Chapter 9: Information and StrategicBehavior

• 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 itsaction,

– 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, Milgrom-Roberts (1982))

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1 Static competition under Asym-metric 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 simultane-ously (Bertrand competition).

Period 2, if entry.• Differentiated products.• Demand curves are symmetric and linear

Di(pi, pj) = a− bpi + dpjfor i, j = 1, 2 and i 6= j where 0 < d < b.

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• The two goods are substitutes (dDidpj= d > 0) and

strategic complements ( d2Πi

dpidpj> 0).

• Marginal cost of firm 2 is c2, and common knowledge.• Marginal cost of firm 1 can take 2 values c1 ∈ {cH1 , cL1}and is private information.

• Firm 2 has only prior beliefs concerning the cost of itsrival, x. Thus

c1 =

(cL1 with probability xcH1 with probability (1− x)

• Firm 1’s expected MC from the point of view of 2 isce1 = xc

L1 + (1− x)cH1

• Ex post profit isΠi(pi, pj) = (pi − ci)(a− bpi + dpj)

• Firm 1’s program is– if c1 = cL1

Maxp1(p1 − cL1 )(a− bp1 + dp∗2)

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– If c1 = cH1

Maxp1(p1 − cH1 )(a− bp1 + dp∗2)

• Firm 2’s programMaxp2{x[(p2 − c2)(a− bp2 + dpL1 )]

+(1− x)[(p2 − c2)(a− bp2 + dpH1 )]}which is equivalent to

Maxp2{(p2 − c2)(a− bp2) + (p2 − c2)pe1}

wherepe1 = xp

L1 + (1− x)pH1

• Best response functions arepL1 =

a + bcL1 + dp22b

= RL1 (p2)

pH1 =a + bcH1 + dp2

2b= RH1 (p2)

p2 =a + bc2 + dp

e1

2b= R2(p

e1)

• Graph

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• Solution of the system of 3 equations gives

p∗2 =2ab + ad + 2b2c2 + dbc

e1

4b2 − d2

• where ∂p∗2∂ce1> 0 and ∂p∗2

∂(1−x) > 0

• Then you plug p∗2 in RL1 (p2) and RH1 (p2) to find thesolution pL1 and pH1 .

• Under asymmetric information, everything is “as if”firm 1 has an average reaction curve

Re1(p2) = xRL1 (p2) + (1− x)RH1 (p2)

=a + bce1 + dp2

2b

• Firm 1 has an incentive to prove that it has a high costbefore engaging in price competition.

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2 Dynamic Game• Assume that direct disclosure is impossible.Timing:Period 1. Price competitionPeriod 2. Price competition

• If entry is not an issue (accommodate), firms want toappear inoffensive so as to induce its rival to raise itsprice.

• 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: thefirm wants to signal a low cost.

• Thus, in first period, low price to induce its rival todoubt about the viability of the market (limit pricingmodel).

• Thus, deterrence calls for top dog strategy.

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3 Accommodation• A firm may rise its price to signal high cost and softenthe behavior of its rival.

• Riordan (1985)’s model• 2 firmsTiming:Period A. Price competitionPeriod B. Price competition

• Marginal cost is 0.• Firm i’s demand is

qi = a− pi + pj• The demand intercept is unknown to both firms, andhas a mean ae.

• In a one-period version of the game, program of firm i

Maxpi{E(a− pi + pj)pi = (ae − pi + pj)pi}

• thuspi =

ae + pj2

,

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• and by symmetry, the Static Bertrand equilibrium isp1 = p2 = a

e.

• 2 period version with same a for each period, and eachfirm observes the realization of its own demand.

• In the symmetric equilibrium,– each firm sets

pA1 = pA2 = α

in the first period.– Thus, each firm learns perfectly a as

DAi = a− α + α = a

– and the second-period is of complete information,and the program of firm i

MaxpBi

(a− pBi + pBj )pBi• thus

pBi =a + pBj2

,

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• and the symmetric equilibrium of second period is

pB1 = pB2 = a.

• Consider a strategic behavior in period A: firm ideviates and chooses

pAi 6= α

• Firm j observes a demand ofDAj = a− α + pAi

• Firm j has a wrong perception of a, and has aperception ea,

a− α + pAi = ea− α + α = eaand thus ea(pAi ) = a− α + pAi

• In the second period, j believes it is playing a game ofperfect information, with intercept ea(pAi ), so it charges

pBj = ea(pAi ) = a− α + pAi

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and thus∂pBj∂pAi

= 1

• A unit increase in the first period triggers a unit increasein the rival’s second period price.

• However i knows the intercept is not the right one, andthe program of i in the second period is

MaxpBi

{ΠBi = (a− pBi + ea(pAi ))pBi }• Thus

pBi =a + ea(pAi )

2= a +

pAi − α

2• The derivative of the second period profit with respectto pAi is

dΠBidpAi

=∂ΠBi∂pBi

∂pBi∂pAi

+∂ΠBi∂pAi

= pBi∂ea(pAi )∂pAi

= pBi

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• Firm i maximizes its expected present discountedprofit, thus the FOC is

EdΠAidpAi

+ δEdΠBidpAi

= 0

• where δ is the discount factor.• Thus, it is equivalent to

ae − 2pAi + α + δ(ae +pAi − α

2) = 0

• In equilibrium pAi = α, thus

α = ae(1 + δ) > ae

• In a dynamic model, a firm may induce its rival to raiseits price.

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4 The Milgrom-Roberts (1982)Model of Limit Pricing• Asymmetric information drives firms to cut their pricein first period.

• 2 risk-neutral firms: firm 1 (incumbent), firm 2(potential entrant)

• Asymmetric information on firm 1’s costs. Firm 2 hasonly prior beliefs concerning the cost of its rival, x.Thus

c1 =

(cL1 with probability xcH1 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.

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• Firm 2 learns 1’s cost immediately after entering.• The incumbent’s profit when price is p1 is

Mt1(p1) = (p1 − ct1)Q(p1)

where t = H,L. (strictly concave function in p1)– Thus pL1 , pH1 are the monopoly prices charged by theincumbent, pL1 < pH1 .

• Duopoly’s payoffs areDti for t = H,L and i = 1, 2.• Assume DH2 > 0 > DL2 : if low cost, no room for 2firms, if high cost, room for duopoly.

• δ Discount factor.• To simplify: only 2 prices pL1 , pH1 and not a continuumof prices.

• Perfect Bayesian Equilibrium concept.• See tree of the game

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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

ML1 (p

L1 ) + δML

1 (pL1 ) > M

L1 (p

H1 ) + δML

1 (pL1 )

⇒ ML1 (p

L1 ) > M

L1 (p

H1 )

which is always satisfied.– if high cost, firm 1 chooses a high price if

MH1 (p

H1 ) + δDH1 > M

H1 (p

L1 ) + δDH1

⇒ MH1 (p

H1 ) > M

H1 (p

L1 )

Result 1. Under symmetric information♦ If c = cL1 , (pL1 , ne) is a Perfect Nash Equilibrium♦ If c = cH1 , (pH1 , e) is a Perfect Nash Equilibrium

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Asymmetric Information• Separating equilibrium?The incumbent does not choose the same price whenits 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 = cH1 , e

• Is it an equilibrium? and under what kind of circum-stances?

• It is an equilibrium if none of the firms deviate.– If c = cL1

ML1 (p

L1 ) + δML

1 (pL1 ) > M

L1 (p

H1 ) + δDL1

⇒ ML1 (p

L1 )−ML

1 (pH1 ) > δ(DL1 −ML

1 (pL1 ))

(1)

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– If c = cH1MH1 (p

H1 ) + δDH1 > M

H1 (p

L1 ) + δMH

1 (pH1 )

⇒ MH1 (p

H1 )−MH

1 (pL1 ) > δ(MH

1 (pH1 )−DH1 )

(2)

– The equation (1) is always satisfied, whereas (2)must be satisfied.

Result 2. If (2) is satisfied, there exists a separatingequilibrium such that♦ the incumbent chooses pL1 and firm 2 does not enter(ne) if c = cL1 ,♦ the incumbent chooses pH1 and firm 2 enters (e) ifc = cH1 .

Pooling equilibrium• Two possible kinds of pooling:P1. the incumbent always chooses pL1 , whatever the cost,P2. the incumbent always chooses pH1 , whatever the cost.

• Updated beliefs equal to prior beliefs.

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P1. (pL1 ) Player 2 stays out if0 > xδDL2 + (1− x)δDH2

⇒ x > ex = DH2DH2 −DL2

• ex ∈ [0, 1]?• ex > 0 ifDH2 > DL2 ,• ex < 1 ifDL2 < 0.• Thus, for x > ex firm 2 prefers to stay out.• Can firm 1 do better?– If c = cL1ML1 (p

L1 ) + δML

1 (pL1 ) > M

L1 (p

H1 ) + δDL1 OK

and ML1 (p

L1 ) + δML

1 (pL1 ) > M

L1 (p

H1 ) + δML

1 (pL1 ) OK

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– If c = cH1MH1 (p

L1 ) + δMH

1 (pH1 ) > M

H1 (p

H1 ) + δDH1 OK

and MH1 (p

L1 ) + δMH

1 (pH1 ) > M

H1 (p

H1 ) + δMH

1 (pH1 ) NO

• Thus, with an out-of-equilibrium prob(e/pH1 ) = 1,there exists a pooling.

Result 3. If (2) is not satisfied, there exists a poolingequilibrium such that♦ the incumbent always chooses pL1 ,♦ and firm 2 does not enter (ne)♦ with an out-of-equilibrium probability

prob(e/pH1 ) = 1.

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P2. (pH1 ) Player 2 enters ifxδDL2 + (1− x)δDH2 > 0

⇒ x < ex = DH2DL2 −DH2

• Then for x < ex firm 2 will enter.• Can firm 1 do better?– If c = cL1ML1 (p

H1 ) + δDL1 > M

L1 (p

H1 ) + δML

1 (pL1 ) NO

and ML1 (p

H1 ) + δML

1 (pL1 ) > M

L1 (p

L1 ) + δML

1 (pL1 ) NO

– Thus firm 1 will always deviate.• There is no pooling P2.

• If (2) is not satisfied, the incumbent manipulates theprice such that its action does not reveal any costinformation.

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• In continuous p ∈ [0,∞[, same results except thatprices are different.

• Single-crossing condition∂2[(p1 − c1)Qm1 (p1)]

∂p1∂c1= −∂Q

m1

∂p1> 0

• It is more costly to the high type to charge low price.Separating equilibrium• – if c = cH1 , pH1 = pHm– if c = cL1 , pL1 ∈ [eep1, ep1] where ep1 < pLm. Low costtype makes pooling very costly to the high cost type.

• There exists a reasonable separating equilibrium where– if c = cH1 , pH1 = pHm and entry occurs,– if c = cL1 , pL1 = ep1and no entry.

• The incumbent does not fool the entrant• But, there exists a limit pricing.

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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.

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