.
Period 2. If firm 2 enters: price competition. If not,
monopoly.
12
• Firm 2 learns 1’s cost immediately after entering.
• The incumbent’s profit when price is p1 is
M1t(p1) = (p1 − ct1)Q(p1)
where t = H, L. (strictly concave function in p1)
– Thus pL1 , pH
1 are the monopoly prices charged by the
incumbent, pL1 < pH
1 .
• Duopoly’s payoffs are Dit for t = H, L and i = 1, 2.
• Assume D2H > 0 > D2L: if low cost, no room for 2
firms, if high cost, room for duopoly.
• δ Discount factor.
• To simplify: only 2 prices pL1 , pH
1 and not a continuum
of prices.
• Perfect Bayesian Equilibrium concept.
• See tree of the game
13
Benchmark case: symmetric information
• Cost is low with probability x = 1
• Cost is high with probability x = 0.
• Decisions of firm 2 to enter?
– if low cost: does not enter,
– if high cost: enters.
• Decision of firm 1?
– if low cost, firm 1 chooses a low price if
L L
M1L(pL1 ) + δM1L(pL1 ) > M1L(pH
1 ) + δM1 (p1 )
⇒ M1L(pL1 ) > M1L(pH
1 )
which is always satisfied.
– if high cost, firm 1 chooses a high price if
H
H L
H
M1H (pH
)
+
δD
>
M
(p
)
+
δD
1
1
1
1
1
H L
⇒ M1H (pH
1 ) > M1 (p1 )
Result 1. Under symmetric information
♦ If c = cL1 , (pL1 , ne) is a Perfect Nash Equilibrium
H
♦ If c = cH
1 , (p1 , e) is a Perfect Nash Equilibrium
14
Asymmetric Information
• Separating equilibrium?
The incumbent does not choose the same price when
its cost is high or low.
• Pooling equilibrium?
The first period price is independent of the cost level.
Separating equilibrium
• Only one possible kind of separating:
– If c = cL1 , ne
– If c = cH
1 ,e
• Is it an equilibrium? and under what kind of circumstances?
• It is an equilibrium if none of the firms deviate.
– If c = cL1
L
M1L(pL1 ) + δM1L(pL1 ) > M1L(pH
1 ) + δD1
(1)
L
L L
⇒ M1L(pL1 ) − M1L(pH
1 ) > δ(D1 − M1 (p1 ))
15
– If c = cH
1
H
H L
H H
M1H (pH
1 ) + δD1 > M1 (p1 ) + δM1 (p1 )
(2)
H L
H H
H
⇒ M1H (pH
1 ) − M1 (p1 ) > δ(M1 (p1 ) − D1 )
– The equation (1) is always satisfied, whereas (2)
must be satisfied.
Result 2. If (2) is satisfied, there exists a separating
equilibrium such that
♦ the incumbent chooses pL1 and firm 2 does not enter
(ne) if c = cL1 ,
♦ the incumbent chooses pH
1 and firm 2 enters (e) if
c = cH
1 .
Pooling equilibrium
• Two possible kinds of pooling:
P1. the incumbent always chooses pL1 , whatever the cost,
P2. the incumbent always chooses pH
1 , whatever the cost.
• Updated beliefs equal to prior beliefs.
16
P1. (pL1 ) Player 2 stays out if
0 > xδD2L + (1 − x)δD2H
•
•
•
•
•
D2H
⇒ x>x
e= H
D2 − D2L
x
e ∈ [0, 1]?
x
e > 0 if D2H > D2L,
x
e < 1 if D2L < 0.
Thus, for x > x
e firm 2 prefers to stay out.
Can firm 1 do better?
– If c = cL1
L
M1L(pL1 ) + δM1L(pL1 ) > M1L(pH
1 ) + δD1
L L
and M1L(pL1 ) + δM1L(pL1 ) > M1L(pH
1 ) + δM1 (p1 )
17
OK
OK
– If c = cH
1
H H
H
M1H (pL1 ) + δM1H (pH
1 ) > M1 (p1 ) + δD1
OK
H H
H H
and M1H (pL1 ) + δM1H (pH
1 ) > M1 (p1 ) + δM1 (p1 )
NO
• Thus, with an out-of-equilibrium prob(e/pH
1 ) = 1,
there exists a pooling.
Result 3. If (2) is not satisfied, there exists a pooling
equilibrium such that
♦ the incumbent always chooses pL1 ,
♦ and firm 2 does not enter (ne)
♦ with an out-of-equilibrium probability
prob(e/pH
1 ) = 1.
18
P2. (pH
1 ) Player 2 enters if
xδD2L + (1 − x)δD2H > 0
D2H
⇒ x<x
e= L
D2 − D2H
• Then for x < x
e firm 2 will enter.
• Can firm 1 do better?
– If c = cL1
L
L H
L L
M1L(pH
)
+
δD
>
M
(p
)
+
δM
1
1
1
1
1 (p1 )
L L
L L
L L
and M1L(pH
1 ) + δM1 (p1 ) > M1 (p1 ) + δM1 (p1 )
NO
NO
– Thus firm 1 will always deviate.
• There is no pooling P2.
• If (2) is not satisfied, the incumbent manipulates the
price such that its action does not reveal any cost
information.
19
• In continuous p ∈ [0, ∞[, same results except that
prices are different.
• Single-crossing condition
m
∂Q
∂ 2[(p1 − c1)Qm
(p
)]
1
1
=− 1 >0
∂p1∂c1
∂p1
• It is more costly to the high type to charge low price.
Separating equilibrium
H
H
• – if c = cH
1 , p1 = pm
– if c = cL1 , pL1 ∈ [e
pe1, pe1] where pe1 < pLm. Low cost
type makes pooling very costly to the high cost type.
• There exists a reasonable separating equilibrium where
H
H
– if c = cH
1 , p1 = pm and entry occurs,
– if c = cL1 , pL1 = pe1and no entry.
• The incumbent does not fool the entrant
• But, there exists a limit pricing.
20
Pooling equilibrium
• The incumbent chooses pLm.
• The incumbent manipulates its price.
• Less entry occurs than under symmetric information.
• High cost type is engaged in limit pricing.
21