Optimal Auction Design in Two-Sided Markets∗ Renato Gomes Toulouse School of Economics [email protected] This Version: July 2011

Abstract This paper investigates the optimal design of advertising auctions such as those used by search engines to sell space to advertisers. A key feature of these auctions is that the value of the advertising position depends on the searchers’ expectation that the selected advertiser is relevant to their needs. The revenue-maximizing mechanism is a scoring auction that works as a crosssubsidization device between searchers and advertisers. Relative to efficiency, profit-maximization selects lower quality advertisers, and searchers make fewer clicks. The implications for competition policy are subtle. Circumstances are characterized in which an increase in competition between auction platforms decreases welfare. JEL classification: D82 Keywords: auctions, two-sided markets, online platforms, adverse selection, incentives.



A previous version of this paper circulated with the title “Mechanism Design in Two-Sided Markets: Auctioning

Users”. I would like to thank my doctoral committee Alessandro Pavan, Marco Ottaviani, Rakesh Vohra and William Rogerson for their support and encouragement, as well as Jacques Crémer, Yassine Lefouili, Carlos Madeira, Shiran Rachmilevitch, James Schummer, Ron Siegel, Michael Whinston and Asher Wolinsky for very helpful conversations. I benefited from discussions at seminars and conferences at Northwestern, Cornell, Bonn, Maastricht, MIT Sloan, Arizona State, Boston University, Toulouse, Bocconi, Collegio Carlo Alberto, Puc-Rio, ZEW Conference on Platform Markets 2010, ESWC 2010 and INTERTIC 2010. The usual disclaimer applies.

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Introduction

In the last decade, a growing number of media companies have turned to auctions for selling advertising space. In the online world, search engines run billions of simultaneous auctions for selling the “sponsored links” displayed alongside the search results associated to searchers’ queries. The three largest contenders in the market for sponsored search advertising (Google, Yahoo! and Microsoft Bing) raised more than $30 billion in 2010 in online auctions for sponsored links. Recently, large online platforms (the so called Ad Exchanges) were created to centralize the trade of display advertising on the web, a market that raised more than $8 billion in 2010. These platforms run large online auctions that sell display advertising in web portals, blogs and commercial websites. In the offline world, a sizable group of newspapers, TV’s and radio stations have switched (at least partially) from the traditional agency-based model for selling advertising space to automated auction processes. In 2006, Google launched its Audio Ads program, which applied the technology of sponsored search auctions to the sale of advertising space in radio stations. Since 2007, Google AdWords runs daily auctions for selling advertising spots in several TV networks and Cable companies in the US. A key feature of these auctions is that the value of the advertising position depends on the searcher’s expectation that the selected advertiser is relevant to their needs: If searchers are not confident that the selected advertiser is worth a click, they make few clicks, and the value of the advertising position (and the platform’s profit) is reduced. In an analogous manner, TV viewers and radio listeners can engage in “ad avoidance” if the selection of advertising brings little value to them. In such a “two-sided” setting, what mechanism should a profit-maximizing platform use to sell advertising positions? What distortions arise relative to the first-best selection of advertisers? Does competition between multiple platforms increase welfare? In order to answer these questions, this paper studies auction design in a two-sided setting where the number of clicks for sale depends on the expectation that searchers have about the quality of the advertisers selected by the mechanism. Model. The baseline model, presented in Section 2, considers a monopolistic platform (let us say, Google) that has a single sponsored link to sell to one of many advertisers. The type of each advertiser is a multi-dimensional vector. The first component is the advertiser’s willingness to pay for a click on her link, reflecting the expected profit generated by an extra visit. The second component is the value that searchers obtain from clicking the advertiser’s link, reflecting the quality or relevance of the advertiser shared by all searchers. The first two components of each advertiser’s type are private information of the advertiser (not observed by the platform). In reality, however, platforms have access to some data about advertisers, such as the content of their websites and their clickthrough rates. To capture this source of information, we let the platform observe (jointly with each advertiser) a signal which is informative about each advertiser’s willingness to pay for clicks and value 1

to searchers. Searchers have heterogeneous opportunity costs of clicking on the sponsored link, and do so only if the expected value generated by the selected advertiser is greater than the opportunity cost of clicking. This leads to an upward sloping supply of clicks: The number of clicks for sale is high when searchers expect the platform to select an advertiser of great value to searchers. The platform’s problem is to design a mechanism to maximize profits, knowing that the number of clicks on the link depends on the searchers’ expectation about the value generated by the selected advertiser. The difficulty faced by the platform is that the advertiser with the highest willingness to pay per click is seldom the one that produces the greatest value to searchers. As a consequence, standard formats such as the first-price or second-price sealed-bid auctions are not profit-maximizing, since the supply of clicks induced by these mechanisms is too small. Cross Subsidization and Auction Design. Section 3 derives the main results of the paper under the fairly natural assumption that the advertisers’ willingness to pay for clicks is positively affiliated with their value to searchers (that is, the advertisers who are eager to pay more for a click are on average of greater value to searchers). An extension to the baseline model discusses how to relax this assumption. The platform’s problem is analyzed in two alternative settings. In the benchmark setting, the platform is allowed to make click-contingent transfers to searchers, so that it can charge or subsidize searchers on a per-click basis. The direct-revelation mechanism that maximizes profits selects the advertiser according to the following scoring procedure. The platform computes a score for each advertiser by linearly combining with equal weights the auction rents per click generated by the advertiser (that is, her “virtual” willingness to pay for a click) with an estimate of the advertiser’s value to searchers (based on her signal and willingness to pay only). The advertiser selected by the platform is the one with the highest score. Having the ability to run click-contingent transfers, the platform charges (or subsidizes) searchers for clicks at the optimum if and only if the auction rents produced by advertisers are smaller (or greater) than the value they generate to searchers, in expectation. For technological reasons, click-contingent transfers to searchers are seldom adopted in practice.1 The main result of this paper offers a characterization of the revenue-maximizing mechanism when click-contingent transfers to searchers are not possible. As in the benchmark setting, the revenue1

There are three main reasons (which are out of the scope of this work) for why most platforms do not

charge/subsidize searchers: First, moral hazard concerns (regarding the opportunistic behavior of searchers) prevent most forms of click-contingent subsidies. Second, many authors argue that setting up "micro payments" (e.g., a penny per click) would sharply decrease the number of clicks due to the decision-making costs (see Szabo (1996)) and the transaction costs involved (see Párhonyi, Lambert and Pras (2005)). Finally, Peha and Khamitov (2004) argue that high rates of identity theft and financial fraud againt searchers have induced many internet platforms to adopt business models that do not require searchers to make payments.

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maximization mechanism selects the winning advertiser according to a scoring rule that linearly combines the auction rents and the value to searchers generated by advertisers. However, the weights in the scoring rule now reflect whether the platform would (if possible) charge or subsidize searchers’ clicks. When subsidizing searchers for clicks is profit maximizing, the scoring rule assigns more weight to the expected value to searchers. Conversely, when charging searchers for clicks is profit maximizing, the scoring rule assigns more weight to the auction rents generated by advertisers. As such, the scoring rule applied by the profit-maximizing mechanism works as a cross-subsidization device that balances the goals of (i) preserving the “click supply” from searchers and (ii) extracting rents from advertisers. Interestingly, starting with Google in 2004, all three major search engines moved to scoring auctions in the spirit of the direct-revelation mechanism described above. Distortions Relative to Efficiency. The analysis of Section 4 reveals that, relative to efficiency, profit-maximization leads to greater exclusion on both sides of the market. On the advertiser side, customized reserve prices exclude advertisers whose auction rents and value to searchers are too small. On the searcher side, the supply of clicks is inefficiently low as a result of three effects. First, there is an informational channel: Because the advertisers’ willingness to pay for clicks is positive affiliated with their value to searchers, the scoring rule followed by the platform selects more frequently advertisers with low value to searchers. Intuitively, the affiliation assumption implies that advertisers with high value to searchers are more likely to have high willingness to pay for clicks. As such, in order to maximize profits, the platform must handicap advertisers with high values to searchers, in order to encourage those with high willingness to pay to submit higher bids. The second effect is the usual monopoly pricing distortion: The revenue-maximizing mechanism does not fully internalize the benefits enjoyed by searchers, but only the marginal revenue appropriated by the platform. Third, the absence of click-contingent transfers further reduces the click supply, since the platform has fewer instruments to cross-subsidize searchers’ clicks. Effect of Competition. Recently, the three major search engines announced plans of collaborating in search technology and advertising.2 The prospect of future mergers brings many new questions to the antitrust debate in online advertising. Namely, what is the welfare-maximizing market structure in a two-sided market where platforms use auctions to select advertisers? Can monopoly result in greater social welfare than duopoly? In order to answer these questions, Section 5 extends the baseline model to a duopolistic setting where advertisers are allowed to post bids on multiple platforms, while searchers click on the link of at most one platform. The timing is as follows. In the first stage, the platforms simultaneously announce what mechanisms they will use to select advertisers. In the second stage, searchers click 2

In June 2008, Google and Yahoo! issued a proposal according to which Google would deliver advertisements next

to Yahoo’s search results. In July 2009, Microsoft and Yahoo! announced a deal to share search technology and to join forces in search-generated advertising. The Department of Justice raised concerns about the anti-competitive effects of such partnerships, which were ultimately aborted.

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on the link offered by the platform that generates greater value (in expectation). As above, consider first the benchmark where click-contingent transfers to searchers are possible. In this case, since advertisers can participate in multiple mechanisms, a “competitive bottleneck” (in the sense of Armstrong (2006)) emerges. The platforms do not need to compete for advertisers, and follow in equilibrium the same mechanism as in a monopolistic market. In contrast, competition is fierce for searchers, who click on at most one link. As a result, platforms give back to searchers in the form of subsidies per click all rents collected in the advertiser side of the market. Because competition leads to an increase in the supply of clicks and to the same selection of advertisers as in a monopoly, a duopolistic market is welfare increasing. The situation is different when platforms cannot make transfers to searchers. In this case, the only instrument to compete for searchers is the mechanism used to select advertisers. In equilibrium, both platforms select the advertiser with the highest value to searchers (in expectation), and enjoy strictly positive profits (since auction rents are no longer dissipated by subsidies). Perhaps surprisingly, duopoly can result in lower welfare than monopoly. The reason is that in equilibrium both platforms forgo selecting advertisers with high willingness to pay in order to generate greater value to searchers. Depending on whether the distribution of advertisers’ willingness to pay for clicks is more dispersed than that of value to searchers, duopoly leads to a reduction in total welfare relative to monopoly. Indirect Implementation and Extensions. In line with the current practice of the major search engines and ad exchanges, Section 6 describes how to implement the mechanisms considered in the previous sections by means of scored versions of the first and second-price auctions. Section 7 discusses a number of important extensions: (i) How to relax the assumption that the advertisers’ willingness to pay and value to searchers are positively affiliated, (ii) how to extend the baseline model to the case where the platform has multiple advertising links to sell, and (iii) how to introduce searcher-customized mechanisms. Section 8 concludes. All proofs are collected in the appendix.

Related Literature This paper brings the theory of optimal auction design (e.g., Myerson (1981) and Riley and Samuelson (1981)) to a two-sided setting where the number of clicks for sale depends on the expectation that searchers have about the quality of the advertisers selected by the mechanism. In addition, this paper is related to the following strands of literature. Two-Sided Markets. The analysis of this paper hinges on the platform’s conflict between inducing participation and extracting rents from both sides of the market. A similar trade-off lies at the heart of the two-sided-markets literature (see Rochet and Tirole (2003, 2006), Armstrong (2006), Caillaud and Jullien (2001, 2003), Evans (2003), Hagiu (2006), Rysman (2009) and Weyl (2010)). This literature focuses on cross-network externalities, where the payoff from agents on one side of the market depends on the participation (total “quantity”) of agents on the other side. In contrast, 4

in this paper, while the advertisers’ payoffs increase with the total number of clicks, the clicking decision by searchers depends on the expected quality of the selected advertiser.3 Moreover, while the two-sided-market literature assumes that platforms choose linear prices or two-part tariffs, this paper studies optimal auction design. Position Auctions. The three major search engines use variations of the Generalized SecondPrice (GSP) auction to sell sponsored links. In its simplest version, this auction works as follows. Each advertiser submits one bid that represents her willingness to pay for a click on her link. Advertising positions are assigned in decreasing order of bids, and each advertiser pays per click the minimum bid necessary to secure her position. In a complete information setting, Aggarwal, Goel, and Motwani (2006), Varian (2007) and Edelman, Ostrovsky and Schwarz (2007) are the first to derive complete information Nash equilibria of this auction.4 Under incomplete information, Edelman, Ostrovsky and Schwarz (2007) modeled the GSP as an ascending clock-auction for multiple goods (they call it the Generalized English auction).5 The current paper departs from this literature by explicitly modeling searchers’ clicking behavior. In the optimal auction design problem analyzed here, the searchers’ expectation play a critical role in the platform’s profits. Online Platforms. Athey and Ellison (2010) derive optimal reserve prices in a model where advertisers bid for sponsored slots in a Generalized English auction, while searchers take clicking decisions rationally and follow a sequential search procedure. The approach of this paper differs from that of Athey and Ellison (2010) in two major aspects. First, Athey and Ellison focus on the Generalized English auction, while this paper studies optimal auction design. Second, in Athey and Ellison’s model, the willingness to pay for a click of advertisers is perfectly aligned to their value to searchers (that is, types are one-dimensional), while in this paper the advertisers’ types are multi-dimensional. As a result, the revenue-maximizing auction derived here is similar to the scoring auctions used by the major search engines, a feature that cannot be replicated by models where the advertisers’ types are one-dimensional.6 Rayo and Segal (2009) study the optimal disclosure rule by an Internet platform that sends signals to searchers about the quality of advertisers under commitment. Instead of information-disclosure 3

See Hagiu (2009), Gomes and Pavan (2011) and Veiga and Weyl (2011) for models of pricing in two-sided markets

that allow for more general forms of cross-network effects. In such models, the participation of agents on a given side is weighted by the “interaction quality” generated to agents on the opposite side. 4 See Borgers, Cox, Pesendorfer and Petricek (2008) for a complete information model where advertisers’ willingness to pay for a click is position-specific. 5 See Edelman and Schwarz (2010) for the optimal reserve prices of the GSP, and Gomes and Sweeney (2011) for a Bayes-Nash analysis of the GSP. 6 Abrams and Schwarz (2008) allow advertisers to choose the “quality” of their landing page while competing in a GSP auction. Abrams and Schwarz introduce a Pigovian tax in the GSP framework that aligns the advertisers’ and the platform’s interests regarding the provision of quality.

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rules, in the current paper the platform determines the selection of advertisers. Hagiu and Jullien (2009) derive the optimal level of search diversion by a platform that trades off consumer traffic with the extra rents that come from “accidental” shopping. Focusing specifically on sponsored search advertising, Chen and He (2006) and Xu, Chen and Whinston (2008) study models that integrate the advertisers’ pricing decisions with their bidding behavior in a sponsored search auction. These papers assume that the platform has complete information about the advertisers’ types, therefore ignoring the mechanism-design issues that are at the core of this work.7 Advertising. Anderson and Coate (2005) study a model of competition between TV channels that, in the first stage, decide which programs to broadcast and, in the second stage, decide the level (and prices) of advertising. The analysis of Anderson and Coate endogenizes the production of broadcasting and characterizes the market distortions (relative to efficiency) on the total level of advertising. Unlike Anderson and Coate (2005), the focus of the current paper is on the optimal mechanism for selling advertising positions, and on the distortions on the selection of advertisers generated by profit-maximization.8

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Model

The mechanism design problem considered in this paper captures important features of many media markets. For expositional reasons, the analysis focuses on the market of sponsored search advertising and, accordingly, describes the model in terms of advertisers and searchers.

2.1

Advertisers

The platform can select one among N advertisers to appear in the sponsored link associated to a particular search query. Advertisers are indexed by j ∈ N ≡ {1, ..., N }, where N is the set of all advertisers that compete for the link.

If a searcher (he) decides to click on the sponsored link when advertiser j (she) is the selected advertiser, he obtains the (possibly negative) value to searchers Uj . In turn, advertiser j has a willingness to pay for a click given by Vj . If the advertiser is an online retailer, one can think of the value to searchers as the consumer surplus and of the advertiser’s willingness to pay as the producer surplus generated by the transaction between the searcher and the advertiser.9 7

See White (2008) and Xu, Chen, and Whinston (2009) for models that include both organic and paid search results.

On the empirical side, see Jeziorski and Segal (2009) for a structural model of searchers’ behavior. Levin (2011) offers a comprehensive survey on the economics of internet markets. 8 A recent literature studies the effects of targeted advertising on prices and welfare. See, for example, Akçura and Srinivasan (2005), Iyer, Soberman and Villas-Boas (2005), Gal-Or, Gal-Or, May, and Spangler (2006), Esteban and Hernandez (2007), Galeotti and Moraga-González (2008), Johnson (2009), Athey and Gans (2010) and Bergemann and Bonatti (2010). 9 More realistically, the searchers associated with a particular query may be heterogeneous on the value they derive

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The realizations of Vj and Uj , vj and uj , are private information of advertiser j (and are not observed by the platform). Yet, in reality search engines observe the click history of advertisers, the content of the advertisers’ websites, and other attributes which contain information about (vj , uj ). To capture this source of information, we let the platform observe for each advertiser j the realization θj of a signal Θj correlated with (Vj , Uj ). The realization θj , which is also observed by advertiser j, captures the information about advertiser j that is available to the platform before the start of the bidding process.10 The triple tj ≡ (θj , vj , uj ) is the type of advertiser j and t ≡ (t1 , ..., tN ) is the profile of advertisers’ types.

The types of advertisers are independent draws from the tridimensional distribution F (θ, v, u), which has full support on T ≡ [θ, θ] × [v, v] × [u, u] and admits a continuous density f . Conditional on the realization of the signal Θ, the marginal distribution of of V is FV (v|θ) (and the density is

fV (v|θ)). As is standard in the mechanism design literature, assume that V satisfies the monotone hazard rate property for each realization of Θ, that is all θ ∈ [θ, θ].

fV (v|θ) 1−FV (v|θ)

is weakly increasing in v ∈ [v, v] for

For expositional convenience, the baseline model considers environments in which the following

property holds: Assumption 1 (Affiliation) (Θ, V, U ) are positively affiliated random variables.11 Assumption 1 implies that the advertisers with higher values to searchers uj are more likely to be willing to pay more for the searchers’ clicks (higher vj ). In particular, the assumption above contemplates the cases where Θj is independent of (Vj , Uj ) (uninformative signals), the case where Vj and Uj coincide (one-dimensional types), and the case where Θj perfectly reveals Uj (complete information on the value to searchers). Subsection 7.1 shows that the results of this paper continue to hold in many environments where Assumption 1 is not satisfied.

2.2

Selection of Advertisers

The platform follows a selection rule Z to select (in a possibly non-deterministic manner) an advertiser as a function of the profile of advertisers’ types. The rule Z can be conveniently represented by the from visiting advertiser j’s website (let searcher i obtain Uji ). In this case, Uj can be interpreted as the average value to searchers associated with advertiser j, that is Uj ≡ E[Uji ]. Analogously, the value Vj can be seen as the average

profit from a searcher visit (as not all visits result in transactions). Accordingly, denoting by Vji advertiser j’s profit following a visit by searcher i, it follows that Vj ≡ E[Vji ]. The results of this paper remain valid under the specification

above up to the qualification that (Uj , Vj ) represent averages in the population of searchers. 10 If only the platform were to observe θj (but not advertiser j), the platform could achieve full rent extraction by running a mechanism in the spirit of Crémer and McLean (1988) and McAfee and Reny (1992). The common knowledge between the platform and advertiser j of the signal θj is an empirically sound assumption that rules out full rent extraction. 11 See Milgrom and Weber (1982) for a formal discussion of the concept of affiliation.

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vector (Z0 (t), Z1 (t), ..., ZN (t)) , where Zj (t) is the probability that advertiser j ∈ N is selected when

the profile of types is t (and Z0 (t) is the probability that the platform sells the link to no advertiser, in which case Θ0 = V0 = U0 ≡ 0).

The payment rule P ≡ (P1 (t), ..., PN (t)) assigns to each advertiser j a payment Pj (t), which is

a function of the profile of types.

An important class of selection rules can be described by scoring rules, which work as follows. Each advertiser j is assigned a score, s(tj ), as a function of her type tj only, and the bidder with the highest score is selected (ties are broken arbitrarily). Formally: Definition 1 A selection rule Z is described by the scoring rule s : t �→ s(t) if for any two bidders j1 , j2 ∈ N∪{0}, s(tj1 ) > s(tj2 ) implies Zj2 (t) = 0.

A selection rule Z naturally induces a probability distribution, F (·|Z), over the type of the winning advertiser. When Z is described by the scoring rule s(t), F (·|Z) takes the form � � � � � F (θ, v, u|Z) = Prob Θj ≤ θ, Vj ≤ v, Uj ≤ u �j = arg max s((Θk , Vk , Uk )) . k∈N∪{0}

With the measure F (·|Z) on hand, one can readily define the expected value to searchers of the winning advertiser, E[U |Z], as well as the expected value of any measurable function of (Θ, V, U ).

2.3

Supply of Clicks

Searchers are indexed by i and face a cost of clicking ci , which captures the value of alternative search resources or the opportunity cost of time. Normalize the population of searchers to one and assume that their costs ci are distributed according to the twice differentiable and log concave function G, with support [0, C].12 Searcher i clicks on the sponsored link if and only if the expected payoff from a click is greater than the cost ci . The expected payoff from clicking the link is E [U |Z] − Q, where Q is the charge/subsidy per click set by the platform.13 The total number of clicks enjoyed by the platform is then G(E [U |Z] − Q). Let us refer to G(E [U |Z] − Q) as the platform’s click supply.

In practice, click-contingent transfers to searchers are infrequent and hard to implement (see

discussion on footnote 1). As a benchmark, let us first consider the case where the platform is able 12 13

See Bagnoli and Bergstrom (2005) for an extensive list of distributions satisfying log-concavity. Mainly in order to economize on notation, the model does not allow for transfers to searchers Q which are contingent

on the profile of bidders’ types t and on the searcher type ci . Because searchers are risk neutral, the dependence on t is redundant (expected transfers are all that matters for optimality). Moreover, the platform has no way to screen the opportunity cost ci , in which case restricting attention to click-contingent transfers which are constant on (t, ci ) is without any loss of generality.

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to make transfers to both advertisers and searchers. Such mechanisms are called mechanisms with two-sided transfers and are denoted by the triple (Z, P, Q). In the more realistic scenario where charging/subsidizing searchers per click is not feasible (that is, Q ≡ 0), mechanisms are described by the pairs (Z, P ), and called mechanisms with one-sided transfers.

2.4

Platform Profits

By the Revelation Principle, the platform can restrict attention to direct-revelation mechanisms which ensure that all advertisers participate and truthfully report their types. Denote by zj (θ, vˆ, u ˆ) ≡

Et−j [Zj ((θ, vˆ, u ˆ), t−j )] the probability that advertiser j is selected by the platform after reporting type (θ, vˆ, u ˆ) when all other advertisers truthfully report their types, and denote by pj (θ, vˆ, u ˆ) ≡

Et−j [Pj ((θ, vˆ, u ˆ), t−j )] the expected payment associated to this strategy. Equipped with this notation,

one can now write the individual rationality (IR) constraints for any advertiser j ∈ N with type (θ, v, u) ∈ T as

Wj (θ, v, u) ≡ G(E [U |Z] − Q) · zj (θ, v, u) · v − pj (θ, v, u) ≥ 0,

(1)

where Wj (θ, v, u) is the payoff associated to an advertiser with type (θ, v, u). Similarly, the incentive compatibility (truth-telling) constraint requires for each advertiser j ∈ N and for any two types (θ, v, u), (θ, vˆ, u ˆ) ∈ T that

Wj (θ, v, u) ≥ G(E [U |Z] − Q) · zj (θ, vˆ, u ˆ) · v − pj (θ, vˆ, u ˆ).

(2)

As is standard in the mechanism-design literature, the selection rule Z is said to be implementable if there is a pair (P, Q) such that (Z, P, Q) satisfies the IR constraint (1) and the IC constraint (2). The platform’s expected profits are equal to its revenue from searchers plus the expected payments from advertisers: G(E [U |Z] − Q) · Q +

N �

E [Pj (t)] .

(3)

j=1

The expression for profits in (3) makes clear the two-sidedness of the platform’s problem. On the one hand, the choice of the selection rule Z affects the click supply G(E [U |Z] − Q). On the other hand,

it affects the rents that can be extracted from advertisers, as the payment rule P (t) has to satisfy constraints (1) and (2).

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Cross-Subsidization and Auction Design

The next lemma characterizes the set of implementable mechanisms. Lemma 1 The mechanism (Z, P, Q) satisfies the IR constraint (1) and the IC constraint (2) if and only if 9

1. for all (θ, v), Wj (θ, v, u) = Wj (θ, v, u ˆ) for all u, u ˆ ∈ [u, u]; 2. for all (θ, u), zj (θ, v, u) is weakly increasing in v; furthermore, except for a measure-zero subset of [θ, θ] × [v, v], zj (θ, v, u) = zj (θ, v, u ˆ) for all u, u ˆ ∈ [u, u]; 3. expected payments are given by � ˆ pj (θ, v, u) = G(E [U |Z] − Q) · zj (θ, v, u) · v −

v

v

� zj (θ, v˜, u)d˜ v − Wj (θ, v, u),

for all (θ, v, u), with Wj (θ, v, u) ≥ 0. To understand condition 1, note that the payoff of advertiser j depends on her reported value to searchers (to the extent that it affects the outcome of the mechanism), but not on her true value to searchers. As a consequence, incentive compatible mechanisms have to deliver identical payoffs to all advertisers who differ on searcher values u but share the same signal and value per click (θ, v). This does not mean, however, that the platform cannot condition the selection probability zj (θ, v, u) on the value to searchers u reported by advertiser j. Indeed, by setting the payment rule pj (θ, v, u) appropriately, the platform can satisfy condition 1 and let zj (θ, v, u) vary with u. As it turns out, condition 2 in the lemma shows that incentive compatibility greatly limits the platform’s ability to discriminate on the basis of the advertisers’ values to searchers: The selection probability zj (θ, v, u) can vary with u at most over a measure-zero subset of [θ, θ] × [v, v].

The remaining condition is the familiar envelope formula that pins down the payments from the

selection rule Z, up to a scalar. Note that this expression defines the payment rule that implements any selection rule Z satisfying condition 2. Therefore, an immediate implication of Lemma 1 is that a selection rule Z is implementable if and only if it satisfies condition 2 above. To economize on notation, let us hereafter describe mechanisms with two-sided transfers by the pair (Z, Q), and mechanisms with one-sided transfers by Z. After plugging the formula for payments from Lemma 1 into the expression for profits (3), and noting that the IR constraints bind at the optimum for those advertisers with willingness to pay v (that is, Wj (θ, v, u) = 0), the platform’s problem can be rewritten as max G(E [U |Z] − Q) · (Q + E [ω(Θ, V )|Z]) , Z,Q

where ω(θ, v) = v −

1 − F (v|θ) f (v|θ)

for all

(4)

(θ, v) ∈ [θ, θ] × [v, v].

The platform’s profits are simply the product of the click supply G(E [U |Z]−Q) with the expected

profit per click. The latter is the sum of the searchers’ transfer Q and the expected rents extracted from advertisers, E [ω(Θ, V )|Z]. As a benchmark for future comparisons, the next subsection considers the setting where the platform can charge/subsidize searchers as well as make transfers to advertisers. 10

3.1

Benchmark Mechanism with Two-Sided Transfers

The profit-maximizing selection rule must internalize the cross-side effects between searchers and advertisers. On the one hand, the click supply increases as the platform selects advertisers with higher values to searchers (as captured by E [U |Z]). On the other hand, the advertisers that generate

the highest rents to the platform (as captured by E [ω(Θ, V )|Z]) are not necessarily the ones that produce the highest values to searchers. As the next lemma shows, solving this trade-off is particularly simple when transfers to searchers are available. In this case, the profit-maximizing selection rule Z II can be described by a linear scoring rule that weighs equally the values to searchers and the auction rents generated by advertisers. Lemma 2 There exists a profit-maximizing mechanism that employs the selection rule Z II described by the scoring rule sII (t) = µ(θ, v) + ω(θ, v), where µ(θ, v) ≡ E[U |Θ = θ, V = v]. Let us refer to Z II as the virtual selection rule. Intuitively, having the ability to make clickcontingent transfers to searchers, the platform selects the advertiser to maximize the total value of the match searcher-advertiser. This can be accomplished by a linear scoring rule that assigns equal weights to µ(θ, v) and ω(θ, v). The platform then chooses transfers QII to set the supply of clicks at its profit-maximizing level. To see how, note that, from Lemma 2, the platform’s problem (4) can be rewritten only in terms of Q: � � � � � � �� max G E U |Z II − Q · Q + E ω(Θ, V )|Z II . Q

(5)

The optimal transfers QII have to satisfy the following first-order condition: �

QII + E ω(Θ, V )|Z where η (x) ≡ x ·

g(x) G(x)

� II

� � E U |Z II − QII = , η (E [U |Z II ] − QII )

(6)

is the elasticity of click supply with respect to searchers’ surplus.

The formula (6) is reminiscent of the standard Lerner formula: The left-hand side captures the

marginal gain from having one extra click, while the right-hand side captures the infra-marginal losses from decreasing the charge to searchers (or increasing the subsidy).

� � The technical assumption that the support of G, [0, C], satisfies E U + ω(Θ, V )|Z II < C guar-

antees an interior solution to problem (5). The next proposition provides a necessary and sufficient condition for the platform to subsidize searchers for their clicks.

11

Proposition 1 The revenue-maximizing mechanism with two-sided transfers employs the virtual selection rule Z II and sets searchers’ transfers QII according to the Lerner formula (6). Moreover, searchers are subsidized (QII ≤ 0) if and only if � � � � E U |Z II II ≤ E ω(Θ, V )|Z . η (E [U |Z II ])

(7)

Searchers are subsidized if and only if, after controlling for supply conditions, the value of adver-

tisers to searchers is smaller than the auction rents generated by advertisers, in expectation. As such, the optimality of charges or subsidies depends on how searchers and advertisers split the total value generated by the match. What is the profit-maximizing mechanism when transfers to searchers are not possible?

3.2

Mechanisms with One-Sided Transfers

The revenue-maximizing mechanism with one-sided transfers solves problem (5) subject to the additional constraint that Q = 0. Incorporating this constraint into the objective function leads to max G(E [U |Z]) · E [ω(Θ, V )|Z] . Z

(8)

For an heuristic solution to problem (8), take an implementable selection rule Z and an arbitrary profile of types t. Pick two advertisers j1 , j2 ∈ N ∪ {0} with types (θj1 , vj1 , uj1 ) and (θj2 , vj2 , uj2 ),

and assume that Z assigns a positive probability to the event that advertiser j1 is selected, that is, Zj1 (t) > 0. Now consider the selection rule Zˆ defined as follows: When Z selects j1 , Zˆ selects j2 instead of j1 with probability q. Otherwise, Z and Zˆ coincide. After differentiating the objective in (8) with respect to q and rearranging, the net effect from moving from Z to Zˆ is positive if and only if � � I � � �� E ω(Θ, V )|Z ω(θj1 , vj1 ) + η E U |Z I · · µ(θj1 , vj1 ) E [U |Z I ] � � � � �� E ω(Θ, V )|Z I ≤ ω(θj2 , vj2 ) + η E U |Z I · · µ(θj2 , vj2 ). E [U |Z I ]

Interestingly, the expression above reveals that the gain from swapping types (θj1 , vj1 , uj1 ) and (θj2 , vj2 , uj2 ) is the same regardless of the types of the advertisers other than j1 and j2 . This suggests a particular scoring rule for selecting the winning advertiser, as established by the next proposition. Proposition 2 The profit-maximizing mechanism with one-sided transfers employs the selection rule Z I described by the scoring rule implicitly defined by � � � � �� E ω(Θ, V )|Z I I s (t) = ω(θ, v) + η E U |Z · · µ(θ, v). E [U |Z I ] I

12

(9)

The proof in the appendix formalizes the discussion above. First, it establishes that there exists a solution to the platform’s problem (8), and applies variational techniques to show that this solution is described by the scoring rule sI (t). It also demonstrates that there exists a unique selection rule Z I consistent with the implicit definition of sI (t). As in the benchmark setting with two-sided transfers, the revenue-maximizing selection rule Z I adopts a scoring rule that linearly combines the advertisers’ bids (as captured by ω(θ, v)) with an estimate of the advertisers’ value to searchers (µ(θ, v)). However, the weight on the values to searchers of advertisers may depend on supply conditions (e.g., the elasticity η(·)), as transfers contingent on clicks are not available. In this case, how does the selection rule Z I compare with the virtual selection rule Z II ? And how is the click supply affected by the platform’s inability to charge/subsidize searchers? The next subsection answers these questions.

3.3

Cross-Subsidization

Under what conditions do the selection rules Z I and Z II coincide? Equation (9) reveals that Z I = Z II if and only if � � � � �� � � E U |Z II = η E U |Z II · E ω(Θ, V )|Z II .

Turning to Proposition 1, one can see that this is precisely the condition under which QII = 0! This implies that Z I and Z II differ when the platform would like to subsidize or charge searchers but lacks the means to do it. In such a case, the platform might benefit from altering the selection of advertisers to compensate for the lack of transfers to searchers. To explore this idea further, consider the following definition. ˆ ≤ E[U |Z]. Definition 2 Call the selection rule Z more searcher-friendly than rule Zˆ if E[U |Z] The next proposition is the main result of this paper. It shows that scoring rules work as a cross-subsidization device between the searcher side and the advertiser side of the market. Namely, Proposition 3 shows that Z I is more searcher-friendly than the virtual efficient rule Z II when the profit-maximizing mechanism with two-sided transfers subsidizes searchers for their clicks, and is less searcher-friendly when the profit-maximizing mechanism with two-sided transfers charges searchers for their clicks. Still, do searchers overall benefit when the platform is able to make transfers contingent on clicks? Proposition 3 provides an affirmative answer to this question. Intuitively, the ability to make transfers contingent on clicks alleviates the platform’s trade-off between extracting rents from advertisers and attracting clicks. As a result, all searchers enjoy higher payoffs when transfers are available (even in the case where searchers are charged per click!), and the click supply is higher. 13

Proposition 3 The selection rule with one-sided transfers Z I is more searcher-friendly than the virtual efficient rule Z II if and only if searchers are subsidized under the profit-maximizing mechanism with two-sided transfers: E[U |Z II ] ≤ E[U |Z I ] ⇐⇒ QII ≤ 0. Moreover, searchers enjoy higher payoffs when the platform can employ mechanisms with two-sided transfers: E[U |Z II ] − QII ≥ E[U |Z I ]. The figure above illustrates the results from Proposition 3. The variable on the X-axis is the searcher’s payoff, x ≡ E [U |Z] − Q, and the variable on the Y-axis is the platform’s profit per click, y ≡ Q + E [ω(Θ, V )|Z], induced by some mechanism (Z, Q). Now consider the curve labelled I. This curve is the frontier of undominated pairs (x, y) that can be implemented by mechanisms with one-sided transfers.14 Analogously, the curve labelled II is the frontier of undominated pairs (x, y) that can be implemented by mechanisms with two-sided transfers. Note that, due to the optimality of Z II established in Lemma 2, the frontier II is linear: y = E[U + ω(Θ, V )|Z II ] − x. Moreover, I and II intersect at a single point a0 where Q = 0 and Z I = Z II . 14

That is, (x, y) ∈ I if (x, y) can be implemented by a mechanism with one-sided transfers and there is no (x� , y � )

that can be implemented by a mechanism with one-sided trasnfers such that x� ≥ x and y � ≥ y with one inequality strict.

14

The platform’s problem is that of choosing the pair (x, y) that maximizes profits G(x) · y. The

profit-maximizing mechanism with one-sided transfers is associated to the point aI where the platform’s isoprofit curve is tangent to the frontier I (the appendix establishes that the frontier I is described by a strictly decreasing and weakly concave function in the space (x, y)). In turn, the profit-maximizing mechanism with two-sided transfers is associated to the point aII where the platform’s isoprofit curve is tangent to the frontier II. All pairs (x, y) on the frontier II that are to the right of a0 involve subsidizing searchers’ clicks (QII < 0), while all pairs (x, y) that are to the left of of a0 involve charging searchers for their clicks (QII > 0). Analogously, all pairs (x, y) at frontier I that are to the right of of a0 apply selection rules that are more searcher-friendly than Z II , and all pairs (x, y) that are to the left of of a0 apply selection rules that are less searcher-friendly than Z II . Geometrically, the first claim of Proposition 3 establishes that aI is to the right of a0 whenever aII is to the right of a0 . Its second claim states that aII is always to the right of aI . Proposition 3 is the analog in an auction setting to the cross-subsidization patterns discussed in the literature of monopolistic pricing in two-sided markets (see, for example, Rochet and Tirole (2003, 2006), Armstrong (2006), Hagiu (2006) and Weyl (2010)).

3.4

Discussion

The results above deliver two prescriptions on how search engines should sell sponsored links. First, as Z I does not depend on u, the revenue-maximizing mechanism does not require advertisers to report information about their value to searchers. As such, the selection rule that maximizes the search engine’s revenue is solely based on the observable characteristics of advertisers (as captured by the the signal θ) and on the advertisers’ bids (as captured by the virtual willingness to pay ω(θ, v)). In the extreme case where Θ is independent of (U, V ), the profit-maximizing mechanism with either one or two-sided transfers boils down to a first-price auction with a reserve price. This was the case of Google’s Audio Ads program, which, being unable to measure the effectiveness of radio advertisements, followed a first-price auction to sell radio advertising slots. Secondly, when Θ is informative about (U, V ), the profit-maximizing way of selecting advertisers employs a scoring procedure that combines the auction rents generated by advertisers (as captured by ω(θ, v)) and the advertisers’ value to searchers (estimated by µ(θ, v)). This prescription mirrors recent developments in the sponsored search industry. In 2007, Yahoo! moved to a scored version of the Generalized Second Price (GSP) auction, in which the bid submitted by each advertiser is combined with a quality measure to generate the advertiser’s score. The other major search engines followed a path similar to Yahoo’s: Google had already moved to a scored version of the GSP in 2004, and Microsoft’s LiveSearch (Bing’s predecessor) did the same in 2006. Section 6 discusses how to implement Z I by scored versions of the first and second-price auctions. 15

The next section analyzes the distortions relative to efficiency brought by revenue-maximization.

4

Distortions Relative to Efficiency

To set a benchmark, let us now derive the implementable mechanism that maximizes ex-ante efficiency. For a mechanism (Z, Q), denote by c¯(Z, Q) ≡ E [U |Z] − Q the threshold such that all searchers with ci ≤ c¯(Z, Q) click on the platform’s advertisement and all searchers with ci > c¯(Z, Q) do not click. The ex-ante efficiency produced by mechanism (Z, Q) is given by

G(¯ c(Z, Q)) · [E [U + V |Z] − E[c|c ≤ c¯(Z, Q)] .

(10)

The term inside the brackets is the expected total value produced by the winning advertiser, E [U + V |Z], net of the average opportunity cost of all searchers who make clicks, E[c|c ≤ c¯(Z, Q)].

The term outside the brackets, G(¯ c(Z, Q)), is the click supply. The next lemma shows that the welfare-maximizing selection rule Z E maximizes the total value E [U + V |Z] and sets the transfer � � QE in a way that all searchers whose opportunity cost is such that ci ≤ E U + V |Z E click on the sponsored link.

Lemma 3 The welfare-maximizing mechanism employs the selection rule Z E described by the scoring rule sE (t) = µ(θ, v) + v, � � and subsidizes searchers by QE = −E V |Z E .

The next proposition compares the outcomes under profit and welfare-maximization on both the

advertiser and the searcher sides of the market.15 Proposition 4 Relative to the welfare-maximizing mechanism, profit-maximization: 1. Precludes socially efficient advertising by setting bidder-specific reserve prices, which are defined as: rm (θ) ≡ inf {v ∈ V : sm (t) ≥ 0} ,

m ∈ {I, II}

2. In the case of two-sided transfers, leads to less searcher-friendly selection rules: E[U |Z II ] ≤ E[U |Z E ]. Moreover, it generates a smaller click supply and yields lower payoffs to searchers: G(E[U |Z I ]) ≤ G(E[U |Z II ] − QII ) ≤ G(E[U |Z E ] − QE ). 15

An alternative efficiency benchmark is the mechanism that maximizes welfare among all mechanisms with one-sided

transfers. One can prove results analogous to Proposition 4 under this alternative formulation.

16

The first distortion originates from reserve prices. Relative to efficiency, the profit-maximizing reserve prices are distortionary since the platform excludes too many advertisers (note that sI (t) < sE (t) for all t). Intuitively, reserve prices have two purposes: First, as in standard auctions, reserve prices encourage the advertisers with high values vj to bid more. Second, reserve prices exclude advertisers whose values to searchers are too low (possibly negative), therefore increasing the click supply. As such, the use of reserve prices reflects the partial alignment between the searchers’ and the platform’s objective. Interestingly, the revenue-maximizing selection rule Z I and Z II induce reserve prices that vary with θ, that is, advertiser-specific reserve prices. In line with this prescription of the model, all major search engines recently adopted advertiser-specific reserve prices.16 The second distortion from Proposition 4 indicates that, relative to efficiency, revenue-maximization under two-sided transfers serves fewer searchers. This result is the sum of two effects. First, Z II is less searcher-friendly than Z E . To see why, note that the signal θj observed by the platform is informative not only about the value to searchers uj of advertiser j, but also about her willingness to pay vj . Because of positive affiliation, advertisers with high signals are more likely to have high willingness to pay. In order to encourage them to bid more, the profit-maximizing platform distorts the selection rule away from efficiency to favor advertisers with low signals. By doing so, as demonstrated by Proposition 4, revenue-maximization introduces a bias on the selection rule Z II towards advertisers with low values to searchers. There is a second effect stemming from the Lerner formula (6). Namely, the standard trade-off in monopoly pricing between enlarging the click supply and increasing the searchers’ transfers per click induces the platform to reduce even further the number of clicks. As a result of these two effects, the click supply (and searchers’ payoffs) are smaller under (Z II , QII ) than under (Z E , QE ). In the case of mechanisms with one-sided transfers, this distortion is magnified by the lack of charges/subsidies contingent on clicks. In this case, as proved in Proposition 3, the choice of Z I produces an even smaller click supply than (Z II , QII ).17 The analysis so far restricted attention to the problem of a two-sided platform that monopolizes the market for sponsored advertising. The next section extends the basic model to analyze the effects of duopolistic competition on the pricing and selection of advertisers. In so doing, it derives novel implications for the antitrust policy in two-sided markets.

16 17

See Even-Dar, Feldman, Mansour and Muthukrishnan (2008). This does not mean however that Z I is less searcher-friendly than Z E . For instance, when the distribution of

opportunity cots G(·) is degenerate at E[U + ω(Θ, V )|Z U ], where Z U is described by the score sU (t) = µ(θ, v), the platform sets Z I = Z U . In this case, E[U |Z I ] = E[U |Z U ] > E[U |Z E ].

17

5

Effect of Competition

Would a merger between the leading search engines be beneficial to searchers or advertisers? What is the impact on social welfare? How does oligopolistic competition (as opposed to a monopolistic platform) affect the pricing and the selection of sponsored search advertisers? In order to answer these questions, it is essential to understand how competition affects the choice of selling mechanisms by two-sided platforms. This section develops a simple extension of the baseline model to analyze duopolistic competition under the extreme assumption that platforms are perfectly homogeneous. As reported by Evans (2008) and in line with casual empiricism, searchers click on the link of at most one platform (single-homing), while advertisers have the option to participate in more than one platform (multi-homing). Indexing platforms by A and B, the timing of the model is as follows: 1. The platforms simultaneously announce mechanisms (ZA , PA , QA ) and (ZB , PB , QB ), 2. Types are realized and advertisers submit reports (θjA , vˆjA , u ˆjA ) and (θjB , vˆjB , u ˆjB ) to the platform(s) in which they want to buy sponsored links, 3. Each platform selects its winning advertiser according to ZA and ZB , 4. Searchers choose which platform to visit, and whether to click or skip the link (knowing the mechanisms but not the realization of the advertisers’ types), 5. Searchers and advertisers make payments according to (PA , QA ) and (PB , QB ). For simplicity, platforms are homogeneous in the eyes of searchers.18 Therefore, a searcher with opportunity cost ci clicks the link displayed by platform l ∈ {A, B} if E [U |Zl ] − Ql ≥ max{E [U |Z−l ] − Q−l , ci },

(11)

and makes no click if the inequality above is violated for l = A, B. If E [U |ZA ]−QA = E [U |ZB ]−QB ≥

ci , the searcher visits either platform (and clicks the sponsored link) with some arbitrary tie-breaking probability. Advertisers, in turn, multi-home: Their decisions to participate in the mechanism of each platform are independent of each other, as are their choices of which types to report. Therefore, the implementable mechanisms available to platforms A and B have to satisfy the conditions of Lemma 18

In reality, searchers might face swithcing costs or have idyiossincratic tastes for either platform. A previous working

paper version introduces horizontal differentiation by allowing searchers to choose between two platforms located in the extreme points of a Hotelling line. This material is available upon request.

18

2. As a result, the profits of each platform are given by an expression analogous to (4), with the qualification that the click supply is now determined by (11).19 As a benchmark, let us study first the case of competition in mechanisms with two-sided transfers. The next proposition characterizes the unique equilibrium of this game. Proposition 5 In the unique Nash equilibrium of duopolistic competition under mechanisms with two-sided transfers, both platforms employ the virtual selection rule Z II . Moreover, searchers’ subsidies per click equal the auction rents generated by advertisers: � � II II QII . A = QB = −E ω(θ, v)|Z

Therefore, platforms enjoy zero profits, and competition is welfare-improving as it increases the click supply relative to monopoly. Since advertisers multi-home and searchers single-home, each platform monopolizes the sale of its “captive” searchers to the advertiser side of the market. In contrast, platforms engage in a Bertrandtype competition for searchers. As a consequence, in equilibrium both platforms apply the virtual selection rule Z II (as in a monopoly), but give back to searchers in the form of subsidies all rents obtained from the advertiser side of the market (as in a perfectly competitive industry). This results in zero profits for both platforms and welfare gains relative to monopoly. The result above bears close relation to the competitive bottleneck model of Armstrong (2006). As in Armstrong, the size of the single-homing side (that is, the click supply) is chosen to maximize the welfare produced by the platform, while the multi-homing side (advertisers) faces the same distortions as in a monopolistic market.

5.1

When Competition Decreases Welfare

More realistically, platforms compete without being able to make click-contingent transfers to searchers. In this case, the selection rule Z is the only instrument that platforms have to combine two goals: Compete for searchers and extract rents from advertisers. In equilibrium, Bertrand-type competition on the searcher side of the market pushes both platforms to select advertisers based on their values to searchers only. As such, platforms A and B employ in equilibrium the selection rule Z U described by the scoring rule sU (t) = µ(θ, v). As a result, moving from monopoly to duopoly has the effect of increasing searchers’ payoffs. 19

Other authors have studied competition between two-sided platforms. McAfee (1993), Peters and Severinov (1997),

Caillaud and Jullien (2001, 2003), Ellison, Fudenberg and Möbius (2004) and Damiano and Li (2007) assume that buyers and sellers have to make exclusive decisions on which platform to join (universal single-homing). More in line with the present paper, Rochet and Tirole (2003) and Armstrong (2006) embed a two-sided market model into a Hotelling duopoly game in which one side single-homes and the other multi-homes. See also White and Weyl (2010).

19

More surprisingly, however, competition may decrease total welfare since, by tilting the selection rule towards advertisers with high values to searchers, both platforms forgo advertisers with high willingness to pay. The welfare-reducing effect of competition can be easily characterized when the click supply does not respond in the extensive margin (that is, the distribution G(·) is degenerate at zero, in which case the number of searchers who click on some link is one). Under this assumption, the mechanism with one-sided transfers that maximizes profits in a monopolistic market employs the selection rule Z W , described by the scoring rule sW (t) = ω(θ, v). This observation reveals a necessary and sufficient condition for competition to reduce welfare, as the next proposition describes. Proposition 6 In the unique Nash equilibrium of duopolistic competition under mechanisms with one-sided transfers, both platforms employ the selection rule Z U described by the scoring rule sU (t) = µ(θ, v). As such, competition leads to an increase in searchers’ payoffs relative to monopoly and both platforms make positive profits in equilibrium. However, competition may decrease social welfare. In the case where the click supply does not respond in the extensive margin, competition decreases social welfare if and only if E[U + V |Z U ] < E[U + V |Z W ],

(12)

where the selection rule Z W is described by the score sW (t) = ω(θ, v). Intuitively, competition is more likely to decrease social welfare when (i) the correlation between V and U is low, and (ii) the distribution of V is more disperse than that of U . In such a case, assigning all weight to searcher values is likely to induce large efficiency losses on the advertiser side of the market (since the relative willingness to pay across advertisers is ignored). The example below evaluates condition (12) in an important special case of the model. Example 1 Let Θ ≡ U (complete information about searcher values) and V and U be independent. In this case, competition decreases social welfare if and only if

E[V 1:N − E[V ]] ≥ E[U 1:N − E[U ]], where V 1:N is the first-order statistic of V from a sample of size N (analogous to U 1:N ). If the marginals of V and U are uniformly distributed with support [v, v] and [u, u], this condition boils down to v − v ≥ u − u. The analysis above reveals that competition under mechanisms with one-sided transfers can generate strictly positive profits, as well as decrease total welfare. In this regard, regulators should exercise caution in applying standard antitrust economics to two-sided industries such as sponsored search advertising. 20

6

Indirect Implementation

The analysis so far has focused on direct-revelation mechanisms, that is, mechanisms in which advertisers truthfully report their types. Yet, direct revelation mechanisms are not employed by search engines and other online intermediaries, since their payment rules are not “simple” in the eyes of advertisers. Instead, search engines often employ variations of the first and second-price auctions, whose rules are significantly easier to explain to advertisers. In line with the business models of the major search engines, the next subsections show that the selection rules Z I , Z II , Z U and Z W discussed in the previous sections can be easily implemented by scored versions of the first and second-price auctions.

6.1

Implementation by Pay-Your-Bid Scoring Auctions

Consider the following pay-your-bid scoring auction: 1. Each advertiser j faces a personalized reserve price rj , and submits a bid bj , 2. Each advertiser j is assigned a bid-score, defined by the bid-scoring rule sj = sˆ(θj , bj ), 3. The advertiser with the highest bid-score sj wins and pays bj per click. The next proposition shows that by appropriately designing the bid-scoring rule sˆ(θ, b) and the reserve prices {rj }j=1,...,N the platform can implement through a pay-your-bid scoring auction any of the selection rules Z I , Z II , Z U and Z W discussed above.

Proposition 7 Consider an implementable selection rule Z described by the scoring rule s(θ, v) with interim probability z(θ, v) strictly increasing in v. Define the function ˆ v z(θ, v˜) b(θ, v) ≡ v − d˜ v, r(θ) z(θ, v)

(13)

where the function r(·) is defined by s(θ, r(θ)) = 0. The rule Z can be indirectly implemented through a pay-your-bid scoring auction with advertiser-specific reserve prices rj ≡ r(θj ). The bid-scoring

−1 rule of this auction is sˆ(θ, b) = s(θ, b−1 v (θ, b)), where the inverse function bv (θ, b) is defined by

b−1 v (θ, b) ≡ sup{v ∈ [v, v] : b(θ, v) ≤ b}. In the unique Bayes-Nash equilibrium of this auction, advertisers bid according to the bidding function b(θ, v).

The mechanism described above resembles the generalized first-price (GFP) auction first adopted by Overture (acquired by Yahoo! in 2003). In 2004, Yahoo! moved to a pay-the-next-bid design. In turn, in the generalized second-price (GSP) auction, each advertiser pays the next lowest bid for each click received (that is, the bid submitted by the advertiser immediately below him). The next subsection derives a necessary and sufficient condition that reveals when a selection rule Z can be indirectly implemented by a scored version of a pay-the-next-bid auction such as the GSP. 21

6.2

Implementation by Pay-the-Next-Bid Scoring Auctions

Consider the following pay-the-next-bid scoring auction: 1. Each advertiser j faces a personalized reserve price rj , and submits a bid βj , 2. Each advertiser j is assigned a bid-score, defined by the bid-scoring rule sj = s˜(θj , βj ), 3. The advertiser with the highest score sj wins and pays the second highest bid per click. The next proposition derives a necessary and sufficient condition for the platform to be able to implement a given selection rule Z by means of a pay-the-next-bid scoring auction. Proposition 8 Consider an implementable selection rule Z described by the scoring rule s(θ, v) with interim probability z(θ, v) strictly increasing in v. The distribution associated to s(θ, v) is Fs (x) ≡

Pr [s(Θ, V ) ≤ x]. Now define the inverse function s−1 v (θ, s) ≡ sup{v ∈ [v, v] : s(θ, v) ≤ s} and the function r(·) according to s(θ, r(θ)) = 0. The rule Z can be implemented through a pay-the-next-bid scoring auction with reserve prices rj ≡ r(θj ) for j = 1, ..., N if and only if the candidate bidding function β(θ, v) defined by the Volterra equation � � ˆ v (Fs (s(θ, v)))N −1 · z(θ, v) · v − z(θ, v˜)d˜ v N −1 r(θ) =

ˆ

θ

˜ θ¯ ˆ s−1 v (θ,s(θ,v)) r(θ)

(14)

� �N −2 ˜ v˜) Fs (s(θ, ˜ v˜)) ˜ v˜))d˜ β(θ, fs (s(θ, v dθ˜

is strictly increasing in v for each θ. In such an auction, scores are computed according to the rule s˜(θ, β) = s(θ, βv−1 (θ, β)), where βv−1 (θ, β) ≡ sup{v ∈ [v, v] : β(θ, v) ≤ β}. When an equilibrium implementing the rule Z exists, advertisers bid according to the bidding function β(θ, v).

Although the Volterra equation above often does not possess an analytic solution, one can easily verify numerically whether the candidate bidding function β(θ, v) is strictly increasing in v for every θ.20 Interestingly, the result above reveals that if this monotonicity condition fails, search engines can increase profits by moving to auction formats that apply payment rules other than the “next-bid” design discussed above.

7

Extensions

7.1

The Role of Positive Affiliation

The analysis of this paper assumed that the advertisers’ willingness to pay for clicks is positively affiliated with their value to searchers (this includes the case of statistical independence). Arguably, 20

A Matlab program that computes β(θ, v) for the case where (i) Θ ≡ U, (ii) (U, V ) have an arbitrary discrete

distribution, and (iii) Z is described by a linear scoring rule s(θ, v) = θ + αv, is available upon request. In all examples computed by the author, an equilibrium implementing Z existed.

22

in many economic situations, it seems more realistic to believe that the advertisers with higher willingness to pay for clicks are the ones with lower value to searchers (e.g., the willingness to pay per click of retailers of a homogeneous good is proportional to the price of the good, and is therefore inversely related to the consumer surplus). It is worth noting that the characterization of the profit-maximizing mechanisms with one and two-sided transfers derived in Propositions 1, 2 and 3 remains valid in many cases where the positive affiliation assumption is not satisfied. To see why, recall that, as established in Lemma 1, a selection rule Z described by the scoring rules s(θ, v) is implementable if and only if s(θ, v) is weakly increasing in v. In the case where U and V are not positively affiliated, the scoring rules associated to Z I and Z II are weakly increasing in v as long as the advertisers’ expected value to searchers µ(θ, v) does not decrease faster in v than the virtual willingness to pay ω(θ, v). Therefore, Z I and Z II still solve the platform’s problem provided that the negative affiliation between U and V is not “too strong”, as captured by the slope of the conditional expectation µ(θ, v). Otherwise, Propositions 1, 2 and 3 can be adapted if one replaces Z by its “ironed” version Z (which can be derived following, for example, the method developed in Toikka (2010)).

7.2

Multiple Advertising Positions

The auctions run by search engines and ad exchanges (as well as by some offline media companies) often sell multiple advertising positions at once. For expositional simplicity, the model of this paper so far has restricted attention to a single advertising position. This subsection shows that one can readily extend the analysis to the case of K advertising positions. To avoid uninteresting cases, assume K < N , where N is the number of advertisers. Consider a selection rule Z that picks (in a possibly non-deterministic manner) K advertisers as a � � � � function of the joint reports submitted by all advertisers. Denote by E U(k) |Z and E ω(Θ(k) , V(k) )|Z

the expected value to searchers and the virtual willingness to pay of the k-th advertiser selected by the platform. By convention, the k-th advertiser appears at the k-th highest sponsored position. For tractability, assume that searchers realize the total payoff associated to browsing the advertisers’ websites only at the end of the search effort. In this setting, the searcher i with clicking cost � � ci clicks on the k-th sponsored position if and only if E U(k) |Z − Qk ≥ ci , where Qk is transfer per

click on position k. Accordingly, the click supply associated to the k-th sponsored position is given � � � � by G E U(k) |Z − Qk . After extending the IC characterization of Lemma 1 to this multi-unit scenario, the platform’s

problem can be recasted in a form analogous to (4): max Z,Q

K � k=1

� � � � � � �� G E U(k) |Z − Qk · Qk + E ω(Θ(k) , V(k) )|Z . 23

In a setting with K positions, the selection rule Z is said to be described by the scoring rules {sk (t)}K k=1 if the assignment of advertisers to positions is done according to the following procedure.

The advertiser in the first position, denoted j(1) , is the one with the highest signal s1 (t) among all advertisers j ∈ N. For all other k ∈ {2, ..., K}, the advertiser in the k-th position is the one with highest signal sk (t) among all advertisers j ∈ N/{j(1) , j(2) , ..., j(k−1) }. If sk (t) = s(t) for all k ∈ {1, ..., K}, the selection rule Z is said to be described by the scoring rule s(t).

Arguments analogous to those in the proofs of Propositions 1, 2 and 3 prove the following result.

Proposition 9 Consider the problem of a platform selling K advertising positions. In the benchmark case of two-sided transfers, the revenue-maximizing mechanism employs the virtual selection rule Z II described by the scoring rule sII (t) = ω(θ, v) + µ(θ, v). Moreover, searchers are subsidized for clicks in the k-th position (QII k ≤ 0) if and only if � � � � E U(k) |Z II � � �� ≤ E ω(Θ(k) , V(k) )|Z II . η E U(k) |Z II

In turn, in the case of mechanisms with one-sided transfers, the revenue-maximizing selection rule � �K Z I (K) is described by the scoring rules sIk (t) k=1 implicitly defined by sIk (t)

� � � � �� E ω(Θ(k) , V(k) )|Z I (K) I � � = ω(θ, v) + η E U(k) |Z (K) · · µ(θ, v). E U(k) |Z I (K)

The selection rule with one-sided transfers Z I (K) is more searcher-friendly in the k-th position than the virtual efficient rule Z II if and only if searchers are subsidized under the profit-maximizing mechanism with two-sided transfers for clicks on the k-th position: E[U(k) |Z II ] ≤ E[U(k) |Z I ] ⇐⇒ QII (k) ≤ 0. The result above shows that the characterization of the revenue-maximizing mechanism extends naturally to the case of K advertising positions. In particular, the insight that the scoring rule with one-sided transfers works as a cross-subsidization device between searchers and advertisers remains valid in this more general setting. Interestingly, the cross-subsidization patterns may change across � � positions, as the elasticity of click supply and the distribution of U(k) , ω(Θ(k) , V(k) ) may vary with

k. The use of position-specific scoring rules is consistent with the practice of search engines to adjust scores by position-specific factors that depend on the identity of the advertiser.

7.3

Searcher-Customized Mechanisms

Online platforms use detailed information about searchers in order to customize the selection of advertisers in both display and sponsored search advertising.21 This information is often used both 21

Such information comes from the click history of searchers as well as from the use of cookies and other spywares.

24

to (i) improve the matching (or targeting) between advertisers and searchers, as well as to (ii) adjust the selection of advertisers to the supply of clicks offered by each searcher (estimated from the searcher’s click history). The model presented in this paper sheds light on (ii). To see how, denote by σ i the previous activity of searcher i (click history, purchasing habits, demographics, etc). The platform uses the posterior G(·|σ i ) on searcher i’s cost of clicking to update the mechanism used to select advertisers as it gets to know more about searcher i. Denote by η(·|σ i ) the elasticity of the click supply of searcher i associated to the posterior G(·|σ i ), and by Z I (σ i ) the profit-maximizing mechanism with one-sided transfers customized to searcher i. From Proposition 2, Z I (σ i ) is described by the scoring rule implicitly defined by � � � � � i � E ω(Θ, V )|Z I (σ i ) I i I i s (t, σ ) = ω(θ, v) + η E U |Z (σ ) |σ · · µ(θ, v). E [U |Z I (σ i )]

Therefore, as the platform gathers information about searcher i’s click supply, it tilts the customized � � selection rule of advertisers to be more (less) searcher-friendly as the estimated elasticity η ·|σ i shifts up (down).

8

Conclusion

This paper studies auction design in advertising markets, where the number of clicks (or attention) for sale depends on the expectations that searchers (or viewers, in the case of TV) have about the relevance (or quality) of the advertisers selected by the mechanism. The analysis shows that the revenue-maximizing mechanism employs scoring rules that combine the auction rents and the value to searchers generated by advertisers. The main result states that the weight assigned to each of these factors reflects a cross-subsidization strategy: If the auction rents generated by advertisers are in expectation greater (smaller) than the impact on the searchers’ click supply, the platform subsidizes (charges) searchers’ clicks by selecting advertisers with greater (smaller) value to searchers. As a byproduct of the characterization discussed above, this paper describes the distortions generated by revenue-maximization relative to efficiency. It shows that profit-maximization leads to greater exclusion on both sides of the market. On the advertiser side, customized reserve prices exclude advertisers whose auction rents and value to searchers are too small. On the searcher side, the click supply is inefficiently low as a consequence of distortions in the selection of advertisers. Finally, motivated by the recent antitrust debate in online advertising, the model is extended to an oligopolistic setting where platforms compete by means of auctions. Perhaps counter-intuitively, the analysis shows that competition can decrease welfare if the advertisers’ willingness to pay and value to searchers are lowly correlated. Advertisers face budget constraints which limit their bids per click as well as their total spending in different auctions. It is an interesting topic of future research to incorporate budget constraints into the platform’s auction design problem. 25

9

Appendix: Proofs Proof of Lemma 1: Standard arguments show that, for any fixed u, the advertiser truthfully

reports her willingness to pay v if and only if zj (θ, v, u) is weakly increasing in v and the envelope formula from condition 3 holds (see, for example, Mas-Colell et al. (1995) Proposition 23.D.2). For condition 1, note that because u ˆ does not directly enter the advertisers’ payoff, truth-telling requires that Wj (θ, v, u) = Wj (θ, v, u ˆ) for all u, u ˆ ∈ [u, u].

To prove that zj (θ, v, u) = zj (θ, v, u ˆ) for all u, u ˆ ∈ [u, u] except for a measure-zero subset of

[θ, θ] × [v, v], take v˜ such that zj (θ, v˜, u ˜) > zj (θ, v˜, u ˆ) for some u ˜, u ˆ ∈ [u, u]. Now note that there is no v¯ > v˜ such that zj (θ, v˜, u ˜) > zj (θ, v¯, u) ≥ zj (θ, v˜, u ˆ) for some u ∈ [u, u], as this would violate

incentive compatibility between types (θ, v˜, u ˜) and (θ, v¯, u). Therefore, zj (θ, v, u) ≥ zj (θ, v˜, u ˜) for all v > v˜ and zj (θ, v, u) ≤ zj (θ, v˜, u ˆ) for all v < v˜.

Define zj (θ, v) ≡ EU Et−j [Zj ((θ, v, U ); t−j )]. By the argument above, it is clear that zj (θ, v) has

a discontinuity point at (θ, v˜). Because zj (θ, v, u) is weakly increasing in v for all u, it follows that zj (θ, v) is weakly increasing in v. As a result, for a given θ, zj (θ, v) is discontinuous in at most a countable subset of [v, v]. Therefore, zj (θ, v, u) = zj (θ, v, u ˆ) in almost every (θ, v) ∈ [θ, θ] × [v, v], as desired. �

Proof of Lemma 2: Consider an implementable mechanism with two-sided transfers (Z, Q). ¯ such that the The idea of the proof is to show that, for any mechanism (Z, Q), there is a transfer Q ¯ than under (Z, Q). platform’s profits are weakly greater under (Z II , Q) First, note that the scoring rule sII (t) is strictly increasing in v, since µ(θ, v) weakly increases in v by Assumption 1 and ω(θ, v) strictly increases in v from the monotone hazard rate condition. Therefore, the interim probability z II (t) implied by Z II is strictly increasing in v, which assures that the condition 2 of Lemma 1 is satisfied and proves that Z II is implementable. Consider the following lemma. Lemma 4 Z II maximizes E[U + ω(Θ, V )|Z] among all implementable selection rules. Proof of Lemma 4: Note that if Z0 (t) > 0 for some positive measure set of type profiles, F (t|Z) is a mixed distribution with a unique mass point at 0 ≡ (0, 0, 0) (which corresponds to the platform selecting no advertiser). For t ∈ T /{0}, F (t|Z) admits a density given by f (t|Z) =

N ˆ � j=1

T N −1

Zj (t, ˆ t−j ) · f (t, ˆ t−j ) · dˆ t−j =

N �

zj (t).

j=1

Take a selection rule Z satisfying the conditions of Lemma 1. Now note that   ˆ ˆ N �  E[U |Z] = u · f (t|Z)dt = u· zj (t) dt   T /{0}

T /{0}

26

j=1

=

ˆ

µ(θ, v) ·

T /{0}

 N � 

zj (t)

j=1

  

dt =

ˆ

µ(θ, v) · f (t|Z)dt = E[µ(Θ, V )|Z],

T /{0}

where the equality from the first to the second line follows from the fact that zj (t) is invariant in u a.e. (condition 2 in Lemma 1). Therefore, it follows from the construction of Z II that E [U + ω(Θ, V )|Z] = E [µ(Θ, V ) + ω(Θ, V )|Z]

(15)

� � � � ≤ E µ(Θ, V ) + ω(Θ, V )|Z II = E U + ω(Θ, V )|Z II ,

i.e., the selection rule Z II , with score sII (t) = µ(θ, v) + ω(θ, v), maximizes E[U + ω(Θ, V )|Z] among all selection rules that are implementable. � Now consider an implementable mechanism with two-sided transfers (Z, Q) where Wj (θ, v, u) = 0, ¯ satisfying and take the supply-preserving transfer Q � � ¯ E [U |Z] − Q = E U |Z II − Q.

¯ than under (Z, Q). Let us now show that the platform’s profits are weakly greater under (Z II , Q) ˜ Indeed, the platform’s profits under (Z, Q), denoted Π(Z, Q), is ˜ Π(Z, Q) = G (E [U |Z] − Q) · (Q + E [ω(Θ, V )|Z]) � � � � � � � � ¯ · E [U + ω(Θ, V )|Z] − E U |Z II + Q ¯ = G E U |Z II − Q � � � � � � � � � � ¯ · E U + ω(Θ, V )|Z II − E U |Z II + Q ¯ ≤ G E U |Z II − Q � � � � � � �� ¯ · Q ¯ + E ω(Θ, V )|Z II = G E U |Z II − Q ˜ II , Q), ¯ = Π(Z

¯ and the inequality in the third line where the second equality follows from the construction of Q follows from Lemma 4. This concludes the proof of Lemma 2. � Proof of Proposition 1: For convenience, define the one-dimensional real function ψ(x) as � � � � G(E U |Z II − x) ψ(x) ≡ − E ω(Θ, V )|Z II . II g(E [U |Z ] − x) Condition (6) can be rewritten as a fixed point of function ψ(x): ψ(QII ) = QII . � � At the optimum −E ω(Θ, V )|Z II ≤ QII , as otherwise the platform’s profits are negative. More� � over, QII ≤ E U |Z II , as otherwise no searchers make clicks. Now note that



ψ(−E ω(Θ, V )|Z

II





) − (−E ω(Θ, V )|Z 27

II



� � G(E U + ω(Θ, V )|Z II ) )= >0 g(E [U + ω(Θ, V )|Z II ])

and

� � � � � � ψ(E U |Z II ) − (E U |Z II ) = −E U + ω(Θ, V )|Z II < 0.

Since ψ(x) − x is a continuous function, one can apply the intermediate value theorem to conclude

that there is a transfer

� � � � QII ∈ (−E ω(Θ, V )|Z II , E U |Z II )

satisfying ψ(QII ) = QII (or, equivalently, condition (6)).

Let us now argue that the solution to (6) is a local maximum. Indeed, the second-order derivative of the objective function (5) with respect to Q is � � � � � � � � ∂ 2 Π(Z, P, Q) = g � (E U |Z II − Q) · E ω(Θ, V )|Z II + Q − 2g(E U |Z II − Q). 2 (∂Q)

Evaluating this condition at QII (by using (6)) leads to � � � � � � G(E U |Z II − QII ) ∂ 2 Π(Z, P, QII ) � II II = g (E U |Z −Q )· − 2g(E U |Z II − QII ), 2 II II g(E [U |Z ] − Q ) (∂Q)

which is strictly negative since G(·) is log concave. This shows that any solution to (6) is a local maximum. Because G(·) is log concave, the function ψ(·) is weakly decreasing. As a consequence, it has only one fixed point, what implies that condition (6) is both necessary and sufficient for the optimum. Finally, its fixed point is weakly smaller than zero if and only if ψ(0) ≤ 0. This is precisely condition (7). �

Proof of Proposition 2: Let us for now ignore the monotonicity constraint implied by incentive compatibility and consider the platform’s relaxed problem of choosing Z = (Z0 (t), Z1 (t), ..., ZN (t)) to maximize the functional       ˆ � ˆ � N N   Π (Z) ≡ G  µ(θj , vj ) · Zj (t) · f (t)dt · ω(θj , vj ) · Zj (t) · f (t)dt   TN  TN  j=0

subject to

�N

j=0 Zj (t)

j=0

= 1 and 0 ≤ Zj (t) ≤ 1 for all t ∈ T N . Because of this last constraint, one can

restrict attention to essentially bounded controls (and therefore square-integrable), and conveniently define the Banach operator Π(·) from the Hilbert space L2 × L2 × .... × L2 (N + 1 times) into R. Lemma 5 A solution to the platform’s relaxed problem exists. Proof of Lemma 5. Because 0 ≤ Zj (t) ≤ 1, the Banach-Alaoglu theorem implies that the

relevant set of controls is compact in the weak-� topology. Because the objective Π(·) is a continuous

composition of linear operators, it follows that that Π(·) is continuous in the weak-� topology. It then follows that a maximum exists. � 28

ˆ To simplify matters, define the functional Π(·) according to   N � ˆ (Z1 (t), ..., ZN (t)) ≡ Π 1 − Π Zj (t), Z1 (t), ..., ZN (t) . j=1

ˆ The platform’s problem is then to choose Z−0 ≡ (Z1 (t), ..., ZN (t)) to maximize Π(·) subject to 0 ≤ Zj (t) ≤ 1

and

0≤

N � j=1

Zj (t) ≤ 1

f or all t ∈ T N .

(16)

ˆ ˆ � [Z−0 ](t), is a N -dimensional The Frechet gradient of the functional Π(·) at point Z−0 , denoted by Π ˆ � [Z−0 ](t) defined by vector of functions Π j

ˆ � [Z−0 ](t) ≡ g (E[U |Z]) · E[ω(Θ, V )|Z] · µ(θj , vj ) · f (t) + G (E[U |Z]) · ω(θj , vj ) · f (t), Π j where Z = (1 −

�N

j=1 Zj (t), Z−0 ).

Lemma 6 Let Z ∗ be a solution to the platform’s relaxed problem. Then Zj∗ (t) ∈ {0, 1} for a.e. t ∈ TN.

Proof of Lemma 6. Consider the analogue of the platform’s relaxed program by setting the domain of Z to be AN , rather than T N , where A is some borelian subset of T . Note that if Z ∗ solves the platform’s relaxed problem, then Z ∗ restricted to the domain AN solves the associated problem with domain AN . If an interior maximum Z ∗ exists for the platform’s problem with domain AN (where A has positive measure), then it has to satisfy for every square-integrable functions h1 (·), ..., hN (·) the following first-order condition: ˆ

AN

 N � 

j=0

ˆ � [Z ∗ ](t) · hj (t) Π j −0

By setting hj (t) = 0 for all j �= k and hj (t) =

1 λ(A) ,

  

dt = 0.

where λ(A) is the Lebesgue measure of A, the

condition above becomes

g (E[U |Z ∗ ]) · E[ω(Θ, V )|Z ∗ ] · E[U |(Θ, V, U ) ∈ A] + G (E[U |Z ∗ ]) · E[ω(Θ, V )|(Θ, V, U ) ∈ A] = 0. The unique Z ∗ that solves the equation above to every borelian set A of positive measure is such that Z0∗ (t) = 1 for a.e. t ∈ T N . Such a mechanism is clearly not a maximum to Π(·), as it can be strictly improved by a mechanism Zˆ that randomizes among all advertisers with min{µ(θj , vj ), ω(θj , vj )} > 0 and sets Zˆ0 (t) = 1 if min{µ(θj , vj ), ω(θj , vj )} ≤ 0 for all j ∈ {1, ..., N }. As a consequence, for any borelian domain A with positive measure, the platform’s relaxed program has no interior solution. Therefore, Zj∗ (t) ∈ {0, 1} for a.e. t ∈ T N . � 29

In light of the last result, let us hereafter restrict attention (without any loss of optimality) to selection rules Z such that Zj (t) ∈ {0, 1} for all t ∈ T N . The next lemma derives a necessary condition that any solution Z ∗ has to satisfy.

Lemma 7 Any solution Z ∗ to the platform’s relaxed problem is such that Zk∗ (t) = 1 if and only if � � E [ω(Θ, V )|Z ∗ ] ∗ k = arg max ω(θ, v) + η (E [U |Z ]) · · µ(θ, v) . (17) j E [U |Z ∗ ] ∗ maximizes the functional Π(·) ˆ Proof of Lemma 7. By standard arguments, if Z−0 subject to ˆ subject to (16) at some open neighborhood (16), then it has to maximize the Frechet differential of Π(·) ∗ . The Frechet differential of Π(·) ∗ is given by ˆ of Z−0 evaluated at Z−0   ˆ � N  � � ∗ � ∗ ∗ ˆ ˆ Π(Z−0 ) + Πj [Z−0 ](t) · Zj (t) − Zj (t) dt.  TN  j=0

The pointwise maximum of the integral above subject to (16) is such that Zk (t) = 1 if and only if � � ∗ ˆ �j [Z−0 k = arg max Π ](t) , j

and Zj (t) = 0 for j �= k. After rearranging, one obtains that any solution to the platform’s relaxed problem is such that Zk∗ (t) = 1 if and only if condition (17) holds. �

Finally, the next lemma establishes that the necessary condition (17) has a unique solution. This implies that the selection rule Z I described by the score sI (·) is well defined. Lemma 8 If Z ∗ and Z � solve (17), then Z ∗ = Z � . Proof of Lemma 8: Consider the selection rule Z(b) described by the scoring rule sb (θ, v) = ω(θ, v) + b · µ(θ, v). With slight abuse of notation, define the function E [ω(Θ, V )|Z(b)] that maps each b to the expected value of ω(Θ, V ) associated to the selected advertiser under the rule Z(b). Analogously, define E [µ(Θ, V )|Z(b)] as the function associated to the expected value of µ(Θ, V ). The function E [ω(Θ, V )|Z(b)] is continuous and strictly decreasing, while E [µ(Θ, V )|Z(b)] is continuous and strictly increasing. Now consider the function ϕ(b) ≡ η (E [µ(θ, v)|Z(b)]) · Because G(·) is log concave, it follows that

η(x) x

E [ω(θ, v)|Z(b)] . E [µ(θ, v)|Z(b)]

is a strictly decreasing function of x, and therefore

ϕ(b) is a strictly decreasing function of b. 30

Any selection rule Z ∗ that solves (17) is described by a score sb∗ (θ, v) such that ϕ(b∗ ) = b∗ . Note that ϕ(0) > 0 and that ϕ(ϕ(0)) < ϕ(0). Therefore, ϕ(·) crosses the 45-degree line at a unique point b∗ , what shows that (17) has a unique solution. � Because (U, V ) are positively affiliated and the hazard rate is monotone for every θ, it follows that sI (θ, v)

is strictly increasing in v. Therefore, the solution to the platform’s relaxed program satisfies

the condition 2 from Lemma 1, and is therefore implementable. As a consequence, Z I maximizes the platform’s profits among all implementable mechanisms with one-sided transfers. This concludes the proof of Proposition 2. � Proof of Lemma 3: It follows from the objective function (10) that the platform’s problem is to choose c¯ and Z to maximize G(¯ c) · E [U + V |Z] −

ˆ



ci g(ci )dci .

0

It is then immediate that the welfare-maximizing selection rule is the one that maximizes E [U + V |Z]

among all implementable selection rules. By the same arguments from the proof of Lemma 2, this is uniquely achieved (up to zero-measure perturbations) by the selection rule Z E described by the score sE (t) = µ(θ, v) + v. After plugging Z E into the objective function above and taking the first-order � � condition with respect to c¯, one obtains that c¯(Z E , QE ) = E U + V |Z E , which is a maximum since (10) is quasi-concave in c¯. It then follows that QE = −E[V |Z E ], as desired. �

Proof of Proposition 3: Consider the family of selection rules Z(b) described by scoring rules of the form sb (θ, v) = ω(θ, v) + b · µ(θ, v). Note that Z II = Z(1) and that Z I = Z(bI ), where the coefficient bI is the unique solution to ϕ(bI ) = bI . Recall that the function ϕ(·) is defined by ϕ(b) ≡ η (E [µ(θ, v)|Z(b)]) ·

E [ω(θ, v)|Z(b)] . E [µ(θ, v)|Z(b)]

The proof of Lemma 8 shows that ϕ(b) is a strictly decreasing function of b. For future reference, note that Z(b) is more searcher-friendly than Z(ˆb) if and only if b ≥ ˆb. Claim 1 E[U |Z II ] ≤ E[U |Z I ] ⇔ QII ≤ 0. Using Proposition 1, this is equivalent to showing that ϕ(bI ) ≥ 1 if and only if ϕ(1) ≥ 1.

“if ” part. In order to obtain a contradiction, let us assume that ϕ(bI ) < 1. This implies that � � � � � � � � bI < 1, and therefore E ω(θ, v)|Z I ≥ E ω(θ, v)|Z II and E µ(θ, v)|Z I ≤ E µ(θ, v)|Z II . Since η(x) x

is strictly decreasing in x (because G(·) is log concave), it follows that � � � � � � �� E ω(θ, v)|Z I � � �� E ω(θ, v)|Z II I I II 1 > ϕ(b ) = η E µ(θ, v)|Z · ≥ η E µ(θ, v)|Z · = ϕ(1), E [µ(θ, v)|Z I ] E [µ(θ, v)|Z II ] 31

contradicting the hypothesis that ϕ(1) ≥ 1.

� � “only if ” part. Let ϕ(bI ) ≥ 1. This implies that bI ≥ 1, and therefore E ω(θ, v)|Z I ≤ � � � � � � E ω(θ, v)|Z II and E µ(θ, v)|Z I ≥ E µ(θ, v)|Z II . Since η(x) x is strictly decreasing, it follows that � � � � � � �� E ω(θ, v)|Z I � � �� E ω(θ, v)|Z II I II · ≤ η E µ(θ, v)|Z · = ϕ(1), 1 ≤ ϕ(b ) = η E µ(θ, v)|Z E [µ(θ, v)|Z I ] E [µ(θ, v)|Z II ] I

as desired.

Claim 2 E[U |Z II ] − QII ≥ E[U |Z I ]. ˆ such that E[U |Z II ] − Q ˆ = E[U |Z I ]. Establishing that E[U |Z II ] − QII ≥ E[U |Z I ] is Define Q ˆ ≥ QII . Let us divide the proof in two cases. equivalent to Q

case 1: QII ≥ 0. From the claim established above, QII ≥ 0 is equivalent to ϕ(bI ) ≥ 1. ˆ < QII . Because G(·) is log concave, this implies that the Suppose towards a contradiction that Q

ˆ is such that first-order condition of problem (5) evaluated at Q � � II ˆ � η E[U |Z ] − Q � � � ˆ < 1. · E ω(Θ, V )|Z II + Q ˆ E[U |Z II ] − Q

� � � � ˆ and the fact that E ω(Θ, V )|Z I ≤ E ω(Θ, V )|Z II + Q, ˆ the inequality Using the definition of Q above implies that

� � �� � � η E U |Z I · E ω(Θ, V )|Z I = ϕ(bI ) < 1. I E [U |Z ]

ˆ ≥ QII . The last inequality contradicts the assumption that QII ≥ 0, and proves that Q

case 2: QII < 0. As in the main text, let us define the searchers’ benefit per click, x ≡ E [U |Z]−Q,

and the platform’s profit per click, y ≡ Q + E [ω(Θ, V )|Z]. Define the implementation frontiers I in the plan X × Y as the set of points (x, y) such that Q = 0 and 1. Z is implementable, 2. there is no (x� , y � ) associated to some Z � implementable such that x� ≥ x and y � ≥ y with one inequality strict.

Define the implementation frontier II in an analogous manner. From Lemma 2, it follows that � � �� II = (x, y) : x + y = E U + ω(Θ, V )|Z II .

Note that I and II intersect at a single point a0 = (x0 , y 0 ) such that (x0 , y 0 ) ∈ II and

x0 η(x0 )

= y0.

Moreover, I and II are strictly decreasing (by definition) and weakly concave curves in the plan X × Y (since the platform can always randomize between any two implementable mechanisms). 32

The platform’s problem is that of maximizing profits G(x) · y subject to (x, y) ∈ I in the case

of one-sided transfers and (x, y) ∈ II in the case of two-sided transfers. The profit-maximizing mechanism with one-sided transfers is associated to the point aI = (xI , y I ) where the platform’s

isoprofit curve is tangent to the frontier I. In turn, the profit-maximizing mechanism with two-sided transfers is associated to the point aII = (xII , y II ) where the platform’s isoprofit curve is tangent to the frontier II (see Figure 1 in the text and Figure 2 above). Showing that E[U |Z II ]−QII ≥ E[U |Z I ] is equivalent to showing that xI ≤ xII . Because QII ≤ 0,

it follows from the argument in Step 1 that xI ≥ x0 . Since I is weakly concave, the slope of the

isoprofit curve G(x)·y = G(xI )·y I is greater than 1 at aI . Therefore, from the Supporting Hyperplane Theorem, there exists a ≥ 1 such that aI = (xI , y I ) maximizes G(x) · y subject to ax + y = b (this is represented in Figure 2 by the curve III).

� � In turn, aII = (xII , y II ) maximizes G(x) · y subject to x + y = E U + ω(Θ, V )|Z II . As such,

in analogy to consumer theory, moving from aI to aII can be decomposed in a “substitution effect”

(where the price of x goes down from a to 1 - represented by the shift in the “budget constraint” � � from III to IV ) and an income effect (where the income goes up from b to E U + ω(Θ, V )|Z II represented by the shift in the “budget constraint” from IV to II). The substitution effect implies

an increase in x at the optimum (as the optimum moves from aI to point aIII in Figure 2). Because G(·) is log-concave, x is a normal good. Therefore, the income effect also implies an increase in x at the optimum (as the optimum moves from aIII to point aII in Figure 2). The Slutsky equation then implies that xII ≥ xI , as desired. � 33

Proof of Proposition 4: The reserve prices come from the fact that the platform sells the sponsored link to no advertiser (that is, leaves it blank) if and only if sm (tj ) < 0 for all j, where m ∈ {1, 2}. Profit-maximization precludes efficient advertising since sII (t) < sW (t) for all t.

It is immediate from equation (6) that G(E[U |Z II ] − QII ) ≤ G(E[U |Z E ] − QE ). Moreover, from

Proposition 3 we know that G(E[U |Z I ]) ≤ G(E[U |Z II ] − QII ).

ˆ vˆ) Let us now show that E[U |Z II ] ≤ E[U |Z E ]. The idea is to argue that for any (θ, v) and (θ, ˆ vˆ), sE (θ, v) > sE (θ, ˆ vˆ) implies sII (θ, v) > sII (θ, ˆ vˆ). Let µ(θ, v) < µ(θ, ˆ vˆ) and such that µ(θ, v) < µ(θ, ˆ vˆ) + vˆ. This obviously implies that v > vˆ. Moreover, assume that µ(θ, v) + v > µ(θ, ˆ v |θ) ˆ vˆ) = v − vˆ − 1 − F (v|θ) + 1 − F (ˆ ω(θ, v) − ω(θ, ˆ f (v|θ) f (ˆ v |θ) � � ˆ 1 − F (ˆ v |θ) 1 − F (ˆ v |θ) = v − vˆ + − ˆ f (ˆ v |θ) f (ˆ v |θ) � �� � A � � 1 − F (ˆ v |θ) 1 − F (v|θ) + − . f (ˆ v |θ) f (v|θ) � �� � B

Assumption 1 (positive affiliation) implies that A > 0 and the monotone hazard rate condition implies that B > 0. One can then conclude that ˆ vˆ) > v − vˆ ω(θ, v) − ω(θ,

ˆ vˆ) − µ(θ, v). > µ(θ,

ˆ vˆ) such that µ(θ, v) < µ(θ, ˆ vˆ), sE (θ, v) > sE (θ, ˆ vˆ) implies sII (θ, v) > Therefore, for any (θ, v) and (θ, ˆ vˆ). This implies that E[U |Z II ] ≤ E[U |Z E ], concluding the proof of Proposition 4. � sII (θ, Proof of Proposition 5: Let platform A post a mechanism with two-sided transfers (ZA , QA ). Note that it is a dominated strategy to offer a mechanism (ZA , QA ) such that E[U − QA |ZA ] > E[U +ω(Θ, V )|Z II ], as that would imply that A makes negative profits. Therefore, let E[U −QA |ZA ] ≤

E[U + ω(Θ, V )|Z II ]. Because advertisers multi-home and searchers single-home, the best-reply of platform B is to post (Z II , Q∗B ), where   −E[ω(Θ, V )|Z II ] if   ∗ II QB = E[U − QA |ZA ] − E[U |Z ] − ε if    QII if

E[U + ω(Θ, V )|Z II ] = E[U − QA |ZA ]

E[U + ω(Θ, V )|Z II ] > E[U − QA |ZA ] ≥ E[U − QII |Z II ] E[U − QII |Z II ] > E[U − QA |ZA ],

where ε is arbitrarily small and positive. In the unique Nash equilibrium of this game, both platform post (Z II , −E[ω(Θ, V )|Z II ]) and enjoy zero profits. � Proof of Proposition 6: Let platform A post a mechanism with one-sided transfers ZA . Because advertisers multi-home and searchers single-home, the best-reply of platform B is to post a selection 34

∗ such that E[U |Z ∗ ] = E[U |Z ] + ε whenever E[U |Z ] ≥ E[U |Z I ], where ε is arbitrarily small rule ZB A A B

∗ = Z I . In the unique Nash equilibrium of this and positive. Else, if E[U |ZA ] < E[U |Z I ], then set ZB

game, both platform post Z U and enjoy profits given by E[ω(Θ, V )|Z U ] > 0. �

Proof of Proposition 7: First, note from (13) that b(θ, v) is strictly increasing in v (which follows from the fact that z(θ, v) is strictly increasing in v). Therefore, for all vj ∈ [rj , v¯], b−1 v (θ, b)

is strictly increasing in b and b−1 v (θj , b(θj , vj )) = vj . Let Q denote the transfers to searchers set by the platform (which are zero when transfers are one-sided). It is an equilibrium that each advertiser j bids according to b(θj , v) if and only if � � � � � � II −1 vj ∈ arg max G E U |Z − Q · (vj − b(θj , vˆj )) · Pr max{s(Θk , Vk )} ≤ s(θj , bv (θj , b(θj , vˆj ))) , vˆj ≥rj

which rewrites

k�=j

� � � � vj ∈ arg max G E U |Z II − Q · (vj − b(θj , vˆj )) · z(θj , vˆj ). vˆj ≥rj

By the Integral-form Envelope Theorem (see Milgrom (2004)), the condition above implies that ˆ vj � � � � � � � � II II G E U |Z − Q · (vj − b(θj , vj )) · z(θj , vj ) = G E U |Z −Q · z(θj , v˜)d˜ v, r(θj )

which is equivalent to (13). Therefore, by construction, the selection vˆj = vj satisfies the Envelope condition. Since the advertisers’ payoff function satisfies strictly increasing differences in (ˆ vj , vj ), one can use the Constraint Simplification Theorem (Milgrom (2004), page 105) to conclude that vˆj = vj best responds all reports in the range vj ∈ [rj , v¯]. This is equivalent to saying that bidding b(θj , vj )

is a best response in the interval [rj , b(θj , v¯)]. As it is clearly not optimal to bid b > b(θj , v¯), one can conclude that b(θj , vj ) best responds all possible bids (and constitutes the unique Bayes-Nash equilibrium) of the pay-your-bid scoring auction considered above. Finally, note that sˆ(θj , b(θj , vj )) ≥ sˆ(θk , b(θk , vk )) if and only if s(θj , vj ) ≥ s(θk , vk ). Moreover,

reserve prices rj are chosen such that some advertiser j wins if and only if maxj {s(θj , vj )} ≥ 0. Therefore, the pay-your-bid scoring auction considered above implements the selection rule Z, concluding the proof of Proposition 7. � Proof of Proposition 8: Denote by (Θmax , V max ) the random variable defined by the maximizer of maxj∈{1,...,N −1} {s(Θj , Vj )}. Note that for any x ≤ y Pr [s(Θ with associated density

max

,V

max

) ≤ x |s(Θ

max

,V

max

) ≤ y] =

(N − 1) (Fs (x))N −2 fs (x) (Fs (y))N −1 35

.



Fs (x) Fs (y)

�N −1

,

By the rules of a pay-the-next-bid scoring auction, the expected payments of an advertiser with signal and willingness to pay (θ, v) equals p(θ, v) = G(E [U |Z] − Q) · E [β(Θmax , V max ) |s(Θmax , V max ) ≤ s(θ, v) ] = G(E [U |Z] − Q) ·

ˆ

θ

˜ θ¯ ˆ s−1 v [θ,s(θ,v)] r(θ)

By the condition 3 in Lemma 1, it follows that

� �N −2 ˜ v˜) Fs (s(θ, ˜ v˜)) ˜ v˜))d˜ ˜ β(θ, fs (s(θ, v dθ.

(18)



(19)

p(θ, v) = G(E [U |Z] − Q) · z(θ, v) · v −

ˆ

v

v

� zj (θ, v˜)d˜ v .

After equalizing (18) and (19), it follows that the bidding function in any equilibrium of a pay-thenext-bid scoring auction that implements the selection rule Z has to satisfy (14). The integral equation (14) is a Volterra equation of the first kind. Applying an existence result from Debnath and Mikusinski (1999), one can conclude that the the candidate bidding function β(θ, v) exists and is unique. In the case where β(θ, v) is strictly increasing in v for all θ, a proof analogous to that of Proposition 7 shows that the selection rule Z is implemented by the proposed pay-the-next-bid scoring auction. Conversely, assume that β(θ, v) is not strictly increasing in v for some θ. By construction, β(θ, v) is the unique candidate bidding function that is consistent with the selection rule Z in any pay-the-next-bid scoring auction. Take v and vˆ such that v > vˆ but β(θ, v) ≤ β(θ, vˆ). In this case, s(θ, v) > s(θ, vˆ) but s˜(θ, b(θ, v)) ≤ s˜(θ, b(θ, vˆ)). As a consequence,

the rule Z cannot be implemented by any pay-the-next-bid scoring auction, concluding the proof of Proposition 8. � Proof of Proposition 9: The proof is analogous to that of Propositions 1, 2 and 3, and is therefore omitted. �

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