Competition, Market Coverage, and Quality Choice in Interconnected Platforms

Transcription

1 Competition, Market Coverage, and Quait Choice in Interconnected Patforms Pau Njoroge Asuman Ozdagar Gabrie Weintraub Nicoas Stier ABSTRACT We stud duopo competition between two interconnected Internet Service Providers (ISP that compete in quait and prices for both Content Providers (CP and consumers. We deveop a game theoretic mode using a two-sided market framework, where ISP s are modeed as interconnected patforms with quait bottenecks; a consumer on a ow quait network accessing content on a high quait patform experiences ow quait. Patforms first pick quait eves from a bounded interva and in the subsequent stages compete in prices for both CP s and consumers. CP s are heterogenous in content quait which is uniform distributed between [γ 1, γ. We first estabish the existence of a price subgame perfect equiibrium (SPE given an asmmetric pair of patform quait choices. We show that the higher the asmmetr the more ike the CP market is to be uncovered if the average content quait (represented b γ is ow. In contrast, if γ is high then the market is awas covered. We then show that a SPE for the whoe game exists and characterize a the equiibrium choices of the quait game. In particuar, we show that the equiibria invove either maxima differentiation or partia differentiation depending on γ. Moreover, we characterize the resuting market configurations in the fina stage and show that the depend on γ and the asmmetr between patforms represented b the ratio of the quaities, I. 1. INTRODUCTION Consumers and Content providers (CP base their choice of ISP not on on prices but aso on other features such as Department of Eectrica Engineering and Computer Science, Massachusetts Institute of Technoog. Department of Eectrica Engineering and Computer Science, Massachusetts Institute of Technoog. Decision, Risk, and Operations Division, Coumbia Business Schoo Decision, Risk, and Operations Division, Coumbia Business Schoo speed of access, specia add-ons ike spam bocking and virus protection [2. These extra features can be abstracted as quait. ISP s have the abiit to infuence quait through upgrade of infrastructure, offering of enhanced services or even traffic-management [4. A ke question that arises in the provision of network access is what quait profit maximizing ISP s wi offer in a competitive environment and its concomitant effect on market coverage. In this paper we stud this question using a mode of duopo competition in interconnected two-sided-market patforms in the presence of quait choice. We deveop a game-theoretic mode where ISP s are represented as profit maximizing twosided interconnected patforms that choose quait eves and then compete in prices for both CP s and consumers. In addition, we mode quait of service effects through a botteneck effect. Whie there is much work on competition modes between two-sided patforms (see for exampe, [9, 1, 10, 6, most existing work focuses on determinants of pricing. These works do not address interconnection between patforms, endogenous quait choice b patforms and market coverage. In this paper we consider these effects in tandem. Our objective is to understand what strategic quait choices interconnected patforms make and their effects on market structure. Our mode consists of two interconnected patforms, a heterogenous mass of CP s, and a heterogenous mass of consumers. Patforms provide connection services to consumers and CP s and charge a fat access fee to both. We mode the interaction between ISP s and end-users 1 as a six-stage game. In the first stage, patforms simutaneous pick a quait eve from a bounded interva. Second, the simutaneous compete in CP prices. Third, the CP s decide which patforms, if an, to connect to. Fourth, the patforms simutaneous compete in consumer prices. In the fifth stage, consumers decide what patforms to join. In the ast stage Consumers decide which CP s to patronize. We first derive resuts reating to the price competition SPE between the patforms given exogenous quait choices and then use these resuts to sove for the quait choice SPE. The resuts show that given an asmmetric quait pair, a subgame perfect equiibrium (SPE in both consumer and CP prices exists. In addition, we show the reationship between the ISPs quait choices and market structure on the 1 End-users refers to both CP s and Consumers

2 CP side. Specifica, we show that the resuting market configuration depends on the average content quait characteristic, γ, of the CP s and the asmmetr between the patforms. We show that if γ is ow, then an increase in the quait ratio, I, defined as the ratio between patform quaities, increases the patforms market power resuting in an uncovered content provider market. On the other hand, when γ is high, the reative difference in content quait amongst CP s diminishes and content providers make simiar connection decisions regardess of quait ratio eves, i.e., the a join the high quait network. The above resuts suggest that patforms which are high differentiated in quait pose a barrier to entr for CP s that have ow content quait. Indeed, the high asmmetr in quait induces market power on the patforms which enabes them to raise prices and thus excude ow quait content CP s from the market. Our second set of resuts pertain to the quait choice SPE. We show that if we assume a fixed cost (or costess quait investment mode, then a SPE in quait choice stage exists. We characterize a such equiibria and show that depending on γ, one of the foowing three tpes of equiibrium exists in the fina stage of the game: amaxima differentiation equiibrium which invoves one patform picking the highest quait and the other the owest, bpartia differentiation equiibrium where one patform picks the highest quait and the other chooses a proportion of the highest quait that depends on γ, cpartia differentiation where one patform picks the owest quait and the other picks some positive fraction of the highest quait. In addition, we aso give the market configurations that arise at the various SPE. In particuar, we show that the CP market is covered when γ is high or when it is ow and the bounded interva from which quait is chosen is sma. In the former case the heterogeneit between CP s is ow. Therefore, if one CP joins a patform a the others make a simiar decision as previous discussed and the market is covered. In the atter case, though the heterogeneit is high, the bounded interva from which quait is chosen is sma and thus the eve of differentiation is imited. This impies that price competition resuts in ow prices such that the market is covered. When γ is ow and the bounded interva from which quait is chosen is arge, the CP market is uncovered. In this case the eve of differentiation is arge. Patforms exercise market power charging higher prices which eads to ess enroment of CP s and an uncovered market. Other than the papers cited above on two sided-markets, our paper buids on contributions to the industria organization iterature on price competition and quait choice in vertica differentiated markets, [5, 13, 3,, 11. Somewhat reated to our work are the competition modes anazed in Mussachio [et a [8. Athough the stud the effect of a network regime on CP investments and network quait, their neutra and non-neutra modes can be independent viewed as modeing competition between network providers where quait choice is endogenous. In both of these modes, the providers are assumed to be smmetric. Moreover, consumers and CP s are viewed as homogenous and the user base on both sides of the market is fixed. The resuting equiibrium in network quait is therefore smmetric with a network providers choosing the same quait. Issues of market coverage are aso not addressed. The distinguishing feature our work is to consider the strategic interactions between quait-picking patforms and heterogenous end-users and their effect on market entr b the CP s 2 The rest of this paper is organized as foows. In Section 2 we present the mode. In Section 3 we anaze the mode soving for the SPE of this game as we as discussing our findings. In Section 4 we concude. Due to space imitations certain proofs have been omitted and those that are essentia have been reegated to the Appendix. 2. MODEL We consider two patforms denoted b α and β, and a continuum of consumers and content providers with a unit voume. Let α and β be the quait-of-service chosen b patforms α and β, respective. We represent the quait of a CP j b the scaar γ j. We assume γ j is uniform distributed with support [γ 1, γ where γ 1. We aso assume γ j are independent identica distributed random variabes across the popuation of content providers. Let φ : [0, 1 {α, β} and φ : [0, 1 {α, β} be mappings from the space of consumers and providers respective to the set of patforms. A consumer i on a patform φ(i {α, β} connecting to a CP j on patform φ(j {α, β} receives utiit, u ij ( φ(i, φ(j, γ j = γ j + min{ φ(i, φ(j }. (1 The consumer utiit impies that a consumer on a high quait patform, connecting to a content provider present on a high quait patform, receives more utiit than if he connected to a content provider of the same quait on the ower quait patform. In essence, utiit captures the fact that service quait depends on the botteneck, [12. A consumer i on patform φ(i connects with CP j if and on if u ij 0. Let F i( φ(i, r α, r β, γ be the quait perceived b consumer i when he joins patform φ(i. Forma, F i ( φ(i, r α, r β, γ = 1 [ E max{u ij ( φ(i, φ(j, γ j, 0} dj. 0 Here r α and r β are the masses of content providers that join patform α and β respective. We assume that consumers have heterogenous preferences represented b a taste parameter θ i which is uniform distributed in the interva [0,1. A consumer i perceives the quait of patform φ(i as his expected utiit, F i ( φ(i, r α, r β, γ. In addition, each consumer has a reservation utiit R. The prices charged b the patforms are p α and p β for patforms α and β respective. Each consumer connects to at most one patform but once connected has access to a content due to the interconnection of the patforms. Therefore, the net utiit of a consumer i connecting to patform φ(i is given b, U i(φ(i = max{r + θ if i( φ(i,rα,r β,γ, p φ(i, 0}. Consumers prefer the patform with the higher perceived quait, ceteris paribus. 2 Market entr is proxied b market coverage.

3 Patforms aso charge a fixed connection fee w α and w β to CP s that connect to them. We assume that CP s make revenues b seing advertising. Let q α and q β denote the mass of consumers ocating on patforms α and β respective. Without oss of generait we assume α β. We aso assume that φ(i > for some > 0, where is some minimum quait eve that patforms have to guarantee. The utiit v j of a CP j is defined to be his profit where, V ( = v j = V (γ j, α, β, q α, q β w φ(j, (2 { g(γ j, α q α + g(γ j, β q β, if φ(j = α, g(γ j, β q β + g(γ j, β q α, if φ(j = β. Here g(γ j, φ(i is a function that represents the advert price and is increasing in both parameters; CP j gets a higher advert price for having a higher content quait and aso for ocating on a patform with higher quait. The function V (γ j, α, β, q α, q β represents the gross revenue earned b a CP j. This function depends on which patform the CP joins as we as the number of consumers on the other side of the market. In particuar, if a CP j joins the high quait patform, it is abe to command a higher advert price for connections arising from consumers on that patform. If a CP joins the ower quait patform its advert price is the same for the two patforms, i.e, the advert price depends on the patform that acts as the botteneck. Fina we consider the paoff functions of the patforms: we assume that patforms incur no cost(or a fixed cost in choosing the quait eve. The paoff of patform α, which we denote b π α, is given b π α = p αq α + w αr α, where q α is the mass of consumers attached to patform α and r α is the mass of CP s attached to patform α. The paoff for patform β is simiar. The mode we have outined corresponds to a dnamic game with the foowing timing of events: 1 Quait Choice Stage: Patforms α and β simutaneous choose quait-of-service from the interva [,. 3 2 Pricing Decisions: Patforms simutaneous choose connection fees w α and w β. 3 Connection Decisions: CPs decide which patform to join. 4 Pricing Decisions: Patforms simutaneous choose prices p α and p β. 5 Connection Decisions: Users decide which patform to join. 6 Consumption Decisions: Consumers decide which CPs to connect. 3 represents a minimum quait that a patform is required to maintain. represents the maximum quait that can be achieved, for instance due to technoogica imits. We sove this game b considering its subgame perfect equiibrium (SPE, which we find using backward induction. Steps 4-6 are simiar to a pricing game with vertica differentiation; steps 1-3 are simiar to a quait choice and pricing game with vertica differentiation. 3. MODEL ANALYSIS Let I = {α, β, [0, 1 j, [0, 1 i} denote the set of paers in the muti-stage game, where α and β are the patforms, [0, 1 j and [0, 1 i are the continuum of content providers and consumers respective, both with unit voume. We denote the information set at stage k of the game for a paer i I b h k i. Let the set of actions avaiabe to a paer i at stage k and information set h k i be denoted as A i (h k i. 3.1 Consumption Decisions. We begin b anazing the ast stage of the game, i.e, the consumption decisions of the consumers. On the consumers make a move in this stage. The choice set of a consumer i [0, 1 i given an information set h k i is A i(h k i 2 [0,1 j. A consumer i on a patform φ(i {α, β} accessing content of a CP j on patform φ(j {α, β} receives utiit u ij which is defined in Eq. (1. As previous discussed, this impies that a consumer connecting to a higher quait patform gets more utiit when he accesses content providers on that patform, compared to when he connects to the same content providers whie connected to the ower quait patform. Consumer i on patform φ(i connects with CP j whenever u ij 0 which impies that i connects with CP j if γ j min{ φ(i, φ(j }. Since γ j > 0, whenever a consumer joins an of the patforms he wi connect to a content providers on that patform and those on the other patform. 3.2 Consumer Patform Connection Decisions. In this stage the consumers are the on movers and the decide which patforms to join. The choice set of a consumer i given an h k i is A i (h k i = {α, β}. Through his information set, a consumer has knowedge of the number of content providers on each patform, the prices that the patforms charge and the quait eve of each patform. Each consumer i soves the foowing utiit maximization probem, maximize s.t. U i (φ(i φ(i {α, β}. A consumer that does not join an patform receives a utiit of zero. We proceed next to give the demand functions faced b each patform based on consumer choices in this stage. We first make the foowing assumption on the reservation price which we invoke through out the anasis of this paper. Assumption 1. R is arge enough that the consumer market is covered. Let α > β, then it foows that F i ( α, > F i ( β,. If θ p α p β F i ( α, F i ( β, consumers with a taste parameter θ i θ join the patform with the higher perceived quait, F i ( α,, since θ i F i ( α, p α θ i F i ( β, p β if and on if θ i θ. Those whose taste parameter θ i < θ wi join patform β if

4 and on if θ i p β R F i ( β. From Assumption 1 we can deduce, that if R is arge enough then a consumers get a positive utiit b participating in the market. In such a market, the demands for the patforms are characterized as foows, q β (p α, p β = ( p α p β, F i ( α, F i ( β, ( q α(p α, p β = 1. p α p β F i( α, F i( β, We wi show in the next stage that if α = β then an aocation of demand across patforms is possibe at the resuting price equiibrium. 3.3 Patform Pricing Decisions for the Consumer Side. In this stage of the game the patforms are the on movers and the decide what prices to charge to the consumers. The choice set of patform i {α, β}, given an h k i, is A i(h k i = p i R. Thus the patforms simutaneous decide what prices p α and p β to charge to consumers. Through his information set, a patform has knowedge of the number of content providers on each patform and the quait eve of each patform. Profit for patform i is given b, π i = p iq i + w ir i, where w i is the price charged to content providers and r i is the mass of content providers on patform i. The demand for patform i denoted b q i is defined b the set of consumers who maximize their utiit when the join patform i. The Nash equiibrium in this price subgame depends on the information set h k i. In particuar, if h k i is such that α > β it can be shown that, p β = 1 (F 3 i( α, F i ( β,, and p α = 2 (F 3 i( α, F i ( β,, and the consumer demands addressed to the patforms at this equiibrium are q α = 2 and q 3 β = 1. If 3 hk i is such that α = β then F i( α, = F i( α,. A Bertrand competition ensues and the resuting subgame Nash equiibrium has p α = p β = 0. The consumer demands addressed to the patforms at this equiibrium price are indeterminate, i.e an aocation such that q α + q β = 1, is a soution to the Bertrand game. 3.4 Content Provider Connection Decisions Given the quait of service offered b patforms α and β and the prices w α and w β, the content providers decide on which patform to ocate. The choice set of a CP j given an h k j is A j (h k j = {α, β}. As mentioned in section 3, γ j is uniform distributed with a support [γ, γ 1 where γ 1. The utiit v j gained b a content provider when he joins a patform is given b Eq. (2. A CP s utiit is zero if he doesn t join an patform. In this stage, CP s take the investment(choice in quait as given. Moreover, the anticipate the mass of consumers on each patform q α and q β. Let g(γ j, φ(j = γ j φ(j, a CP j perceives the quait of patform α to be αq α + β q β and that of patform β to be β q α + β q β. For the rest of this section we assume α > 4 β. A CP j maximizes the utiit v j and is indifferent between the two patforms if and on if γ j( αq α + β q β w α = γ j( β q α + β q β w β. Let γ j = w α w β q α( α β, then the content providers 4 We wi show ater that α = β is not a SPE with quait exceeding γ j join the high quait patform α. Those whose content quait is ower than γ j, but arger than w β /( β (q β + q α, join the ower quait patform β. The others do not join an patform. Since α > β there s a possibiit of patform α preempting the market with a imit price w α = w β + (γ 1(q α ( α β. The mass of content providers r α (r β is defined b those content providers who maximize v j when the join patform α(β. It foows that given the n-tupe (γ, α, β, w α, w β, there are four possibe market configurations that ma arise depending on the demands addressed to the patforms. We next describe the market configurations of content providers at different CP prices 1. Uncovered Market: r α (w α, w β < 1, r β (w α, w β = 0. We denote this configuration as CI. 2. Uncovered Market: r α(w α, w β + r β (w α, w β < 1, 0 < r α (w α, w β < 1, 0 < r β (w α, w β < 1. We denote this configuration as CII. 3. Covered market: r α(w α, w β +r β (w α, w β = 1, r α(w α, w β > 0 and r β (w α, w β > 0. We denote this configuration as CIII. 4. Preempted covered market: r α (w α, w β = 1, r β (w α, w β = 0. We denote this as configuration CIV. 3.5 Patform Pricing Decision for the Content Provider Side In this stage of the game the patforms are the on movers and the decide what prices to charge to the CPs. The choice set of patform i {α, β} given an h k i is A i (h k i = w i R. Thus the patforms simutaneous decide what prices w α and w β to charge to CPs. Before proceeding we make the foowing definition of a subgame price equiibrium. Definition 1. A (subgame perfect Nash price equiibrium pair (w α, w β is a pair of price strategies such that π α(w α, w β π α(w α, w β for a w α R and π β (w α, w β π β (w α, w β for a w β R. At the price subgame Nash equiibrium each patform i maximizes its own profit, π i = p iq i + r iw i, given the other patform s price strateg and has no incentive to deviate to another price. In this section, we provide resuts showing that given a tupe ( α, β, γ such that α > β there exists a pure strateg price subgame Nash equiibrium pair (wα, wβ 5. In addition we characterize the market configurations that resut. Specifica, we show the conditions under which particuar market configurations arise depending on the parameters γ, α, and β. Our resuts show that the uncovered market configuration, (CI, does not occur at a subgame price equiibrium. On the other hand, we show that given a tupe ( α, β, γ one of the other configurations, CII, CIII or CIV, wi emerge. In 5 The actua price characterizations can be found in the LIDS report.

5 doing so, we determine the set of parametric vaues (γ, α, β for which these different configurations exist. We prove the existence of the price SPE b a construction argument. The proofs, which are omitted due to space imitations 6, invove first identifing candidate equiibrium price pairs in each possibe market configuration. We then check to see whether these price equiibrium pairs are indeed Nash equiibria of the price subgame. We do so b verifing that the equiibrium price candidates are best repies on the whoe domain of strategies: That is, not on are the best responses in their respective market configurations but that the are aso best repies if the other market configurations are taken into account. For ease of presenting our first theorem that summarizes the above resuts we define the foowing sets of prices which we use in the theorem, R II = {(w α, w β r α + r β < 1, r α > 0, r β > 0}, R III = {(w α, w β r α + r β = 1, r α > 0, r β > 0}, R IV = {(w α, w β r α + r β = 1, r α = 1, r β = 0}. The sets R II, R III and R IV consists of price pairs (w α, w β that resut in configuration CII, CIII and CIV respective. We next present the Theorem showing that for an tupe (γ, α, β a price subgame Nash equiibrium exists and on one market configuration is feasibe. In addition, for market configurations CII and CIII, the price characterizations are unique. Theorem 1. Let Assumption 1 hod. Given (γ, α, β there exists a Nash equiibrium pair (wα, wβ in the pricesubgame. Moreover, the resuting market configuration is unique and the foowing hod: 1. If 1 < γ < 5 β +22 α 9( β +2 α, then the equiibrium price pair is unique and (wα, wβ R II. { } 2. If 5 β +22 α γ min β +8 α, 1 β +10 α 9( β +2 α 3(2 α + β 3(2 α + β then the equiibrium price pair is unique and (wα, wβ R III If β +10 α < γ < 3(2 α+ β then the equiibrium price pair is 6 unique and (wα, wβ R III. 4. If max { }, β +8 α 6 3(2 α+ β γ < then (w p α, w p β RIV. Figure 1 shows the resuting market configurations for different vaues of the investment ratio, α / β = I, and the average CP quait characteristic, γ. In particuar, given a I and γ, figure 1 shows the distinct resuting market configuration. For a fixed I as the quait characteristic γ increases the covered market is more ike. At the extreme, when γ is high, the CPs content quaities are reative cose to each other since as γ, the ratio γ 1. Thus CPs are ess γ 1 distinguishabe from each other; a decision made b a CP wi be mirrored b the other cose CPs and a covered market is ike. On the other hand, for a fixed vaue of ow γ, 6 The proofs can be found in the LIDS technica report Quait γ 6 Covered Market (Interior Soution Pre-empted Market 5 2 Covered Market (Corner Soution Uncovered Market Investment Ratio I Figure 1: Content Quait Characteristic γ versus Investment Ratio α / β = I and Resuting Market Configurations. as the investment ratio increases the two patforms become more differentiated. This means that the patforms can exert some market power. Specifica, one patform serves the high quait CP market and the other the ow quait CP market. Price competition becomes ess intense as the two patforms focus on different markets. This softening of price competition resuts in an uncovered market because the patforms pricing strategies invove them pricing above the utiit that woud be derived b the owest quait CP. However, for a fixed high γ, the reative coseness of the content providers quait, dominates the differentiation effects of the patforms and a preempted covered market is reaized as a CP s fock to one patform. 3.6 Quait Choice In this stage of the game the patforms are the on movers and the decide what quait to set. We assume that once patforms are in operation quait choice is costess. The choice set of patform i {α, β} given an h k i is A i(h k i = i where i [,. Thus the patforms simutaneous decide what quait to choose. We find the equiibrium quait choices b considering the best rep responses of the two patforms. We find the set that contains patform β s best repies to patform α s choices and vice versa. We then anaze the points where these sets intersect and show that the indeed are the subgame perfect equiibria. Due to space imitations the anasis is provided in Appendices A.1 and A.2. For ease of presenting the Theorem that characterizes the subgame perfect equiibrium of the quait choice game, and the Coroaries that characterize the resuting market configurations, we make the foowing cassifications: C.1 1 < γ <, C.2 γ < 24 and, C.3 γ < and <, C.4 γ 8 The above cassifications 6 foow from the anasis in the Appendix where we partition the range in which γ ies into three sections depending on the tpes of market configurations that are possibe in each partition. From a quaitative view, the partitions represent the ranges in which the average content quait is ow, medium or high. In the medium range we make two further cassifications that depend on the bounded interva from which quait is chosen. For a given γ in the medium range, if the bounds satisf condition C.2(C.3 then we have a sma(arge quait choice range.

6 We wi now present the theorem that characterizes the resuts of the quait choice game. Theorem 2. Given (γ,, there exists a subgame perfect Nash equiibrium (SPE in the quait choice game. Moreover, the foowing hod: (iif C.1 hods then the SPE entais maxima differentiation where one patform chooses the best quait,, and the other chooses the owest quait,. (ii If C.2 or C.4 hods then the SPE entais partia to maxima differentiation where one patform chooses a quait, ỹ [f(,,, and the other chooses the owest quait,. (iii If C.3 hods then the SPE entais one patform choosing the highest quait,, and the other one choosing a proportion of that depends on the average quait characteristic, γ. In particuar, the ow quait patform picks =. In genera, the above resuts suggest that the patforms differentiate in patform quait to soften price competition. If the patforms are undifferentiated both patforms earn zero profits due to Bertrand price competition on both sides of the market. Therefore, patforms have incentive to choose different quait eves in equiibrium. The eve of the differentiation depends on the average CP quait characteristic, γ. When γ is ow, patforms soften price competition through maxima differentiation, i.e. one patform picks the highest quait and the other the owest. This enabes the patforms to corner different segments of the market and exert market power. When γ is in the medium range the resuting equiibrium depends on the quait choice interva, [,. If the quait choice interva is arge, patforms differentiate between themseves with one picking the highest quait whie the other picks a fraction that is a function of γ. As γ increases this fraction diminishes and patforms become more differentiated. Since the CP s are ess heterogenous for higher vaues of γ, there is a more intense competition for them b the patforms. To soften this competition patforms aso increase the eve of differentiation in quait. Hence the positive correation between γ and the eve of differentiation. In contrast, when the quait choice interva is sma patforms sti differentiate between themseves but with one picking the owest quait whie the other picks either the highest quait or some fraction of it(which is higher than the owest quait. Since the quait choice interva is sma the eve of differentiation is bounded. Indeed, no amount of differentiation is abe to segment the market. Therefore a fierce competition ensues and a pre-empted market configuration where a CP s join one patform resuts. Since there are mutipe price equiibria in this configuration there aso exists mutipe quait choice equiibria. When γ is high, there is partia differentiation simiar to that when condition C.2 is met and a simiar expanation hods. We next present the coroaries that show which market configurations resut given the quait choice interva [, and γ. Coroar 1. If C.1 hods both patforms enjo positive market share in the content provider market with the resuting market configuration depending on the investment ratio I and γ. In particuar, 1. If 1 < I < 21γ 1 then a covered market with an 2(3γ 5 interior soution is the outcome. 2. If 21γ 1 I < 9γ 5 then a covered market with 2(3γ 5 2(9γ 11 a corner soution is the outcome. 3. If 9γ 5 I < Φ, where Φ <, then an uncovered 2(9γ 11 market is the outcome. Coroar 2. If C.2 or C.3 or C.4 hods, one patform has a the market share in the content provider market, i.e., a pre-empted market is the outcome. This market configuration is independent of the investment ratio, I. The first coroar shows that when γ is ow, an of the three market configurations can occur depending on the quait choice range. Since there is maxima differentiation, the quait choice range can be proxied b the eve of differentiation between the patforms at the SPE, characterized b the Investment ratio.when I is ow, a covered market resuts since the eve of asmmetr is sma and price competition is intense. In contrast, high vaues of asmmetr resut in an uncovered market since the differentiation eve is high and price competition is reaxed. The second coroar shows that for medium to high vaues of γ on a preempted market resuts regardess of the asmmetr between patforms. In this case, the reative difference in content provider quait is too ow to be abe to distinguish amongst them via patform differentiation. Therefore, the fierce price competition resuts in a pre-empted market where a CP s fock to the high quait patform. 4. CONCLUSIONS We stud duopo competition between two interconnected ISP s in the presence of quait choice and service botte neck effects. We show that given two asmmetric patforms (i.e. with different quait eves, a price SPE on both sides of the market exists. Moreover, we show that the higher the asmmetr the more ike the CP market is to be uncovered if γ is ow. This suggests that high differentiated patforms provide a barrier to entr because of their market power. Our fina resuts show that a SPE exists for the Quait choice game and the equiibrium invoves differentiation in quait. The eve of differentiation depends on γ and the quait choice interva. In addition, we aso show the tpes of market configurations that resut from the quait choices. A imitation of our mode is the assumption a fixed investment cost for quait (or costess quait choice. Nevertheess we beieve that with ow margina costs of investment the effects captured in this mode wi sti hod. Moreover, in certain cases ISP s can increase or ower their quait without cost. For exampe once an ISP invests a fixed amount on a router it can increase or decrease bandwidth(hence atering quait for certain traffic tpes b simp setting parameters at no additiona costs.

7 5. REFERENCES [1 M. Armstrong. Competition in two-sided markets. Rand Journa of Economics, 3(3: , [2 S. Berger. How to choose an isp? [3 C.-J. Choi and H. S. Shin. A comment on a mode of vertica product differentiation. Science, 40(2: , June [4 D. Davis. Contro unwanted traffic on our cisco router using car:. [OnineAvaiabe: [5 J. Gabszewicz and J. Thisse. Price competition, quait and income disparities. Journa of Economic Theor, 20(3: , June 199. [6 J. J. Gabszewicz and X. Y. Wauth. Two-sided markets and price competition with muti-homing. CORE Discussion Paper No. 2004/30, Ma [ S. Moorth. A comment on a mode of vertica product differentiation. Marketing Science, (2: , [8 J. Musacchio, G. Schwartz, and J. Warand. Network neutrait and provider investment incentives. Review of Network Economics, 8(1:22 39, March [9 G. G. Parker and M. W. V. Astne. Two-sided network effects: A theor of information product design. Management Science, 51(10, October [10 R. Roson. Two-sided markets: A tentative surve. Review of Network Economics, 4(2: , June [11 A. Shaked and J. Sutton. Reaxing price competition through product differentiation. Review of Economic Studies, 49(1, Januar [12 A. Technoogies. Internet bottenecks [13 X. Wauth. Quait choice in modes of vertica differentiation. Journa of Industria Economics, 44(3: , September APPENDIX A. A.1 Best Repies We first show that a smmetric equiibrium is not feasibe. Let j, i {α, β} and B i ( j be the set of i [, such that, i arg max i [, π j ( i, j. Lemma 1. Let j [, then j / B i( j. Proof. We show that given j patform i never chooses i = j and therefore a smmetric equiibrium is not possibe. A smmetric argument appies for the other patform. Assume j B i ( j so that i = j, then both patforms woud make zero profits because of Bertrand competition on both sides of the market. We show that there exists profitabe deviations for patform i. We consider two cases: Case I i = j <. Let patform i increase its quait to i + δ <, where δ > 0; patform i becomes the high quait patform. Resuts from Theorem 1 imp that the resuting equiibrium price w i for the high quait patform given the subgame (γ, i+δ, j is ess than the utiit earned The fu version of this Theorem gives the price characterizations and can be found in the LIDS report. b the highest quait content provider under a the market configurations, i.e., w i < γ(q j j + q i i for a γ > 1, therefore r i > 0. Moreover, from Theorem 1, we can show that w i > w j 0. This impies that π i > 0. Thus patform i woud prefer to set quait i = j + δ instead of i = j. Case II < i = j =. Let patform i decrease its quait to i δ < ; patform i becomes the ow quait patform. Since the patforms are now differentiated, the price charged to consumers b patform i is p i > 0 and q i = 1/3. Therefore p i q i +w i r i = π i > 0 since w i 0. We concude that patform i woud prefer to set quait i = j instead of i = j =. Given quait choice α, patform β can choose a best rep that depends on whether it acts as a high quait or a ow quait patform. In the former case it chooses a rep in the domain ( α, and in the atter case it chooses a rep in the domain [, α. In order to avoid confusion when patform β is the high quait firm we wi change notation as foows; we abe the high(ow quait patform as h( and the quait associated with it as h(. A.1.1 Best rep in the domain (, We wi first anaze the best rep when patform β chooses a rep in the domain ( α,. In this case patform β is the high quait patform and is abeed as h. We wi show that the profit of the high quait firm is increasing in quait in ever configuration. In Theorem 1 we have characterized the conditions for various market configurations to occur in the price subgame as a function of γ, and h. These characterizations can be viewed as restrictions on h given γ,. When viewed as such the foowing hod, 1. Market is uncovered, with positive masses of consumers on both patforms, in the in the price subgame whenever, h > 9γ 5 2(9γ 11. (3 2. Market is covered and a corner soution appies in the price subgame whenever, [ 21γ 1 h 2(3γ 5, 9γ 5, (4 2(9γ 11 h if 1 < γ, 6 ( 3γ 1 2(3γ 4, 9γ 5, (5 2(9γ 11 if 6 < γ < Market is covered and an interior soution appies in the price subgame whenever, ( 21γ 1 h,. (6 2(3γ 5 4. Market is preempted whenever, ( 3γ 1 h,, if 21 2(3γ 4 γ < 24, ( γ 24 (8 One can aso deduce from the above that the uncovered configuration is possibe on if 1 < γ < 22 ; a covered mar-

8 ket configuration with a corner soution is possibe on if 1 < γ < 24 ; a covered market configuration with an interior soution is possibe on if 1 < γ < ; and preempted market 6 configuration is possibe on if γ. The foowing emma 6 shows that the profit of a high quait patform is increasing in its quait in a configurations. Lemma 2. Given, γ and the domain (,, the profit function π h (, h is increasing in h for the market configurations CI, CII, and CIV. Proof. We show that for each configuration the profit function π h (, h is increasing in h. Uncovered Configuration CI: configuration is given b, The profit function in this π u h = ( h(24γ (12γ 1 2 ( h 54( + 8 h 2. The derivative of the above function is positive for γ > 1 and h >, hence the profit function in this configuration is increasing in h. Covered Configuration with corner soution CIII: The profit function in this configuration is given b, π cc h = ( h(16γ (3γ ( h The second derivative of the above function is given b, 2 π cc h 2 h = 32 (1 2γ + γ 2 4( h 3. The above derivative is positive for γ > 1 and h >, which impies that the function is convex under these restrictions. Moreover, the profit function { π h has a singe root} at ỹ = (3. Since ỹ < max 2(3γ+2, 21γ 1 hence 2(3γ 5 the profit function in this configuration is increasing in h. Covered Configuration with interior soution CIII: The profit function in this configuration is given b, πh ci = (6γ ( h. 486 The derivative of the above function is given b, π ci h h = (6γ and the derivative is positive. Therefore the profit function is increasing in h. The set of prices in these two regions given the tupe, h and γ are given beow. a. b. γ < 24 and ( 6, (9γ 8. We deduce from Theorem 1 8 that given γ in the above range and h in the above partition man price equiibria exist. In particuar, patform h wi offer the price, where, c [ w h = 1 3 (γ 1( + 2 h c, (9γ h, (3γ (6γ 8 h. 9 The choice of c depends on the price that patform wi pick. Specifica, the highest price charged b patform h for a particuar h is b 1 ( 9 h (6γ 5 and the owest price is given b 1 (5 3γ( 9 h. [ γ < 24 and 6 (9γ 8, In the second partition, we deduce from Theorem 1 that the highest price that patform h can charge is w h = 1 (γ 1( 3 +2 h. The owest price that patform h can charge at equiibrium for a particuar h is given b 1 (5 3γ( 9 h. When γ 24 patform h wi offer the price, w h = 1 (γ 3 1( + 2 h c, where, c [ { (9γ 8 max 9 { (3γ 1 min } 9 h, 0, } (6γ 8 h, (γ 1, 9 where c depends on the price offered b patform. Given (, γ the correspondence Π : R + R + gives the set of profit vaues that patform h can attain for each choice ỹ (,. We define the set of profit functions that have a maximum over the domain (,. Let P ( h = {π p h ( h ( h π p h ( h g( h }. where g( h = max{ 1 3 (γ 1( + 2 h, 1 9 ( h (6γ 5} and ( h = max{ 1 9 (5 3γ( h, 2 3 (γ 1( h }. We wi now show that if more than one market configuration is possibe, for a given interva (,, given γ, patform h wi prefer to pick h = Lemma 3. Given, γ and an interva (, assume that either (i γ, or (ii γ > 6 6 and, then the best response, B h ( =. From Theorem 1 we can deduce that the preempted market configuration CIV is possibe whenever γ. Severa price 6 equiibria exist in this configuration depending on the tupe ( h,, γ. Given, and γ the correspondence W : R + R + gives the set of best response prices for each choice of ỹ (, γ. When γ < 24, we can partition the set in 6 which h ies into two. These are, [ 3γ 1 (, (9γ 8 and (9γ 8,. 2(3γ 4 Proof. It is sufficient to show that the profit function is increasing across the different market configurations under the above assumptions. We partition the domain in which γ is defined according to the tpes of market configurations that are possibe for each γ. We wi then show that for each of these partitions the profit function is non decreasing in h. 8 The fu version of this Theorem gives the price characterizations and can be found in the LIDS report.

9 Case I: 1 < γ < 21 ; As previous stated, three market configurations are possibe depending on the vaue of h and ; these are uncovered, CI, covered with a corner soution and covered with an interior soution, both of which are in CIII. Given a γ in the above range, the domain (, in which h ies can be partitioned into three sets; each of which corresponds to one of the three market configurations. These partitions are captured b the sets (3, (4 and (6. B emma 2, we know that profits are increasing in h for each partition. We wi first show that the vaue of the profit function in the partition defined in (3, is arger than an profit attained in the partition defined in (4 and then show that an profit attained in the partition defined in (6 is ess than that attained in (4. To show the first resut we compare the infimum vaue of the profit function in the uncovered configuration to the highest profit attainabe when patform h chooses h such that a covered market with a corner soution resuts (i.e, h is in the set specified b (4. Let cc 9γ 5 =, it foows that im h cc πu h( h, = πh cc ( cc,. B emma 2, 2(9γ 11 πh( u h, > πh( u cc, whenever h ies in the set specified b the constraint (3. It aso foows that πh( u h, > πh cc (ỹ, whenever ỹ is in the set specified b (4 since b emma 2, πh cc ( cc, πh cc (ỹ,. To show the second resut we compare the owest vaue of the profit function in the covered configuration with a corner soution, to the supremum profit vaue attained when patform h chooses h such that a covered market with an interior soution resuts. The interva over which the covered configuration with an interior soution, CIII, is defined is open. Let ci 21γ 1 =, since πci 2(3γ 5 h (, is right continuous, the supremum of πh ci (, over the range in which this configuration hods is given b πh ci ( ci,. Moreover, it is the case that πh cc ( ci, = πh ci ( ci,. Therefore, it foows from emma 2, that πh cc ( h, > πh ci (ỹ, whenever h is in the set specified b (4 and ỹ is in the set specified b (6. Case II: γ < 22 ; When γ < 22 three market configurations are possibe depending on the vaue of h and 6 6 ; these are uncovered, CI, covered with a corner soution, CIII, and a pre-empted market, CIV. Given a γ in the above range, the domain (, in which h ies can be partitioned into three sets each of which corresponds to one of the three market configurations. These partitions are captured b the sets (3, (5 and (. We proceed in a simiar manner as we did for case I. B emma 2 we know that profits are increasing in h for each partition. We wi first show that the vaue of the profit function in the partition defined b the inequait (3, is arger than an profit attained in the partition defined b (5 and then show that an profit attained in the partition defined b ( is ess than that attained in (5. The first resut is proved in the same wa as it is done in case I. To show the second resut we compare the infimum vaue of the profit function in the covered configuration with a corner soution to the profit vaue attained when patform h chooses h such that a pre-empted market resuts. Note the interva over which the covered configuration with a corner soution, CIII, is defined is open on its ower imit. Let p =, since πcc h (, is eft continuous the infimum of πh cc (, over the range in which this configuration is defined is πh cc ( p,. B pugging in h = p into πh cc ( h, and π p h ( h, we determine that the im h p πcc h ( p, = π p h (p,. Note that for an π p h ( h P ( h if h then πp h ( h is singe vaued, see the price characterization in Theorem 1. Moreover, π p h (p, π p h ( h, where π p h ( h, P ( h and h. Therefore, it foows from emma 2, that πh cc ( h, > π p h (ỹ, when h is in the set specified b (5 and ỹ is in the set specified b (. Case III: 22 γ < 24 ; When γ fas in the above range two market configurations are possibe depending on the vaue of h and ; these are covered with a corner soution, CIII and a pre-empted market, CIV. The domain (, in which h ies can be partitioned into two sets each of which corresponds to one of the two market configurations. These partitions are captured b sets (5 and (. B emma 2, we know that profits are increasing in h in the partition where a configuration CIII is defined, i.e in the set (5. Showing that the vaue of the profit function in the partition defined b (5, is arger than an profit attained in the partition defined in ( empos the same proof that is used to show the second resut for case II above. Therefore, given that patform picks < and patform h chooses to be a high quait patform, patform h chooses its best response to be. Lemma 4. Given, γ and an interva (,. Assume that i. γ < 4 then the best response, B 6 3 h( [f(,, where f(, = { (9 3γ+(6γ 4γ+3 9γ if ( 3γ+2, 2, γ 3 (9 3γ ( 4γ+3 9γ if [ 2 3γ+1, γ 3. ii. γ 4 3 then the best response, B h( [f(,, where f(, = { (1+3γ+ 6γ 1+2γ if ( 3γ+2, 2, 3γ+1 (1+3γ (γ 19 6γ 1+2γ if [ 2 3γ+1,. 3γ+1 Proof. We give a genera outine on how to prove each of the above cases. Given [C, K, where C and K are the reevant constants given in the hpothesis. The east profit that patform h can make is given b ( where ( h P ( h. This foows from the fact that ( h is an increasing function of h. Let P ( h = {π p h ( h max h π p h ( h (}. We find the minimum vaue of ỹ in the domain (, such that max h π p h ( h ( where π p h ( h P ( h, i.e. ỹ = min h s.t. π p h ( h P ( h 1 π p h ( h ( > 0 Since the correspondence Π is convex vaued (this foows from the convexit of W we find that ỹ = h where g( h = (. Therefore the best response B h ( = f(, ies in the set given b [ỹ,, where,f(, = arg max π p h ( h and π p h ( h P ( h.

10 A.1.2 Best rep in the domain [, h We wi foow a simiar approach to that used in section A.1.1. Given h, we wi compute firm s best rep. We wi show that the profit for the ow quait firm is decreasing in across a configurations which are possibe whenever 1 < γ /6 or γ > 4/3. This wi hep us infer that the ow quait patform chooses as its best response in those ranges. For the range, 21/ γ < 24/, we show that the ower quait patform chooses the maximum between and a fraction of the quait chosen b the high quait patform. Since the choice of b the ow quait firm determines the market configuration we define the critica imits for which the various configurations exist given h. 1. Market is uncovered, with positive masses of consumers on both patforms, in the in the price subgame whenever, 2(9γ 11 < h. (9 9γ 5 2. Market is covered and a corner soution appies in the price subgame whenever, [ 2(9γ 11 2(3γ 5 h, h 9γ 5 21γ 1,, (10 if 1 < γ, 6 [ 2(9γ 11 2(3γ 4 h, h, (11 9γ 5 3γ 1 if 6 < γ Market is covered and an interior soution appies in the price subgame whenever, ( 2(3γ 5 h 21γ 1, h. (12 4. Market is preempted whenever, [ 2(3γ 4 h 3γ 1, h, if 21 γ < 24, (13 γ 24 (14 Lemma 5. Given h, γ and [, h, the profit function π (, h is decreasing in for a market configurations for which it is defined. Proof. We show that for each configuration the profit function π (, h is decreasing in. Uncovered Configuration CII: The denote the profit function in this configuration b π u, One can show that if the quait parameter is in the range 1 < γ < 22 π u < 0. Hence the profit function in this configuration is decreasing in. Covered Configuration with corner soution CIII: The profit function in this configuration is denoted b π cc, One can show that the above derivative is negative when ies in the set specified in (10 is satisfied and γ > 1. Hence the profit function in this configuration is decreasing in. Covered Configuration with interior soution CIII: The profit function in this configuration is given b, π ci = (103+36γ 84γ 486( h. π ci The derivative of the above function is given b, = γ γ. The above derivative is negative therefore the profit function is decreasing in when ies in the 81 set specified in (12. Pre-empted Configuration CIV : The profit function in this configuration is given b, π p = 2 9 h 2 9. The derivative of the above function is given b, πp = 2. The above deriva- 9 tive is negative therefore the profit function is decreasing in when this configuration is defined. We wi now show that given h and γ, and the strateg space E = [, h patform wi prefer to pick whenever 1 < γ <. 6 Lemma 6. Given h, 1 < γ 6, and a strateg space E then B ( h =. Proof. As previous stated in section A.1.1 three market configurations are possibe when 1 < γ < 21 ; these are uncovered (CI, covered with a corner soution and covered with an interior soution (both of which are in configuration CIII. Given a γ in the above range, the domain [, h in which ies can be partitioned into three sets, each of which corresponds to one of the three market configurations. These partitions are captured in (9, (10 and (12. B emma 5, we know that profits are decreasing in for each partition. We wi first show that the vaue of the profit function in the partition defined in (9, is arger than an profit attained in the partition defined in (10 whenever both partitions are defined given and h. Simiar, we show that an profit attained when ies in the partition defined b the constraint in (12 is not greater than that attained when ies in the partition specified b (10. To show the first resut we compare the infimum vaue of the profit function in the uncovered configuration to the highest possibe profit attained when patform chooses such that a covered market with a corner soution resuts (i.e, is in the set specified in (10. Let cc 2(9γ 11 = h, it foows 9γ 5, h (Since π u is right, h that im cc π u (, h = π cc ( cc continuous, the imit exists. Since π u (, h > π u ( cc when satisfies the inequait in (9, it aso foows from emma 5 that π u (, h > π cc (ỹ, h when ỹ ies in the set specified in (10. To show the second resut, we compare the owest vaue of the profit function in the covered configuration with a corner soution to the supremum profit vaue attained when patform chooses such that a covered market with an interior soution resuts. The interva over which the covered configuration with an interior soution, CIII, is defined is open., we define the supremum of π ci (, h over the range in which this configuration is defined as π ci ( ci, h. We note that ci is the infimum of the interva over which this configuration is defined, therefore im ci π ci (, h = π ci ( ci, h since π ci ( h, is Let ci = 2(3γ 5 21γ 1