Faculty Research Working Papers Series

Vertical Networks, Integration, and Connectivity

Pinar Dogan

November 2005

RWP05-061

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Vertical Networks, Integration, and Connectivity�

P¬nar DO¼GANy

November 11, 2005

Abstract

This paper studies competition in a network industry with a stylized two layered network structure,and examines: (i) price and connectivity incentives of the upstream netwoks, and (ii) incentives forvertical integration between an upstream network provider and a downstream �rm. The main resultof this paper is that vertical integration occurs only if the initial installed-base di¤erence between theupstream networks is su¢ ciently small, and in that case, industry is con�gured with two verticallyintegrated networks with neither of the upstream �rm having an incentive to degrade the quality ofinterconnection. When the installed-base di¤erence is su¢ ciently large, there is no integration in theindustry, and the equilibrium quality of interconnection is lower compared to the equilibrium with twovertically integrated �rms.

Keywords: Vertical Integration, Interconnection, Network Externalities.

JEL Classi�cation: L13; L22; L86.

�I thank Jacques Crémer, Patrick Rey, Dani Rodrik, and Yeon-Koo Che for their extremely useful suggestions. An earlierversion of this paper was circulated as "Vertical Integration in the Internet Industry."

yJohn F. Kennedy School of Government, Harvard University, 79 JFK Street, Cambridge, MA, 02138 USA. E-mail:pinar_dogan@ksg.harvard.edu.

1

1 Introduction

In this paper, I consider an industry with a stylized two-layered network structure. Within a given ge-ographical area, two horizontally di¤erentiated downstream �rms compete for supplying connectivity toend-users. In the upstream market, two interconnected networks with di¤erent installed-bases compete toprovide connectivity to downstream �rms. The quality of interconnection between the networks is perfect,unless an upstream network engages in strategic quality degradation. I analyze competition in such an in-dustry, focusing on the incentives for vertical integration and the implications for the equilibrium quality ofinterconnection between upstream networks.The main result of this paper is that vertical integration occurs only if the initial installed-base di¤erence

between the upstream networks is su¢ ciently small, and in that case, industry is con�gured with twovertically integrated networks with neither of the upstream �rm having an incentive to degrade the qualityof interconnection. An interesting feature of this case is that vertical integration improves consumer welfare,through both the elimination of double-marginalization and better interconnectivity. As in the compatibilityliterature, similar-sized networks are more likely to generate perfect connectivity, but in a vertical industrythis requires both networks to integrate vertically.When the installed-base di¤erence is su¢ ciently large, on the other hand, there is no integration in

the industry, and the equilibrium quality of interconnection is lower compared to the equilibrium with twovertically integrated �rms. That is because the larger upstream network does not have the incentive tointegrate vertically when its installed-base is su¢ ciently larger than its rival�s. When this di¤erence is largeenough, the large upstream network can provide connectivity to both downstream �rms at a higher priceby engaging in strategic quality degradation. Even though the incentive for quality degradation would havebeen even larger in a con�guration where the large upstream network integrates with one of the downstream�rms and its rival does not, this is never an equilibrium outcome.Although highly stylized, the network model presented in this paper is relevant for the Internet, where

the interconnection structure is hierarchical. Individuals and businesses sit at the bottom of the hierarchy,and connect their end-systems (PCs, workstations, etc.) to local Internet Service Providers (ISPs), that arein turn connected to the Internet Backbone Providers (or to regional ISPs) which are interconnected. Thequality of the interconnection within the backbone layer is determined in part by the choices made by theInternet Backbone Providers. In contrast to the earlier papers on the connectivity in the Internet, Crémeret al. (2000) and Foros and Hansen (2001), in which both the quality of interconnection and the prices forthe end-users are decided by the same agent, this paper captures the multi-layered structure of the Internet.The analysis of the paper also applies more generally to vertical markets that are characterized by

network externalities. The choice over the quality of interconnection is analogous to a choice over the degreeof compatibility.1 The compatibility incentives of �rms in the presence of network externalities has beenstudied extensively in the literature. However, the question has not been addressed in a vertical settingwhere compatibility decisions are made by upstream providers.2

The organization of the paper is as follows. In Section 2, I present the model for a vertically separatedindustry, and solve for the equilibrium. In Section 3, I introduce an initial stage to the game, where theupstream networks sequentially decide whether or not to vertically integrate with one of the downstream�rms. After characterizing the equilibrium price and quality of interconnection for each subgame, that isfor each industry con�guration, I analyze incentives for vertical integration. Following the presentation ofthe main results, I provide the intuition behind them in greater detail in Section 4. In this section I alsoprovide a benchmark where there is no installed-base advantage in order to illuminate how the installed-basedi¤erence and network externalities play a role in the integration decision. Finally, before concluding, Iprovide a brief discussion on consumer welfare.

1Katz and Shapiro (1985) is one of the most in�uential paper in the general literature on compatibility with presence ofnetwork externalities. See also Farrell and Saloner (1986), (1992), Economides and White (1994), and Economides (1996). Anoverview of this literature is provided by Liebowitz and Margolis (2002), Farrell and Klemperer (2002).

2The framework that is provided in this paper is di¤erent than "systems competition" where end consumers make thepurchase decisions for each of the complementary component that makes a system. For example, Church and Gandal (2000)look at how vertical mergers and foreclosure in systems markets (hardware-software), a¤ect incentives to produce compatiblecomponents to the rival systems, where consumers decide both for their hardware and software purchases, and there are nodirect network externalities. See also Economides and Salop (1992), where authors assume full compatibility and analyzevertical integration in network markets.

2

2 Vertically Separated Industry

Upstream Networks

UA UBInstalled­base βAInstalled­base βA

Installed­base βBInstalled­base βB

θ = min{θA, θB}

D1D1 D2D2

wA wA wBwB

Final Consumers0 1

Downstream Firms

p1 p2

Figure 1: Competition Layout with Vertical Separation

Upstream Networks In the stylized network structure represented on Figure 1, two upstream net-works, UA and UB , with installed-bases �A and �B , respectively, provide connectivity to the downstreammarket. UA has a larger installed-base, �A > �B � 0.3The quality of connectivity is denoted with �. All users within the same upstream network enjoy perfect

connectivity (�in-net = 1). The quality of interconnection between the networks is also perfect (�o¤-net = 1),unless an upstream network engages in strategic quality degradation. Strategic quality degradation ofinterconnection entails a cost, and is de�ned as

& (�) = (1� �)2 =2:

The cost of degrading quality is borne by the network who e¤ectively sets the quality of interconnection, i.e.,by the network that has a lower preference over the quality. Since o¤-net connectivity is a strategic choicevariable, whereas in-net connectivity is not, for expositional simplicity, I refer to �o¤-net 2 [0; 1] as �.Finally, upstream networks have symmetric cost of providing unit connectivity, which is normalized

to zero. They compete à-la Bertrand and charge wA and wB , respectively, for unit connectivity to thedownstream �rms.

Downstream Firms In the downstream market, two �rms, D1 and D2, purchase connectivity fromone of the upstream networks and compete à-la Bertrand with di¤erentiated services. They are located atthe extremities of a segment of length 1. I assume that D1 is located at 0, and D2 at 1.

End-users End-users are distributed uniformly with density 1 on the segment at whose extremities thedownstream �rms are situated. Each end-user has a unit demand for connectivity that is provided by thedownstream �rms. There are positive network externalities, and hence, an end-user�s utility depends on thenumber of other users in the entire network. Parameter, �, with � 2 (0;b�) and b� < 1, captures the bene�tof being connected to another customer. Since in-net connectivity is perfect, and since the quality of o¤-netconnectivity is set by the upstream networks, the end-users utility depends on both (i) their downstreamprovider�s choice of the upstream network, and (ii) the rival provider�s choice of the upstream network.

3The focus of the paper is a local market, and the installed bases of the networks (customers in other markets) are exogenous.

3

The end-user located at x 2 [0; 1], who is connected to the network through the service provided by Di,which is in turn connected to the upstream network Un, with n = A;B, derives the following utility

Ui;n = v � t jx� xij � pi +��(�n + qi + �(��n + (1� qi))) if D�i connects to U�n�(�n + qi + (1� qi) + ���n) if D�i connects to Un

where xi stands for the location of Di, t is the standard transportation cost, qi and (1� qi) are the marketshares of Di and D�i; respectively, and pi is the access price charged by Di. Fixed utility derived byconnecting to any of the downstream �rms, denoted with v, is assumed to be su¢ ciently large so that allend-users buy service from one of the two downstream �rms.Let � denote the installed-base di¤erence between the upstream networks, i.e.,

� = �A � �B ,

and let' � �(1� �) < b�: (1)

We can interpret '� as the large upstream network�s installed-base advantage. This is because for a givenpositive installed-base di¤erence (� > 0), two upstream networks are perceived as completely symmetricfrom the end-users point of view if

i. connectivity between the upstream networks is perfect (� = 1), and/or

ii. if there are no network externalities (� = 0).

Either of the two condition implies that the large network has no installed-base advantage in competition(i.e., ' = 0). For any given level of imperfect connectivity, and positive network externalities, a greatermagnitude of externalities and/or larger in installed-base di¤erences imply a larger installed-base advantage.Finally, for any given positive � and �, the installed-base advantage is higher when connectivity is lower,since a lower quality of connectivity increases the relative quality of service provided by the large upstreamnetwork, compared to that of the smaller one.I assume that

� < (t� �)=�. (A)

This assumption limits the asymmetry in the size of the upstream networks, and assumes away �marketcornering�4 at the �nal market. It implies

t > � � ': (2)

Timing of the Game The game consists of four stages.

Stage 1 � Each upstream network noncooperatively decides on the quality of interconnection, �n, and thee¤ective quality of interconnection is determined by the lowest of these two qualities, that is � = min f�A; �Bg.

Stage 2 � The upstream networks set the prices of unit connectivity, wA and wB .

Stage 3 � The downstream �rms choose their upstream network provider.

Stage 4 � The downstream �rms set pi, compete for the end-users, and pro�ts are realized.

4Market cornering happens, for example, when the downstream �rms connect to di¤erent upstream networks that chargethe same price w, and all end-users subscribe to the downstream �rm which is connected to the large network.

4

2.1 The Equilibrium in the vertically separated industry

The main result of this subsection is Proposition 1, which shows that whenever the connectivity betweenthe upstream networks is imperfect, in the risk dominant equilibrium both downstream �rms connect tothe large upstream network, and the large network obtains a positive pro�t margin that amounts to itsinstalled-based advantage. Proposition 2 shows that in the vertically separated industry, the equilibriumlevel of interconnection between the upstream networks is imperfect, and is decreasing with both the size ofnetwork externalities and installed-base di¤erence. This result is basically an extension of the interconnectiondegradation result in Crémer et al. (2000) to a two-layered network structure. Before the formal proof, Ipresent the intuition behind these results.For all �A > �B , perfect interconnection between the upstream networks (� = 1) makes the two networks

equivalent from the viewpoint of the end-users, and drives prices to marginal cost, which is zero. Degradingthe quality of interconnection, the larger upstream network vertically di¤erentiates its otherwise undi¤er-entiated product (unit access), and obtains a quality advantage. Both downstream �rms have a preferencefor the better quality product. Hence, the large upstream network covers the market and obtains a markupwhich re�ects its installed-base advantage, and price competition drives the smaller network�s price to itsmarginal cost.Let w denote the di¤erence between the unit price of connectivity charged by the upstream networks,

i.e.,w = wA � wB .

Lemma 1 Whatever w and � < 1; in the pure strategy Nash equilibrium of the subgame the two downstream�rms connect to the same upstream network.

Proof. Table 1 summarizes the payo¤s of the subgame where downstream �rms choose their upstreamnetwork provider. The computation of the payo¤s can be found in Appendix A.1.

D2

UA UBD1 UA (t=2; t=2) (h; l)

UB (l; h) (t=2; t=2)

Table 1: Payo¤s of the subgame where the downstream �rms choose their upstream network, withh = (3(t�')�w+'�)2

18(t�') and l = (3(t�')+w�'�)218(t�') .

The two downstream �rms connect to di¤erent upstream networks, only if both h and l are greater thant=2. That is, 3(t�')�w+'� > 3

p(t� ')t and 3(t�') +w�'� > 3

p(t� ')t. Adding two inequalities

yield 6(t � ') > 6p(t� ')t, and by inequality (2) this implies, '2 > 't, which in turn implies ' > t;

establishing the contradiction.Before I provide the intuition for Lemma 1, I brie�y discuss the case in which upstream networks are

perfectly connected, i.e., � = 1. Let (U1,U2) denote the strategy pro�le of the downstream �rms, withthe �rst and second terms describing D1�s and D2�s upstream network choice, respectively, with 1; 2 =A;B. If upstream networks are perfectly connected, and if they charge the same price for connectivity,the downstream �rms�choice of upstream network provider is irrelevant in terms of their potential marketshare, and hence their pro�ts. This is because potential customers of each downstream �rm would enjoyperfect connectivity in the entire network, regardless of their choice of downstream provider. Therefore,when � = 1, there are four pure strategy Nash equilibria, (UA,UA), (UB ,UB), (UA,UB), and (UB ,UA); andthe downstream �rms share the market equally and obtain same pro�ts under all equilibria. If the upstreamnetworks charge di¤erent prices, then the equilibrium is unique; both downstream �rms get connected to theupstream network which sets a lower price for connectivity.Lemma 1 shows that whenever the interconnectivity is imperfect between the upstream networks (� < 1),

there are two pure strategy equilibria of the subgame where the downstream �rms�choose their upstreamnetwork providers. In both equilibria, (UA,UA) and (UB ,UB), the downstream �rms connect to the to sameupstream network. Downstream competition is softer when the downstream �rms are connected to the samenetwork than when they are connected to di¤erent upstream networks, for the following reasons. When

5

the downstream �rms connect to the same upstream network, the end-users of both downstream �rms areperfectly interconnected. Since each downstream �rm pays the same price for the unit connectivity, �rms endup sharing the local market equally. However, when the interconnection is imperfect between the networks,if each �rm connects to a di¤erent upstream network, the �rm which is connected to the large upstreamnetwork has a competitive advantage with respect to its rival. Since all end-users value a better connectivityto a larger installed-base, this induces the downstream �rms to engage in a tougher competition to gainmarket share than when both of them are connected to the same upstream network.Finally note that, the unit cost of connectivity is entirely borne by the end-customers, and hence, the

downstream �rms get the same pro�ts when they both connect to the small network and when they bothconnect to the large network, regardless of w. Therefore, neither of the equilibrium payo¤s dominate theother. I use the risk dominance of Harsanyi and Selten (1988)5 as the equilibrium concept for this subgamein which �rms choose their network providers, and with the following Lemma, I state the conditions for eachof the two equilibria to constitute a risk dominant equilibrium.From now on, I normalize t = 1 for expositional simplicity.

Lemma 2 For all � < 1 , in the subgame where the downstream �rms choose their upstream networks,(UA,UA) is a risk dominant equilibrium if w < '�, and (UB,UB) is a risk dominant equilibrium if w > '�.

Proof. By the de�nition of Harsanyi and Selten (1988), in this 2� 2 symmetric game (UA,UA) is a riskdominant equilibrium if and only if

1

2

�1

2

�+1

2h >

1

2l +

1

2

�1

2

�; (3)

which implies h > l.Given the assumption stated in (A), this inequality holds if and only if w < '�: Similarly, one can show

that (UB ,UB) is a risk dominant equilibrium if and only if w > '�.Risk dominance provides a very intuitive prediction in this game.6 There are two pure strategy Nash

equilibria, and the players cannot predict to which equilibrium their opponent will lean, as both equilibriayield the same payo¤. Assume that 1=2 > h > l: Then by choosing to connect to UB , a downstream �rmtakes the risk of getting payo¤ l; which is the smallest payo¤ in this game. However, by connecting to UAit guarantees itself a payo¤ of at least h. Therefore, provided that w < '�, i.e., h > l, connecting to UA isless risky than connecting to UB in this subgame. The reverse is true if w > '�.

Proposition 1 In the second stage of the game UA sets wA = '�, and UB sets wB = 0, and both downstream�rms connect to UA:

Proof. It is straightforward from the pro�t functions of UA and UB :

�A =

�wA if w < '�;0 if w > '�:

�B =

�wB if w > '�;0 if w < '�:

Bertrand competition yields prices wA = '� and wB = 0:The installed-base advantage of the large upstream network is analogous to a cost advantage in price

competition. The price of the small upstream network is driven to its marginal cost, whereas the largeupstream network holds a markup which amounts to its installed-base advantage, and covers the marketalone.7

5 In a symmetric 2�2 game risk dominant equilibrium is de�ned by the pro�le where both players playing their risk dominantstrategy in which they strictly prefer the same action when they predict that their opponent randomizes 1

2� 1

2:

6However, when advantage from the installed base, '�, is very small (when (n�m)! 0), risk dominance may not providea very good prediction.

7Here, networks make their price o¤ers publicly and simultaneously. These results do not hold when sequential contractingis allowed. In particular, commitment of any of the downstream �rms to buy connectivity from, say, the UA, would result witha higher unit price o¤ered by the UA to the next downstream �rm in line.

6

Proposition 2 In the vertically separated industry, the equilibrium level of connectivity is imperfect, and isdecreasing with both the installed-base di¤erence and the network externalities.

Proof. The equilibrium level of connectivity,

� = 1� ��,

follows from UA�s optimization problem,

max�

(�(1� �)� � (1� �)

2

2

),

since UB has no incentives to degrade quality of interconnection. It is also straightforward to see that � isdecreasing with both � and �.For any positive level of network externalities, larger installed-base di¤erence leads to poorer quality of

connectivity, as it pays the network to engage in degradation. When the installed-base advantage approachesto zero (because of either a negligible installed-base di¤erence or insigni�cant network externalities), qualityof connectivity becomes perfect. The equilibrium payo¤s in the vertically separated industry are �A =1=2 (��)

2, �B = 0, and �1 = �2 = 1=2.

3 Vertical Integration

In this section I consider upstream networks sequentially deciding whether or not to integrate with one of thedownstream �rms. Whenever integration occurs I assume that the integrated upstream network providesconnectivity to its downstream subsidiary at marginal cost (zero). I do not consider the case where theintegrated upstream network commits to charge a positive price to its downstream subsidiary, as the internalprice is not observable by the non-integrated downstream rival.8

Timing of the game: The game consists of six stages.

Stage 1 � UA decides whether to integrate with one of the downstream �rms (say D1)

Stage 2 � Following UA�s decision, UB decides for integration. If integration does not occur at Stage 1,UB decides whether to integrate with one of the downstream �rms (say D2). If integration occurs at Stage1, UB decides whether to engage in counter-merger (with D2).

Stage 3 � The upstream networks decide on the level of connectivity, �.

Stage 4 � The upstream networks set the price for connectivity, wA and wB .

Stage 5 � The non-integrated downstream �rm (if there is any) chooses the upstream network for connec-tivity.

Stage 6 � The downstream �rms set prices and compete for end-users.I assume that upstream networks decide to integrate with one of the downstream �rms only if pro�ts

under integration is larger than the sum of pro�ts when there is no integration. The following �gure depictsthe extensive form of the game tree with some abuse of presentation (of stages 3,4 and 6).

8Otherwise, with vertical integration �rms can always replicate the no vertical integration outcome.

7

UA

UBUB

integrateswith D1

does not integratewith D1

does not integratewith D2

does not integratewith D2

integrateswith D2

integrateswith D2

(1) (2) (3) (4)vertical separation(See Section 2)

UB –D2integration

UA –D1integration

two integratednetworksStage 3 –θA and θ B

Stage 4 –wA and wBD1 D2

(I) (II) (III) (IV) (V) (VI)

Stage 6 – D1 and D2 compete for end­customers

connectsto UB

connectsto UB

connectsto UA

connectsto UA

exits exits

Figure 2: Game Tree with Vertical Integration

In this setting, there are four industry con�gurations (subgames),

(1) vertical separation,

(2) the small upstream network is integrated with one of the downstream �rms (UB �D2),

(3) large upstream network is integrated with one of the downstream �rms (UA �D1),

(4) two integrated networks.

Equilibrium prices, connectivity level and pro�ts of subgame (1) are as in Section 2. For subgames (2)and (3), in which there is one integrated �rm, possible outcomes regarding the non-integrated downstream�rm�s choice of upstream provider are denoted with (I) to (III) for subgame (2), and (IV) to (VI) for subgame(3). For a given wA, wB and �, payo¤s of D1, D2, UA, and UB ; that are realized after the competition forend-users are denoted with �i1;�

i2;�

iA;�

iB , respectively, for i = I � V I. Computations regarding the payo¤s

of subgames (2) and (3) can be found in Appendix A.2 and A.3, and are summarized in Appendix B.

3.1 The Equilibrium

Before I compute the equilibrium for subgames (2), (3), and (4), I rule out outcomes (II) and (V), wherenon-integrated downstream �rm exits the market.

Claim 1 When one of the upstream network integrates with a downstream �rm, the non-integrated down-stream �rm does not withdraw from the market.

Proof. See Appendix C.Consider both cases (II) and (V); if the integrated �rm refuses to deal with the non-integrated downstream

�rm, or equivalently, charges too high a price to it, non-integrated upstream network can improve its payo¤byselling connectivity at a price with which the non-integrated downstream �rm makes positive pro�ts. Later,

8

with Claims 2 and 3, I show that the integrated upstream network is also willing to supply connectivity tothe non-integrated downstream �rm, and hence upstream networks compete to provide connectivity.I proceed with characterizing the equilibrium price and the quality of interconnection for each subgame.

Then, I move backwards and study incentives for vertical integration.

SUBGAME (4) Two integrated networks. Each vertically integrated upstream network providesconnectivity to its own subsidiary at zero price, and hence, double-marginalization is completely mitigatedin this industry con�guration. For a given level of quality of interconnection, the gross pro�ts can be foundbelow, and are computed as in Appendix A.1.2, by replacing zero for both wA and wB .

�4A +�41 =

(3(1� ') + '�)2

18(1� ') (4)

�4B +�42 =

(3(1� ')� '�)2

18(1� ') ; (5)

The only source of pro�ts in this subgame is from provision of service to the end-users. The integrated large(small) upstream network�s pro�ts are increasing (decreasing) with the installed-base di¤erence.

Quality of Interconnection In this industry con�guration the equilibrium connectivity is determined bythe preference of the large upstream network. Gross pro�ts of the small integrated network is decreasingwith ' (and hence increasing with �), whereas the large network�s preferences over the quality of intercon-nection depends on the installed-base di¤erence. When the installed-base di¤erence is su¢ ciently small,the large upstream network prefers perfect connectivity. Lemma 3 characterizes the equilibrium quality ofinterconnection in this subgame.

Lemma 3 In the industry con�guration with two vertically integrated networks, the equilibrium quality ofinterconnection

(i) is determined by the large upstream network,

(ii) is perfect for all � � b�; whatever the level of network externalities are,(iii) belongs to (0; 1] for all � > b�; and is decreasing with the size of the installed-base di¤erence.Proof. See Appendix C.When there are two integrated networks in the industry, both downstream �rms obtain positive markets

shares, and hence positive pro�ts, even if the networks are completely disconnected. This is by virtue of theinitial assumption on the installed-base di¤erence, (A), which rules-out market cornering. The large upstreamnetwork faces a trade-o¤when it decides for the connectivity level: a higher quality of interconnection impliesa higher willingness to pay for its end service, but it also makes the large upstream network�s quality of serviceless di¤erentiated from that of the small network�s.If the initial installed-base di¤erence is small, the marginal gain in terms of quality advantage to the

large network is also small (and zero in the limit as this di¤erence goes to zero) while the marginal cost ofdegrading quality in terms of reduced willingness to pay by end-customers is strictly positive. Hence, evenin the absence of monetary cost of degradation, marginal cost of degradation exceeds the marginal gain ofdegradation. This is the reason why if the size of the installed-base di¤erence is su¢ ciently small, the largeupstream network is better o¤ by providing its end-users with a perfect connectivity to the whole network.If the installed-base di¤erence is su¢ ciently high, the large upstream network �nds it pro�table to degrade

quality of interconnection. For a given large installed-base di¤erence, higher network externalities imply lowerquality of interconnection, since it further elevates the large network�s competitive advantage.Note that the negative impact of the reduced quality di¤erentiation (due to a better connectivity) on

the large network is more severe in other industry con�gurations, where the upstream networks compete forproviding connectivity to at least for one of the downstream �rms. This is why connectivity incentives inthis industry con�guration are higher than in any other con�guration.

9

SUBGAME (3) UA �D1 integration. In this subgame, where the large upstream network is integratedwith one of the downstream �rms and the small upstream network is not, the non-integrated downstream�rm, D2, chooses its upstream network provider. I begin by showing that both upstream networks preferto sell connectivity to D2, and hence there is an e¤ective upstream competition. Then, with Lemma 4, Ishow that D2 ends up buying connectivity from the large upstream network, and �nally, with Lemma 5 Icharacterize the equilibrium quality of interconnection.

Claim 2 In Subgame (3), both upstream networks prefer to sell connectivity to the non-integrated down-stream �rm.

Proof. This is straightforward for UB , and is proved with Claim 1. UA obtains positive pro�ts even if itdoes not sell connectivity to D2, therefore, we need to show that UA is better o¤ by selling connectivity to D2at a price such that D2 buys connectivity from it. D2 buys connectivity from UA if (i) �IV2 (wB) � �V I2 (wA),i.e., if

wA63�(3(1� ')� '� � wB)p

1� ' : (6)

and if (ii) it obtains non-negative pro�ts by doing so, i.e., if wA 2 [0; 3). For all wB 2 (0; 3(1� ')� '�) ; wehave the second term of the right-hand side of the inequality (6) positive, and hence, (i) implies (ii). It is easyto verify that UA�s pro�ts under (VI) are increasing with wA for all wA 2 (0; 3) (since the unconstrainedmaximization problem yields w�A = 3:75). This, in turn, implies that optimal price is the highest pricethat satis�es inequality (6). Given wB 2 (0; 3(1� ')� '�), when UA charges the optimal price, we have�V IA +�V I1 > �IVA +�IV1 for all � 2 (0; (1� �)=�], which concludes that in subgame (3) UA is better o¤ byselling connectivity to D2.

Lemma 4 In the industry con�guration where the large upstream network is integrated with one of thedownstream �rms, and the small upstream network is not, the non-integrated downstream �rm ends upbuying connectivity from the integrated network, given that the connectivity between the upstream networksis imperfect.

Proof. It is straightforward to conclude that price competition ends with w3A = 3�(3(1� ')� '�) =p1� ';

and w3B = 0; since w3A is the optimal price for UA and is strictly positive for all ' 2 (0; 1] (for all � 2 [0; 1)).

Finally, D2 buys connectivity from UA since w3A and w3B satisfy the constraint de�ned in inequality (6) with

equality.At the price equilibrium of this subgame, for any given � < 1, UB obtains zero pro�t, D2 obtains

�32 =(3 (1� ')� '�)2

18(1� ') ; (7)

and the integrated UA obtains a pro�t of �3A (�; �; �), which is de�ned in Appendix C, in the proof of Lemma5.The equilibrium price at which the large upstream network sells connectivity to the non-integrated

downstream �rm is increasing with the installed-base di¤erence. This means that even if both downstream�rms�end-users experience the same in-net quality of connection, the non-integrated downstream �rm willhave a disadvantage in downstream competition. Therefore, the latter�s pro�ts are also decreasing withthe size of the installed-base di¤erence. The presence of network externalities enhances the large upstreamnetwork�s ability to raise its downstream rival�s cost. When there are positive network externalities, theintegrated upstream network, even with no initial installed-base di¤erence (� = 0) can charge a positiveprice, thanks to its added installed-base that its subsidiary obtains in this local market.

Quality of Interconnection Similar to the industry con�guration with two integrated networks, theequilibrium connectivity is determined by the preference of the large upstream network, since the smallupstream network obtains zero gross pro�ts for any level of imperfect connectivity. However, the equilibriumconnectivity level is never perfect under this subgame, since upstream networks are competing for the non-integrated downstream �rm, and price competition between the upstream networks results in marginal costpricing when networks are perfectly interconnected.

10

Lemma 5 In the industry con�guration where the large upstream network is integrated with one of thedownstream �rms and the small upstream network is not, the equilibrium quality of interconnection

(i) is determined by the large upstream network,

(ii) is zero for all � > �, whatever the installed-base di¤erence is,

(iii) belongs to [0; 1) for all � < �, and is decreasing with the size of the installed-base di¤erence.

Proof. See Appendix C.By degrading the quality of connectivity the large upstream network di¤erentiates its quality of service

from that of UB , and increases the non-integrated downstream �rm�s opportunity cost of buying connectivityfrom the small downstream �rm. This increases wA, which in turn provides the subsidiary �rm with acompetitive advantage in downstream competition. Incentives for the large upstream network to degradeconnectivity increases both with the size of the installed-base di¤erence and network externalities. Unlikethe industry con�guration with two integrated networks, no connectivity can be an equilibrium outcome inthis con�guration; this happens when the magnitude of the network externalities is su¢ ciently large.9

Before I move on to subgame (2), I compare the equilibrium connectivity in subgames (3) and (4), whichwill prove useful when I study the incentives for vertical integration.

Lemma 6 Equilibrium connectivity in subgame (3) is strictly lower than in subgame (4).

Proof. See Appendix C.

SUBGAME (2) UB �D2 integration. In this subgame, where the small upstream network is integratedwith one of the downstream �rms, and the large upstream network is not, similar to subgame (3), the nonintegrated downstream �rm, D1, chooses its upstream network provider. With the following Claim, I showthat both upstream networks prefer to sell connectivity to the non-integrated downstream �rm, and hence,they e¤ectively compete for it. While this is straightforward for the large upstream network who would getzero pro�ts otherwise, it is not straightforward for the integrated small upstream network. This is becausethe integrated small network obtains positive pro�ts from its downstream division, and may choose notto compete with the large upstream network for that it may require lowering the price of connectivity toomuch (in particular if the initial installed-base di¤erence is very large), which in return intensi�es downstreamcompetition. Then, I show that the upstream network that sells connectivity at the equilibrium is determinedby the size of the installed-base di¤erence. Finally, with Lemma 8, I characterize the equilibrium quality ofinterconnection in this subgame.

Claim 3 In subgame (2) both upstream networks prefer to sell connectivity to the non-integrated downstream�rm.

Proof. See Appendix C.It follows from Claim 3 that in subgame (2), upstream networks compete for D1. Which upstream

network ends up selling connectivity at the equilibrium depends on the size of the installed-base di¤erence.This is because unlike in subgame (3), the initially small upstream network obtains additional installed-basethrough its subsidiary, and hence, for a su¢ ciently small initial installed-base di¤erences, it can have acompetitive advantage in providing connectivity to the non-integrated downstream �rm.

Lemma 7 In subgame (2), where the small upstream network is integrated with one of the downstream �rmsand the large upstream network is not, whatever the level of network externalities are, the upstream networkwhich ends up selling connectivity to the non-integrated downstream �rm is

(i) the large upstream network if � > b� for all � 2 [0; 1),9When no connectivity occurs, similar to the e¤ect of exclusive dealing in Mathewson and Winter (1987), the large upstream

network eliminates the rival downstream �rm from the market, and potential competition replaces actual competition (in thatthe large upstream network can not charge a too high price to the non-integrated downstream �rm). However, (3) is the onlyindustry con�guration where no connectivity can be an equilibrium outcome, and connectivity incentives are is higher in allother con�gurations, which also implies a lower price for connectivity.

11

(ii) the small integrated network if � 2�e�; b�i for some � 2 [0; 1),

(iii) the small integrated network if � � e� for all � 2 [0; 1).Proof. See Appendix C.When the installed-base di¤erence between the upstream networks is su¢ ciently large, the small inte-

grated upstream network can not attract the non-integrated downstream �rm with any positive price, givenany level of imperfect connectivity. The same is true for the large upstream network, in that, for su¢ cientlysmall installed-base di¤erences the large upstream network can not attract the non-integrated downstream�rm with any positive price, unless the connectivity is perfect. For a relatively small range of installed-basedi¤erences, it requires a very degraded quality of interconnection for the large upstream network to attractthe non-integrated downstream �rm. With Lemma 8, I show that it does not pay the large upstream networkto degrade quality of interconnection in this range of installed-base di¤erence. The equilibrium prices andpayo¤s of the price equilibrium can be found in Appendix C.

Quality of Interconnection Unlike in any subgame, in subgame (2), small upstream network can endup determining the equilibrium connectivity, and this happens when the initial installed-base di¤erence issu¢ ciently small.

Lemma 8 In subgame (2), where the small upstream network is integrated with one of the downstream �rmsand the large upstream network is not, equilibrium quality of interconnection

(i) is always imperfect,

(ii) is determined by the large upstream network for all � > b�, and is decreasing with both installed-basedi¤erence,

(iii) is determined by the small integrated network for all � � b�, and is increasing with the size of theinstalled-base di¤erence.

Proof. See Appendix C.The intuition why the incentives of the large upstream network for degradation increases with the

installed-base di¤erence is similar to the that of other subgames. The main di¤erence is that, if the installed-base di¤erence is su¢ ciently small, it is the small upstream network who determines the e¤ective level ofconnectivity. This is because, for su¢ ciently small installed-base di¤erences the large upstream networkcan not attract the non-integrated downstream �rm with a positive price, and hence obtains zero pro�ts.Although for � small but su¢ ciently close to b� it can attract the non-integrated downstream �rm with apositive price, the marginal gain of degradation is very small, and hence it does not pay to engage in qualitydegradation. Therefore, for � � b�, the equilibrium quality of interconnection is determined by the smallupstream network, and it increases with �. This is because the small integrated network prefers to degradequality of connectivity more when � is smaller, since for small � small upstream network is e¤ectively notsmall when one accounts for the prospective end-users of its subsidiary. Prospective end-users in this localmarket add to UB �s installed-base, and compensate it for its initial installed-base disadvantage. On theother hand, as in any subgame, lower externalities imply higher connectivity incentives.

Incentives for Vertical Integration All the analysis presented so far holds regardless of which upstreamnetwork decides to integrate �rst, since the connectivity prices and quality of interconnection is set after theintegration decisions. In the following, I determine the SPNE of the game where the large upstream networkdecides �rst for integration. I then consider what happens if the small upstream network goes �rst.

Proposition 3 There is no vertical integration in the industry if � > b�, and there are two subsequentvertical mergers otherwise.

Proof. See Appendix C.If the small upstream network moves �rst, for a very large set of parameters, the equilibrium industry

con�guration is with two integrated �rms. It is only for very high installed-base di¤erences, where the large

12

upstream network obtains higher pro�ts under (1) compared to (3), the small upstream network does notimprove total pro�ts with integration (and hence, there is no integration in the industry). Otherwise, twoconsecutive mergers occur, and equilibrium connectivity is determined as in Lemma 3.

4 Discussion

Vertical integration occurs only if the initial installed-base di¤erence between the upstream networks issu¢ ciently small, and in that case, industry is con�gured with two vertically integrated networks withneither of the upstream �rm having an incentive to degrade the quality of interconnection. When theinstalled-base di¤erence is su¢ ciently large, on the other hand, there is no integration in the industry, andthe equilibrium quality of interconnection is lower compared to the equilibrium with two vertically integrated�rms. I now explain the intuition behind these results. To develop the intuition I proceed as follows. First, Iwill describe the equilibrium outcome in the industry con�guration with vertical separation. Then, I describethe di¤erence that allowing vertical integration makes. In that latter case the equilibrium outcomes dependcritically on the size of the installed-base di¤erence between the two upstream networks.

An industry structure with no vertical integration. By degrading the quality of interconnection, thelarge upstream network can introduce an asymmetry in an otherwise symmetric environment, and obtainsa positive markup in the intermediate good market which is increasing in the installed-base di¤erence.10

This is because the impact of degradation on a particular downstream �rm depends not only on its ownupstream network choice, but also on the its rival�s network choice. If both downstream �rms are connectedto the small network, the impact is relatively small since downstream �rm that compete for the same localmarket provide perfect connectivity in that local market, and customers will experience poor quality forthe o¤-net communications regardless of their choice of provider. Degradation has a more severe impacton a downstream �rm that is connected to the small upstream network, when its rival is connected to thelarge network. This is because the downstream �rm, that is connected to the small upstream network,provides end-customers perfect connectivity for a smaller network (in-net communications), which puts itat a disadvantage when competing with its rival that is connected to the large network. Since downstream�rms make their choices non-cooperatively, the large upstream network can obtain a positive markup in therisk-dominant equilibrium (and sells connectivity to both downstream �rms).

An industry structure with vertical integration Now consider the possibility of vertical integration.Integration decisions are made sequentially, with the large upstream network moving �rst. For any givensize of installed-base di¤erences, vertical integration changes both pricing and connectivity incentives. Avertically integrated network, whether it is initially small or large, increases its installed-base through thenew local customers of its subsidiary. Although this does not have much signi�cance for the small integratedupstream network if the initial installed-base di¤erence is very large, it does become signi�cant otherwise.

Large installed-base di¤erence If the installed-base di¤erence between the upstream networks issu¢ ciently large, there is no dominant strategy for the small network. If the large upstream network isnot integrated, integration between the small upstream network and a downstream �rm does not improvetotal pro�ts. This is because the integrated small upstream network can not attract the non-integrateddownstream �rm with any positive price when installed-base di¤erence is su¢ ciently large, and hence, theonly source of its pro�ts is its subsidiary �rm�s customers. The subsidiary �rm which is competing with theother downstream �rm (which is connected to the large upstream network) has a competitive disadvantage,and obtains lower pro�ts compared to the case under which both downstream �rms buy connectivity from thelarge upstream network. However, if the large upstream network is integrated, the small upstream networkand the non-integrated downstream �rm are better-o¤ with a counter-merger. The reason is that when the�rms remain non-integrated, the integrated large network engages in a "raising rival�s cost" type of strategy;

10Beard et al. (2001) show that a vertically integrated dominant �rm may have incentives to engage in "sabotage", that is tointentionally degrade quality of inputs sold, to una¢ liated downstream rivals. In this setting, the downstream �rms that buyconnectivity from the upstream �rm are subject to sabotage, and this is true even in the absence of vertical integration. Thesabotage is channeled through the upstream interconnection, and targeted to those who connect to the rival network.

13

it degrades the quality of interconnection, which increases the non-integrated downstream �rm�s opportunitycost of buying connectivity from the small upstream network, which in turn increases the connectivity pricethat the large upstream network can charge.11 A larger initial installed-base di¤erence implies a higher inputprice (and hence, lower market share) for the non-integrated downstream �rm. Since the large upstreamnetwork expects the counter-merger, which intensi�es downstream competition, the Subgame Perfect NashEquilibrium (SPNE) is one in which the large upstream network chooses not to vertically integrate with adownstream �rm.

Small installed-base di¤erence If the initial installed-base di¤erence between the upstream networksis su¢ ciently small, there is a dominant strategy for the small upstream network, which is to integrate witha downstream �rm. If the large upstream network is not integrated, the small network can improve pro�tswith vertical integration, because in this case it sells connectivity to the non-integrated downstream �rm ata positive price, thanks to its added installed-base brought by its subsidiary. If the large upstream networkis integrated, the small network again chooses vertical integration. This is both because the otherwise-non-integrated downstream �rm can enjoy marginal cost pricing for the input and because this equilibriuminvolves one with high connectivity incentives for the large (integrated) network. The reason the largeupstream network chooses high connectivity in this instance is that when the initial installed-base di¤erenceis small the marginal gain in terms of quality advantage to the large network is also small (and zero inthe limit as this di¤erence goes to zero) while the marginal cost of degrading quality in terms of reducedwillingness to pay by end-customers is strictly positive (even if the di¤erence is zero).In order to illuminate how the installed-base di¤erence and network externalities play a role in the

integration decisions, in the following section I provide a benchmark in which there is no installed-baseadvantage.

4.1 A Benchmark: No installed-base advantage

No installed-base advantage, i.e., ��(1 � �) = 0 refers to the case where either there are no network exter-nalities and/or upstream networks are symmetric in their installed-bases. However, no network externalitiesand symmetric installed-bases have di¤erent impacts on incentives for integration.

4.1.1 Absence of Network Externalities

When there are no network externalities, and there is no integration in the industry, the installed-basedi¤erence brings no advantage to the large upstream network, and upstream networks yield zero pro�ts.The downstream �rms share the market and obtain the same pro�ts irrespective of which upstream networkthey are connected to. The upstream networks have no incentive to degrade quality of connection. In thiscase, integration between any upstream network and the downstream �rm does not improve total pro�tscompared to separation. Regardless of the industry con�guration (e.g., no integration, one integrated �rm,or two integrated �rms), the quality of interconnection is perfect.

4.1.2 Symmetric installed-bases

When the upstream networks have symmetric installed-bases, but there are positive network externalities, theconsequences are the same for the industry con�guration with vertical separation. The upstream networksare symmetric, and compete à-la Bertrand. However, when only one of the upstream network integrateswith a downstream �rm, this grants the integrated network an added installed-base advantage throughthe potential end-users of its downstream subsidiary. Under such an industry con�guration, the integratedupstream network has an incentive to degrade quality of connectivity, since it obtains a positive markupover the price of connectivity by doing so. A larger magnitude of network externalities shifts the marginal

11Unlike Ordover et al. (1990), integrated upstream �rm neither needs to commit not to supply downstream rival nor needs toannounce its price prior to its upstream rival in order to restrain upstream prices. In the presence of network externalities, anda positive installed-base di¤erence, the large upstream network is able to restrain prices through degradation of connectivity,and is always willing to supply to the non-integrated downstream �rm. Any imperfect level of connectivity implies a positivemark-up over the connectivity and hence, supplying connectivity to the non-integrated rival does not neccesarily imply intensedownstream competition.

14

bene�t curve of strategic quality degradation upwards (which is due to the upstream network�s increasedability to charge the non-integrated the downstream a higher price of connectivity), and for su¢ ciently largeexternalities, marginal bene�t of strategic degradation exceeds its marginal cost. For any positive networkexternalities, connectivity is imperfect under this con�guration, and su¢ ciently large network externalities,the integrated upstream network prefers no interconnection.Nevertheless, the industry con�guration where only one upstream network is integrated does not consti-

tute a subgame perfect equilibrium outcome, regardless of the sequence of integration decisions. No matterwhat integration decision is taken by the �rst-mover upstream network, an integration between the second-mover upstream network and a downstream �rm improves pro�ts. This is also true for the �rst-moverupstream network, therefore the industry con�guration is the one with two vertically integrated networks.

4.2 Quality of Interconnection and Consumer Welfare

It is di¢ cult to derive direct policy conclusions from a stylized network model. In practice there may be moresophisticated contracts between the upstream networks and downstream �rms. Furthermore, since I assumefull market coverage in this model, higher prices constitute a transfer from consumers to the producers and donot a¤ect social welfare. With these caveats in mind, what I have shown is that consumers are best o¤ whenthe equilibrium industry con�guration is the one with two integrated networks. Connectivity incentives arehighest in this con�guration, and double-marginalization is eliminated. This is the equilibrium con�gurationonly if the installed-base di¤erence is small.It is also remarkable that the industry performs better with smaller asymmetries, regardless of the indus-

try con�guration. When there is no integration, a small installed-base di¤erence implies a better connectivityand a lower price in both upstream and downstream markets. The essential point here is that, for a givensmall installed-base di¤erence, connectivity is higher and prices for end-users lower when there are two in-tegrated �rms, compared to when there is none. Therefore, consumers are better o¤ under the industrycon�guration with two integrated networks, for any small installed-base di¤erence.12 Consumer bene�ts inthis instance are not solely due to small asymmetries in installed-bases, but also derive from the existenceof two competing vertically integrated networks.Finally, the worst industry con�guration in terms of consumer welfare is the one in which the large up-

stream network is integrated and the small upstream network is not. However, regardless of which upstreamnetwork moves �rst on integration, this is not an equilibrium outcome.

5 Conclusion

In this paper, using a stylized network structure, I investigated incentives for quality of interconnection (ordegree of compatibility) between two upstream networks, incentives for pricing, and how those incentiveswould change with vertical integration, if it appears, in the industry.When network externalities are present in a vertically structured market, installed-bases of the competing

upstream networks are determinant of their pricing and connectivity decisions. Vertical integration in thosemarkets implies an increase in the installed-base, and hence, depending on which upstream network isintegrated (given that the other is not), it can either reduce or increase the initial asymmetry between theupstream networks. An integration between the large upstream network and a downstream �rm harms thenon-integrated downstream �rm, since the large upstream network increases the opportunity cost of buyingconnectivity from the small upstream network, through strategic degradation of connectivity. In general, fora given level of network externalities and installed-base di¤erence, connectivity incentives are worse whenthe large upstream network is integrated with a downstream �rm (and others remain non-integrated), andit is best when there are two consecutive vertical mergers. However, I show that the former con�guration isnot an equilibrium outcome.I �nd that when the asymmetry between the networks is small, there are two consecutive vertical mergers

in the equilibrium industry con�guration. Unlike in any other industry con�guration, equilibrium quality ofinterconnection is perfect. Finally, when the installed-base di¤erence is su¢ ciently large, there is no vertical

12This is true also for larger installed-base di¤erences, but then two integrated �rms do not appear as an equilibriumcon�guration.

15

integration in equilibrium. The equilibrium quality of interconnection is imperfect and, for a given level ofnetwork externalities, it is decreasing with the size of the installed-base di¤erence.

References

[1] Beard, R. T., D. L. Kaserman and J. W. Mayo, 2001, Regulation, Vertical Integration and Sabotage,Journal of Industrial Economics, 49(3), pp. 319-333.

[2] Church, J. and N. Gandal, 2000, Systems Competition, Vertical Merger, and Foreclosure, Journal ofEconomics & Management Strategy, 9(1), pp. 25-51.

[3] Crémer, J., P. Rey, P. and J. Tirole, 2000, Connectivity in the Commercial Internet, Journal of IndustrialEconomics, 48, pp. 433-472.

[4] Economides, N., 1996, The Economics of Networks, International Journal of Industrial Organization,14 (6), pp. 673-99.

[5] Economides, N. and S. C. Salop, 1992, Competition and Integration among Complements, and NetworkMarket structure, Journal of Industrial Economics, 40(1), pp.105-23.

[6] Economides, N. and L. J. White, 1994, Networks and Compatibility: Implications for Antitrust, Euro-pean Economic Review, 38 (3-4), pp. 651-62.

[7] Farrell, J. and P. Klemperer, 2004, Coordination and Lock-In: Competition with Switching Costs andNetwork E¤ects, forthcoming in �Handbook of Industrial Organization�, Vol.3.

[8] Farrell, J. and G. Saloner, 1992, Converters, Compatibility, and the Control of Interfaces, Journal ofIndustrial Economics, 40 (1), pp. 9-35.

[9] Farrell, J. and G. Saloner, 1986, Installed Base and Compatibility: Innovation, Product Preannounce-ments, and Predation, American Economic Review, 76 (5), pp. 940-55.

[10] Foros, Ø. and B. Hansen, 2001, Competition and Compatibility among Internet Service Providers,Information Economics and Policy, 13(4), pp. 411-425.

[11] Harsanyi, J. C. and R. Selten, 1988, A General Theory of Equilibrium Selection on Games, MIT Press,Cambridge, Massachusetts.

[12] Katz, M. and C. Shapiro, 1985, Network Externalities, Competition and Compatibility, American Eco-nomic Review, 75(3), pp. 424-440.

[13] Liebowitz, S. J. and S. Margolis, 2002, Network E¤ects, in: Cave M., Majumdar S., Vogelsang, I. (Eds.),Handbook of Telecommunications Economics, North Holland, Amsterdam.

[14] Mathewson, F. G. and R. A. Winter, 1987, The Competitive E¤ects of Vertical Agreements: Comment,American Economic Review, 77(5), pp.1057-62.

[15] Ordover, J. A., G. Saloner and S. C. Salop, 1990, Equilibrium Vertical Foreclosure, The AmericanEconomic Review, 80(1), pp. 127-142.

16

Appendix

A Computation of payo¤s for a given wA, wB, and �.

A.1 Subgame (1)

In this section I compute the pro�ts of the ISPs when there is no vertical integration. There are four possiblecases regarding the upstream network choice of the ISPs: (UA, UA); (UA, UB); (UB , UA); and (UB , UB),where the initial U stands for D1�s, the second U for D2�s choice.

A.1.1 Downstream competition when both downstream connect to the same upstream net-work

Consider the case in which both ISPs connect to UA. This case yields perfect connection between thedownstream �rms. The utility to the end user at location x from having access to the end-service is

U =

�v � t jx� 0j+ � (�A + q1 + (1� q1) + ��B)� p1 if connects to D1;v � t j1� xj+ � (�A + q1 + (1� q1) + ��B)� p2 if connects to D2:

(A.1.1a)

The marginal consumer who is indi¤erent between connecting to D1 and D2 satis�es �tx�p1 = �t+ tx�p2:Therefore, D1 has a market share of x = 1=2 + (p2 � p1)=2t; and its pro�t is (p1 � wA)(1=2 + (p2 � p1)=2t):Similarly D2 has a market share (1�x) = 1=2� (p2� p1)=2t, and its pro�t is (p2�wA)(1=2� (p2� p1)=2t):Let �IJi denote pro�ts of Di when D1connects to UI and D2 connects to UJ . The Nash equilibrium of

the subgame yields p1 = p2 = wA + t, which implies �AAA = wA, �AAB = 0, and �AA1 = �AA2 = 12 t:

Now consider the case in which both downstream �rms connect to UB . This case yields perfect connectionbetween the downstream �rms. The utility of the end user at location x from having access to the end-serviceis

U =

�v � t jx� 0j+ � (�B + q1 + (1� q1) + ��A)� p1 if connects to D1;v � t j1� xj+ � (�B + q1 + (1� q1) + ��A)� p2 if connects to D2:

(A.1.1b)

The symmetric case to the initial one yields p1 = p2 = wA + t, which implies �BBA = 0, �BBB = wB , and�BB1 = �BB2 = 1

2 t:

A.1.2 Downstream competition when each downstream �rm connects to a di¤erent upstreamnetwork

Now consider the case in which D1 connects to UA and D2 to UB . The utility to the end user at location xfrom having access to the end-service is

U =

�v � t jx� 0j+ � (�A + q1 + �(�B + (1� q1)))� p1 if connects to D1;v � t j1� xj+ � (�(�A + q1) + �B + (1� q1))� p2 if connects to D2:

(A.1.2)

The marginal consumer who is indi¤erent between connecting to D1 and D2 satis�es

�tx+ ��A + �x+ ���B + ��(1� x)� p1 = �t+ tx+ ���A + ��x+ ��B + �(1� x)� p2:D1 has a market share of

x =1

2+

'�

2 (t� ') +p2 � p12 (t� ') :

Pro�ts of D1 and D2 are

�AB1 = (p1 � wA)�1

2+

'�

2(t� ') +p2 � p12(t� ')

�;

and

�AB2 = (p2 � wB)�1

2� '�

2(t� ') +p1 � p22(t� ')

�;

17

respectively. The Nash equilibrium of the subgame yields p1 = (t � ') + (2wA + wB) =3 + '�=3; andp2 = (t� ') + (2wB + wA) =3� '�=3; which implies

�AB1 =(3(t� ')� w + '�)2

18(t� ')

and

�AB2 =(3(t� ') + w � '�)2

18(t� ') :

Similarly, when D1connects to UB and D2 to UA, pro�ts of D1 and D2, are

�BA1 =(3(t� ') + w � '�)2

18(t� ') ;

and

�BA2 =(3(t� ')� w + '�)2

18(t� ') :

respectively.

A.2 Subgame (2)

(I) UB integrates with D2, and D1 connects to UB: The utility to the end user at location x fromhaving access to the end-service is de�ned as in equation (A.1.1b). The Nash equilibrium of the subgameis computed similarly, except that the UB charges zero price to D2, and charges wB to D1. Equilibriumprices are pI2 = 1 + wB=3; and pI1 = 1 + 2wB=3, which implies for all wB 2 [0; 3], �IA = 0 (as UA sells noconnectivity in this market), �IB = (3� wB)wB=6, �I1 = (3�wB)2=18, �I2 = (3+wB)2=18. The integrated�rm�s pro�t is composed of �I2 and the pro�t obtained by selling connectivity to D1, hence obtains

�IB +�I2 =

(3 + wB)2

18+(3� wB)

6wB .

(II) UB integrates with D2, and D1 exits: Consider the case under which UB does not sell connectivity(either charges a too high price, or refuses to supply) to D1, and D1 gains a negative market share if it buysconnectivity from UA: In this case, D1 withdraws from the market, and the integrated upstream networkbecomes a monopolist in the downstream market. Therefore, �II1 = 0, �

IIA = 0, and the integrated upstream

network obtains monopoly pro�ts denoted with �IIB + �II2 = �(v; �; �B) > 0, where �(v; �; �B) > 0; is

increasing in all its arguments.

(III) UB integrates with D2, and D1 connects to UA: The utility of the end user at location x fromhaving access to the end-service is de�ned as in equation (A.1.2). The Nash equilibrium of the subgame iscomputed similarly, except that the UB charges zero price to D2, and UA charges wA to D1. Equilibriumprices are pIII1 = (3(1� ') + '� + 2wA) =3 and pIII2 = (3(1� ')� '� + wA) =3, which implies that for allwA 2 [0; 3(1� ') + '�],

�IIIA =

�3(1� ') + '� � wA

6(1� ')

�wA;

�IIIB = 0;

�III1 =(3(1� ') + '� � wA)2

18(1� ') ;

and

�III2 =(3(1� ')� '� + wA)2

18(1� ') :

18

A.3 Subgame (3)

(IV) UA integrates with D1, and D2 connects to UB: The utility to the end user at location x fromhaving access to the end-service is de�ned as in equation (A.1.2). The Nash equilibrium of the subgame iscomputed similarly, except that the UA charges zero price to D1, and UB charges wB to D2. Equilibriumprices are pIV1 = (3(1� ') + '� + wB) =3 and pIV2 = (3(1� ')� '� + 2wB) =3, which implies that for allwB 2 [0; 3(1� ')� '�],

�IVA = 0;

�IVB =

�3(1� ')� '� � wB

6(1� ')

�wB ;

�IV1 =(3(1� ') + '� + wB)2

18(1� ') ;

and

�IV2 =(3(1� ')� '� � wB)2

18(1� ') :

(V) UA integrates with D1, and D2 exits: Consider the case under which UA does not sell connectivity(either charges a too high price, or refuses to supply) to D2, and D2 gains a negative market share if itbuys connectivity from UB : In this case, D2 withdraws from the market, and the integrated downstream�rm becomes a monopolist in the downstream market. Therefore, �VB = 0 (as UA sells no connectivityin this market), �V2 = 0, and the integrated upstream network obtains monopoly pro�ts denoted with�V1 +�

VA = �(v; �; �A), where �(v; �; �A) > 0; and is increasing in all its arguments.

(VI) UA integrates with D1, and D2 connects to UA: The utility to the end user at location xfrom having access to the end-service is as in equation (A.1.1a). The Nash equilibrium of the subgame iscomputed similarly, except that the UA charges zero price to D1, and charges wA to D2. Equilibrium pricesare pVI1 = 1+wA=3; and pVI2 = 1+2wA=3, which implies �VIA = (3� wA)wA=6, �VIB = 0, �VI1 = (3+wA)

2=18,and �VI2 = (3�wA)2=18 for all wA 2 [0; 3] for wA � 3: The integrated �rm�s pro�t is composed of �VI1 andthe pro�t obtained by selling connectivity to D2, hence obtains

�VIA +�VI1 =(3 + wA)

2

18+(3� wA)

6wA.

A.4 Subgame (4)

The utility to the end user at location x from having access to the end-service is de�ned as in equation(A.1.2). The Nash equilibrium of the subgame is computed similarly, except that the UA charges zeroprice to D1, and UB charges zero price to D2. Equilibrium prices are pVII1 = (3(1� ') + '�) =3 and pVII2 =(3(1� ')� '�) =3, which implies �VIIA = 0, �VIIB = 0,

�VII1 =(3(1� ') + '�)2

18(1� ') ;

and

�VII2 =(3(1� ')� '�)2

18(1� ') :

19

B Summary of Payo¤s

�A �1 �B �2

(I) 0 (3�wB)218

(3+wB)2

18 + (3�wB)wB6

(II) 0 0 �(v; �; �B)

(III) (3(1�')+'��wA)wA6(1�')

(3(1�')+'��wA)218(1�')

(3(1�')�'�+wA)218(1�')

(IV) (3(1�')+'�+wB)218(1�')

(3(1�')�'��wB)wB6(1�')

(3(1�')�'��wB)218(1�')

(V) �(v; �; �A) 0 0

(VI) (3+wA)2

18 + (3�wA)wA6 0 (3�wA)2

18

(VII) (3(1�')+'�)218(1�')

(3(1�')�'�)218(1�')

C Proofs of Claims and Lemmas

.Proof of Claim 1. In subgame (2) we have �IIIA > �IIA = 0 for wA 2 (0; 3 (1� ') + '�) and in

subgame (3) we have �IVB > �VB = 0 for wB 2 (0; 3 (1� ')� '�), which implies that given the integratedupstream network refuses to deal with the non-integrated downstream �rm, the non-integrated upstreamnetwork is strictly better o¤ by selling connectivity to the non-integrated downstream �rm.Proof of Lemma 3.

(i) One can verify that the gross pro�ts of the small integrated upstream network,��4B +�

42

�; de�ned

in (5) is decreasing with ', and hence is increasing with �, which implies that it has no incentivesto degrade quality of interconnection. Replacing ' = � (1� �) in (4) one can show that the sign of@ (�A +�1) =@� is the same as (3� 3� + 3�� � 2� + �� � ���), and the latter is decreasing with �,and is positive for � = 0. Therefore, there exists a treshold � above which UA�s gross pro�ts aredecreasing with the quality of connectivity.

(ii) The optimal connectivity for the large integrated upstream network is de�ned by the following problem,

max�2[0;1]

n�4A +�

41 � (1� �)

2=2o

(8)

where @��4A +�

41

�=@� is decreasing with � , and @

��4A +�

41

�=@� > 0 for � = 0, and cost of degrading

quality is independent of �. Therefore there exists a treshold beta, b� such that for all � � b� we have� = 1 as the solution to the problem de�ned in (8). We have

@��4A +�

41 � (1� �)

2=2�

@�

�������=1

= � (3� 2�) =6

which implies that for all �; optimal connectivity is perfect for � = 1:5. Furthermore, one can showthat the sign of @

��4A +�

41

�=@� is determined by the sign of (3� 3� + 3�� � 2� + �� � ���) which is

decreasing in �. This also implies that for all � < 1:5 connectivity is also perfect for all �. Therefore,the treshold is de�ned as b� = 1:5.

(iii) For all � > b�, equilibrium quality of interconnection belongs to (0; 1], as connectivity level never getszero, since

@��4A +�

41 � (1� �)

2=2�

@�

�������=0

= 0

implies � = 3 (1� �)�� +

p5� 2� � 1

�=� (2� �), which can not be true for � 2 (0; (1� �) =�). Fur-

thermore, the sign of@2��4A +�

41 � & (�)

�=@�@�

20

is determined by the sign of �� (1� �) (� � 3) (2 + �� � �)� 3, which is decreasing with �; and is neg-ative for � = 0. This demonstrates that @2

��4A +�

41 � & (�)

�=@�@� < 0, and the optimal connectivity

is decreasing with the size of the installed-base di¤erence.

Proof of Lemma 5. Let �3A (�; �; �) = �V IA + �V I1 � & (�). Substituting w3A which is de�ned in theProof of Lemma 4 into �3A (�; �; �) yields

+2� (1� �)

�6��� + 9�� + ��2� + 6� � ��2 � 6�� � 9�

�+ 18

18 (1� � (1� �)) (9)

�3p(1� � + ��) (3� 3� � �� + 3�� + ���)

18 (1� � (1� �))

� (1� �)2

2:

(i) UB obtains zero gross pro�ts in this case, and hence, has no incentives to degrade quality of connectivity.

(ii) �V IA (�; �; �) is quadratic in � (�rst increasing and then decreasing in �), and @�V IA (�; �; �) =@� isdecreasing in both � and _�. Therefore, if there exists � such that

@�3A��; �; 0

�@�

������=0

= 0,

then for all � > � we have @�3A (�; �; �) =@� < 0 for all � and �. And hence for all � > � equilibriumlevel of connectivity is zero. We have

@�3A (�; �; 0)

@�

�����=0

= (1� �)� �

4p(1� �)

;

and hence � = 0:69:

(iii) Since @�3A (�; �; �) =@� is decreasing in beta, and � 2 (0; (1� �) =�) ; if there exists � such that

@�3A (�; �; (1� �) =�)@�

�����=0

= 0;

then for all � < �, we have@�3A (�; �; �)

@�

�����=0

> 0

for all �: Since pro�ts are quadratic in �; then there exists � 2 (0; 1) such that pro�ts are maximized.We have

@�3A (�; �; (1� �) =�)@�

�����=0

=4p(1� �) (5� 4�) (1� �)2 � 6 (� + 1) (1� �)2

36 (1� �)52

;

and hence � = 0:48. Finally, I show that optimal connectivity is decreasing with both � and �. At theoptimum, we have

@�3A (��; �; �)

@�= 0:

Since @�3A (�; �; �) =@� is decreasing in �, for �0 > �, we have

@�3A���; �0; �

�@�

< 0:

As �3A (�; �; �) is concave, then �� ��0; �� < �� (�; �). The same proof applies for �.

21

Proof of Lemma 6. I have shown that under (4) connectivity is perfect if � < 1:5, and we know thatwe never observe perfect connectivity under (3). Therefore, for all � < b� = 1:5 we have �3 < �4. For � � b�,equilibrium connectivity is decreasing with � in both subgames. One can verify that, �3

�b�� < �4 �b�� and�3 ((1� �) =�) < �4 ((1� �) =�), for all � 2

�0; 1=

�1 + b���, which concludes that �3 < �4 for all � � b�.

Proof of Claim 3. I begin with showing that UB strictly prefers selling connectivity to the non-integrated downstream �rm, that is, its payo¤ under (I) is strictly greater than its payo¤ under (III) .Under imperfect connectivity (' > 0), for a given wB 2 [0; 3) and a given wA 2 [0; 3(1� ') + '�) (so

that D1 obtains a positive market share when it buys connectivity from any of the upstream networks) D1buys connectivity from UA if �III1 (wA) � �I1 (wB), which implies

wA � 3(1� ') + '� � (3� wB)p(1� '); (10)

and buys from UB otherwise. When D1 buys connectivity from UA; the integrated upstream network obtains

�IIIB +�III2 = 0 +(3(1� ')� '� + wA)2

18(1� ') ;

which is increasing with wA (since a higher connectivity price at which connectivity is sold to the non-integrated downstream �rm implies higher pro�ts for the integrated downstream �rm). Given the constraintde�ned with inequality (10), the maximum level of pro�ts the integrated �rm can obtain under (III) is foundby replacing wA = 3(1� ') + '� � (3� wB)

p(1� ') in its pro�ts, and is�

6 (1� ')�p(1� ') (3� wB)

�218 (1� ') :

When wA is su¢ ciently high (inequality (10) holds in the opposite direction) so that the integrated upstreamnetwork sells connectivity to D1, and it obtains

�IB +�I2 =

(3 + wB)2

18+(3� wB)2

6wB :

One can verify that for all wB 2 (0; 3), we have

(3 + wB)2

18+(3� wB)2

6wB >

�6 (1� ')�

p(1� ') (3� wB)

�218 (1� ') ;

since it simpli�es to

2'� 16(3� wB)

�w2B � 3wB + 4

�+2

3

p(1� ') (3� wB) > 0;

and this is true for ' 2 (0; 1) and wB 2 (0; 3). This concludes that the integrated small upstream networkprefers to sell connectivity to the non-integrated downstream �rm. The non-integrated large upstreamnetwork also prefers to sell connectivity to the non-integrated downstream �rm , since �IIIA > 0 = �IA forany wA > 0 such that inequality (10) holds.Proof of Lemma 7. Claim 3 shows that in this subgame the upstream networks compete for pro-

viding access to the non-integrated downstream �rm. Let m(�; �) = 3p(1� � (1� �))� 3 (1� � (1� �))�

� (1� �)�, and �rst assume that the installed-base di¤erence is su¢ ciently small so that price competitionbetween the upstream networks derive prices to wA = 0, and

wB =m(�; �)p

(1� � (1� �)): (11)

22

However, if the installed-base di¤erence is su¢ ciently high, we can have m (�; �) < 0, and hence, wB de�nedin (11) negative. This implies that UB can not attract D1 with any non-negative price, sets wB = 0, andUA sells connectivity at the highest price de�ned with the constraint in (10). Formally, we have

w2B =

�m(�; �)=

p(1� � (1� �)) for m(�; �) � 0

0 otherwise;

and

w2A =

��m(�; �) for m(�; �) < 00 otherwise

Finally, D1 buys connectivity from UB if m(�; �) � 0, and buys from UA otherwise.

(i) Remark that m (�; �) is decreasing with �; and is negative for � = (1� �) =� for all � < 0:4, and it ispositive for � = 0. Therefore there exists a treshold � above which m (�; �) is negative. Since m (�; �)is decreasing with � for all � > 1:5 (which necessarily means that we have � < 0:4) this treshold isdetermined with m (�; �) = 0 with � ! 0 , and hence b� = 1: 5. The equilibrium of this subgame ischaracterized by (I) for all � > b�:

(iii) Now I show that for all � � 1:29 we havem (�; �) > 0 for all �, and hence the equilibrium of this subgameis characterized with (I). For � < 1:5; we have m (�; �) decreasing with �. Therefore, minimum valueof m (�; �) is obtained for � = 0. It can be veri�ed that for � = 0, we have m (�; �) > 0 for all � � 1:29,and hence, e� = 1:29. It remains to show that for � 2 (1:29; 1:5] the equilibrium is characterized with(I) for some values of �.

(ii) Finally for � 2 (1:29; 1:5], for all � there exists a � (�; �) 2 (0; 1) for which we have m (�; �) > 0, andthis holds for higher level of connectivity except � = 1. One can show that for a given �, higher �implies a higher � (�; �) that is necessary to have m (�; �) > 0.

Proof of Lemma 8.

(ii) We know that for all � > b�, we have m (�; �) < 0 for all �. In this range of installed-base di¤erence,UB has no incentives to degrade connectivity, since its gross pro�ts

�IIIB +�III2 =

�6 (1� � (1� �))� 3

p1� � (1� �)

�218(1� � (1� �)) ;

is increasing with �; for all � > 0:75 (and when � > b� we have � < 0:4), therefore we can conclude that@��2B +�

22

�=@� > 0 for all � > b�. For those range of installed-base di¤erence, net pro�ts of UA are

�2A =

�p1� � (1� �)

��3 + � (1� �) (� � 3)� 3

p(1� � (1� �))

�2(1� � (1� �)) � (1� �)

2

2:

Remark that �A = 0 for � = 1, and is positive for � = 1� �. We have

@2��2A�

@�@�= � (2� � (1� �)) �

4 (1� � (1� �))p1� � (1� �)

< 0:

Hence, optimal connectivity is decreasing with the installed-base di¤erence.

(iii) We know that for all � � b�, we have m (�; �) > 0 for all �. In this range of installed-base di¤erence, UAhas no incentives to degrade connectivity, since its gross pro�ts are zero, independent of the imperfectlevel of connectivity. We have gross pro�ts of UB increasing with wB for all � � b�, and we have,

@2 (wB)

@�@�=

� (2� � (1� �))2�p

1� � (1� �)�(1� � (1� �))

> 0:

23

Since the cost of degradation is independent of � and �, optimal connectivity is increasing with �.Finally, for � 2 (1:29; 1:5] the higher the externalities are, the more UB needs to keep quality ofconnectivity high so that m (�; �) remains positive. One can check that m (�; �) > 0 for � = 1 � �,and that UB obtains positive pro�ts. Note also that , unless the connectivity is degraded su¢ ciently,we can not have m (�; �) < 0. For UA when � 2 (1:29; 1:5] marginal cost of degrading quality ofconnectivity exceeds the marginal gain of degrading it, and hence, it does not engage in degradation.

(i) (ii) and (iii) together imply that the quality of interconnection is always imperfect at the equilibriumof this subgame.

Proof of Proposition 3. I begin with Stage two, where UB decides for integration. Given that UAhas integrated with D1 in the �rst stage, UB integrates with D2 if and only if

�4B +�42 > �

3B +�

32:

Remark that �4B +�42, as de�ned in equation (5), has the same functional form as �3B +�

32 de�ned in (7).

However, in each subgame equilibrium connectivity is di¤erent, and as is shown in Lemma 6, equilibriumconnectivity is always strictly higher in subgame (4) than in subgame (3). This shows that �4B + �

42 >

�3B +�32, and hence UB engages in counter-merger whenever it observes integration in the �rst stage.

Given that UA has not integrated in the �rst stage, UB integrates if and only if

�2B +�22 > �

1B +�

12:

On the one hand, for all � > b�, we have�2B +�

22 = 0 +

�6 (1� ')� 3

p(1� ')

�218(1� ') ;

and �2B +�22 > �

1B +�

12 implies p

(1� ')� (1� ') < 0;

which does not hold for any ' 2 [0; 1). This shows that when � > b�, given that UA has not integrated, UBdoes not integrate. On the other hand, for all � � b�, we have

�2B +�22 > �

1B +�

12

if9p(1� ') + 6 (1� ') + 2'� > 0

which is true for all ' 2 [0; 1). This demonstrates that when � � b�, given that UA has not integrated, UBintegrates with D2:Going backwards, I now study UA�s decision for integration. If UA integrates with D1, it gets

�4A +�41 =

(3(1� ') + '�)2

18(1� ')

If it does not integrate it gets

�2A +�21 =

(3(1� ') + '�)2

18 (1� ')

if � � b�, and gets�1A +�

11 =

�2�2

2+1

2

24

if � > b�. We have perfect connectivity under (4), when � � b� and imperfect connectivity under (2), andhence we have �2A+�

21 < �

4A+�

41 for all � � b�, and hence UA integrates when the size of the installed-base

di¤erence is in this range. And for all � > b�, we have �1A +�11 > �4A +�41, since(3(1� � (1� �)) + � (1� �)�)2

18(1� � (1� �)) � (1� �)2

2<�2�2

2+1

2

is true for all � > b�. This concludes that if � > b�, UA does not integrate.

25