Joerey Drouard
email: joef-frey.drouard@gmail.com.

Content-Distribution Strategies in Markets With Locked-in Customers

Keywords: Distribution of content, Exclusivity, Switching costs, Locked-in customers, Customer base

We study how the presence of locked-in customers in a downstream market aects the distribution choice of an upstream content provider. Two asymmetric distributors compete in a mature market and the content provider sells its rights using lump-sum fees. A higher number of locked-in customers reduces the need to resort to exclusivity to relax downstream competition. The content provider therefore sells its rights to both distributors when there is a suciently-high proportion of locked-in customers. We show that an exclusive aliation with the smaller rather than the larger distributor facilitates distributors' rent-extraction, in particular for low-quality content. When there are few locked-in customers, the content provider sells low-quality content to the smaller distributor and high-quality content to the larger distributor. Our results suggest that competition authorities should cautiously evaluate the eects of lower switching costs on consumer welfare. By encouraging exclusive distribution, a lower proportion of locked-in customers may reduce consumer welfare.

Introduction

Digital-content distributors need to acquire premium content as a way of increasing the value of their services. They in particular would like to acquire this content on an exclusive basis in order to strengthen their market position. Distributors can be thought of as pay-TV, telecom or over-the-top providers that compete for sports and movie rights or premium channels. Apart from the distributors' need to sign exclusive deals in order to dierentiate their product from that of their rivals, these markets are also characterized by switching costs that can produce consumer lock-in. These costs can be represented in monetary terms but also include time, eort and psychological hurdles. This lock-in eect partly explains why switching rates are relatively low in these industries.

1

The size of the customer base and the extent of consumer lock-in are drivers of protability in content-distribution markets [START_REF] Chen | Information Technology and Switching Costs, volume 1 of Handbook on Economics and Information Systems[END_REF]. However, with the exception of [START_REF] Weeds | TV Wars: Exclusive Content and Platform Competition in Pay TV[END_REF], the existing theoretical work analyzing the determinants of content exclusivity make no reference to switching costs. We here analyze the content-distribution strategies employed in markets with locked-in customers. We consider a mature market with two asymmetric downstream distributors and one upstream content provider. The upstream provider sells the rights for its content using lump-sum fees, and can be exclusive to one distributor or multi-home (Stage I). Distributors then compete on price (Stage II). While some consumers are fully locked-in due to prohibitively-high switching costs, others can switch supplier at no cost.

The allocation of content across distributors that emerges from Stage I is the one that maximizes the sum of distributors' prots in Stage II. The eect of Stage-I exclusive distribution on these prots is as follows. On the one hand, some customers will be denied access to the content if the content provider aliates exclusively with one distributor. Consumers' average willingnessto-pay for distributors' services is therefore greater when the content provider multi-homes rather than aliating exclusively with one distributor. On the other hand, exclusivity allows for verti-1 See, for instance, [START_REF] Grzybowski | Estimating Switching Costs in Mobile Telephony in the UK[END_REF], Grzybowski and Pereira (2011) and [START_REF] Grzybowski | Estimating Consumer Inertia in Repeated Choices of Smartphones[END_REF] for enjoy monopolistic power over some customers. Even if they both have content access, distributors extract part of the surplus created by the content and to a larger extent if the mass of locked-in customers is high. When there is a sucient number of locked-in customers, multi-homing yields both full content exposure and high rent-extraction and thus maximizes the sum of the distributors' prots.

Our result highlights that increased competition in the downstream market measured as a lower number of locked-in customers promotes exclusivity. [START_REF] Bagchi | Selling licences for a process innovation: the impact of the product market on the selling mechanism[END_REF][START_REF] De Bettignies | The Eects of Downstream Competition on Upstream Innovation and Licensing[END_REF] examine the eect of downstream competition on licensing strategies when licenses are sold on a lump-sum basis.

5 They both show that the number of licenses falls with the degree of downstream competition. In addition, using a large panel of traded U.S. rms, De Bettignies et al.

(2018) provide empirical evidence supporting that downstream competition encourages exclusive licensing. The theoretical result in De [START_REF] De Bettignies | The Eects of Downstream Competition on Upstream Innovation and Licensing[END_REF] relies on allocative ineciency and is sensitive to small changes in the licensing-environment assumptions.

6 In contrast with the present article, [START_REF] Bagchi | Selling licences for a process innovation: the impact of the product market on the selling mechanism[END_REF] considers a downstream market in which the sum of the rms' demands is elastic. The overall consumer demand therefore rises with the number of licenses granted.

The magnitude of this demand eect, which increases with the degree of product dierentiation, encourages the innovator to grant more licenses.

7 [START_REF] Muto | On licensing policies in Bertrand competition[END_REF], [START_REF] Poddar | On Patent Licensing in Spatial Competition[END_REF] and [START_REF] Stamatopoulos | Licensing of a quality-improving innovation[END_REF] study various types of price competition settings with little to no eect of market expansion. They show that a single license is granted when contracts do not include a per-unit royalty.

Most of previous work on content distribution and patent licensing shows that exclusivity never occurs with the rm at a disadvantage in the downstream market (e.g., [START_REF] Armstrong | Competition in the Pay-TV Market[END_REF] 5 Licensing an innovation is akin to selling the rights to a value-added product. The literature on patent licensing mostly deals with cost-reducing innovation. However, cost-reducing and quality-enhancing innovations can be seen as two sides of the same coin, as they both provide a competitive advantage to the licensee.

6 De [START_REF] De Bettignies | The Eects of Downstream Competition on Upstream Innovation and Licensing[END_REF] assume that, when the patentee chooses to sell an exclusive license, it makes the innovation specic to one rm and afterward determines the level of the exclusive fee. This commitment weakens the bargaining power of the innovator as it can no longer threaten to sell the content to the other rm, and hence reduces the value of an exclusive licensing strategy. However, with other selling mechanisms that can sustain an ecient allocation outcome, exclusivity will always be the preferred choice of the patentee.

7 This suggests that multi-homing could maximize the sum of the distributors' prots if content-distributors were competing in uncovered markets.

5

Accepted manuscript / Final version [START_REF] Wang | On technology transfer to an asymmetric Cournot duopoly[END_REF][START_REF] D'annunzio | Vertical Integration in Two-Sided Markets: Exclusive Provision and Program Quality[END_REF][START_REF] Jiang | Competition of Content Acquisition and Distribution Under Consumer Multipurchase[END_REF]. Two noteworthy exceptions include [START_REF] Stamatopoulos | On the superiority of xed fee over auction in asymmetric markets[END_REF] and [START_REF] Schmidt | Price competition and innovation in markets with brand loyalty[END_REF]. [START_REF] Stamatopoulos | On the superiority of xed fee over auction in asymmetric markets[END_REF] study the licensing of an innovation that only reduces the marginal cost of the less ecient rm. Under a xed-fee policy, the ecient rm has no incentive to buy a license as it cannot prevent its rival from buying one, and hence the less ecient rm acquires the innovation exclusively. [START_REF] Schmidt | Price competition and innovation in markets with brand loyalty[END_REF] analyzes rms' incentives to invest in a cost-reducing innovation in a market with lockedin customers. The rm with the smallest customer base is the most eager to obtain a marginal innovation when there are few locked-in customers. We nd a similar result: the distributor with the smallest customer base secures exclusive content rights when the mass of locked-in customers and content quality are not too high. Our setup diers from that in [START_REF] Schmidt | Price competition and innovation in markets with brand loyalty[END_REF] in one key way. He measures the rms' incentives to innovate by comparing the partial derivatives of their prot with respect to their marginal cost, and therefore neglects the possibility that both rms simultaneously innovate. In our framework, the distributors' willingness-to-pay for content is contingent on the type of allocation (exclusive or not).

Finally, [START_REF] Hagiu | Pricing and commitment by two-sided platforms[END_REF] examines how commitment by platforms aects the existence of both exclusive and multi-homing equilibria in two-sided markets with locked-in buyers. Sellers are assumed to enter the market before buyers, and platforms can potentially commit to the price for buyers from the beginning. Platforms charge access prices to the buyers and sellers, and a transaction fee on each buyer-seller transaction. However, they cannot charge dierent prices to sellers who are exclusive from those who multi-home. In addition, the aliation decision of sellers is exogenously given, not endogenously determined. He shows that commitment makes the existence of exclusive equilibria less likely, but does not aect multi-homing equilibria if these exist.

3

The model

We consider a market with two downstream distributors and one upstream content provider -denoted by C. The content provider may be thought of as a movie studio or a sports organization, and the distributors as two xed-network operators, over-the-top providers, mobile internet service providers or pay-TV operators. At the start of the game, distributors may be asymmetric with respect to the size of their customer base (reecting the distribution of previous market shares).

The subscript "s" is used to refer to the smaller distributor, while the larger distributor is designated by subscript "l". We denote by σ ∈ [0.5, 1) the customer base of the larger distributor.

Firms are assumed to be risk neutral and to have marginal production costs of zero.

Demand side. There is a unit mass of consumers. They have a common utility, v > 0, generated from access to a distributor, and purchase access to (at most) one distributor. Consumers value the distributor's package more highly when the content is included: their utility rises by τ ≥ 0 when the distributor's oer includes the content (where τ can be interpreted as a measure of content quality). Due to suciently high switching costs, some consumers are fully locked-in, while other consumers can switch at no cost. We denote by α ∈ (0, 1) the mass of locked-in customers.8

Timing. We study the subgame-perfect equilibrium of the following two-stage game.

-Stage I: the upstream provider sells the rights for its content using lump-sum fees. It makes sequential oers to the distributors, and either aliates with one distributor exclusively or multi-homes.

-Stage II: distributors set their price simultaneously and consumers decide from which distributor to buy. Locked-in customers continue to buy from the same distributor (as long as the distributor price lies below the consumer's reservation value). Price-sensitive customers buy from the distributor oering them the higher net utility.

The equilibrium

We solve our model by backward induction. We start by deriving the equilibrium pricing in Stage II given the content-distribution strategy (Section 4.1), and then go back to Stage I to determine the content provider's distribution choice (Section 4.2). Finally, we analyze how the contracting stage alters the value of a large customer base in the distribution market (Section 4.3).

Stage II: The pricing game

At the start of Stage II, the content-distribution strategy is given. We use the superscript c to refer to the content-distribution strategy: c = m if C multi-homes, c = l if C is exclusive to the larger distributor, and c = s if C is exclusive to the smaller distributor. We occasionally use the superscript 0 to refer to the situation where none of the distributors has content access (e.g., if the content provider did not reach any agreement with the distributors). We denote by u c i the consumer's reservation value for distributor i:

u c i = v + τ if c ∈ {m, i} and u j i = u 0 i = v with i = j ∈ {l, s}.
The net utility oered to consumers by distributor i is w c i = u c i -p c i with p c i the price of distributor i, i ∈ {l, s} and c ∈ {m, l, s}. To simplify the presentation, we assume that distributors compete in terms of net utility rather than price. This makes it easier to compare and discuss the distributors' pricing strategies. Provided that they oer positive net utility, both smaller and larger distributors have a captive-customer segment of size K s = α (1 -σ) and K l = ασ respectively. Price-sensitive customers, of mass 1 -K l -K s , buy from distributor i if w c i > w c j , with i = j ∈ {l, s}. Distributors attract one-half of the price-sensitive customers whenever they oer the same net utility.

By focusing on its locked-in customers and setting the monopoly price (i.e. that with net utility of 0), distributor i can guarantee itself a prot equal to u c i K i . Distributor i can however attract the group of price-sensitive customers and make a prot slightly lower than u c i -w c j (1 -K j ) by quoting a net utility slightly higher than w c j . The presence of price-sensitive customers therefore produces tension between the distributors' incentives to i) reduce net utility in order to extract rents

Accepted manuscript / Final version from locked-in customers and ii) increase net utility to attract price-sensitive customers. As such, no equilibrium exists in pure strategies. However, there is a mixed-strategy Nash equilibrium, the existence of which is proved by construction (Appendix A of the Online Supplementary Material provides the complete formal proof ).

9 Before providing the intuition for the derivation of this equi- librium, we introduce one piece of additional notation. Let ŵc i be the highest net utility distributor i is willing to oer to attract price-sensitive customers; ŵc i is such that

u c i K i = (u c i -ŵc i ) (1 -K j ),
and thus ŵc

i = (1-K i -K j )u c i 1-K j
with i = j ∈ {l, s} and c ∈ {m, l, s}.

Suppose now that distributors i and j assign a net utility no higher than w with probabilities F c i (w) and F c j (w) and that ŵc i ≤ ŵc j .

(1)

Condition (1) means that distributor j is willing to oer a higher net utility to attract price-sensitive customers than distributor i. Put dierently, distributor j gives relatively greater prominence to price-sensitive customers than distributor i does. Let w c = 0 and wc be the lower and upper bounds of the support of the equilibrium pricing strategies. Assume that distributor j has no mass point at w c and that neither distributor has a mass point at w ∈ (w c , wc ]. By quoting the monopoly price, distributor i can guarantee itself a prot u c i K i . ŵc i insures distributor i a similar prot when it succeeds in attracting the price-sensitive customers. Since distributor i must be indierent between all values in its support, wc = ŵc i and the equilibrium expected prot of distributor i is π c i = u c i K i . By choosing wc , distributor j obtains a prot of π c j =

u c j -wc (1 -K i ) = u c i K i + u c j -u c i (1 -K j ) × 1-K i 1-K j
, which is its equilibrium expected prot since it must be indierent between all values in its support. In equilibrium, the following condition 9 The use of mixed strategies for economic modeling is somewhat controversial, but does not necessarily mean that distributors choose their price by a coin toss. A mixed strategy can also be viewed as a representation of a pure strategy in a larger (unmodeled) game with incomplete information [START_REF] Harsanyi | Games with randomly disturbed payos: A new rationale for mixedstrategy equilibrium points[END_REF]. The idea here is that a rm's mixed-strategy expresses the uncertainty of the other rms about that rm's pure strategy choice. Besides, in reality, content-distributors oer catalogs of movies or services that change over time due to new content acquisition, new home productions or losses in distribution exclusivity. As content-distributors do not necessarily adjust their price when they add new content or remove former content, consumers' net utility may vary with changes in service quality.

9

Accepted manuscript / Final version must be satised

(u c k -w) (1 -K q ) F c q (w) + (u c k -w) K k 1 -F c q (w) = π c k (2)
for k = q ∈ {i, j} and c ∈ {m, i, j}. From condition (2), it follows that F c j (w) =

wK i (1-K i -K j )(u c i -w) and F c i (w) = wK j (1-K j )+(1-K i -K j )((1-K j )u c j -(1-K i )u c i ) (1-K j )(1-K i -K j )(u c j -w)
, with F c j ( wc ) = F c i ( wc ) = 1 and F c j (0) = 0.

Condition (1) implies that F c i (0) ≥ 0 and thus F c i and F c j are proper cumulative distribution functions. When we replace subscript i by l and j by s, condition (1

) holds if (i) c = m, (ii) c = s or (iii) c = l and τ ≤ (K l -Ks)v 1-K l ≡ τ .
Replacing the subscript i by s and j by l, condition (1) holds if c = l and τ ≥ τ . Lemma 1 characterizes the mixed-strategy Nash equilibrium in Stage II.

Lemma 1 Given the content-distribution strategy c, with c ∈ {m, l, s}, there is a mixed-strategy Nash equilibrium in Stage II where the larger and smaller distributors propose net utility no higher than w with probabilities F c l (w) and F c s (w) with supports in which the lower bound is 0 and the upper bound is wc = min { ŵc l , ŵc s }.

• When c ∈ {m, s} or when c = l and τ < τ :

F c l (w) = wK s (1 -K s ) + (1 -K l -K s ) ((1 -K s ) u c s -(1 -K l ) u c l ) (1 -K s ) (1 -K l -K s ) (u c s -w) F c s (w) = wK l (1 -K l -K s ) (u c l -w) and wc = (1 -K l -K s ) u c l 1 -K s ≡ ŵc l .
• When c = l and τ ≥ τ :

F l s (w) = wK l (1 -K l ) + (1 -K l -K s ) (1 -K l ) u l l -(1 -K s ) u l s (1 -K l ) (1 -K l -K s ) u l l -w F l l (w) = wK s (1 -K l -K s ) (u l s -w) and wl = (1 -K l -K s ) u l s 1 -K l ≡ ŵl s .
Distributors' pricing strategies are aected by two (potential) sources of asymmetry: those in quality and market shares. Recall that the best reply for distributor i is either to fully exploit its 10

Accepted manuscript / Final version locked-in customers or to oer a net utility slightly higher than w c j . Distributor i can guarantee itself a prot equal to u c i K i if it quotes the monopoly price, and obtain a prot slightly lower than u c i -w c j (1 -K j ) if it undercuts its rival. Since K i < 1 -K j , distributor i nds it relatively more protable to undercut its rival as u c i rises. A distributor therefore gives greater prominence to price-sensitive customers if it has content access (exclusively or not) than if it does not. Then, greater asymmetry in market shares aects the base of locked-in customers of both distributors in an opposing way. Other things being equal, the larger distributor has more incentive to exploit its locked-in customers with more unevenly-distributed market shares. At the same time, as it has less to lose by cutting price, the smaller distributor is willing to oer higher net utility to attract price-sensitive customers when σ rises.

As a result, in equilibrium, the smaller distributor competes more ercely than its larger rival when the content provider multi-homes. Both eects, the asymmetries in market shares and quality, make the smaller distributor more aggressive in terms of price undercutting when the content provider aliates exclusively with the smaller distributor. It follows that a small exclusive distributor competes more ercely than a large excluded distributor. The two eects will be opposing if the content provider aliates exclusively with the larger distributor. The asymmetry in quality osets that in market shares, and a large exclusive distributor competes more ercely than a small excluded distributor if content quality is suciently high. This leads to the following proposition.

Proposition 1 Equilibrium pricing strategies have the following characteristics:

(i) F m s (w) ≤ F m l (w)
and F m l (w) has a mass point at 0 if σ = 0.5;

(ii) F s s (w) ≤ F s l (w) and F s l (w) has a mass point at 0 if σ = 0.5 or τ = 0;

(iii) F l l (w) ≤ F l s (w) and F l s (w) has a mass point at 0 if τ > τ .

(iv) F l l (w) has a mass point at 0 if τ < τ .

Proof. See Appendix A 11

Accepted manuscript / Final version

It is worth bearing in mind that competition takes place in terms of net utility. A lower value of F corresponds to a distributor that more frequently oers high net utilities, or stated dierently, to a distributor that oers higher net utilities in a stochastic sense.

10 A sucient condition for price-sensitive customers to be more likely to buy from distributor i rather than distributor j is that F c i (w) ≤ F c j (w), with i = j ∈ {l, s} and c ∈ {m, l, s}. Items (i) and (ii) in Proposition 1 show that the larger distributor quotes the monopoly price with positive probability when the smaller distributor has content access (exclusively or not). In this case, the smaller distributor oers higher net utilities and is more likely to attract price-sensitive customers than its larger rival.

Item (iii) shows that the reverse comes about when the larger distributor has exclusive high-quality content (τ > τ ): a small excluded distributor charges the monopoly price with positive probability and oers high net utilities with a lower probability than does a large exclusive distributor. The situation in which the larger distributor has exclusive low-quality content is less clear-cut: the large exclusive distributor quotes the monopoly price with positive probability but there is no stochastic dominance (item (iv)). Since the probability that a large exclusive distributor attracts price-sensitive customers rises with τ , there exists a value τ ∈ (0, τ ) such that price-sensitive customers are more likely to purchase from a small excluded distributor if τ < τ and from a large exclusive distributor otherwise.

We now discuss how content quality and asymmetry in market shares aect the intensity of competition according to the type of content distribution. We prove the following proposition.

Proposition 2 Content quality and initial market split aect the equilibrium pricing strategies as follows:

(i) Equilibrium pricing strategies & content quality

• F m l and F m s decrease with τ ;

• F l l and F l s decrease with τ if τ < τ , otherwise F l l is independent of τ and F l s rises with τ ;

• F s s is independent of τ and F s l rises with τ .

(ii) Equilibrium pricing strategies & initial market split

• ∀c ∈ {m, s}, F c l and F c s increase with σ;

• F l l and F l s increase with σ if τ < τ , otherwise F l l and F l s decrease with σ.

Proof. See Appendix B

Two elements are particularly relevant for understanding the results of Proposition 2. The rst element is the identity of the distributor that places the least emphasis on price-sensitive customers. Assume that ŵc i ≤ ŵc j , meaning that distributor i is not willing to oer such a high net utility as its rival to attract price-sensitive customers. The upper bound of the support of the equilibrium pricing strategies is then equal to ŵc i . When distributor i's interest in attracting price-sensitive customers rises, the support of the equilibrium pricing strategies becomes wider.

Inversely, the upper bound of this support decreases when distributor i places even less emphasis on attracting price-sensitive customers. The second element is that the pricing game exhibits strategic complementarity.

Item (i) in Proposition 2 highlights the relationship between the intensity of competition and content quality. Greater content quality raises the benet for a distributor that has content access (exclusively or not) to undercut its rival, but does not aect the best response of an excluded distributor. In other words, the highest net utility a distributor is willing to oer to attract price-sensitive customers rises with τ only if it has content access. When the content provider multi-homes, both distributors have more incentive to compete for price-sensitive customers as τ rises. They both increase their price but not as much as the rise in content quality, and thus oer higher net utilities as τ rises. As a result, part of the surplus from content-quality improvement is captured by customers if content is present on both distributors. Whenever the smaller distributor has exclusivity or the larger distributor has exclusive high-quality content (τ ≥ τ ), the upper bound of the support of the equilibrium pricing strategies is equal to the highest net utility the excluded distributor is willing to oer to attract price-sensitive customers. The support of the equilibrium pricing strategies is therefore independent of content quality. However, higher content 13

Accepted manuscript / Final version quality reduces the possibility for the excluded distributor to win price-sensitive customers, which in turn focuses more on its captive customers and thus oers more frequently low net utilities.

The exclusive distributor, meanwhile, oers a similar level of net utility as τ rises. 11 This means that the exclusive distributor appropriates the entire surplus from content-quality improvement and becomes relatively more aggressive than its rival as τ rises. Otherwise, when the larger distributor has exclusive low-quality content, the upper bound of the support of the equilibrium pricing strategies is equal to the highest net utility the exclusive distributor is willing to oer to attract price-sensitive customers. It follows that the support of the equilibrium pricing strategies becomes wider as τ rises. Because the pricing game exhibits strategic complements, both distributors oer higher net utilities as τ rises. Several lessons can be drawn from the above discussion. First, an increase in content quality reduces competition and harms consumers, unless the content is present on both distributors or the larger distributor has exclusive low-quality content. Second, an exclusive aliation with the smaller rather than the larger distributor facilitates distributors' rent-extraction.

12 In particular, price-sensitive customers enjoy, on average, lower net utility if the content provider aliates exclusively with the smaller rather than the larger distributor (see Appendix C). Third, once content quality is above τ , the relative dierence in competition intensity between the two types of exclusive allocation vanishes as content quality rises. Indeed, when τ ≥ τ , E (p s s ) /E p l l and E (p s l ) /E p l s are greater than one but decrease with τ : the rent extracted by the distributors, according to whether the content provider aliates exclusively with the smaller or the larger distributor, becomes (relatively) more similar as content quality rises (see Appendix 11 This follows from the characteristics of the mixed-strategy Nash equilibrium. Assume that distributor j has exclusivity and ŵj i ≤ ŵj j : distributor i quotes the monopoly price with positive probability and can guarantee itself a prot vK i . Since distributor i must be indierent between all values in its support, F j j must be such

that (v -w) (1 -K j ) F j j (w) + (v -w) K i (1 -F j j (w)) = vK i ; τ is not part of this expression, and hence, F j j is independent of τ .
12 The intuition is as follows: the upper bound of the support of an exclusive equilibrium pricing is determined by the highest net utility the exclusive distributor is willing to oer to attract price-sensitive customers if the content provider is exclusive to the larger distributor and τ is lower than τ . Otherwise, the upper bound is equal to the highest net utility the excluded distributor is willing to oer to attract price-sensitive customers. Recall that ŵj i is independent of τ and ŵi i increases with τ , with i = j ∈ {l, s}. Thus, the upper bound of the support of the equilibrium pricing strategies is independent of τ if the smaller distributor has exclusivity. By contrast, when the larger distributor has exclusivity, the upper bound increases with τ as long as τ is lower than τ , and is independent of τ otherwise. It follows that wl ≥ ws . The support of the equilibrium pricing strategies is wider, and thus competition ercer, if the content provider is exclusive to the larger rather than the smaller distributor.

14

Accepted manuscript / Final version D).

Item (ii) in Proposition 2 shows the relationship between the intensity of competition and the asymmetry in market shares. Recall that a distributor places more weight on exploiting locked-in customers than on attracting new ones when its customer base increases. It follows that ŵc l falls in σ, while ŵc s increases with σ, with c ∈ {m, l, s}. When the larger distributor has exclusive high-quality content, the smaller distributor is the least inclined to oer a high net utility to attract price-sensitive customers. Hence, the upper bound of the support of the equilibrium pricing strategies is equal to ŵl s and increases with σ. To withstand the increased competitive pressure exerted by its rival, the larger distributor reacts likewise by oering higher net utilities. As a result, more unevenly-distributed market shares strengthen competition if the content provider is exclusive to the larger distributor and τ ≥ τ . Otherwise, the larger distributor is the least inclined to oer a high net utility to attract price-sensitive customers. The larger distributor's interest in attracting price-sensitive customers and thus the upper bound of the support of the equilibrium pricing strategies decrease with σ. Because of strategic complementarity, the smaller distributor reacts by oering lower net utilities and greater asymmetry in market shares weakens competition. To summarize, increased asymmetry between distributors' market shares softens competition unless the larger distributor has exclusive high-quality content.

Equilibrium expected prots in Stage II. Given the content-distribution strategy c ∈ {m, l, s}, the equilibrium expected prots in Stage II are: 13

• π c s = (u c l K l + (u c s -u c l ) (1 -K s )) × 1-K l 1-Ks and π c l = u c l K l if c ∈ {m, s} or if c = l and τ < τ • π l s = u l s K s and π l l = u l s K s + u l l -u l s (1 -K l ) × 1-Ks 1-K l if τ ≥ τ
We show that the following condition holds:

π i i ≥ π m i ≥ π 0 i ≥ π j i (3)
13 Depending on the mass of locked-in customers, equilibrium expected prots in Stage II are between those from full collusion and competition. Collusive-type outcomes come about as α approaches one and the equilibrium converges to Bertrand competition as α approaches zero.

with i = j ∈ {l, s}. Condition (3) implies that a distributor in Stage II prefers to have content exclusivity rather than C to multi-home, and prefers C to multi-home rather than have C exclusive to its rival. It is important to stress that these expressions are the expected payments collected in

Stage II; the expected total prots for each industry participant also include the payments made between distributors and the content provider in Stage I.

Stage I: Content distribution

We now move back to Stage I to determine the outcome of the contracting game. Following [START_REF] Armstrong | Competition in the Pay-TV Market[END_REF], we assume that the content provider makes sequential oers. It faces three possibilities: sell the rights to both distributors, sell the rights exclusively to the larger distributor, or sell the rights exclusively to the smaller distributor. The negotiations are such that the content provider tries each of these possibilities one after the other, in an order that it has previously dened. Negotiations cease when a contract is accepted or, at the end of these three stages, if no agreement is reached.

We denote by T i i and T m i , with i ∈ {l, s}, the lump-sum payments required by the content provider from distributor i in exchange for exclusive and non-exclusive aliation respectively. The content provider can aliate exclusively with one of the distributors only if this distributor accepts the lump-sum payment required for exclusive distribution; and multi-home only if they both accept the lump-sum payments required for non-exclusive distribution. Without loss of generality, we assume that distributors accept an oer when they are indierent between accepting or refusing it. Clearly, the content provider could lower the relevant fee by some ε > 0 so that distributors will strictly prefer to accept the oer.

The maximum tari the upstream provider can ask of a distributor depends on both the prot this distributor can gain from the content (according to the type of distribution under negotiation)

and on its outside option. By threatening to give the exclusive rights to its rival, the content provider minimizes the value of a distributor's outside option and can thus obtain the maximum payo for each of the dierent forms of distribution. We discuss below the way of doing this 16

Accepted manuscript / Final version according to whether the content provider prefers exclusive or non-exclusive distribution.

• Assume that the content provider wants to induce exclusivity with distributor i. First, the content provider oers non-exclusive contracts to both distributors at prohibitive prices, which are rejected. Distributor i then receives an exclusive oer. It knows that if it rejects the exclusive oer, its rival will be granted the content exclusively. Indeed, as a last resort, the content provider can induce distributor j to accept an exclusive oer by setting an exclusive fee lower than or equal to π j j -π 0 j . Distributor i is therefore better o accepting the exclusive

contract if π i i -T i i ≥ π j i .
It follows that the content provider can induce distributor i to pay a lump-sum fee T i i = π i i -π j i for exclusivity.

• Assume that the content provider wants to induce non-exclusive distribution. First, the content provider oers non-exclusive contracts to both distributors. Each distributor knows that if it rejects the non-exclusive oer, the content provider will then make an exclusive oer to one of the distributors. In that case, and following the same arguments as above, the distributor receiving the exclusive oer will accept it and both distributors will end up with their excluded Stage-II prot. The larger and smaller distributors are therefore better o accepting the non-exclusive oer submitted to them in the rst place if π m l -T m l ≥ π s l and π m s -T m s ≥ π l s . It follows that the content provider can induce both distributors to accept a non-exclusive contract by setting T m l = π m l -π s l and T m s = π m s -π l s .

To summarize, the content provider's revenue is π m l + π m s -π s l + π l s if it sells the rights to both distributors and max π l l -π s l , π s s -π l s if it sells the rights exclusively to one distributor. The content provider therefore opts for a non-exclusive distribution if π m l +π m s ≥ max π l l + π l s , π s l + π s s and for an exclusive aliation with distributor i if π i l + π i s > max π m l + π m s , π j l + π j s , with i = j ∈ {l, s}. We can now state the following lemma.

Lemma 2 The industry structure that emerges from Stage I is that which maximizes expected industry prots in Stage II. Distributor i will pay

T i i = π i i -π j
i in exchange for exclusive content and T m i = π m i -π j i for non-exclusive content, with i = j ∈ {l, s}.

Proof. See Appendix E

The threshold values in Lemma 3 are such that α = 2σ+1-

√ 8σ 2 -4σ+1 2σ(2-σ) , ᾱ = 1- √ σ(1-σ) 1+σ 2 -σ , τ d 1 = 2-3α+α 2 1-α+α 2 σ-α 2 σ 2 × v and τ d 2 = α(2-α)(σ(1-ασ)-(1-σ)(1-α+ασ)) (1-ασ)((1-σ+σ 2 )α 2 -2α+1)
× v. They have the following properties:

0 < α ≤ ᾱ < 1, τ d
1 falls in α and is higher than τ over the interval (0, α), τ d 2 rises with α and is higher than τ over the interval (α, ᾱ),

τ d 1 = τ d 2 if α = α, τ d 1 α→0 = 2v, τ d 2 α→ ᾱ = ∞.
When choosing its distribution strategy, the upstream provider faces a trade-o between ensuring that a large number of customers have access to the content and that distributors have sucient market power to extract the surplus created by the content. On the one hand, multihoming guarantees that every customer has access to the content. Total surplus is therefore greater when the content provider multi-homes rather than aliating exclusively with one distributor. On the other hand, exclusivity allows for vertical dierentiation among distributors and can facilitate distributors' rent-extraction.

14 Indeed, whenever the smaller distributor has exclusivity or the larger distributor has exclusive high-quality content, an increase in content quality reduces competition and harms consumers. By contrast, under multi-homing, customers always capture part of the surplus resulting from an increase in content quality. To provide the intuition for Lemma 3, we distinguish three dierent situations according to the fraction of locked-in customers. In each 14 A well-known principle in economics is that dierentiation is needed to relax price competition (e.g., [START_REF] Shaked | Relaxing Price Competition Through Product Dierentiation[END_REF][START_REF] Champsaur | Multi-product Duopolists[END_REF].

18

Accepted manuscript / Final version case, we then discuss how distributors appropriate the surplus from content-quality improvement according to the type of content distribution.

(i) α ≥ ᾱ. When there is a high proportion of locked-in customers, distributors mostly quote net utility close to 0 and extract a large fraction of consumer surplus -regardless of whether the content provider is exclusive or not. Indeed, a greater number of locked-in customers makes the demand faced by distributors more inelastic, thereby strengthening distributors' market power. The need to resort to exclusivity to relax downstream competition is therefore limited. To obtain broader content exposure, the upstream provider has to supply its content non-exclusively. It follows that multi-homing maximizes expected industry prots.

15

(ii) α ∈ [α, ᾱ). As the number of locked-in customers falls, distributors adopt more aggressive pricing strategies. Vertical dierentiation through exclusivity may be necessary to avoid that customers capture a too large part of the value created by the content. The mass of locked-in customers is nevertheless not negligible, and an exclusive aliation with the smaller distributor will prevent a large fraction of the consumers (at least those who are locked-in with the larger distributor) from having access to the content. When the smaller distributor has exclusivity, the loss of content exposure is too large and cannot be oset by the fact that downstream competition is less intense. As a result, expected industry prots are greater if the content provider multihomes rather than aliating exclusively with the smaller distributor. When the content is of low-quality, an exclusive aliation with the larger distributor also results in a signicant loss of content exposure, 16 without mitigating downstream competition. 17 However, the probability that the exclusive distributor attracts price-sensitive customers rises with τ : a large exclusive distributor will serve a signicant part of the demand if content is of high-quality. In addition, once content quality is above τ , a large exclusive distributor appropriates the entire surplus from content-quality improvement (dE p l l /dτ = 1 if τ ≥ τ ). A small excluded distributor, meanwhile, makes less attractive oers as τ rises. As such, for high-quality content, an exclusive aliation with the larger distributor guarantees both high rent-extraction and a broad content diusion (albeit incomplete).

By contrast, under multi-homing, part of the surplus from content-quality improvement is always captured by customers. Overall, higher content quality increases the appeal of exclusivity to the larger distributor relative to non-exclusivity. There is thus a cuto value of content quality, τ d 2 ,

above which exclusivity to the larger distributor maximizes expected industry prots, and below which multi-homing maximizes expected industry prots. Consistent with multi-homing being more likely to maximize expected industry prots as the number of locked-in customers rises, the content quality threshold τ d 2 increases with α if α ∈ [α, ᾱ).

(iii) α < α. When few customers are locked-in and both distributors have content access, better content quality very partially translates into higher prices and mostly benets consumers. Although every customer has access to the content, expected industry prots weakly increase with τ if the content provider multi-homes. Vertical dierentiation is necessary to relax downstream competition and prevent distributors from dissipating most of the surplus through erce competition.

Thus, either exclusivity to the larger or to the smaller distributor maximizes expected industry prots.

18 The segment of the market of price-sensitive customers being large, the demand served by a distributor mainly depends on its ability to attract those customers, and relies less on whether it has a large or small customer base. The dierence in content exposure, according to whether the upstream provider is exclusive to the larger or the smaller distributor, is therefore of small magnitude (especially for low-quality content).

19 In addition, an exclusive aliation with the smaller 18 This appears clearly when the mass of locked-in customers approaches zero: expected industry prots tend to zero if the content is present on both distributors and to τ if the content is exclusive to one distributor.

19 With poor quality content, a large exclusive distributor places relatively little importance on attracting pricesensitive customers. Price-sensitive customers are more likely to purchase from a small excluded distributor, and the advantage of having a large customer base to insure a wide distribution of content is therefore undermined. On the other extreme, when τ is really high, the exclusive distributor attracts price-sensitive customers with a probability close to 1, regardless of whether it has a large or small customer base. In Appendix B of the Online Supplementary Material, we depict the expected market share of the exclusive distributor, according to whether it has a large or small customer base, as a function of content quality, for dierent levels of the initial market split and of the mass 20 Accepted manuscript / Final version rather than the larger distributor signicantly softens downstream competition if τ < τ . 20 This explains why, even if content exposure is slightly lower, expected industry prots are higher when the upstream provider sells low-quality content to the smaller rather than the larger distributor. However, once content quality is above τ , the relative dierence in competition intensity between these two types of exclusive allocation vanishes as content quality rises. In particular, regardless of whether the content provider is exclusive to the larger or the smaller distributor, an increase in content quality is fully passed on in the exclusive distributor's price (dE p l l /dτ = dE (p s s ) /dτ = 1 if τ ≥ τ ). An increase in content quality being monetized over a larger number of customers, expected industry prots will increase more sharply with τ if the content provider is exclusive to the larger rather than the smaller distributor. Consequently, there exists a threshold quality level τ d 1 , strictly higher than τ , above which an exclusive aliation with the larger distributor will yield higher expected industry prots than an exclusive aliation with the smaller distributor.

To summarize, exclusivity to the smaller distributor maximizes expected industry prots if content quality is lower than τ d 1 . Otherwise, exclusivity to the larger distributor maximizes expected industry prots. Lastly, a rise in the mass of locked-in customers positively aects the base of locked-in customers of both distributors, but to a greater extent that of the larger distributor. As a result, τ d 1 falls in α if α < α: an exclusive aliation with the smaller distributor is less likely to maximize expected industry prots as α rises.

We now discuss how greater asymmetry in market shares aects expected industry prots according to the type of content distribution. More unevenly-distributed market shares generate two conicting eects. On the one hand, greater asymmetry in market shares broadens the base of the larger distributor's locked-in customers at the expense of the smaller distributor. The distributor with the larger customer base can more easily insure a broad content diusion with more unevenly-distributed market shares. Thus, other things being equal, it would reduce the need to resort to multi-homing to guarantee broad content diusion and make exclusivity with of locked-in customers. These numerical examples illustrate that the loss of content exposure that comes with an exclusive aliation with the smaller rather than the larger distributor is lower when content is of low-quality.

20 Indeed, when τ < τ , F l l and F l s decrease with τ , while F s s is independent of τ and F s l rises with τ .

Proposition 3 shows that a low level of content quality or of downstream competition is less conducive to exclusivity. [START_REF] Bagchi | Selling licences for a process innovation: the impact of the product market on the selling mechanism[END_REF] and De Bettignies et al. ( 2018) point to a similar link between competition and licensing strategies. In [START_REF] Bagchi | Selling licences for a process innovation: the impact of the product market on the selling mechanism[END_REF], exclusivity reduces overall consumer demand and is less likely to occur when products are more dierentiated in the downstream market.

In De Bettignies et al. ( 2018), the business stealing eect from which an exclusive licensee benets is undermined by the degree of horizontal product dierentiation. A higher degree of horizontal dierentiation therefore decreases the appeal of exclusivity relative to non-exclusivity. Weeds ( 2016) studies the trade-o a vertically-integrated distributor faces between supplying its retail rival and increasing its subscriber base. Exclusivity allows the integrated distributor to increase its current market share, but leads to a loss of wholesale revenue. Poor content quality and more dierentiated distributors make consumers harder to attract, and therefore dissuade the integrated distributor from choosing exclusivity in the initial phase of market development.

When there are few locked-in customers, Proposition 3 shows that the content provider opts for an exclusive distribution. It sells high-quality content to the larger distributor and low-quality content to the smaller distributor. This result contrasts with [START_REF] Armstrong | Competition in the Pay-TV Market[END_REF][START_REF] D'annunzio | Vertical Integration in Two-Sided Markets: Exclusive Provision and Program Quality[END_REF] and [START_REF] Jiang | Competition of Content Acquisition and Distribution Under Consumer Multipurchase[END_REF], which nd that the content provider never aliates exclusively with the distributor who has an ex-ante competitive disadvantage.

To illustrate our results, we propose a numerical example for four dierent levels of the initial market split, σ ∈ {0.55, 0.70, 0.85, 0.99} and v = 1. 23 When there are few locked-in customers, greater asymmetry in market shares slightly encourages exclusivity with the smaller rather than the larger distributor.

Our results are consistent with many of the contractual agreements in digital markets. For instance, broadband providers oer internet access with a bundle of television channels and services. Basic channels are mostly made available via non-exclusive agreements, and are therefore proposed by all broadband providers. However, some of the latter also acquire exclusive content, mainly premium channels, popular movies or major sports events. While switching from one 23 In the exceptional case where τ is very high, a rise in σ promotes exclusivity over multi-homing.

24

Accepted manuscript / Final version broadband provider to another can be a laborious and costly process, subscription and cancellation are usually easier for over-the-top streaming platforms, almost via a single click and with no termination fee.

24 These streaming platforms oer catalogs of lms and TV-series that are mainly exclusive content.

25 This is in line with our results that multi-homing is less likely in markets with low switching costs, and that when customers face high switching costs mostly high-quality content is sold on an exclusive basis.

The value of a large customer base

Let Π c i be the equilibrium expected total prot of distributor i ∈ {l, s}, where c ∈ {m, l, s} is the content-distribution strategy. Π c i is composed of the payments collected from consumers in Stage II and (potentially) the lump-sum fee paid in Stage I to have access to the content. We dene 24 Switching from one broadband provider to another can be costly due to minimum contract periods or the payment of a cancellation fee, contacting a customer-service representative, and having to return the internet box.

Streaming platforms do not lock customers into long-term contracts, but employ other strategies to retain customers, by for example constantly rening their algorithms to oer a list of recommendations specically tailored to each user. Keeping users watching prevents them from switching. 25 The data from Reelgood provides good descriptive evidence. Reelgood gives information on the catalog of movies and TV-series available on the main streaming platforms. In addition, Reelgood ranks lms and TV-series, using a score that takes into account audience, critical reviews, recent popularity and other quality criteria such as the actors' reputation. Among the top 1000 lms listed by Reelgood and that are available on Amazon Prime Video, Disney+, HBOmax, Hulu or Netix, more than 90% of them are available on only one of these platforms in the United States. Similarly, among the top 500 TV-series listed by Reelgood and that are available on these streaming platforms, more than 85% of them are available on only one of these platforms. We collected the data from reelgood.com in May, 2020. 4) is the value of a large customer base if there were no content provider. Since ∆ 0 ≥ 0, distributors would be better o with the largest customer base if there were no content provider.

∆ = ∆ 0 + Σ (4) with ∆ 0 = vK l (K l -Ks) 1-Ks , Σ = (1 -K l ) τ if τ < τ and Σ = (1-K l -Ks)(K l -Ks)v 1-Ks if τ ≥ τ . The rst term in (
The second term in (4) expresses the way competition for content access alters the value of a large customer base. Σ ≥ 0 and therefore the presence of a content provider, which has all the bargaining power, further deteriorates a distributor's situation if it has a small customer base.

With a larger customer base, a distributor can guarantee itself a higher minimum prot and thus limit the harmful eects of competition against a rival of higher quality. The value of a large customer base is therefore greater than if there were no content provider. More specically, ∆ increases with τ up to v (K l -K s ) and then stays constant at this level. 27 5 Locked-in customers and welfare

Switching costs, which are often perceived as producing less-intense competition and being detrimental to consumers, have raised strong regulatory concerns. For instance, in November 2009

the European Commission adopted a directive that (among other things) facilitated the change 27 Once content quality is above τ , π j i = vK i with i = j ∈ {l, s}. In this case, π j i corresponds to the prot the excluded distributor can guarantee itself by charging the monopoly price and is independent of τ . Thus ∆ =

v (K l -K s ) if τ ≥ τ .
of provider.

28 More recently, the former Federal Communications Commission Chairman Tom Wheeler conceded that once consumers choose a broadband provider, they face high switching costs and as a result even though there may be competition, the market place may not be oering consumers competitive opportunities to change providers.

29 In the following, we discuss the eect of a fall in the mass of locked-in customers on consumer and total welfare.

Locked-in customers and consumer welfare. In our model, captive customers who are locked-in with the larger or smaller distributor obtain net utility of w c l and w c s respectively. Price- sensitive customers buy from the distributor oering higher net utility, and obtain a net util-

ity of w c h = max {w c l , w c s }. Consumer welfare is CW c = K l wc 0 wdF c l (w) + K s wc 0 wdF c s (w) + (1 -K l -K s ) wc 0 wdJ c (w), where J c = F c l × F c s is the cumulative distribution function of w c
h and c ∈ {m, l, s}. As long as the content provider's distribution choice remains unchanged, consumer welfare falls with the proportion of locked-in customers. Exclusivity is however more likely to come about when there is only a small proportion of locked-in customers. This produces a dilemma for competition authorities between reducing switching costs so as to stimulate price competition, and a less-restrictive switching-costs policy to encourage multi-homing. By promoting exclusive distribution, lower switching costs may reduce consumer welfare.

To illustrate this dilemma, we propose a numerical example in which we compare consumer welfare with a low and high percentage of locked-in customers. More specically, we study the consumer-welfare eect of a fall in α (from 0.6 to 0.4), which we will henceforth refer to as lower α. For the sake of clarity and ease of comprehension, we rst depict in Figure 2 There are three areas in Figure 3. In the white area, lower α strengthens competition without aecting the content-provider choice to be exclusive or non-exclusive, and so is consumer-welfare enhancing. In the dark-and light-grey areas, lower α encourages the content provider to choose exclusive rather than non-exclusive distribution. Lower α increases consumer welfare in the lightgrey area, but reduces it in the dark-grey area. When α = 0.4 and the market split is not too asymmetric, the content provider is exclusive to the smaller distributor if τ is below τ d,0.4 1 (which Extensions and robustness checks now relax some of the assumptions and consider a number of extensions of the baseline model.

We below summarize the main results displayed in the Online Supplementary Material.

Price discrimination

The baseline model assumes that distributors adopt non-discriminatory pricing strategies. In this section, distributors can price discriminate between their loyal customers and the others (see Section I of the Online Supplementary Material for the proof ). The trade-o between setting a high price to extract surplus from locked-in customers and a discounted price to win price-sensitive customers disappears when distributors can price discriminate. Indeed, distributors compete à la Bertrand in the segment of the market of price-sensitive customers and extract the full amount of surplus from their locked-in customers. As a result, the intensity of competition does not depend anymore on the initial market split. It follows that the content provider can no longer soften competition in the downstream market by aliating exclusively with the smaller rather than the larger distributor. Since an exclusive aliation with the larger distributor guarantees that a larger number of consumers have access to the content, the upstream provider never aliates exclusively with the smaller distributor. Finally, price discrimination hurts industry protability more strongly when the content provider multi-homes and makes multi-homing less likely.

Alternative contracting games

In the baseline model, the content provider has all the bargaining power and makes take-it-orleave-it oers to the distributors. In addition, the contract species whether the agreement is exclusive or not. We here study two alternative contracting games.

Bidding game

In this section, we consider bidding game similar to that used in [START_REF] Hagiu | Exclusivity and Control[END_REF] (see Section II of the Online Supplementary Material for the proof ). Distributors compete for content access by oering menus of prices contingent on whether content rights are sold on an exclusive or non-exclusive basis. Since each of them can unilaterally rule out multi-homing, distributors face a coordination problem; both exclusive and non-exclusive equilibria exist when multi-homing maximizes expected industry prots. We select among the multiple equilibria via the Paretoundominated solution (for the distributors) and show that the allocation of content across distributors is similar to that of the baseline model. However, the lump-sum transfers, and hence the benet from being the dominant distributor, are dierent. We show that the value of a large customer base is equal to the Stage II expected prot of a large exclusive distributor minus the Stage II expected prot of a small exclusive distributor. Since an exclusive distributor faces more intense competition when it has a larger customer base, the larger distributor may value the exclusivity less. Competition over content access can therefore reduce the value of having a large customer base. This negative eect can be strong enough for the largest customer-base distributor to be worse o. Specically, we show that distributors are better o with the smallest customer base when the captive-customer base of the larger distributor is not too large and the content is of low-quality. [START_REF] Klemperer | Entry Deterrence in Markets with Consumer Switching Costs[END_REF] and Schmidt (2010) have already pointed out several reasons to explain why rms can be made worse o having a larger customer base.

30 Our result stems purely from the type of competition over content rights. Indeed, a larger customer base is always benecial when the content provider makes the initial oers to the distributors. However, the reverse can be true when it is the distributors who initiate contract oers.

30 [START_REF] Klemperer | Entry Deterrence in Markets with Consumer Switching Costs[END_REF] shows that it can be protable for an incumbent to reduce the size of its customer base in order to deter entry. Schmidt (2010) underlines that, as a rm's customer base grows, its rival competes more ercely in order to attract new customers. As prices are strategic complements, the intensity of competition rises when the ex-ante market split is more skewed. A rm may therefore have less interest in building up its customer base, as a way of moderating its rival's pricing behavior.

Discriminatory xed-fee policy

In this section, content rights are sold using a xed-fee policy (see Section III of the Online Supplementary Material for the proof ). The content provider announces two fees (T l , T s ), which do not specify whether the agreement is exclusive or not. Distributor i obtains the content rights if it is willing to pay the fee T i , with i ∈ {l, s}. When it unilaterally deviates from a non-exclusive equilibrium, a distributor will be left with its individual excluded Stage-II prot.

In that case, the cost of rejecting an oer is similar to that in the baseline model. By contrast, any exclusive equilibrium must be such that only one of the distributors accepts the oer. Given that distributor j rejects the oer T j , distributor i will make a prot π 0 i and no content will then be distributed if it also rejects the oer. Since π 0 l = π s l and π 0 s ≥ π l s , this selling format reduces the bargaining power of the content provider only when it wants to aliate exclusively with the smaller distributor. Indeed, the cost of a deviation from an exclusive equilibrium for the smaller distributor is now lower. Compared to the baseline model, the smaller distributor is therefore less likely to obtain exclusive rights to content.

Advertising revenues

In the baseline model, neither the content provider nor distributors collect advertising revenue from end users. Advertising is, however, an important source of revenue for entertainment and media-content providers. For instance, sports organizations earn revenues from advertisements placed directly in arenas or through mobile App advertising. Movie studios and channel networks have prolically used product placements to target consumers. In addition, even with considerable heterogeneity across pay-TV networks, most distributors earn a combination of aliation and advertising revenues. We here study two extensions, allowing for either the content provider or the distributors to earn revenue from advertising.

31 To simplify the analysis, the per-user advertising revenue is assumed to be constant.

31 [START_REF] Bagwell | The Economic Analysis of Advertising, volume 3 of Handbook of Industrial Organization[END_REF] provides a survey on the economics of advertising.

32

Accepted manuscript / Final version

Advertising revenue collected by the content provider

In this section, the content provider receives advertising revenue depending on exposure to end (see Section IV of the Online Supplementary Material for the proof ). Its advertising revenue increases (linearly) with the number of customers with access to the content. When choosing its content-distribution strategy, the upstream provider must now take account of the relative loss in terms of market exposure to some end users. Higher per-user advertising revenue raises the value of reaching the entire market, and therefore promotes multi-homing. Since content exposure is greater if the upstream provider is exclusive to the larger rather than the smaller distributor, higher per-user advertising revenue fosters exclusive aliation with the larger rather than the smaller distributor.

Advertising revenue collected by the distributors

In this section, distributors nance themselves through subscription fees and advertising (see Section V of the Online Supplementary Material for the proof ). Since the market is fully covered, aggregate advertising revenues are independent of the form of distribution. However, distributors have more incentive to compete for price-sensitive customers as the per-user advertising revenue increases. We show that this competition-enhancing eect is stronger when the larger distributor has exclusivity. It follows that expected industry prots increase with the per-user advertising revenue, but to a lower extent when the content provider is exclusive to the larger distributor.

Greater per-user advertising revenue therefore makes exclusivity with the larger distributor less likely.

Vertical Integration

The baseline model studies the incentives for an independent content provider to deal exclusively or non-exclusively with distributors. In this section, a vertically-integrated distributor chooses whether to supply its content to its downstream rival (see Section VI of the Online Supplementary Material for the proof ). The vertically-integrated distributor can deny access to its content, sell its 6.5.2 Small switching costs

In the baseline model, consumers are either fully locked-in or free to switch supplier at no cost. In section, we consider an alternative approach in which some consumers may pay to switch in equilibrium (see Section VIII of the Online Supplementary Material for the proof ). Distributors still oer homogeneous products but the switching costs are now continuously distributed over a range including zero. Consistent with the baseline model, increased competition in the downstream market measured as a lower average switching cost promotes exclusivity.

Heterogeneity in the value of regular content

In the baseline model, excluding content, distributors oer services of similar quality. In this section, we consider distributors with dierent values of regular content (see Section IX of the Online Supplementary Material for the proof ). A rise in v i has a greater eect on expected industry prots when distributor i can monetize this quality improvement over more customers. Since exclusive content access strengthens the distributor's ability to reach more customers, distributor i can more easily monetize this quality improvement if it has exclusivity. It follows that a rise in v i positively aects expected industry prots to a greater extent when distributor i has exclusivity, and thus promotes exclusivity with distributor i.

Independent captive-customer bases

In the baseline model, a distributor increases its base of locked-in customers either if its initial market share or if the mass of locked-in customers rises. In both cases, the base of locked-in customers of its rival is also altered. In other words, K l and K s are interdependent. However, in reality, a distributor could increase its base of locked-in customers without aecting that of its rival. In this section, we characterize the equilibrium outcome in Stage I as a function of K l and K s , assuming that the bases of locked-in customers are not interdependent (see Section X of the Online Supplementary Material for the proof ). This extension can encompass two dierent 35

Accepted manuscript / Final version scenarios.

• Heterogeneity in lock-in eciency. Distributors can dier in their eciency in retaining customers (i.e., i = α j ). Greater eciency in retaining customers allows a distributor to broaden its own base of locked-in customers without aecting that of its rival.

• New uncommitted consumers. There is a unit mass of consumers, but with an inow of new uncommitted consumers, µ, such that µ = 1 -(σ i + σ j ) and σ i + σ j ≤ 1. An increase in the initial market share of one of the distributors no longer necessarily reduces the base of locked-in customers of its rival.

The main ndings of the baseline model are qualitatively unchanged. The comparative statics analysis is, however, slightly dierent. In the baseline model, greater asymmetry in market shares strengthens competition when the larger distributor has exclusive high-quality content, which in turn encourages exclusivity with the smaller rather than the larger distributor. Here, an increase in the base of locked-in customers of one distributor reduces the number of price-sensitive customers, without aecting the base of locked-in customers of its rival. When the base of locked-in customers of the larger distributor rises, the smaller distributor has therefore less incentive to compete for price-sensitive customers. In this case, the larger distributor can more easily supply the content on a larger scale, without facing stier competition. As a result, when there are few locked-in customers, the larger distributor is more likely to obtain exclusive rights to content if its base of locked-in customers rises.

6.8 Initial Period -competition for market share.

In this section, we study an emerging-market phase in which there is no content provider and distributors are symmetric (see Section XI of the Online Supplementary Material for the proof ).

Distributors anticipate the future entry of a content provider and its content-distribution strategy.

In equilibrium, rst-period prices decrease with the value of having a large customer base in the mature market phase. When distributors benet from being the dominant player in the mature 36

Accepted manuscript / Final version market, prices and prots in the initial phase of market development are expected to be lower than they would be in a without switching costs. The reverse holds when competition for content exclusivity in the mature market makes the value of a large customer base negative.

Conclusion

This paper has analyzed content-distribution strategies in the presence of locked-in customers in the downstream market. An upstream provider sells the rights for its content using lump-sum fees and either aliates with one distributor exclusively or multi-homes. We have shown that the content provider multi-homes if there is a suciently-high proportion of locked-in customers. For intermediate values of the percentage of locked-in customers, the content provider multi-homes if content quality is not too high and otherwise aliates exclusively with the larger distributor.

When few customers are locked-in, the content provider sells its rights on an exclusive basis.

Because competition among distributors is ercer when the content provider aliates exclusively with the larger rather than the smaller distributor, exclusivity to the smaller distributor may arise in equilibrium. More specically, the content provider aliates exclusively with the smaller distributor if the mass of locked-in customers and content quality are not too high. Finally, we have also shown that the value of a large customer base is positive and greater than if there were no content provider.

From a policy perspective, our results bear two policy implications. They rst suggest that lower switching costs will not necessarily be consumer-welfare enhancing. Indeed, lowering switching costs may harm consumers by inducing the content provider to switch from non-exclusive to exclusive distribution. Second, they underline that a dominant player will not necessarily monopolize all the content items.

The nding that a greater number of locked-in customers promotes multi-homing is robust to a number of changes in our baseline model, but other results crucially rely on specic assumptions. In particular, in the baseline model, distributors adopt uniform pricing and the content provider makes if w ≤ ws . Since ws ≤ v, each term in ( 5) is positive and M (w) 0 if w ≤ ws . Therefore, M (w) ≥ 0 for any w.

D Relative dierence in competition intensity under exclusivity

Assume that τ ≥ τ and let F s-l i-j (w) = F s i (w)-F l j (w), with i = j {l, s}. Since ws ≤ wl , integration by parts implies that E w l j -E (w s i ) = ws 0 F s-l i-j (w) dw + wl ws 1 -F l j (w) dw.

(6)

Since F l j (w) ≤ 1, F s-l i-j (w) ≥ 0 is a sucient condition to guarantee that 0 ≤ E w l j -E (w s i ) for i = j ∈ {l, s}. We rst prove that F s-l i-j (w) ≥ 0 for i = j ∈ {l, s}.

(i) F s-l s-l (w) =

(K l -Ks)w (1-K l -Ks)(v-w)

, and thus F s-l s-l is positive.

(ii) F s-l l-s (w) =

Bv×v+Bw×w (1-K l )(1-Ks)(1-K l -Ks)(v+τ -w) , with B v = (K l -K s ) (2 -K l -K s ) (1 -K l -K s ) ≥ 0 and B w = -(1 -K l ) (K l -K s ) (1 -K s ) ≤ 0. B v × v + B w
× w decreases with w and B v × v + B w × ws = (K l -K s ) (1 -K s ) (1 -K l -K s ) v ≥ 0. Thus, F s-l l-s (w) is positive.

F s-l i-j (w) ≥ 0 and thus E (w s i ) ≤ E w l j for i = j ∈ {l, s}. Since p i i = v + τ -w i i and p j i = v -w j i , E (p s i ) /E p l j ≥ 1 with i = j ∈ {l, s}. Let E s-l i-j = E (p s i ) /E p l j . We now show that E s-l i-j decreases with τ for i = j {l, s}.

(iii) E s-l s-l = v+τ -E(w s s ) v+τ -E(w l l )

. E (w s s ) and E w l l are independent of τ and E w l l ≥ E (w s s ). It follows that E s-l s-l decreases with τ . Accepted manuscript / Final version that d E w l s -E (w s l ) /dτ ≤ 0 will imply that E s-l decreases with τ . We now show that d E w l s -E (w s l ) /dτ ≤ 0.

Using equation ( 6) and given that ws and wl are independent of τ , d E w l s -E (w s l ) /dτ = ws 0 dF s-l l-s (w) dτ dw -wl ws dF l s (w)

dτ dw, with dF s-l l-s (w) /dτ =

Av×v+Aw×w

(1-K l -Ks)(v+τ -w) 2 , A w = (K l -K s ) ≥ 0

and A v = (1 -K l -K s ) 1-K l 1-Ks -1-Ks 1-K l ≤ 0 . Since A v × v + A w × w increases with w and

A v × v + A w × ws = v × Ks(1-Ks)-K l (1-K l )
1-K l ≤ 0, dF s-l l-s (w) /dτ ≤ 0. In addition, F l s increases with τ . It follows that d E w l s -E (w s l ) /dτ ≤ 0 and thus E s-l l-s decreases with τ .

Items (iii) and (iv) imply that E (p s s ) /E p l l and E (p s l ) /E p l s decrease with τ .

E Lemma 3

First, we analyze the conditions under which expected industry prots are higher when the content provider (a) aliates exclusively with the larger rather than the smaller distributor and (b) multihomes rather than providing exclusive access to one distributor. Then, we determine which contentdistribution strategy maximizes expected industry prots.

(a) Exclusivity for the larger or smaller distributor. Let Λ c 1 ,c 2 = π c 1 l + π c 1 s -(π c 2 l + π c 2 s )

with c 1 = c 2 ∈ {l, s}, τ d 1 =

(2-3α+α 2 )v 1-α+α 2 σ-α 2 σ 2 and α x = 1+σ- √ 5σ 2 -4σ+1 σ(3-2σ)

. Expected industry prots are higher when the content provider aliates exclusively with the larger rather than the smaller distributor only if Λ l,s ≥ 0. Λ l,s is continuous in τ , Λ l,s τ =0 = 0 and dΛ l,s dτ =

     σ(2σ-3)α 2 +2(σ+1)α-2 1+ασ-α ≡ Υ inf Λ l,s if τ < τ (2σ -1) α ≡ Υ sup Λ l,s if τ > τ ,
with Υ sup Λ l,s ≥ 0. If α ≤ α x : Υ inf Λ l,s < 0, Λ l,s ≥ 0 if τ ≥ τ d 1 and Λ l,s < 0 otherwise. If α > α x :

Υ inf Λ l,s ≥ 0 and thus Λ l,s ≥ 0. Accepted manuscript / Final version and α < α x . Expected industry prots are higher when the content provider multi-homes only if Λ l,m and Λ s,m both negative. Λ s,m > 0 if α < α and Λ s,m ≤ 0 otherwise. Therefore, multi-homing never maximizes expected industry prots if α < α.

Assume that α ≥ α and so Λ s,m ≤ 0: multi-homing maximizes expected industry prots if Λ l,m is also negative. To analyze the sign of Λ l,m , we need to do the distinction between smaller and larger values of τ .

• Assume that τ < τ . Λ l,m ≤ 0. (1-ασ)((1-σ+σ 2 )α 2 -2α+1) .

• Assume that τ ≥ τ . Λ l,m = c τ × τ + c v × v with c v = α(1-σ)(2-α)
The content-distribution strategy that maximizes expected industry prots.

  Figure 1 depicts the equilibrium outcome in Stage I as a function of the mass of locked-in customers α (on the horizontal axis) and the level of content quality τ (on the vertical axis). The dashed line represents ᾱ. L, S and M correspond to exclusivity with the larger distributor, exclusivity with the smaller distributor and multi-homing, respectively.

Figure 1 :

 1 Figure 1: The equilibrium outcome in Stage I

  the value of having a large customer base as ∆ c = Π c l -Π c s . Distributors are better o with the largest customer base if ∆ c ≥ 0. Since the content provider can threaten distributors to award the content rights to their rival if they reject the oer submitted to them, distributors are left with their individual excluded Stage-II prot regardless of the content-distribution outcome. It follows that Π c i = π j i for i = j ∈ {l, s} and c ∈ {m, l, s}. The value of a large customer base is therefore equal to the Stage II prot of a large excluded distributor minus the Stage II prot of a small excluded distributor, ∆ c = π s l -π l s ≡ ∆. We can decompose the value of a large customer base as follows:

  the equilibrium outcome in Stage I as a function of market-share asymmetry (on the horizontal axis) and the level of content quality (on the vertical axis) for α ∈ {0.4, 0.6} and v = 1: L, S and M correspond to exclusivity with the larger distributor, exclusivity with the smaller distributor and multi-homing, respectively. Then, Figure 3 shows the consumer-welfare eect of a fall in α, as a function of market-share asymmetry (on the horizontal axis) and the level of content quality (on the vertical 28 See Directive 2009/136/EC, Article 30. 29 See https://apps.fcc.gov/edocs_public/attachmatch/DOC-329161A1.pdf for the full formal speech from September 2014. 27 Accepted manuscript / Final version axis), with v = 1.

Figure 2 :Figure 3 :

 23 Figure 2: The equilibrium outcome in Stage I for = 0.4 and α = 0.6

.

  Since 0 < E p l s ≤ E (p s l ) and dE (w s l ) /dτ ≤ 0, dE(w s l ) dτ E (p s l ) -E p l s ≤ 0. It follows 43

  (b) Exclusive distribution vs. Non-exclusive distribution. Let Λ c 1 ,c 2 = π c 1 l + π c 1 s -(π c 2 l + π c 2 s ) with c 1 = c 2 ∈ {m, l, s}, α = 2σ+1- √ 8σ 2 -4σ+1 2σ(2-σ) and ᾱ = 1-√ σ(1-σ) 1+σ 2 -σ with 0 < α ≤ ᾱ < 1 44

  < 0 and c τ = α 2 (1-σ+σ 2 )-2α+1 1-α+ασ . When α ≥ ᾱ, c τ ≤ 0 and so Λ l,m ≤ 0. When α ∈ [α, ᾱ), c τ > 0, Λ l,m increases in τ and is negative if τ = τ . It follows that Λ l,m ≤ 0 if τ ∈ τ , τ d 2 and Λ l,m > 0 if τ ≥ τ d 2 with τ d 2 = -cv×v cτ = α(2-α)(σ(1-ασ)-(1-σ)(1-α+ασ))v

  Multihoming never maximizes expected industry prots if α < α and always maximizes expected indus- try prots if α ≥ ᾱ. When α ∈ [α, ᾱ), either multi-homing or exclusive aliation with the larger distributor maximizes expected industry prots. The content quality threshold that determines which content-distribution strategy maximizes expected industry prots is τ d 1 if α < α and τ d 2 if α ∈ [α, ᾱ). It follows that multi-homing maximizes expected industry prots when α ∈ [α, ᾱ) and τ ≤ τ d 2 or when α ≥ ᾱ. Exclusive aliation with the larger distributor maximizes expected industry prots when α ∈ (0, α) and τ > τ d 1 or when α ∈ [α, ᾱ) and τ > τ d 2 . Finally, exclusive aliation with the smaller distributor maximizes expected industry prots when α ∈ (0, α) and τ ≤ τ d 1 . This completes the proof of Lemma 3.

  Lemma 2 shows that the content provider chooses the content-distribution strategy that maximizes expected industry prots in Stage II. As the content provider makes no revenue in Stage II, expected industry prots are simply the sum of distributors' prots in Stage II. The next lemma characterizes the content-distribution strategy that maximizes expected industry prots in Stage II.Lemma 3 Multi-homing maximizes expected industry prots if α ≥ ᾱ or if α ∈ [α, ᾱ) and τ ≤ τ d 2 . Expected industry prots are maximized when the content provider aliates exclusively with the larger distributor if α ∈ [α, ᾱ) and τ > τ d 2 or if α ∈ (0, ᾱ) and τ > τ d 1 ; and when the content provider aliates exclusively with the smaller distributor if α ∈ (0, α) and τ ≤ τ d 1 .

  Finally, a number of examples conrm that the dominant players do not monopolize all the content items. For instance, beIN Sports USA (a global network of sports channels), which was launched in 2012 in the United States, acquired the exclusive rights to broadcast sport events such as Soccer, Women's Tennis and Motorsports road racing (which are secondary sports in the United States). This strategy of exclusive content acquisition has allowed

beIN Sports USA to progressively catch up with its competitors. In mobile telecommunications markets, providers often oer exclusive content delivered through carrier-specic Apps. Leading mobile internet service providers secure most of the exclusive high-quality content rights, they however do not monopolize all of the content.

26

We explore a new motive that can promote multi-homing, namely the presence of locked-in customers. In our model, distributors compete in a covered market and 3 Following[START_REF] Segal | Contracting with Externalities[END_REF], we dene a contract as ecient if the allocation of content that arises in equilibrium is the one that maximizes industry prots subject to the pricing game.4[START_REF] Armstrong | Competition in the Pay-TV Market[END_REF] shows that exclusivity (resp., multi-homing) occurs if the content provider sells its rights on a lump-sum (resp., per-subscriber) basis. The other articles assume that content rights are sold on a lump-sum basis. D'Annunzio (2017) studies the eects of vertical integration on investment in content quality. Exclusivity is always the preferred choice of the content provider, regardless of the vertical structure of the industry.[START_REF] Hagiu | Exclusivity and Control[END_REF],[START_REF] Weeds | TV Wars: Exclusive Content and Platform Competition in Pay TV[END_REF] and[START_REF] Stennek | Exclusive quality -Why exclusive distribution may benet the TV-viewers[END_REF] show that the content is provided non-exclusively if the upstream provider collects suciently high revenue from end users. For instance, the content provider can receive additional revenue from selling its goods directly to consumers or from advertising.[START_REF] Jiang | Competition of Content Acquisition and Distribution Under Consumer Multipurchase[END_REF] demonstrate that consumer's multi-purchase promotes exclusivity. In[START_REF] Armstrong | Competition in the Pay-TV Market[END_REF],[START_REF] Weeds | TV Wars: Exclusive Content and Platform Competition in Pay TV[END_REF] and D'Annunzio (2017), the content provider has all the bargaining power and makes sequential take-it-or-leave-it oers to the distributors. In case the distributor that receives the former oer rejects it, the content provider grants the exclusive rights to the other distributor. As it may not be in its interests to do so, a strong commitment power by the content provider is required. In[START_REF] Hagiu | Exclusivity and Control[END_REF], distributors oer exclusivity-contingent contracts to the content provider.Distributors face a coordination problem, and both exclusive and non-exclusive equilibria exist when multi-homing maximizes industry prots. They select among multiple equilibria via the Pareto-undominated solution.[START_REF] Stennek | Exclusive quality -Why exclusive distribution may benet the TV-viewers[END_REF] analyzes a bargaining game with alternating oers in which contracting parties have outside options. In[START_REF] Jiang | Competition of Content Acquisition and Distribution Under Consumer Multipurchase[END_REF], the content provider sells its rights with a xed-fee contract. At the exclusive equilibrium point, only one of the distributors accepts the oer. This distributor knows that if it rejects the oer, the content rights will not be allocated to any distributor. This selling format reduces the bargaining power of the content provider when it wants to induce exclusivity, and may lead to inecient allocation of content.

This setup, where some consumers have suciently high switching costs such that they are locked-in, is commonly used in the literature (e.g.,[START_REF] Padilla | Mixed pricing in oligopoly with consumer switching costs[END_REF][START_REF] Hagiu | Pricing and commitment by two-sided platforms[END_REF][START_REF] Schmidt | Price competition and innovation in markets with brand loyalty[END_REF][START_REF] Villas-Boas | A Short Survey on Switching Costs and Dynamic Competition[END_REF].7Accepted manuscript / Final version

For the sake of brevity, the precision in a stochastic sense will not be specied anymore hereafter.12 Accepted manuscript / Final version

The value of non-exclusive distribution is obvious when we consider the limiting case as the mass of locked-in customers approaches one: expected industry prots tend to v + τ if content is present on both distributors and are below v + στ if content is exclusive to one distributor.

For instance, with poor content quality, price-sensitive customers are more likely to purchase from a small excluded distributor rather than from a large exclusive distributor.

As long as τ is lower than τ , a large exclusive distributor and a small excluded distributor oer higher net utilities as τ rises.

When content is of low-quality, exclusivity with the larger distributor never maximizes expected industry prots. The case where the larger distributor has exclusive low-quality content (lower than τ ) is therefore of no importance.

tends to innity as α approaches ᾱ, there exists α * such that dτ d 2 /dα < 0 if α ∈ (α * , ᾱ).

For instance, in 2019, Verizon holds exclusive deals with major U.S. Sports Leagues (e.g. the NFL and the NBA), while T-mobile, the third-largest wireless carrier in the United States (with a customer base only half that of Verizon or AT &T ), has signed a content-distribution deal with Viacom Media Networks.

Accepted manuscript / Final version the oers to the distributors in the contracting stage. When distributors can price-discriminate between their loyal and the others, exclusivity with the smaller distributor never occurs.

In addition, distributors can be worse o with the largest customer base if they are bidding and competing for content access in the contracting stage.

I thank participants to the ANR-DFG workshop in Düsseldorf (2015) and to the conference on the economics of information and communication technologies in Paris (2015). I gratefully acknowledge support from the Agence Nationale de la Recherche and the Deutsche Forschungsgemeinschaft for the French-German cooperation project "Competition and Bargaining in Vertical Chains".

Accepted manuscript / Final version the smaller distributor less valuable. On the other hand, greater asymmetry in market shares strengthens competition when the larger distributor has exclusive high-quality content (higher than τ ), and undermines competition otherwise. Contrary to the former eect, the latter reduces the value of an exclusive aliation with the larger distributor.

21 Overall, the latter eect is likely to prevail over the former. The threshold values identied in Lemma 3 vary with the marketshare asymmetry as follows: (i) α falls with σ and ᾱ rises with σ, (ii) τ d 1 and τ d 2 increase with σ, except when α is close to ᾱ. 22 Unless τ is very high, multi-homing is more likely to maximize expected industry prots with more unevenly-distributed market shares. Exclusivity with the smaller distributor maximizes expected industry prots within a tighter interval of α when σ rises.

Within this interval, however, there is a wider range of content-quality levels for which exclusivity with the smaller distributor maximizes expected industry prots. When there are few locked-in customers, exclusivity with the smaller distributor is therefore more likely to maximize expected industry prots with more unevenly-distributed market shares. Finally, note that α = ᾱ = 2 3 if σ = 0.5: in case the market split is symmetric, multi-homing maximizes expected industry prots if there are two thirds or more locked-in customers, otherwise exclusivity maximizes expected industry prots.

We dene τ d such that

ᾱ). As we know from

Lemma 2, the upstream provider chooses the content-distribution strategy that maximizes expected industry prots. Hence, we can state the following:

Proposition 3 There exist values 0 < α ≤ ᾱ < 1 and τ d such that if:

• α ∈ (0, ᾱ) and τ > τ d , C will aliate exclusively with the larger distributor in equilibrium;

• α ∈ (0, α) and τ ≤ τ d , C will aliate exclusively with the smaller distributor in equilibrium;

• α ∈ [α, ᾱ) and τ ≤ τ d or α ≥ ᾱ, C will multi-home in equilibrium.

Accepted manuscript / Final version is represented by the dashed line), and otherwise the larger distributor.

In the following discussion, we focus on the scenario in which lower α induces the content provider to switch from multi-homing to exclusivity. Lower α reduces the distributors' market power over locked-in customers, but promotes vertical dierentiation (through exclusivity).

Whether lower α reduces or increases consumer welfare depends on content quality, and also on the initial market split. When exclusivity produces little vertical dierentiation among distributors, the intensity of price competition is mostly aected by α: lower α increases consumer welfare when content quality is low (this corresponds to A 1 in Figure 3). For mid-quality content, the benets of lower α for consumers are more than oset by the higher rent-extraction power of distributors induced by exclusivity, and hence lower α reduces consumer welfare (see A 2 in Figure 3). Once content quality is above τ d,0.4 1 , lower α leads to exclusive aliation with the larger distributor.

Although the transition from multi-homing to exclusivity generates high vertical dierentiation among distributors, lower α may benet consumers (see A 3 and A 4 in Figure 3). In particular, lower α enhances consumer welfare if the initial market split is suciently asymmetric. This is because greater asymmetry in market shares strengthens competition and thus limits the rentextraction power of distributors when the content provider is exclusive to the larger distributor.

Locked-in customers and total welfare. We now turn to the analysis of total welfare, which is much simpler. Total welfare, dened as the sum of consumer surplus and industry prots, is

Since the market is fully covered and switching does not induce any costs (either customers are fully locked-in or can switch at no cost), total welfare only depends on the average value of the services oered by both distributors. Total welfare is therefore maximized when the mass of locked-in customers is suciently high to bring about multi-homing.

To summarize, a decrease in α improves consumer welfare without changing total welfare when the content provider does not switch from multi-homing to exclusivity. However, when it induces the content provider to switch from multi-homing to exclusivity, a decrease in α deteriorates total welfare, and can either benet or harm consumers. In this latter case, a fall in switching costs is

Accepted manuscript / Final version less likely to be consumer-welfare enhancing with high-quality content. content non-exclusively (i.e., retaining the rights for itself as well) or give up its content rights.

the latter case, its rival becomes the exclusive distributor. The industry structure that emerges is still that which maximizes expected industry prots. The allocation of content across distributors is therefore unaected by the vertical structure of the industry.

Alternative consumer demand specications

We here consider alternative demand specications, assuming that distributors are symmetric with respect to their initial market share.

Multi-homing consumers

In the baseline model, consumers purchase from exactly one of the two distributors. Consumers, however, frequently buy multiple competing services or products. In this section, consumers can purchase from both distributors (see Section VII of the Online Supplementary Material for the proof ). Under non-exclusive distribution, joint consumption does not provide any incremental surplus and expected industry prots are unchanged compared to the baseline model. By contrast, an exclusive distributor can now induce consumers that are locked-in with its rival to purchase its service. It can then serve the entire demand by oering a suciently high net utility. Indeed, the locked-in customers of the excluded distributor multi-home if the exclusive distributor oers a net utility of v (or more). In this case, there is no opportunity for its rival to attract pricesensitive customers, which in turn competes less ercely. Compared to the baseline model, the excluded distributor oers more frequently low net utility, while the demand served by the exclusive distributor is likely to be greater. Exclusivity can therefore yield higher expected industry prots when consumer multi-purchase. In line with [START_REF] Jiang | Competition of Content Acquisition and Distribution Under Consumer Multipurchase[END_REF], consumer multi-purchase promotes exclusivity.

Appendix

A Proposition 1 Let F c s-l (w) = F c s (w) -F c l (w), with c {m, l, s}.

Item (i). F m l (0) > 0 if σ = 0.5 and F m l (0) = 0 if σ = 0.5. Hence, the larger distributor has a mass point at

Item (ii). F s l (0) > 0 if σ = 0.5 or τ = 0 and F s l (0) = 0 otherwise. Hence, the larger distributor has a mass point at 0 if σ = 0.5 or τ = 0. Assume that w ∈ (0, ws ]:

. Since w ≤ ws < v,

Hence, the smaller distributor has a mass point at 0 if τ > τ . Assume that w ∈ 0, wl :

Item (iv). F l l (0) > 0 and the larger distributor has a mass point at 0 if τ < τ .

This completes the proof of Proposition 1.

B Proposition 2

Item (i).

• dF m l /dτ = -w (v+τ -w) 2

Ks 1-K l -Ks + K l -Ks 1-Ks < 0 and dF m s /dτ = -wK l

(1-K l -Ks)(v+τ -w) 2 < 0.

• Assume that τ < τ . dF l l /dτ = -(1-K l ) (1-Ks)(v-w) < 0 and dF l s /dτ = -wK l

(1-K l -Ks)(v+τ -w) 2 < 0.

This completes the proof of Proposition 2.

C Competition for price-sensitive customers under exclusivity

Price-sensitive customers buy from the distributor oering the higher net utility and so obtain a net utility w c h = max {w c l , w c s }. Let J c (w) be the cumulative distribution of w c h given that the distribution of content is c ∈ {m, l, s}: J c (w) = F c l (w) × F c s (w). A higher J means that pricesensitive customers receive a high net utility with a higher probability. Let M (w) = J s (w)-J l (w).

In order to prove that price-sensitive customers enjoy higher net utility if the content provider aliates exclusively with the larger rather than the smaller distributor, we show that M (w) ≥ 0 for any w. Since ws ≤ wl , it is straightforward that M (w) = 0 if w > wl . If w ∈ ws , wl , J l (w) ≤ 1 and J s (w) = 1 and thus M (w) ≥ 0. τ < τ α(2σ-1)w(τ (1-α+σ(1-σ)α 2 )+v((1-α)α+2σ(1-σ)α 2 ))