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  merican Economic ssociation

Naked Exclusion: CommentAuthor(s): Ilya R. Segal and Michael D. WhinstonSource: The American Economic Review, Vol. 90, No. 1 (Mar., 2000), pp. 296-309Published by: American Economic AssociationStable URL: http://www.jstor.org/stable/117295 .

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Naked Exclusion:Comment

By ILYAR. SEGALAND MICHAELD. WHINSTON*

The ability of an incumbent firm to deterentry by writing exclusionary contracts withcustomers has been a subject of contention inthe antitrust iterature.The courts' concern withsuch exclusionary contracts has been chal-lenged by those who argue that an incumbent,faced with buyers whose interest is to promoteentry andcompetition,would have to pay buy-ers more for the inclusion of exclusionary pro-visions than it could possibly gain fromexclusion.

In a provocative article, Eric B. Rasmusen etal. (1991) (henceforth,RRW)have arguedthatan incumbentmay in fact be able to excluderivals profitably using such contractualprovi-sions because it can exploit buyers' lack ofcoordination. n essence, if buyers expect otherbuyers to sign such provisions, then they maysee little reason not to do so themselves. TheRRW argument s potentiallyan importantonebecause most alleged instances of entry deter-rence through exclusionary contracts with cus-tomers occur in situations with multiple

buyers. 1,2

Unfortunately, however, RRW' two mainresults contain errors. In this Note, we recon-sider the RRW model, providing correct char-acterizations of the likelihood and cost ofexclusion for an incumbent firm. Our resultsindicate that while the intuition suggested byRRW is valid, the equilibrium likelihood andcost of exclusion differ from what RRW derive.Moreover,ouranalysis illuminatessome furtheraspectsof exclusionary contractingwith multi-ple buyers. Among these issues, we focus on

how an incumbentcan use discriminatory ffersto exploitthe externalities hat exist among buy-ers in the provisionof competition.3

In Section I, we review the basic assumptionsof the RRW model. In Sections II and III wethen consider,in turn, he cases of simultaneousand sequential offers studied by RRW. For thesimultaneous model, we distinguish betweensettingsin which the incumbentcan and cannotdiscriminatein its offers to different buyers.(This distinction s the source of the problem nRRW's analysis: their simultaneous-offer

model assumesno discrimination,but the proofof theirProposition2 assumes-at times-thatdiscrimination s feasible.)We show thatabsentthe ability to discriminate,the incumbentcanexclude profitablyonly when buyersfail to co-ordinate on their most preferredcontinuationequilibrium.In contrast, we show that whendiscrimination s possible, the incumbentneednot rely on a lack of buyer coordination to

* Segal: Departmentof Economics, Landau Building,Stanford University, Stanford, CA 94305 (e-mail:ilya.segal@stanford.edu);Whinston: Department of Eco-nomics, Northwestern University, 2003 Sheridan Road,Evanston, IL 60208 (e-mail: mwhinston@nwu.edu).Wethank Eric Rasmusen, John Wiley, and participants n theBerkeley-Stanford1Ofest, the University of CopenhagenVertical RestraintsConference, and a seminar at the Anti-trust Division of the U.S. Departmentof Justicefor helpfulcomments. We thankFederico Echeniquefor excellent re-search assistance.

' For example, a recent Departmentof Justiceinvestiga-tion concerned he use of exclusive contractsby the leadingprovider of computerized icketing services, Ticketmaster.Most major cities have severallargeconcert/sportsvenues.In many cities, Ticketmasterhas exclusive contractswith avery large share of these venues.

2 The other leading response to the argument hat con-tractswith buyers cannot be a profitablemethod of exclu-sion is given in Philippe Aghion and PatrickBolton (1987Section I). One factor limiting the applicability of theAghion and Bolton theory, however, is that it involvespartial exclusion: the contractssigned are not fully exclu-

sionary (they involve finite stipulated damages) and the

profitability f the strategyarises fromextractingrents froman entrant when entry occurs. Indeed, in the Aghion andBolton model, contracts hat fully exclude offer no benefitsover writingno contractat all. In Section III of theirpaper,AghionandBolton also discuss a two-buyerversionof theirmodelin which externalitiesariseacross buyers.In contrastto the analysis here and in RRW,they allow the incumbentto make an offer to a buyer i that is conditional on theacceptancedecision of the otherbuyer.

3 R. Innes and R. J. Sexton (1994) also discuss the use ofdiscriminatoryoffers in the context of a model in whichbuyerscan form coalitions with the entrant.

296

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VOL. 90 NO. I SEGALAND WHINSTON:NAKEDEXCLUSION,COMMENT 297

excludeprofitably:discrimination llows the in-cumbent o successfully exploit theexternalitiesthat exist across buyers.

For the sequentialmodel, we provide a cor-rect characterization of equilibrium play(RRW's analysis failed to complete the neces-sary backwardinduction). Our results confirmRRW's finding that the incumbent'sability todeal with buyers sequentially strengthensitsability to exclude. However, our result for thesequentialmodel also has importantdifferencesfrom the characterizationgiven in RRW. Weshow, for example,thatas the numberof buyersbecomes large, the externalitiesacross buyersbecome so severe thatthe incumbent s always

able to exclude for free.In Section IV we introduce he possibilityforthe incumbent to offer partially exclusionarycontracts stipulateddamages)as in AghionandBolton (1987). We show thatalthough his low-ers the likelihood of total exclusion, the basicthrust of our previous analysis remains un-changed.Section V concludes.

I. TheModel

The model has three setsof agents:anincum-

bent firm(I), a potentialrival(R), and a set of Nbuyers. There are three basic periods to thegame:In period 1, the incumbentoffers buyersexclusionarycontracts. Theprecise structure fthis period will be specified in later sections.)Following RRW,we assume that an exclusion-ary contractcommits the buyer to purchasingonly from the incumbent.4 n period 2, firmRdecides whether to enter or not. In period 3,active firms name prices.5 The incumbent isable to discriminatebetween those buyerswhohave signed an exclusive contract and those

who have not (the "free"buyers). The formerareoffereda (unit) pricep5, while the latter areoffereda pricePf* The rival,if it has entered, sable to makeoffersonly to freebuyers.It offersthem a price Pr Each buyer has a demandfunction q() with q'(Q) < 0. We denote the

numberof buyers who have signed an exclu-sionarycontractby S. We also define CS(p) =

fp q(s) ds to be a buyer's surplus at pricep.

RRWassumethattheincumbentandthe rivalhave the same average cost functionc(), withc(Q) = c if Q ? Q*, and c'(Q) < 0 at allQ < Q*. FirmR entersin period2 if andonlyif it can make nonnegativeprofitssplittingthefree marketat a price of c and,if firm R enters,the prices offeredto the free buyersin period 3arepf = Pr = c.6 Following RRW, we definepm = arg maxp(p - O)q(p) to be firm I'soptimal price to a buyer who has signed anexclusionarycontract, as well as to those whohave not in the event of no entryby firmR. We

also let u - (pm - c)q(pm) denote firm I'smonopoly profitper buyer,andx* = CS(c) -

CS(pm) denote the extraconsumersurplusen-joyed by a free buyerin the event of entry. Thedifferencebetweenx* and IT is due to the dead-weight "triangle"oss from monopoly pricing;we will assume thatthis loss is strictlypositive(with the exception of our analysis in SectionIV), hence 7T < X*.7

The presence of scale economies means thatthere is an integerN* such thatfirm R entersifand only if the number of buyers that have

signed exclusive contracts(S) is less than N*.Specifically, letting[zl denote the smallest in-teger greater hanor equalto z, we have N* =

[N - 2Q*lq(c)1. When N* equals N, firm Ican exclude firm R only by signingallbuyerstoexclusionary contracts. In this case, RRW's

4 We consider the possibility of partiallyexclusionarycontracts i.e., stipulateddamagesfor purchasing rom firmR) in Section IV.

S RRWalso allow for sales by firmI in period 1, but wedrop these because they play no role in the analysis.

6 There are some problemswith RRW's specific formu-lation of periods2 and 3 (e.g., firm R's optimalentry andpricingstrategiesare actuallyindeterminate).Here we sim-ply adopt the outcome of periods2 and3 used by RRW. InSection IV,

however, we providean alternativemodel thatleads to these same period 2 and 3 outcomes.7 If the incumbentwere able to write arbitrary ontracts

with buyers, it would be able to eliminate the distortionunder an exclusionarycontract(i.e., we would have x* =

w). To rule this out, we assume that(1) contracts n period1 canincorporate nexclusivityprovision,but no agreementon thepriceof the good in period3, and(2) offers in period3 can specify only a per unitprice. The firstassumptioncanbe justified in circumstances n which the precise natureofthe good to be deliveredin period3 is not knownin period1. The second assumptioncan be justified formally wheneach buyer wants at most one unit of the good and has arandomreservationprice [q(p) is then the probability hatthe consumerbuys at pricep].

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298 THE AMERICANECONOMICREVIEW MARCH2000

"triangle oss argument"applies and profitableexclusion is impossible: since every buyer ispivotal for whetherentry occurs, eachrequiresa

payment of at least x* to sign an exclusivecontract.But, if so, exclusion is unprofitableorthe incumbentsince it earns at most Xrn period3 from each buyer it signs to an exclusive con-tract.8In the remainderof this Note, we focuson cases in which N* E (0, N).

II. SimultaneousOffers

In this section, we study the potential forexclusion when period 1 consists of firm I si-multaneouslyannouncinga set of offersto buy-

ers, and the N buyers then simultaneouslyaccepting or rejecting the offers made. For-mally, we let xi ' 0 denote the amount ofmoney firmI offers to pay buyeri for signinganexclusive contract, and we let si E {0, 1denotebuyeri's response,with si 1 denotingacceptanceand si = 0 denotingrejection.Thenumber of buyers who accept is S = Ei si.

In the RRW model the incumbentmakes asingle (nondiscriminatory)ffer x to all buyers.9Webeginbyconsideringhiscase,which amountsto therestrictionhatxi- j for all i t j:

PROPOSITION1: Whenthe incumbentmakessimultaneousoffers to buyersand is unable todiscriminate, period 1 play in a subgame-perfect equilibrium can take the followingforms:

(i) EXCLUSION EQUILIBRIA:x E [0, ',T]

and S > N*, with S = N wheneverx > 0;(ii) NO EXCLUSIONEQUILIBRIA:x 6 [0,

x*] and S 0.

PROOF:Consider the continuationplay following an

offer of x. If x > x*, the unique continuationequilibriumhas all buyers accept firmI's offersince the most a buyer could ever gain by re-jecting is x* (thisoccurs when entryfollows thebuyer's rejection).When x < x*, we get mul-tiple continuation equilibria, with exclusionsucceedingin some continuationequilibria,andfailing in others. In particular,we have the

following set of buyer acceptancesin continu-ationequilibria:

Offer (x) Numberaccepting(S)

x = x* SE [O,N*);S=Nx C (O, x*) S O; S = Nx = O S O; S E (N*, N].

Note that for any x ? x*, there is always acontinuationequilibrium n which S 0.

In any subgame-perfect quilibrium n whichexclusion succeeds we must have x ' w, since

otherwisefirm I would earnnegativeprofits (itcan assure itself nonnegativeprofitsby offeringx 0). Also, in any subgame-perfectequilib-rium in which exclusion fails, we must havex ? x*. This establishesthe boundson x givenin the two types of equilibria describedin theproposition. The number of buyers who canacceptin these equilibria ollows fromthe tableabove [thefact that S 0 in any NO EXCLU-SIONequilibrium ollows because we can haveS E (0, N*) only if x- x, in whichcase firmI would be betteroff offeringx = 0]. Finally,

to verifythat the configurationsof period1 playdescribedin the proposition all constitutesub-game-perfectequilibriumplay, note that firmIearns N( r - x) ? 0 in an EXCLUSION

equilibrium,and zero in a NO EXCLUSIONequilibrium.Thus, all of these equilibriacan besustained by having the continuationequilib-rium following any offer x / x with x E [0,x*] be such that S = 0 (in which case firm Iearnszero).

Accordingto Proposition 1, when offers must

8This result can be overturnedwith other models of ex

post competition. Suppose, for example, that there is asingle buyer and that stage 3 competitionis such thatfirm

R's entry in the absence of an exclusive contract s socially

inefficient [as in N. GregoryMankiwand Whinston 1986)].In that case, firm I and the buyer have an incentive to signan exclusive since their joint surplus must fall whenever

firmR findsentry profitable.Here these issues do not arisebecause firm R's entry generatesa positive aggregateex-ternality. Similar issues arise when the buyer can form acoalition with firm R, as Innes and Sexton (1994) haveshown.

9As we mentionedearlier,thereis actually some ambi-guity here, because although RRW's model assumes no

discrimination, he proof of their Proposition2 assumes, atsome points, thatdiscrimination s possible [specifically, inestablishing hat exclusion must occur under heir condition(3), RRW allow firm I to make the exclusionary contractoffer to only a subset of buyers,a form of discrimination].

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VOL. 90 NO. I SEGALAND WHINSTON:NAKEDEXCLUSION,COMMENT 299

be nondiscriminatory,here are always both ex-clusionary and nonexclusionary equilibria.Note, however, that since the payment in an

exclusionary equilibrium s bounded above byv < x*, all buyers would be better off if allinstead rejected firm I's offer. Thus, it appearsthat when offers must be nondiscriminatoryfirm I is able to successfully exclude only ifbuyers fail to coordinateon their most preferredcontinuationequilibrium.

This point can be made more formally byrestrictingattentionto equilibria n which, fol-lowing firm I's offer, subsets of buyers are ableto make nonbinding agreements among them-selves concerningtheir accept/rejectdecisions.

This idea is capturedby the concept of a per-fectly coalition-proof Nash equilibrium(PCPNE) [B. Douglas Bernheim et al. (1987)].This concept requires hat equilibriabe immuneto self-enforcing coalitional deviations. For-mally, we then have the following.

PROPOSITION2: Whenthe incumbentmakessimultaneousoffers to buyers and is unable todiscriminate, exclusion cannot occur in anyperfectly coalition-proofNash equilibrium.

PROOF:We argue that any PCPNE must be a NOEXCLUSION subgame-perfect equilibrium.Any PCPNE must be a subgame-peifect equi-librium and must involve coalition-proofNashequilibriumbehavior in the continuation sub-games following firmI's offer [see Bernheimetal. (1987)]. Since x < x* in any EXCLUSIONequilibrium,every buyermust receive a payoffstrictlyless thanCS(c). Thus, a joint deviationin which every buyer rejectsfirmI's offer [andthereby earns exactly CS(c) ], would strictly

increaseevery buyer's payoff. Moreover,sincethis is every buyer'smaximalpossible payoff inthe subgame following firm I's offer, this devi-ationis necessarily self-enforcing. Hence, onlyNO EXCLUSION equilibriacan be perfectlycoalition-proof.

Thus, once we allow buyers to coordinatetheir responses to firm I's exclusionaryoffers(in a nonbindingmanner),exclusion never suc-ceeds if offers must be nondiscriminatory.

Typically, however, we would expect dis-

crimination to be feasible whenever informa-tionalconditionsaresuch that firm I canenforceexclusivity provisions (at least absentany laws

prohibitingdiscrimination). n the remainderofthis section we explore how allowing discrimi-natory offers affects this result. Once we allowfor discrimination, a question arises as towhether a buyercan observe the offers made tootherbuyers priorto makingits decision (in theabsence of discrimination,a buyercan infer thisdirectlyfromits own offer).Inwhatfollows, wefocus on the case of observable offers; afterderivingresults for this case, we comment onhow these resultschangewhen offers are unob-servable.

PROPOSITION3: Whenthe incumbentmakessimultaneous offers to buyers and is able todiscriminate, period 1 play in a subgame-perfect equilibrium can take the followingforms.'0

Case A: N-r > N*x*e

(i) S > N*, Ei sixi ' N*x*, xi 0 whenever

si = 0.

(ii) S = N*, xi = 0 when si - 0, xi = X*

otherwise.

Case B: NrT< N*x*.

(i) S > N*, Ei sixi c NT, xi 0 wheneverSi = 0.

(ii) S = 0, xi E [0, x*] for all i.

Moreover, only the period 1 equilibriumplays describedin CasesA(ii) andB(ii) arise ina perfectlycoalition-proofNash equilibrium.

PROOF:We begin with classifying all the continua-

tion equilibria ollowing firm I's offers (xl, ...

XN) into three types:

1. S < N* (exclusion fails). Then for everybuyer i we can have si = 1 only if xi ? x*

10 When N7r = N*x*, all of the listed equilibria aresubgame-perfectquilibria,and the set of perfectlycoalition-proofequilibriaconsists of the equilibriadescribed n CasesA(ii) and B(ii).

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300 THEAMERICANECONOMICREVIEW MARCH2000

(otherwise it could profitablydeviateby re-jecting), and si = 0 only if xi ? x* (other-wise it could profitablydeviate by signing).

Observe that this continuation equilibriumcan only arise when I{i: xi > x*II < N*.Firm I' s payoff in this equilibrium is

i Si(T - Xi) ? S(W - X*) C 0.

2. S = N* (exclusion occurs). Then everybuyer i with si 1 is pivotal: if it rejects.exclusion fails. For every such buyer wemust have xi- x*, since otherwiseit couldprofitably deviate by rejecting. Also, everybuyeri with si = 0 musthavexi = 0, sinceotherwise t could profitablydeviate by sign-ing. Firm I's payoff in this equilibriumis

N7T- i sixi ? N7T- N*x*.3. S > N* (exclusionoccurs).Theneverybuyer

withsi = 0 musthavexi = 0, since otherwiseit could profitablydeviateby signing.

Now we can show thatany play described nthe propositioncan indeed arise in a subgame-perfect equilibrium.To see this, consider anequilibriumn whichbuyers respond o any firrmI's deviation (xl,--, NX) # (X1, XN) byplaying a type 1 (nonexclusive) continuationequilibrium f I i: xi > x* }I < N, andplaying

a type 3 continuation equilibrium otherwise.Given thatbuyersuse such strategies,a devia-tion never brings firm I more than max{ 0,NTr - N*x*}. But the firm earns at least asmuch in all the plays described in the proposi-tion, hence any suchplay can be sustained n asubgame-perfect quilibrium.

We nextargue hatno otherperiod1subgame-prefect equilibriumplay exists in the two casesgiven in the proposition:

Case A: In this case firm I can earn a profit

arbitrarilylosetoNi - N*x* > 0 by offeringx*+ e to N* buyersfor s > 0 sufficientlysmall,which forcesbuyersto play an exclusivecontin-uationequilibrium.Thisrules out type 1 (nonex-clusive) equilibria,and implies that Ei sixi cN*x*.Theremaining ype2 andtype3 equilibriasatisfyCasesA(ii) andA(i) respectively.

Case B: Since firm I is always assuredof earn-ing 0 by offeringxi = 0 for all i, this rules outtype 2 equilibria, n which firmI's payoff doesnot exceed Nw - N*x* < 0. Type 1 andtype

3 equilibria in which firm I earns nonnegativeprofits satisfy Cases B(ii) and B(i) respectively.

Finally, consider perfectly coalition-proofequilibria. Here we shall argue only that playfollowing firm I's equilibrium offers in CasesA(i) and B(i) is not coalition-proof; hence,these equilibriaare not PCPNEs;the argumentshowing that the outcomes describedin CasesA(ii) and B(ii) are supportableas PCPNEs iscontained in our working paper [Segal andWhinston(1996)]. Note that in both Case A(i)and Case B(i) fewer than N* buyersareoffered

xi > x*. Since exclusion succeeds in bothcases, eachof thebuyerswho is offeredxi < x*

receives a payoff strictly less than CS(c). Ifthese buyers arrangea joint deviationin whichthey all reject,exclusion will fail andeach willreceive a payoff of exactly CS(c). Since this iseach of these buyers' greatestpossible payoff inthe subgame (holding fixed firm I's offers andthe other buyers' responses), this deviation isnecessarily self-enforcing.

Proposition3 shows thatwhendiscriminationis feasible exclusion necessarily occurs in asubgame-perfect equilibrium in Case A, and

may or may not occur in Case B (dependingonwhetherbuyers coordinate).The result makesclear that when firmI can make discriminatoryoffers, it need not rely (in Case A) on a lack ofbuyer coordination to exclude profitably.Rather,it can turnbuyers against one another,offeringan exclusionarycontract o only a sub-set of the buyers, who then impose the exter-nality of no entry on the otherbuyers. This isprofitable f NTI-,he amountearnedfromthe Nbuyers under exclusion, exceeds N*x*, theminimumamount hat must be paid (with buyer

coordination) o exclude.Comparing our Propositions 2 and 3 to

RRW's Proposition2, we can see elements ofboth of our results in theirProposition2. Theirbounds on payoffs coincide with those we de-rive in Proposition 2, because their model as-sumes that offers are nondiscriminatory.ButtheirProposition2, like our Proposition 3, alsoasserts thatexclusion necessarilyoccurs in anysubgame-perfect equilibrium when NwT >

N*x*, because at one point in theirproof theyimplicitly allow the incumbentto discriminate.

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VOL.90 NO. I SEGALAND WHINSTON:NAKEDEXCLUSION,COMMENT 301

Propositions 2 and 3 are also similar to thefindings of Innes and Sexton (1994 Section II,subsection A). Instead of using a coalitional

refinement to model buyers' coordination ontheir preferred equilibrium, they obtain thesame result by letting buyers make their accep-tance decisions sequentiallyratherthan simul-taneously. They find that inefficient exclusionmay occur if and only if the incumbentfirm isallowed to make discriminatoryoffers.

Finally, we note that the results of Proposi-tion 3 change very little when a buyer does notobserve other buyers' offers: the only alter-ations arethat we musthave xi = 0 for all i inCases A(i) and B(i). The formal analysisof this

case can be found in our working paper [Segaland Whinston(1996)].

III. Sequential Offers

In this section we analyze the game in whichthe incumbent approaches buyers sequentiallyin period 1 with offers of exclusionary con-tracts. Each buyer's acceptancedecision is per-manent,and t is observedby all otherplayers nthe game. Following RRW, we number the

N stages of the period 1 game in reverse orderby T = 1, ... , N, where T is the stage at which

T buyers remain to be offered contracts. FirmI's decision whether to exclude at stage T de-pends on the comparisonof continuationbene-fits and continuation costs at this stage. Bothnumbersdependon the numberS of buyerswhohave already signed exclusionarycontractsbystage T.

The continuation ost of exclusion to firmI atstage T depends on buyers' willingness to ac-cept exclusionary contracts, which is in turn

determinedby their expectationof the likeli-hood of exclusion if they reject. The simplestcase to consider is the one in which each re-maining buyeris crucial for exclusion, i.e., S +T = N*. Then each remaining buyer will notaccept an exclusionaryoffer for less than x*.Thus, the continuation cost of exclusion is(N* - S)x*. The continuationbenefit of ex-clusion to firm I is (N - S) i, since the S

"captured" ustomers will be chargedthe mo-nopoly price in period 3 whetheror not exclu-sion occurs. Firm I will choose to excludeif and

only if the continuationcost of exclusion doesnot exceed the continuationbenefit:"

(1) Tx* = (N* - S)x* :-(N - S)T.

To develop some intuitionfor this condition,observe that it can be rewrittenas

(2) (Nw - N*x*) + S(x* - I) 0.

The first term is firmI's net benefit of exclusionfrom the ex ante perspective, taking into ac-count that each buyer signing an exclusionarycontractwill be paidx*. The second termis thenet "sunkcost" of exclusion, assuming that S

buyers have alreadybeen paid x* for signingexclusionarycontracts,and firmI will earnIToneach of them whether or not exclusion actuallyoccurs. This condition demonstrates hat whenS is large, a substantialportionof the cost ofexclusion is already sunk, and firm I is morelikely to proceed with exclusion. Therefore,firm I may be able to comimit o exclusion bysinkingsome of its cost in the early stagesof thegame.

It turnsout, moreover, that firmI's commit-ment to exclusion may allow it to exclude at

zero continuationcost. For example, considerthe situationin which there is one more buyerleft than is necessary for exclusion, i.e., S +T = N* + 1, and where S < N*. If S >

(N*x* - Nw)l(x* - rr),then condition(1) issatisfied, and the next remaining buyer knowsthat even if it rejects an exclusionarycontract,firm I will proceed with exclusion. Therefore,this buyer is willing to sign an exclusionarycontract or free. The samereasoningwill applyto subsequentbuyers, so firm I will achieveexclusion at a continuationcost of zero.

In contrast, f S < (N*x* I Nw)l(x* - I),then condition(1) is violated, and the next re-maining buyer knows that his rejection of anexclusionarycontractpreventsexclusion. Thisbuyerwill not sign an exclusionarycontract orless thanx*. In order o exclude, firm I needs tosign up sufficiently many buyersfor x* each so

1 FFor simplicity we assume in what follows that, at anystage in the game, firm I chooses to exclude if it is indif-ferentbetween excluding and not.

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302 THEAMERICANECONOMICREVIEW MARCH2000

that condition (1) becomes satisfied. Once it issatisfied, the remaining buyers sign for free.Depending on how many buyers firmI needs tosign up for x* each, it may or maynot choose toexclude.

The main result of this section extendstheseobservations o the generalcase. We findthat, ngeneral, the continuationcost of exclusion de-pends on two numbers:one is the numberS of"captured" uyers, and the other is the numberk = T + S - N* of buyersleft in excess of thatnecessary for exclusion.As we have seen, undersome conditionsfirmI will be able to exclude atzero continuationcost. Our analysis will dem-onstratethat the minimumnumberof capturedbuyers required or such costless exclusion at astage where there are k more buyers left thannecessaryfor exclusion is given by a sequence

{Skk }k 0 that can be recursivelydefined as fol-lows:

SO= N*,

Sk+l

SX

- fork ?0.

In the simplest case where k = 0 each re-

maining buyer is crucial for exclusion, so freeexclusion is impossible unless S ' N* So.Our previous observations demonstrate thatwhen k = 1, free exclusion can be achieved ifand only if S satisfies condition (1), which isequivalent to S ? S1. Before we proceed toestablishthe generalresult, we establish someuseful propertiesof the sequence Sk.

LEMMA 1: For all k ? 1, if

(3) N-N*?x*la -1,

then Sk < Sk- ? N*; otherwise Sk -

Sk l = N* 12

PROOF:

By induction on k. For k = 1 we can write

N- N ]SI - so - x*I/ir - 1I

This implies that S1 f S0 + F-il < so=when N - N* ' xx*& - 1, and S1 = S0-

N* otherwise. This establishes the inductivestatementfor k- 1.

Supposethe statement s truefor k ? 1, thenfor k + 1 we can write

FSkN-NN -N]Sk+-1 Sk X*/

-a

1 _ x*I/T - 1 .

This implies that Sk+ 1 Sk + F-i < Sk

N* when N - N* ' x*I/vr - 1, and Sk+ 1

Sk N* otherwise.This establishes the induc-tive statementfor k + 1.

We now establish the main result of thissection.

PROPOSITION : Whenthe incumbentmakes

offers to buyers sequentially, exclusion occursin a subgame-perfectequilibrium f and only ifSN - N* + 1 ' 0. If exclusion occurs, the cost of

exclusion is max {X8SN- N*, 0 } Otherwise, no

buyer signs an exclusionarycontract.

PROOF:The proof is based on the following lemma.

LEMMA 2: Suppose there are T ? 0 stagesleft in thegame, S ? 0 buyershave signed, andS + T ? N*. Then exclusion occurs in a

subgame-perfectequilibrium f and only if SSS+ T- N* + I- If exclusion occurs, the continu-ation cost of exclusion is max( (SS + T- N*

S) x*, 01. Otherwise,no buyer signs an exclu-sionary contract.

PROOF OF LEMMA 2:By induction on S + T ?: N*. We have

established above that when each remainingbuyer is crucial for exclusion (i.e., S + T =

N*), the continuation cost of exclusion is

12 Condition (3) can be interpretedas saying that when

N*- 1 buyers have alreadysigned exclusionarycontracts,the incumbent s willing to pay x* for the last buyer to sign

[to see this, substitute S =- N* - 1 in condition (2)].Holding the ratio N*IN fixed, this condition is satisfied

when N is sufficiently large.

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VOL. 90 NO. 1 SEGALAND WHINSTON:NAKEDEXCLUSION,COMMENT 303

(N* - S)x* = (So - S)x* - (SS+?TN* -

S)xW? 0. Exclusion occurs in this case if andonly if condition(1) holds, which is equivalentto S ? SI = SS+T-N*+1- If this condition

fails, exclusion will not occur, and no new ex-clusionarycontractswill be signed.This verifiesthe inductive statementfor S + T - N*.

Suppose the inductivestatement s known tobe true for S + T = n - 1 - N*, and consider

the situationwhere S + T - n. There aretwocases to consider:

(a) S ? SnfN - S(n-1)-N*+l. Then thenext buyerknows thateven if it rejectsthecontract which reduces the value of S + Tto n - 1), according to the inductive hy-

pothesis exclusion would still occur.Thus,it is willing to sign for free. If it signs, S +T will stayconstantand the samereasoningwill applyfor the next remainingbuyer, etc.Thus,in any continuation quilibrium irmIachieves exclusion at zero continuationcost. 13

(b) S < Sn-N* - S(n-1)-N*+I- Then thenext buyer is "pivotal":f it refuses to signthe contract, hen S + T is reduced to n -

1, and according to the inductive hypothe-sis exclusiondoes not occur.Thus, the nextbuyerwill only be willing to sign forx*. Inorder to exclude, firm I needs to sign up

(Sn -N* S) buyersin a row for x* each.Once the threshold is crossed, we willswitch to case (a), and theremainingbuyerssign for free. On the otherhand, when firmI chooses notto exclude, it will not sign anyadditionalbuyers (it makes at most x* -ir < 0 on each buyer it signs). Thus, thecontinuationcost of exclusion is (S -N*

-

S)x*. Firm I will choose to exclude if andonly if the continuationbenefit of exclu-sion, (N - S)7r, is greater than or equal to

the continuationcost:

(Sn,N* -S)x '* (N - S)ir.

Since S is an integer,this inequalitycan berewrittenas S ?

rF(Sn-NX*- N7r)I(X* IT)1

Sn-N*+1 - SS + T-N* + I If this conditionholds,exclusionwill occurat thecontinuationcost of (Sn-N* - S)x* > 0, otherwiseitwill fail and no new exclusionarycontracts

will be signed.

Combining the two cases, we see that the in-ductive statement s truefor S + T = n.

The proposition ollows by applyingLemma2 for S - 0, T= N.

The exclusionaryconditionobtained n Prop-osition 4 differs from the incorrect conditionobtainedby RRW, whose Proposition3 statesthatexclusion occurs in the sequentialgame if

and only if N*IN ' (irlx*)[2 - (,W1X*)].14

The difference can be seen in the followingnumericalexample considered by RRW.

Example 1: Suppose that N- 100, r = 10,x* =- 14. The conditionobtainedby RRWsaysthat exclusion will occur if and only if N* <

92. Take N* - 94. Then we have So = 94,

Si = 79, S2- r53/21= 27, S3 [-311/21 < 0.This implies that SN-N* = S6 ' S3 < 0, SO

our Proposition4 predicts that exclusion doesoccur in this case, and it is costless to firmI.

Unfortunately,it is impossible to obtain aclosed-formsolution for Sk, and so the exclu-sionaryconditionobtained n Proposition4 can-not be expressed n terms of primitivevariables.However, by investigating the propertiesof thesequence {SkJ we are able to establish somequalitativepropertiesof our exclusionarycon-dition. While differentfrom the condition ob-tained by RRW, our exclusionary conditionshares some of its qualiltativeproperties. For

13 Differentcontinuationequilibriamay differ in whichof the remainingT customerssign exclusionarycontracts,aslong as at least N* - S of them do so.

14 While the derivation of RRW's Proposition 3 beginscorrectly,they cut the inductiveargumentafter the secondstep. Indeed, ignoring the integer problem (and this is asecond problem with theirargument),RRW's exclusionarycondition is equivalentto S2 ? 0. The error s contained nthe last statementof their"Situation6." The firstbuyersarenot necessarily afe in refusing o sign:if T > 7** 2 (N* -

S) t 1, each remaining buyer is not crucial and will signfor free. When N > N* + 1, firm I can achieve thissituationby signing sufficientlymany buyers for x8. Thispossibilitybringsus to the next inductionstep,and it makesexclusion more likely thanRRW describe.

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304 THEAMERICANECONOMICREVIEW MARCH2000

example, our exclusionary condition, like thatof RRW, is weaker than the conditionN*x* -

NT ? 0 which is necessaryfor exclusion in a

perfectly coalition-proof equilibrium with si-multaneousoffers (to see this, note thatthe lastconditionis equivalentto S1 s 0). Also, bothconditions say that given N andN*, the likeli-hood of exclusion is a function of the ratioi-/x*. Ourexclusionarycondition, ust like thatof RRW, implies that the closer this ratio is toone (i.e., the less distortionary xclusion is), themore likely we are to observe exclusion.

COROLLARY 1: For a fixed N and a fixedN* E (0, N), there exists E E&0, 1) such that

exclusion occurs if and only if rrlx* ? P.

PROOF:In the Appendix.

To get a feeling for how our result qualita-tively differsfrom thatof RRW,we can obtainan approximationto Sk by ignoring integerproblems. Define Sk by:

So -N*

SkX* -NSk+l _gX Tr fork-0.

Then successive substitution nables us to write

(4) Sk=N[ ( N)(* )1I

for k ? 0.

This expression allows us to consider the fol-

lowing experiment.Start with a model with Nbuyers, and consider subdividing each buyerinto m identical smallerbuyers,so thatthe totaldemand does not change: each new buyer de-mandsqJ(p) = q(p)lm at any price p, and thetotal numberof buyersis Nm= mR. DenotebyN* the numberof buyersfirmI needs to sign toexclude when there areNm buyers.Then in theRRW model we have Nj = I aNml for all m,where a = 1 - 2Q*I(Nq(c)). (Note that

N% Nm -> a as N -* oo.) Substituting N

Nm = mN and N* = NFc= FmaN] n expres-

sion (4), we can see that since x*/(x* - w) >

1, we must have SN,m ; < 0 for m large

enough. If Sk is a good enough approximation

to Sk, we can therefore use Proposition4 toestablish that exclusion occurs for free for mlarge enough. The following result establishesthat this is indeed the case.

COROLLARY2: As each buyer is subdividedinto m identical buyers keeping total demandconstant, or m sufficiently arge the incumbentexcludesfor free.

PROOF:In the Appendix.

According to RRW's condition, when a2Q*I(Nq(c3)) (which equals the limit of 1N* INm as buyers are subdivided) is smallenough, exclusion does not occur even as buy-ers become infinitelysmall relative to the mar-ket. In contrast,our Corollary2 establishes thatfor any level of a exclusion occurs costlesslywhen buyersare sufficientlysmall.

To understand he result intuitively, observethatas buyersbecome very small relative to themarket, he numberof buyersin excess of those

necessary for exclusion goes to infinity. Thefirst buyer offered an exclusionarycontract inthis situationknows for sure that it is not piv-otal, so it will accept such a contract for free.This acceptancewill keep the numberof buyersin excess of those necessaryfor exclusion un-changed. The next remaining buyer will thenalso realize that it is not pivotal, so it will alsoaccept an exclusionarycontract for free. Con-tinuing this reasoning, we see that firm I ex-cludes for free.

The analysis of this section shows that se-

quential offering of contracts exacerbates thefree-riderproblem among buyers. However, itleaves unclear whether the incumbent cancommit to make buyers choose sequentially ifall buyers are always present in the market. Inthe sequential model, a buyer turns down anexclusionary offer for x E (0, x*) only if itexpects that this decision prevents exclusion.If, however, firmI could instead make furtheroffers to this buyer, the buyer might haveanother reason to turn down such an offer:after the rejection, firm I may be tempted to

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VOL.90 NO. 1 SEGALAND WHINSTON:NAKEDEXCLUSION,COMMENT 305

make this buyer an exclusionary offer onbetter terms. In our working paper [Segal andWhinston (1996)], we analyze a game with T

stages, where ineach stage the incumbentcan

make exclusive offers to all the buyers whohave not yet signed. We show that this temp-tation does not undermine the result of thesequential model; rather, the opportunity to

approach all buyers in each of the N stagesactually helps the incumbent to exclude. In-tuitively, the reason is that when firm I isrestricted to approach buyers sequentially,there are situations in which firm I is nolonger able to exclude because many buyershave already rejected its offers, and this may

put early buyers in a position to prevent ex-clusion by rejecting firm I's offers. This isless likely when firmI can reapproachbuyers.

IV. PartiallyExclusionaryContracts

Up to this point, we have assumed that firmI's contract offers take the form of a purely("naked") exclusionary contract. One mightwonderto what extent our conclusions are ro-

bust to the possibility that firm I could use

partially exclusionary contracts and benefitfrom the presenceof the entrant,as in Aghionand Bolton (1987). In this section we explorethis issue in the context of the simultaneous-offer model, assuming that buyers are able tocoordinate.

To consider this issue we need to introducea model of period 3 competition in which firmR is more efficient than firm I (to provide a

possible rent-extraction motive for partialexclusion). In what follows, we focus on amodel in which firm R has a constant mar-

ginal productioncost of c and there is a sunkcost of entryf that firm R must pay in period2 if it decides to enter. Firm I, in contrast,hasa constant marginal cost of -c > c (it has

already paid its sunk cost) satisfying the con-dition that (_c - c)q(c) > (p -- c)q(p) for allp E [c, c). Price offers in period 3 followingentry are made simultaneously. With theseassumptions, absent any penalties, followingentry firm R makes all sales to free buyersat price c. Assuming that firm R enters ifindifferent, all of our previous analysis of

fully exclusionary contracts would then stillapply with respect to N* N - Ff/i(Cc)q(c)]l + 1.15

We now allow firmI to make contractoffersof the form (ti, xi) to buyer i, where ti is thepenaltythatbuyer i pays firm I if it buys fromfirm R in period 3 and, as before, xi is thepayment hatfirm I makes to buyeri for signingthe contract. (Fully exclusionary contractscor-respond to ti = oo.) We assume that if firm Renters, t is ableto discriminaten period3 in itspriceoffers to differentbuyers (allowing firmRto discriminateamong free buyers would havehad no effect on the results in previous sec-tions).

To keep matters as simple as possible, inwhat follows we shall focus on the special casein which each buyer has a demand for at mostone unit,and has a known reservationvalue v >

c (our assumptionof linearpricing n period 3 isfully justified in this case; see footnote 7). Inthis case, q(p) = 1 if p ? v and = Oif p > v.Note thatwe now have Ir = x* v - c, andso absentthe abilityto engage in partialexclu-sion, firm I wouldalwaysexclude as long as N*< N. We shall see, however, thatwhen partialexclusion is feasible, firm I will sometimes (but

not always) choose to accommodate firm R'sentry n order o extractsome of the surplus irmR bringsto the market,miuchas in Aghion andBolton (1987).

Consider first the competitionfor buyer i inperiod3 after firm R enters.If ti < v - c, firmI's equilibriumpriceoffer to buyeri will be c +

ti (at this price firm [ is indifferentbetweenwinning and losing the buyer's business), firmR's equilibriumprice will be c, and the buyerwill buy from firm R. The buyer's total cost ofpurchasewill be c + ti.16 FirmI and firmR's

profits from buyer i are then ti and c - c

15 Note that in this model firm R's optimal entry and

pricing strategiesare well defined (recall footnote 6). Noticealso thatby consideringa sequence of cases {Cm fm }m = , in

which m Cm 0, andN -Ffm[(c -- c m)q(C)]11 = N* for all m, we can allow the efficiency differencebetweenthe firmsto grow arbitrarily mall while maintain-ing the same thresholdnumberof free buyers necessary forentry,in essence approaching he case consideredby RRW.

16 This is not the unique equilibrium. Just as in the

standardBertrandmodel withdifferingcosts, there are otherequilibria in which firm R wins the buyer's business at a

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306 THEAMERICANECONOMICREVIEW MARCH2000

respectively. If, instead, we have ti C [v -c,v - c), thenfirm R's equilibriumprice to buyeri will be v - ti, firm I's price to this buyer will

be v, and buyer will buy from firmR. Note thatin this case firm R's price is forcedbelow c andits profit is less thanc - c, its profiton a freebuyer. Finally, if ti > v - c, then firm I wins

buyer i's business at a price of v and firm Rearns 0.

Now consider firm R's entry decision. Sup-pose that the vector of damage penaltiesfor theN buyersis (t1, ..., tN) (wherea buyerwho hasrejected firm I's offer is taken to have ti 0).Then firm R enters if and only if

E max{0, min{c, v-

tj} - c} 'f.

Note thatif buyeri expects entryto occur,itis willing to accept any contract (ti, xi) thatgives it a surplus of v - c. It is immediate,therefore,that firm I can earn an amount arbi-

trarily close to N(c c) - f by offering every

buyeri a contract(t, x) with penaltyt (v -

c) - flN and payment x = v - c (note that

t > v-c S by virtue of the fact that -c >

flN andthat, given these offers,entryis assured

regardlessof buyers' acceptancedecisions, andso all buyers accept).This is preciselythe pointmadeby Aghion andBolton (1987), but hereina nonstochastic etting:by being a firstmoverincontractingwith the buyers, firm I can appro-priate the efficiency gain that firm R brings tothe market.

However, firm I also has anotherpossiblestrategy that it can follow: it can attempttoexclude firm R. Thekey pointis thatwith buyercoordinationt still costs firmR exactly N*x* todo so (thisis established ormally n theproofof

Proposition 5). In particular,we then get thefollowing result.17

PROPOSITION5: Suppose thatfirm I can of-fer partiallyexclusionarycontracts and that allbuyershave a reservation value v. Thenin anyperfectly coalition-proofNash equilibriumex-clusion occurs if and only if

N - N*x* =(N - N*)(v -

>N(c - c) -f.

If exclusion occurs, firm I offers N* buyerscontracts with ti > v - c and a payment of

xi = v - c, and earns exactly NIT - N*x* =

(N - N*)(v - c). If exclusion does not occur,firm I earns exactly N(c -

c)-f, firm R earns

0, and every buyer has a surplus of CS(c)v - c.

PROOF:In the Appendix.

To understandProposition5's condition forexclusion, note that ignoring the integer prob-lem (alternatively,assuming that firm R is al-most indifferentaboutenteringwhen N* buyershave signed fully exclusionary contracts),we

have N* N-

fl(c-

c), and the conditionin Proposition5 is approximately

v - c N* f>- = 1-

v-c N N(c-c)

The condition shows that firm R is excludedwhen its fixed costf is sufficientlyhigh (corre-spondingly,N* is sufficiently ow). Thus,in oursimple case the availabilityof partiallyexclu-sionary contracts does not eliminate the possi-bility that firm I will exclude firmR. Note that

this may happeneven thoughthe joint welfareof firm I and the buyers is maximized by ap-propriating firm R's efficiency advantagethroughpartiallyexclusive contracts [yieldingthem a joint surplus of N(v - c) - f]. As

before, it is the presenceof externalities acrossbuyers that may make exclusion a profitablestrategyfor firm 1.18

price below c. Theseequilibria nvolve firm I pricingbelowc + ti, which is a weakly dominatedchoice for firm I.

17 Note that in the limiting case described n footnote 15in which we let c -c ~andf -O 0, while keeping N*constant, heprofitsobtainableby firmI fromallowingentryapproach 0, and the exclusionary condition identified inProposition5 becomes identical to that in Proposition3.

18 With more general demand functions q(O),exclusionis detrimental to the joint welfare of firm I and buyers

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VOL.90 NO. I SEGALAND WHINSTON:NAKEDEXCLUSION,COMMENT 307

V. Conclusion

In this Note we put the RRW insight on afirmer foundation, offering correct results for

their model and illuminating some furtheras-pects of exclusionarycontractingwith multiplebuyers. We showed that when an incumbentfirm can make discriminatoryoffers of exclu-sive contracts to buyers, it need not rely onbuyers'disorganizationo successfullyexclude.Rather, by exploiting the externalities presentamong buyers, an incumbent firm can oftenprofitablyexclude potentialrivals.

The RRW argumentcan be nicely related tothe historyof debateson the use of exclusionarycontracts.Suppose that we denote by U(s) the

incumbent's gross payoff given buyers' re-sponse profile s = (sI, ... , SN), and by ui(s)buyer i's gross payoff given this profile.Then,in the absence of buyer coordination, o gener-ate buyersresponse profiles with simultaneousoffers, the incumbent must pay each buyer ipreciselyui(O,s_) - ui(s). Giventhis fact, theincumbent will induce buyer responses thatsolve

max U(s) - E {uj(O, s_j) - ui(s)},sE{O,I}N

or with a slight rewriting:

max [U(s) + ui(s) - E u(O, sJ).sE{O,t}N[ J i

The pre-Coasian/pre-Chicagoview can bethought of as saying that the incumbent willexclude in order to maximize U(). The Chi-cago School, however, correctly pointed outthat the incumbentwould have to compensatethe buyers for signing, and asserted that thiswould imply that surplusis maximized,which

is precisely the expression in square bracketsabove. But we see that the last term, >i uj(0,sj), is not accounted for by this argument.Itrepresents he sum of buyers' reservationutili-

ties when faced with the incumbent'soffers andit arisesprecisely because of the externality hatbuyers impose on one another: when manyother buyers sign exclusionarycontracts,buyeri's achievableutilitywill be low andthe incum-bent will not have to pay much to get this buyerto sign also. Indeed,when each buyersees him-self as nonpivotalfor exclusion, ui(si, sj) =

ui(0, s- ) for si E {0, 1 , andso the seller canexclude for free. In this case, he will excludewhenever exclusion maxiinizes U(s)-the pre-Coasianresult With buyercoordination, irmI

may need to pay more than Ei {uj(0, sj) -ui(s)l to generateresponse profiles, but, as wehave seen, using a "divide-and-conquer"trat-egy, firmI is still able to exploitthe externalitiesacross buyers to raise its payoff.

This featureof the exclusionary contractingproblem s in fact sharedby manyothersettings nwhich a single partycontractswith many otheragents among whom extemalitiesexist, such asproblems of takeovers, licensing, mergers, anddebt bailouts. In a recent paper, Segal (1999)addresseshis issue in a moresystematicway.

APPENDIX: PROOFS OF COROLLARY 1,

COROLLARY 2, AND PROPOSITION 5

PROOF OF COROLLARY 1:The proof is based on the following three

claims.

Claim 1: For any k ' 0, Sk is a nonincreasingand continuous romtheright unctionof lIx* E

(0, 1).

PROOF:By induction on k. The statement s trivially

truefor SO- N*. Supposethe statement s truefor a certaink ' 0. WriteSk(r) to denote Skasa function of r = rIx*. Then we can write

(Al)kI ) LN NfSktrine) Sk+ l(r) n N by rth i

Since Sk( ) is nonincreasing by thieinductive

because of the deadweight loss from monopoly pricing

considered n the previoussection,as well as becauseof the

loss of firm R's appropriablesurplus considered in this

section. While the qualitative conclusions of this section

continue to be true in this case, the analysis of period 3

pricingin the generalcase is substantiallymoredifficult (in

particular, irmR may receive a strictly positive surplus n

a nonexclusionaryequilibrium).

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308 THEAMERICANECONOMICREVIEW MARCH2000

hypothesis, the fraction [N - Sk(r)]I(l - r) is

nondecreasing n r, which implies thatSk+ 10

is a nonincreasing unction. Take any sequence

r, X r^C (0, 1). Since by the inductive

hypothesis Sk(-) is nonincreasingand continu-ous from the right, we must have Sk(rj) A

Sk(P). This implies that N - [N - Sk(rf)]I

(1 - r) / N - [N - Sk(P)]I(l - r). Using

(Al) and the fact thatr?1s continuous romtheleft, we have Sk+I(rj) rN [N - Sk(rf)]I

(1 - r)] / Sk l(r). Thus, Sk+ I( ) is alsocontinuous from the right.

Claim2: Foranyk ' 0, Sk- N* when ur/x*E[0o,(N - N* + 1)].

PROOF:See Lemma 1.

Claim3: Foranyk? l,Sk -ooas aTrx* >1.

PROOF:From (Al) we see that SI(r) = N - (N -

N*)I(l - r)] -> -oo as r -- 1. Since by

Lemma 1 we have Sk ? SI for k ' 1, thestatementfollows.

As Proposition4 shows, exclusion occurs ifand only if SN-N*+1 ? 0. Claims 1-3 showthatSN N* + 1 is a nonincreasingand continu-ous from the right function of lTIx* E (0, 1),that t is positive for an/x* lose enoughto zero,andthatit is negativefor wrrx*close enoughtoone. This implies the statementof Corollary1

PROOF OF COROLLARY 2:Let SW and S' be the sequences Sk and

Sk defined for the model where each of the Noriginal buyersis subdivided nto m, i.e., whereN = Nm = mN, N* = N* = FamNA.

We show that for a given k, gm is a"good

enough approximation"o Smwhen m is suffi-ciently large. Namely, we establishthat for anyfixed k, we have (Sm - Sm)Im -> 0 as m -- o.

We do this by induction.The statement s triv-ially true for k = 0. Suppose it is true for acertain k ' 0. Using the fact that Fxl E [x,x + 1), we can then write

mSk+SI k-SMk

m1 FSk x 1X

m x* m > - WT

z sm - ,m shk k X

1 Sm - gm X*

Ml zn x*- .

By the inductivehypothesisboth bounds go tozero as m ??9>o so we must have (Skm

gm+ )Im --> 0. Therefore,the inductive state-ment is true for all k ? 0.

SubstitutingN- Nm m=N, N* - NroamNl into (4), we obtain

[F1 F( mJVI_mNx* ) |

ObservethatFamk]/(mNT)- a E (0, 1) asm oo, therefore we can find 8 E (0, 1) andm ? 0 such that ramNAT]I(mN) ( for any

m ' .Using the above expressionfor (lIm)S' and

the fact thatx(x*-x w) > 1, for anym ? mi

we can write

-Sm-? NI- Jj8 -_cm L\x* J

as k- oo.

Hence, (lIm)3m -> -oo uniformly on m > mt

as k -> ?. In particular,here exists k ' 0 such

that (l1m)3m c -1 for all m ? mh. Since(Sm - Sm)Im -O 0 as m -o co, there exists

m 0 such that(llm)Sm ? (lIm)3m + 1 forall m ? mh.Combiningthe two inequalities,wesee that(llm)Sm ? 0 for all m 2 max{,mi,m}Using in addition Lemma 1, for all m

max{tm, ii, r(k + 1)/(l - a)N]} we canwrite:

SN/II m-N= anNl? S(I-la)mNl Sk-I

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VOL.90 NO. 1 SEGALAND WHINSTON:NAKEDEXCLUSION,COMMENT 309

Applying Proposition4, we see that the in-cumbent excludes costlessly for all m largeenough.

PROOF OF PROPOSITION5:It follows from the discussionin the text that

firm I can earn N(c - c)- f by allowing entry.

Note thatit cannotearnmorethanthis if firm Renters:aggregatesurpluswhen firm R enters isat most N[CS(c) + (c - c)] - f = N(v -

c)- f (since the cost of purchase s at least cfor each buyer) and, if entry is occurring,eachbuyercan assureitself at least CS(c) v -c

by rejectingfirmI's offer.Now supposethatthere is a PCPNEin which

entrydoes not occur and firmI pays less than

N*x* = N*(v - c) in aggregate to buyers who

sign contracts.Let S 2 N* denote the set of

buyerswho sign andlet S* = { i C S: xi ' v- c}. Then we must have |S*| < N* and so(N - S*I)(C - c) ' f. Now, since

(N-|JSI)(C - c)

+ E max{O,min{c, v - t} - c} -f < O,

there exists a set of buyers J C S\S* such that

(N- |S + JI)(C-c)

+ E max{O,min{c, v-tj} - c}-f '?Oi E S\J

and such that for all J' C J we have

(N- SI+ IJ'I)(- c)

+ > max{O,min{c, v - tj} - c} - f < O.iES\J'

But buyers in set J have a self-enforcingPareto-improvingdeviation in which they all

reject firm I's offer. By doing so, they eachearn CS(c) = v - c, which is the most theycan earn given other buyers' acceptance de-cisions, and which strictly exceeds their pay-

offs in the purportedequilibrium (which is anamount strictly less than v - c). This yieldsa contradiction: we conclude that firm I's costof exclusion must be at least N*(v - c) inany PCPNE in which firm R does not enter.That firm I can exclude for a total cost ofN*(v - c) follows from the same argumentsas in Proposition 3.

REFERENCES

Aghion, Philippe and Bolton, Patrick. "Contracts

as a Barrier to Entry." American Economic

Review, June 1987, 77(3), pp. 388-401.

Bernheim, B. Douglas; Peleg, Bezalel and Whin-

ston, Michael D. "Coalition-Proof Nash Equi-libria: Concepts." Journal of EconomicTheory,June 1987, 42(1), pp. 1-12.

Innes, R. and Sexton, R. J. "Strategic Buyers

and Exclusionary Contracts." AmericanEconomic Review, JuIne 1994, 84(3), pp.

566-84.Mankiw, N. Gregory and Whinston, Michael D.

"Free Entry and Social Inefficiency." RandJournal of Economics, Spring 1986, 17(1),pp. 48-58.

Rasmusen, Eric B.; Ramseyer, J. Mark and

Wiley, John Shepard Jr. "Naked Exclusion."

American Economic Review, December1991, 81(5), pp. 1137-45.

Segal, Ilya R. "Contractingwith Externalities."QuarterlyJournalof Economics, May 1999,114(2), pp. 337-88.

Segal, Ilya R. and Whinston, Michael D. "Naked

Exclusion and Buyer Coordination."Discus-sion Paper No. 1780, Harvard Institute ofEconomic Research, September 1996.