Internet Advertising and the Generalized Second-Price Auction: Selling Billions of Dollars Worth of Keywords

By BENJAMIN EDELMAN,MICHAEL OSTROVSKY, AND MICHAEL SCHWARZ*

We investigate the “generalized second-price” (GSP) auction, a new mechanism used by search engines to sell online advertising. Although GSP looks similar to the Vickrey-Clarke-Groves (VCG) mechanism, its properties are very different. Unlike the VCG mechanism, GSP generally does not have an equilibrium in dominant strategies, and truth-telling is not an equilibrium of GSP. To analyze the properties of GSP, we describe the generalized English auction that corre- sponds to GSP and show that it has a unique equilibrium. This is an ex post equilibrium, with the same payoffs to all players as the dominant strategy equilibrium of VCG. (JEL D44, L81, M37)

This paper investigates a new auction mech- term (“query”) into a search engine, he gets anism, which we call the “generalized second- back a page with results, containing both the price” auction, or GSP. GSP is tailored to the links most relevant to the query and the spon- unique environment of the market for online sored links, i.e., paid advertisements. The ads ads, and neither the environment nor the mech- are clearly distinguishable from the actual anism has previously been studied in the mech- search results, and different searches yield dif- anism design literature. While studying the ferent sponsored links: advertisers target their properties of a novel mechanism is often fasci- ads based on search keywords. For instance, if a nating in itself, our interest is also motivated by travel agent buys the word “Hawaii,” then each the spectacular commercial success of GSP. It is time a user performs a search on this word, a the dominant transaction mechanism in a large link to the travel agent will appear on the search and rapidly growing industry. For example, results page. When a user clicks on the spon- Google’s total revenue in 2005 was $6.14 bil- sored link, he is sent to the advertiser’s Web lion. Over 98 percent of its revenue came from page. The advertiser then pays the search engine GSP auctions. Yahoo!’s total revenue in 2005 for sending the user to its Web page, hence the was $5.26 billion. A large share of Yahoo!’s name—“pay-per-click” pricing. revenue is derived from sales via GSP auctions. The number of ads that the search engine can It is believed that over half of Yahoo!’s revenue show to a user is limited, and different positions is derived from sales via GSP auctions. As of on the search results page have different desir- May 2006, the combined market capitalization abilities for advertisers: an ad shown at the top of these companies exceeded $150 billion. of a page is more likely to be clicked than an ad Let us briefly describe how these auctions shown at the bottom. Hence, search engines work. When an Internet user enters a search need a system for allocating the positions to advertisers, and auctions are a natural choice. Currently, the mechanisms most widely used by * Edelman: Department of Economics, Harvard University, Cambridge, MA 02138 (e-mail: [email protected]); search engines are based on GSP. Ostrovsky: Graduate School of Business, Stanford University, In the simplest GSP auction, for a specific Stanford, CA 94305 (e-mail: [email protected]); keyword, advertisers submit bids stating their Schwarz: Yahoo! Research, 1950 University Ave., Suite 200, maximum willingness to pay for a click. When Berkeley, CA 94704 (e-mail: [email protected]). We a user enters a keyword, he receives search thank Drew Fudenberg, Louis Kaplow, Robin Lee, David McAdams, Paul Milgrom, Muriel Niederle, Ariel Pakes, David results along with sponsored links, the latter Pennock, and Al Roth for helpful discussions. shown in decreasing order of bids. In particular, 242 VOL. 97 NO. 1 EDELMAN ET AL.: INTERNET ADVERTISING AND THE GSP AUCTION 243 the ad with the highest bid is displayed at the pected revenue to the seller is at least as high top, the ad with the next highest bid is displayed as in the dominant-strategy equilibrium of the in the second position, and so on. If a user VCG auction. subsequently clicks on an ad in position i, that In Section IV, we present our main result. We advertiser is charged by the search engine an introduce the generalized English auction with amount equal to the next highest bid, i.e., the independent private values, which corresponds bid of an advertiser in position (i ϩ 1). If a to the generalized second-price auction and is search engine offered only one advertisement meant to capture the convergence of bidding per result page, this mechanism would be behavior to the static equilibrium, in the same equivalent to the standard second-price auc- spirit as taˆtonnement processes in the theory tion, coinciding with the Vickrey-Clarke- of general equilibrium and the deferred- Groves (VCG) mechanism (William Vickrey acceptance salary adjustment process in the theory 1961; Edward H. Clarke 1971; Theodore Groves of matching in labor markets. The generalized 1973), auction. With multiple positions available, English auction has several notable features. GSP generalizes the second-price auction (hence Although it is not dominant-strategy solvable, it the name). Here, each advertiser pays the next has a unique, perfect Bayesian equilibrium in highest advertiser’s bid. But as we will demon- continuous strategies. In this equilibrium, all strate, the multi-unit GSP auction is no longer players receive VCG payoffs. Moreover, this equivalent to the VCG auction and lacks some of equilibrium is ex post, i.e., even if a particular VCG’s desirable properties. In particular, un- player learned the values of other players before like the VCG mechanism, GSP generally does the game, he would not want to change his not have an equilibrium in dominant strategies, strategy. This, in turn, implies that the equilib- and truth-telling is not an equilibrium of GSP. rium is robust, i.e., it does not depend on the In Section I, we describe the evolution of the underlying distribution of values: the profile of market for Internet advertisements and the strategies that we identify is an ex post Bayesian unique features of the environment in this mar- Nash equilibrium for any set of distributions of ket. In Section II, we introduce a model of advertisers’ private values. sponsored search auctions, and we begin our There are several recent theoretical and em- analysis of the model in Section III. Since ad- pirical papers related to sponsored search auc- vertisers can change their bids frequently, spon- tions. Gagan Aggarwal and Jason D. Hartline sored search auctions can be modeled as a (2005), Aranyak Mehta et al. (2005), and Mo- continuous or an infinitely repeated game. By hammad Mahdian, Hamid Nazerzadeh, and the folk theorem, however, such a game will Amin Saberi (2006) propose computationally have an extremely large set of equilibria, and so fast, near-optimal mechanisms for pricing and we focus instead on the one-shot, simultaneous- allocating slots to advertisers in the presence of move, complete information stage game, intro- budget constraints and random shocks. Christo- ducing restrictions on advertisers’ behavior pher Meek, David M. Chickering, and David B. suggested by the market’s dynamic structure. Wilson (2005) describe incentive-compatible We call the equilibria satisfying these restric- auctions with stochastic allocation rules, gener- tions “locally envy-free.” alizing Vickrey auctions, and argue that such We then proceed to show that the set of auctions can be useful for selling Internet ad- locally envy-free equilibria contains an equi- vertising despite being inefficient. Note that, in librium in which the payoffs of the players are contrast to these papers, we study the mecha- the same as in the dominant-strategy equilib- nisms actually used by the search engines. rium of the VCG auction, even though both Xiaoquan Zhang (2005), Kursad Asdemir the bids of the players and the payment rules (2006), and Edelman and Ostrovsky (forthcom- in the mechanisms are very different. More- ing) present empirical evidence of bid and rank- over, this equilibrium is the worst locally ing fluctuations in both generalized first-price envy-free equilibrium for the search engine and generalized second-price auctions. They ar- and the best locally envy-free equilibrium for gue that history-dependent strategies can give the advertisers. Consequently, in any locally rise to such fluctuations. However, Hal R. Var- envy-free equilibrium of GSP, the total ex- ian (forthcoming) empirically analyzes GSP 244 THE AMERICAN ECONOMIC REVIEW MARCH 2007 auction data from Google and reports that lo- corresponds to a pricing model in which an cally envy-free Nash equilibria “describe the advertiser is charged every time its link is basic properties of the prices observed in shown to a potential customer. “Pay-per-click” Google’s ad auction reasonably accurately.”1 is a middle ground between the two models: the advertiser pays every time a user clicks on the I. The Structure and Evolution of Sponsored link. All three payment models are widely used Search Auctions on the Internet.3 The specific sector of Internet advertising that we study, sponsored search auc- A. Notable Features of the Market for tions, has converged to pay-per-click pricing. Internet Advertising Since GSP evolved in the market for online advertising, its rules reflect the environment’s A combination of features makes the market for unique characteristics. GSP insists that for each Internet advertising unique. First, bids can be keyword, advertisers submit a single bid—even changed at any time. An advertiser’s bid for a though several different items are for sale (dif- particular keyword will apply every time that key- ferent advertising positions). GSP’s unusual word is entered by a search engine user, until the one-bid requirement makes sense in this setting: advertiser changes or withdraws the bid. For ex- the value of being in each position is propor- ample, the advertiser with the second highest bid tional to the number of clicks associated with on a given keyword at some instant will be shown that position; the benefit of placing an ad in a as the second sponsored link to a user searching higher position is that the ad is clicked more, for that keyword at that instant. The order of the but the users who click on ads in different ads may be different next time a user searches for positions are assumed to have the same values that keyword, because the bids could have to advertisers (e.g., the same purchase probabil- changed in the meantime.2 ities). Consequently, even though the GSP en- Second, search engines effectively sell flows vironment is multi-object, buyer valuations can of perishable advertising services rather than be adequately represented by one-dimensional storable objects: if there are no ads for a partic- types. For some advertisers, one bid per key- ular search term during some period of time, the word may not be sufficiently expressive to fully “capacity” is wasted. convey preferences. For example, a single bid Finally, unlike other centralized markets, ignores the possibility that users who click on where it is usually clear how to measure what is position 5 are somehow different from those being sold, there is no “unit” of Internet adver- who click on position 2; it does not allow for the tisement that is natural from the points of view possibility that advertisers care about the allo- of all involved parties. From the advertiser’s cation of other positions, and so on. Nonethe- perspective, the relevant unit is the cost of at- less, these limitations are apparently not large tracting a customer who makes a purchase. This enough to justify added complexity in the bid- corresponds most directly to a pricing model in ding language. Nico Brooks (2004) finds only which an advertiser pays only when a customer moderate differences in purchase probabilities actually completes a transaction. From the when ads are shown in different positions. Fol- search engine’s perspective, the relevant unit is lowing search engines’ approaches and Brooks’ what it collects in revenues every time a user empirical findings, we likewise assume the performs a search for a particular keyword. This value of a click is the same in all positions.

1 Varian discovered envy-free Nash equilibria indepen- 3 A prominent example of “pay-per-transaction,” and dently and called them “Symmetric Nash Equilibria” in his even “pay-per-dollar of revenue” (“revenue sharing”), is paper. Amazon.com’s Associates Program, www.amazon.com/gp/ 2 For manual bidding through online advertiser centers, browse.html?&nodeϭ3435371 (accessed June 10, 2006). both Google and Yahoo! allow advertisers to make unlim- Under this program, a Web site that sends customers to ited changes. In contrast, the search engines impose restric- Amazon.com receives a percentage of customers’ pur- tions on the behavior of software bidding agents: e.g., chases. “Pay-per-impression” advertising, in the form of Yahoo! limits the number of times an advertiser can change banner ads, remains popular on major Internet portals, such his bid in a given period of time. as yahoo.com, msn.com, and aol.com. VOL. 97 NO. 1 EDELMAN ET AL.: INTERNET ADVERTISING AND THE GSP AUCTION 245

One important possibility that we abstract B. Evolution of Market Institutions away from is that advertisers differ along di- mensions other than per-click value, i.e., have The history of sponsored search auctions is of different probabilities of being clicked when interest as a case study of whether, how, and placed in the same position. (These probabilities how quickly markets come to address their are known in the industry as “click-through structural shortcomings. Many important mech- rates,” or CTRs.) Different search engines treat anisms have recently been designed essentially this possibility differently. Yahoo! ignores the from scratch, entirely replacing completely dif- differences, ranks the advertisers purely in de- ferent historical allocation mechanisms: radio creasing order of bids, and charges the next- spectrum auctions (Paul Milgrom 2000; Ken highest advertiser’s bid.4 Google multiplies Binmore and Paul Klemperer 2002), electricity each advertiser’s bid by its “quality score,” auctions (Robert Wilson 2002), and others. In which is based on CTR and other factors, to contrast, reminiscent of the gradual evolution of compute its “rank number,” ranks the ads by medical residency match rules (Alvin E. Roth rank numbers, and then charges each adver- 1984), sponsored search ad auctions have tiser the smallest amount sufficient to exceed evolved in steps over time. In both medical the rank number of the next advertiser.5 In our residency and search advertising, flawed mech- analysis, we assume that all advertisers are anisms were gradually replaced by increasingly identical along dimensions other than per- superior designs. Notably, the Internet advertis- click value, which eliminates this difference ing market evolved much faster than the medi- between the mechanisms used at Google and cal matching market. This may be due to the Yahoo!. As we discuss at the end of Section competitive pressures on mechanism designers III, the analysis would remain largely the present in the former but not in the latter, much same if there were advertiser-specific differ- lower costs of entry and experimentation, ad- ences in CTRs and “quality scores,” although vances in the understanding of market mecha- the equilibria under Google and Yahoo! nisms, and improved technology. mechanisms would not be identical.6,7 We proceed with a brief chronological review of the development of sponsored search mechanisms. 4 See help.yahoo.com/help/us/performance/customer/ dtc/bidding/, link to “How do I figure out my cost?” and Early Internet Advertising.—Beginning in searchmarketing.yahoo.com/srch/index.php, link to “How Sponsored Search Works” (accessed June 10, 2006). 1994, Internet advertisements were largely sold 5 See www.google.com/adwords/learningcenter/#section1, on a per-impression basis. Advertisers paid flat links to “Ad Ranking” and “Cost Control” (accessed June fees to show their ads a fixed number of times 10, 2006). Initially, Google’s pricing mechanism was more (typically, 1,000 showings or “impressions”). transparent: quality score was equal to the estimated click- Contracts were negotiated on a case-by-case through rate. 6 The analysis would have to change considerably if basis, minimum contracts for advertising pur- there were specific advertiser-position effects. The magni- chases were large (typically, a few thousand tude of these advertiser-position effects is ultimately an dollars per month), and entry was slow.8 empirical question, and we do not have the kind of data that would allow us to answer it; however, judging from the fact that the two major search engines effectively ignore it in Generalized First-Price Auctions.—In 1997, their mechanisms (Yahoo! ignores CTRs altogether; Google Overture (then GoTo; now part of Yahoo!) in- computes an advertiser’s estimated CTR conditional on the troduced a completely new model of selling advertiser attaining the first position), we believe it to be small. 7 Another important difference between the search en- gines’ implementations of GSP is the amount of information From this information, they can back out estimates of their available to the advertisers. On Yahoo!, advertisers can competitors’ rank numbers. Moreover, advertisers can ex- directly observe the bids of their competitors (uv.bidtool. periment with changing their bids, which can also give them overture.com/d/search/tools/bidtool). On Google, they can- independent and relatively accurate estimates. Web sites not. For any keyword and any bid amount, however, they accessed on June 10, 2006. can get an estimated average position and average cost- 8 See www.worldata.com/wdnet8/articles/the_history_ per-click they can expect (adwords.google.com/select/ of_Internet_Advertising.htm and www.zakon.org/robert/in- KeywordToolExternal, “Cost and ad position estimates”). ternet/timeline (both accessed June 10, 2006). 246 THE AMERICAN ECONOMIC REVIEW MARCH 2007

Internet advertising. In the original Overture seller. Moreover, if the “speed” of the robots var- auction design, each advertiser submitted a bid ies across advertisers, revenues can be very low reporting the advertiser’s willingness to pay on even if advertisers’ values are high. For instance, a per-click basis, for a particular keyword. The in the example above, suppose advertiser 1 has a advertisers could now target their ads: instead of robot that can adjust the bid very quickly, while paying for a banner ad that would be shown to advertisers 2 and 3 are humans and can change everyone visiting a Web site, advertisers could their bids at most once a day. In this case, as long specify which keywords were relevant to their as advertiser 3 does not bid more than his value, products and how much each of those keywords the revenues of a search engine are at most $2.02 (or, more precisely, a user clicking on their ad per click. Indeed, suppose advertiser 3 bids $2.00. after looking for that keyword) was worth to If advertiser 2 bids $2.01, he will be in the second them. Also, advertising was no longer sold per position paying $2.01. If he bids any amount 1,000 impressions; rather, it was sold one click greater than that but lower than his value, he will at a time. Every time a consumer clicked on a remain in the second position and will pay more sponsored link, an advertiser’s account was au- per click, because the robot of advertiser 1 will tomatically billed the amount of the advertiser’s quickly outbid him. The revenue would not most recent bid. The links to advertisers were change even if the values of advertisers 1 and 2 arranged in descending order of bids, making were much higher. highest bids the most prominent. The ease of use, the very low entry costs, and the transparency of Generalized Second-Price Auctions.—Under the mechanism quickly led to the success of Over- the generalized first-price auction, the advertiser ture’s paid search platform as the advertising pro- who could react to competitors’ moves fastest vider for major search engines, including Yahoo! had a substantial advantage. The mechanism and MSN. However, the underlying auction therefore encouraged inefficient investments in mechanism itself was far from perfect. In partic- gaming the system, causing volatile prices and ular, Overture and advertisers quickly learned that allocative inefficiencies. Google addressed these the mechanism was unstable due to the fact that problems when it introduced its own pay-per-click bids could be changed very frequently. system, AdWords Select, in February 2002. Google also recognized that an advertiser in Example. Suppose there are two slots on a page position i will never want to pay more than one and three advertisers. An ad in the first slot bid increment above the bid of the advertiser in receives 200 clicks per hour, while the second position (i ϩ 1), and adopted this principle in its slot gets 100. Advertisers 1, 2, and 3 have newly designed generalized second-price auc- values per click of $10, $4, and $2, respectively. tion mechanism. In the simplest GSP auction, Suppose advertiser 2 bids $2.01, to guarantee an advertiser in position i pays a price per click that he gets a slot. Then advertiser 1 will not equal to the bid of an advertiser in position (i ϩ want to bid more than $2.02—he does not need 1) plus a minimum increment (typically $0.01). to pay more than that to get the top spot. But This second-price structure makes the market then advertiser 2 will want to revise his bid to more user friendly and less susceptible to $2.03 to get the top spot, advertiser 1 will in gaming. turn raise his bid to $2.04, and so on. Clearly, Recognizing these advantages, Yahoo!/Over- there is no pure strategy equilibrium in the ture also switched to GSP. Let us describe the one-shot version of the game, and so if adver- version of GSP that it implemented.9 Every tisers best respond to each other, they will want to revise their bids as often as possible. Hence, advertisers will make socially ineffi- 9 We focus on Overture’s implementation, because cient investments into bidding robots, which can Google’s system is somewhat more complex. Google ad- also be detrimental for the revenues of search justs effective bids based on ads’ click-through rates and engines. David McAdams and Schwarz (forth- other factors, such as “relevance.” But under the assumption that all ads have the same relevance and click-through rates coming) argue that in various settings, the costs conditional on position, Google’s and Yahoo!’s versions of that buyers incur while trying to “game” an auc- GSP are identical. As we show in Section III, it is straight- tion mechanism are fully passed through to the forward to generalize our analysis to Google’s mechanism. VOL. 97 NO. 1 EDELMAN ET AL.: INTERNET ADVERTISING AND THE GSP AUCTION 247 advertiser submits a bid. Advertisers are ar- position i is equal to the difference between the ranged on the page in descending order of their aggregate value of clicks that all other advertis- bids. The advertiser in the first position pays a ers would have received if i were not present in price per click that equals the bid of the second the market and the aggregate value of clicks that advertiser plus an increment; the second adver- all other advertisers receive when i is present. tiser pays the bid of the third advertiser plus an Note that an advertiser in position j Ͻ i is not increment; and so forth. affected by i, and so the externality i imposes on her is zero, while an advertiser in position j Ͼ i Example (continued). Let us now consider the would have received position (j Ϫ 1) in the payments in the environment of the previous absence of i, and so the externality i imposes on example under the GSP mechanism. If all ad- her is equal to her value per click multiplied by vertisers bid truthfully, then bids are $10, $4, the difference in the number of clicks in posi- and $2. Payments in GSP will be $4 and $2.10 tions j and (j Ϫ 1). Truth-telling is indeed an equilibrium in this example, because no advertiser can benefit by Example (continued). Let us compute VCG changing his bid. Note that total payments of payments for the example considered above. advertisers 1 and 2 are $800 and $200, The second advertiser’s payment is $200, as respectively. before. However, the payment of the first ad- vertiser is now $600: $200 for the externality Generalized Second-Price and VCG Auc- that he imposes on advertiser 3 (by forcing him tions.—GSP looks similar to the VCG mecha- out of position 2) and $400 for the externality nism, because both mechanisms set each that he imposes on advertiser 2 (by moving him agent’s payments based only on the allocation from position 1 to position 2 and thus causing and bids of other players, not based on that him to lose (200 Ϫ 100) ϭ 100 clicks per hour). agent’s own bid. In fact, Google’s advertising Note that in this example, revenues under VCG materials explicitly refer to Vickrey and state are lower than under GSP. As we will show that Google’s “unique auction model uses Nobel later (Remark 1 in Section II), if advertisers Prize–winning economic theory to eliminate were to bid their true values under both mech- ... that feeling that you’ve paid too much.”11 anisms, revenues would always be higher under But GSP is not VCG.12 In particular, unlike the GSP. VCG auction, GSP does not have an equilib- rium in dominant strategies, and truth-telling is C. Assessing the Market’s Development generally not an equilibrium strategy in GSP (see the example in Remark 3 in Section II). The chronology above suggests three major With only one slot, VCG and GSP would be stages in the development of the sponsored identical. With several slots, the mechanisms search advertising market. First, ads were sold are different. GSP charges the advertiser in po- manually, slowly, in large batches, and on a sition i the bid of the advertiser in position i ϩ cost-per-impression basis. Second, Overture 1. In contrast, VCG charges the advertiser in implemented keyword-targeted per-click sales position i the externality that he imposes on and began to streamline advertisement sales others by taking one of the slots away from with some self-serve bidding interfaces, but them: the total payment of the advertiser in with a highly unstable first-price mechanism. Next, Google implemented the GSP auction, which was subsequently adopted by Overture (Yahoo!). 10 For convenience, we neglect the $0.01 minimum in- crements. Interestingly, Google and Yahoo! still use 11 See https://www.google.com/adsense/afs.pdf (accessed GSP, rather than VCG, which would reduce June 10, 2006). incentives for strategizing and make life easier 12 Roth and Axel Ockenfels (2002) describe another for advertisers. We see several possible reasons example in which the architects of an auction may have tried to implement a mechanism strategically equivalent to for this. First, VCG is hard to explain to typical the Vickrey auction, but did not get an important part of the advertising buyers. Second, switching to VCG mechanism right. may entail substantial transition costs: VCG 248 THE AMERICAN ECONOMIC REVIEW MARCH 2007 revenues are lower than GSP revenues for the position to g(2), and so on, down to position same bids, and advertisers might be slow to stop min{N, K}. Note that each advertiser gets at shading their bids. Third, the revenue conse- most one object. If a user clicks on an advertis- quences of switching to VCG are uncertain: er’s link, the advertiser’s payment per click is even the strategic equivalence of second-price equal to the next advertiser’s bid. So advertiser (i) ␣ (iϩ1) and English auctions under private values fails g(i)’s total payment p is equal to ib for to hold in experiments (John Kagel, Ronald M. i ʦ {1, ... , min{N, K}}, and his payoff is equal ␣ Ϫ (iϩ1) Harstad, and Dan Levin 1987). And, of course, to i(sg(i) b ). If there are at least as many simply implementing and testing a new system positions as advertisers (N Ն K), then the last may be costly—imposing switching costs on advertiser’s payment p(K) is equal to zero.15 advertisers as well as on search engines. It is also useful to describe explicitly the rules that the VCG mechanism would impose in this II. The Rules of GSP setting. The rules for allocating positions are the same as under GSP: position i is assigned to Let us now formally describe the rules of a advertiser g(i) with the i-th highest bid b(i). The sponsored search auction. For a given keyword, payments, however, are different. Each adver- there are N objects (positions on the screen, tiser’s payment is equal to the negative exter- where ads related to that keyword can be dis- nality that he imposes on others, assuming that played) and K bidders (advertisers).13 The (ex- bids are equal to values. Thus, the payment of pected) number of clicks per period received by the last advertiser who gets allocated a spot is the advertiser whose ad was placed in position i the same as under GSP: zero if N Ն K; ␣ ␣ (Nϩ1) Ͻ is i. The value per click to advertiser k is sk. Nb otherwise. For all other i min{N, Advertisers are risk-neutral, and advertiser k’s K}, payment pV induced by VCG will be dif- ␣ payoff from being in position i is equal to isk ferent from payment p induced by GSP. V,(i) ϭ ␣ Ϫ ␣ (iϩ1) ϩ V,(iϩ1) minus his payments to the search engine. Note Namely, p ( i iϩ1)b p . that these assumptions imply that the number of In the following two sections, we will con- times a particular position is clicked does not sider two alternative ways of completing the depend on the ads in this and other positions, model: as a simultaneous-move game of complete and also that an advertiser’s value per click does information, resembling a sealed-bid second-price not depend on the position in which its ad is auction, and as an extensive-form game of incom- displayed. Without loss of generality, positions plete information, resembling an ascending En- are labeled in descending order: for any i and j glish auction. Before moving on to these models, Ͻ ␣ Ͼ ␣ such that i j, we have i j. let us make a few observations about GSP and We model the GSP auction as follows. Sup- VCG. pose at some time t a search engine user enters a given keyword, and, for each k, advertiser k’s REMARK 1: If all advertisers were to bid the last bid submitted for this keyword prior to t same amounts under the two mechanisms, then was bk; if advertiser k did not submit a bid, we each advertiser’s payment would be at least as ϭ (j) set bk 0. Let b and g(j) denote the bid and large under GSP as under VCG. identity of the j-th highest advertiser, respec- tively. If several advertisers submit the same This is easy to show by induction on adver- bid, they are ordered randomly.14 The mecha- tisers’ payments, starting with the last advertiser nism then allocates the top position to the ad- who gets assigned a position. For i ϭ min{K, (i) ϭ V,(i) ϭ ␣ (iϩ1) Ͻ vertiser with the highest bid, g(1), the second N}, p p ib . For any i min{K,

13 In actual sponsored search auctions at Google and 15 Although we set the reserve price to zero, search Yahoo!, advertisers can also choose to place “broad match” engines charge the last advertiser a positive reserve price. bids that match searches that include a keyword along with We also assume that a bid can be any nonnegative real additional search terms. number, while in practice bids can be specified only in $.01 14 The actual practice at Overture is to show equal bids increments. Finally, we assume that advertiser g(i)is according to the order in which the advertisers placed their charged the amount b(iϩ1) per click, while search engines bids. typically charge one cent more, (b(iϩ1) ϩ $.01). VOL. 97 NO. 1 EDELMAN ET AL.: INTERNET ADVERTISING AND THE GSP AUCTION 249

V,(i) Ϫ V,(iϩ1) ϭ ␣ Ϫ ␣ (iϩ1) Յ N}, p p ( i iϩ1)b We therefore focus on simple strategies and ␣ (iϩ1) Ϫ ␣ (iϩ2) ϭ (i) Ϫ (iϩ1) ib iϩ1b p p . study the rest points of the bidding process: if the vector of bids stabilizes, at what bids can it stabi- REMARK 2: Truth-telling is a dominant strat- lize? We impose several assumptions and restric- egy under VCG. tions. First, we assume that all values are common knowledge: over time, advertisers are likely to This is a well-known property of the VCG learn all relevant information about each other’s mechanism. values. Second, since bids can be changed at any time, stable bids must be best responses to each REMARK 3: Truth-telling is not a dominant other—otherwise, an advertiser whose bid is not a strategy under GSP. best response would have an incentive to change it. Thus, we assume that the bids form an equilib- For instance, consider a slight modification of rium in the simultaneous-move, one-shot game of the example from Section I. There are still three complete information. Third, what are the simple advertisers, with values per click of $10, $4, and strategies that an advertiser can use to increase his $2, and two positions. However, the click-through payoff, beyond simple best responses to the other rates of these positions are now almost the same: players’ bids? the first position receives 200 clicks per hour, and One clear strategy is to try to force out the the second one gets 199. If all players bid truth- player who occupies the position immediately Ϫ fully, then advertiser 1’s payoff is equal to ($10 above. Suppose advertiser k bids bk and is as- ϭ Ј Ͼ ء $4) 200 $1,200. If, instead, he shades his bid signed to position i, and advertiser k bids bkЈ bk and bids only $3 per click, he will get the second and is assigned to position (i Ϫ 1). Note that if k position, and his payoff will be equal to ($10 Ϫ raises his bid slightly, his own payoff does not ϭ $1,592 Ͼ $1,200. change, but the payoff of the player above him 199 ء ($2 decreases. Of course, player kЈ can retaliate, and III. GSP and Locally Envy-Free Equilibria the most she can do is to slightly underbid adver- tiser k, effectively swapping places with him. If Advertisers bidding on Yahoo! and Google can advertiser k is better off after such retaliation, he change their bids very frequently. We therefore will indeed want to force player kЈ out, and the think of these sponsored search auctions as con- vector of bids will change. Thus, if the vector tinuous time or infinitely repeated games in which converges to a rest point, an advertiser in position advertisers originally have private information i should not want to “exchange” positions with the about their types, gradually learn the values of advertiser in position (i Ϫ 1). We call such vectors others, and can adjust their bids repeatedly. In of bids “locally envy-free.”17 principle, the sets of equilibria in such repeated games can be very large, with players potentially DEFINITION 1: An equilibrium of the punishing each other for deviations. The strategies simultaneous-move game induced by GSP is required to support such equilibria are usually locally envy-free if a player cannot improve quite complex, however, requiring precise knowl- edge of the environment and careful implementa- tion. In theory, advertisers could implement such 17 An alternative interpretation of this restriction is as fol- strategies via automated robots, but in practice lows. With only one slot, GSP coincides with the standard they may not be able to: bidding software must second-price auction, and the restriction of local envy-freeness first be authorized by the search engines,16 and simply says that the losing advertiser bids at least his own search engines are unlikely to permit strategies value, ruling out various implausible equilibria. Likewise, with multiple slots, the local envy-freeness restriction is equivalent that would allow advertisers to collude and sub- to saying that the bid of the advertiser who gets position i and stantially reduce revenues. thus “loses” position (i Ϫ 1) is such that his “marginal bid” (i.e., the difference between the highest amount that he could have paid if he had “won” position (i Ϫ 1) and the amount he actually pays for position i) is at least as high as the marginal 16 See, e.g., help.yahoo.com/help/us/performance/ value for the extra clicks he would have received in position customer/dtc/bidding, questions 7 and 8 (accessed June (i Ϫ 1). To see this, simply rearrange equation (1) to get (iϪ1) Ϫ (i) Ն ␣ Ϫ ␣ 10, 2006). p p sg(i)( iϪ1 i). 250 THE AMERICAN ECONOMIC REVIEW MARCH 2007 his payoff by exchanging bids with the player then any stable assignment is an outcome of a ranked one position above him. More for- locally envy-free equilibrium of auction ⌫. mally, in a locally envy-free equilibrium, for any i Յ min{N ϩ 1, K}, We will now construct a particular locally envy-free equilibrium of game ⌫. This equilib- ␣ Ϫ ͑i͒ Ն ␣ Ϫ ͑i Ϫ 1͒ (1) i sg͑i͒ p i Ϫ 1 sg͑i͒ p . rium has two important properties. First, in this equilibrium, advertisers’ payments coincide Of course, it is possible that bids change over with their payments in the dominant-strategy time, depending on the players’ strategies and equilibrium of VCG. Second, this equilibrium is information structure. However, if the behavior the worst locally envy-free equilibrium for the ever converges to a vector of bids, that vector search engine and the best locally envy-free should correspond to a locally envy-free equi- equilibrium for the advertisers. Consequently, librium of the simultaneous-move game ⌫ in- the revenues of a search engine are (weakly) duced by GSP. Consequently, we view a locally higher in any locally envy-free equilibrium of envy-free equilibrium ⌫ as a prediction regard- GSP than in the dominant-strategy equilibrium ing a rest point at which the vector of bids of VCG. stabilizes. In this section, we study the set of Consider the following strategy profile B*. locally envy-free equilibria. Without loss of generality, assume that adver- We first show that the set of locally envy-free tisers are labeled in decreasing order of their Ͻ Ն equilibria maps naturally to a set of stable as- values, i.e., if j k, then sj sk. For each ʦ ϩ signments in a corresponding two-sided match- advertiser j {2, ... , min{N 1, K}}, bid b*j is V,(jϪ1) ␣ V,(jϪ1) ing market. The idea that auctions and two- equal to p / jϪ1, where p is the sided matching models are closely related is not payment of advertiser j Ϫ 1 in the dominant- new: it goes back to Vincent P. Crawford and strategy equilibrium of VCG where all adver- 18 Elsie M. Knoer (1981), Alexander S. Kelso and tisers bid truthfully. Bid b*1 is equal to s1. Crawford (1982), Herman B. Leonard (1983), and Gabrielle Demange, David Gale, and THEOREM 1: Strategy profile B* is a locally Marilda Sotomayor (1986), and has been stud- envy-free equilibrium of game ⌫. In this equi- ied in detail in a recent paper by John W. librium, each advertiser’s position and payment Hatfield and Milgrom (2005). Note, however, are equal to those in the dominant-strategy that in our case the nonstandard auction is very equilibrium of the game induced by VCG. In different from those in the papers noted above. any other locally envy-free equilibrium of game Our environment maps naturally into the ⌫, the total revenue of the seller is at least as most basic assignment model, studied first by high as in B*. Lloyd S. Shapley and Martin Shubik (1972). Consider each position as an agent who is look- To prove Theorem 1, we first note that pay- ing for a match with an advertiser. The value of ments under strategy profile B* coincide with ␣ a position-advertiser pair (i, k) is equal to isk. VCG payments and check that B* is indeed a We call this assignment game A. The advertiser locally envy-free equilibrium. This follows makes its payment pik for the position, and the from the fact that, by construction, each adver- ␣ Ϫ advertiser is left with isk pik. The following tiser is indifferent between remaining in his pair of lemmas shows that there is a natural positions and swapping with the advertiser one mapping from the set of locally envy-free equi- position above him. Next, from Lemma 1 we libria of GSP to the set of stable assignments. know that every locally envy-free equilibrium All proofs are in the Appendix. corresponds to a stable assignment. The “core elongation” property of the set of stable assign- LEMMA 1: The outcome of any locally envy- ments (Shapley and Shubik 1972; Crawford free equilibrium of auction ⌫ is a stable and Knoer 1981) implies that there exists an assignment.

LEMMA 2: If the number of advertisers is 18 This bid does not affect any advertiser’s payment and greater than the number of available positions, can be set equal to any value greater than b*2. VOL. 97 NO. 1 EDELMAN ET AL.: INTERNET ADVERTISING AND THE GSP AUCTION 251

“advertiser-optimal” assignment A in that set, locally envy-free equilibrium if and only if, for ␣ ␤ Ϫ ␥ ␥ Ն such that in any other stable assignment, each any i and j, i g(i)(sg(i) g(iϩ1)bg(iϩ1)/ g(i)) ␣ ␤ Ϫ ␥ ␥ advertiser pays at least as much to the search j g(i)(sg(i) g(jϩ1)bg(jϩ1)/ g(i)). Dividing both ␤ ␥ engine as he does in A. Moreover, Leonard sides by g(i) and multiplying by g(i),weget ␣ ␥ Ϫ ␥ Ն ␣ ␥ Ϫ (1983) and Demange, Gale, and Sotomayor i( g(i)sg(i) g(iϩ1)bg(iϩ1)) j( g(i)sg(i) ␥ (1986) show that in general assignment games, g(jϩ1)bg(jϩ1)). Hence, the set of bids {bk}isa payoffs of “buyers” in the buyer-optimal stable locally envy-free equilibrium under Google’s ver- ␣ assignment coincide with their VCG payoffs, sion of GSP with position-specific factors { i}, ad- ␥ which is sufficient to complete the proof. This is vertiser-specific quality scores { k}, and per-click ␥ particularly easy to show in the specific envi- values {sk}, if and only if the set of bids { kbk}isan ronment that we consider, and so for complete- equilibrium of our basic model with position-specific ␣ ␥ ness we include a short independent proof. CTRs { i}, per-click values { ksk}, and no quality In the model, we assume that all advertisers scores or advertiser-specific factors in CTRs. are identical along dimensions other than per- click value, and in particular have identical IV. Main Result: GSP and Generalized English click-through rates. The analysis remains Auction largely the same if, instead, we assume that the CTRs of different advertisers are multiples of In the model analyzed in the previous section, one another, i.e., if any advertiser k assigned to we assume that advertisers have converged to a ␣ ␤ ␣ any position i receives i k clicks, where i is a long-run steady state, have learned each other’s ␤ position-specific factor and k is an advertiser- values, and no longer have incentives to change specific factor. In this case, the versions of GSP their bids. But how do they converge to such a implemented by Yahoo! and Google differ. situation? In this section, we introduce the gen- Under Yahoo!’s system, advertisers are still eralized English auction, an analogue of the ranked by bids, and each of them is charged the standard English auction corresponding to GSP, next-highest advertiser’s bid. Then, bids form a to help us answer this question. locally envy-free equilibrium if and only if, for In the generalized English auction, there is a ␣ ␤ Ϫ (iϩ1) Ն ␣ ␤ Ϫ any i and j, i g(i)(sg(i) b ) j g(i)(sg(i) clock showing the current price, which contin- b(jϩ1)). Dividing both sides by the positive num- uously increases over time. Initially, the price ␤ ␣ Ϫ (iϩ1) Ն ␣ Ϫ ber g(i),weget i(sg(i) b ) j(sg(i) on the clock is zero, and all advertisers are in b(jϩ1)), i.e., the necessary and sufficient condition the auction. An advertiser can drop out at any for a locally envy-free equilibrium in the case time, and his bid is the price on the clock at the ␤ where all k are equal to one. Hence, under Ya- time when he drops out. The auction is over hoo!’s version of GSP, equilibria are not affected when the next-to-last advertiser drops out. The by changes in ␤s. ad of the last remaining advertiser is placed in Under Google’s system, advertisers are ar- the best position on the screen, and this adver- ranged by “rank numbers.” Advertiser k’s rank tiser’s payment per click is equal to the price at number is the product of his bid and “quality which the next-to-last advertiser dropped out. ␥ 19 score” k. Thus, under Google’s system, g(1) The ad of the next-to-last advertiser is placed is the advertiser with the highest rank number, second, and his payment per click is equal to the g(2) is the advertiser with the second highest third-highest advertiser’s bid, and so on.20 In rank number, and so on. Per-click payment of other words, the vector of bids obtained in the advertiser g(i) is equal to the smallest amount generalized English auction is used to allocate (i) ␥ (i) x , such that g(iϩ1)x is greater than or equal to the next highest advertiser’s rank number, (i) ϭ ␥ ␥ i.e., x g(iϩ1)bg(iϩ1)/ g(i). Then, bids form a 20 If several advertisers drop out simultaneously, one of them is chosen randomly. Whenever an advertiser drops out, the clock is stopped, and other advertisers are also 19 Initially, Google simply used click-through rates to allowed to drop out; again, if several advertisers want to ␥ ϭ ␤ determine quality scores, setting k k. Later, however, it drop out, one of them is chosen randomly. If several adver- switched to a less transparent system for determining qual- tisers end up dropping out at the same price, the first one to ity scores, incorporating such factors as the relevance of an drop out is placed in the lowest position of the still available ad’s text and the quality of an advertiser’s Web page. ones, the next one to the position right above that, and so on. 252 THE AMERICAN ECONOMIC REVIEW MARCH 2007 the objects and compute the prices according to is the price at which he drops out, i is the the rules of GSP. With one object, the general- number of advertisers remaining (including ad- ϭ ized English auction becomes a simple English vertiser k), and h (biϩ1, ... , bNϩ1)isthe auction.21 history of prices at which previous advertisers We view the generalized English auction in have dropped out. (As a result, the price that the same light as the taˆtonnement processes in advertiser k would have to pay per click if he the theory of general equilibrium (see, e.g., An- dropped out next is equal to biϩ1, unless the dreu Mas-Colell, Michael D. Whinston, and history is empty, in which case we say that ϵ Jerry R. Green 1995, sect. 17.H) and the salary biϩ1 0.) The following theorem shows that adjustment process in the theory of matching in this game has a unique perfect Bayesian equi- labor markets with heterogeneous firms and librium with strategies continuous in advertis- workers (Crawford and Knoer 1981).22 While ers’ valuations.23 The payoffs of all advertisers all these processes, taken literally, happen in in this equilibrium are equal to VCG payoffs. “imaginary time,” they are meant to resemble the underlying dynamics of the actual markets, THEOREM 2: In the unique perfect Bayesian to help us distinguish more plausible equilibria equilibrium of the generalized English auction from less plausible ones, characterize their sta- with strategies continuous in sk, an advertiser bility and other properties, and examine the with value sk drops out at price significance of the underlying assumptions. As in the case of the taˆtonnement and salary ad- ␣ justment processes, there are many features of ͑ ͒ ϭ Ϫ i ͑ Ϫ ͒ (2) pk i, h, sk sk ␣ sk bi ϩ 1 . real markets not captured by the generalized i Ϫ 1 English auction, but we believe that it provides a natural and useful approximation. In this equilibrium, each advertiser’s result- To define the game formally, assume that ing position and payoff are the same as in the there are N Ն 2 slots and K ϭ N ϩ 1 advertis- dominant-strategy equilibrium of the game ers. (Cases with K N ϩ 1 require only minor induced by VCG. This equilibrium is ex post: modifications in the proof.) Click-through rates the strategy of each advertiser is a best re- ␣ ␣ ϵ i are commonly known, with Nϩ1 0. Ad- sponse to other advertisers’ strategies re- vertisers’ per-click valuations sk are drawn from gardless of their realized values. a continuous distribution F٪ on [0; ϩϱ) with a ,continuous density function f٪ that is positive The intuition of the proof is as follows. First everywhere on (0, ϩϱ). Each advertiser knows with i players remaining and the next highest his valuation and the distribution of other ad- bid equal to biϩ1, it is a dominated strategy for vertisers’ valuations. a player with value s to drop out before price p The strategy of an advertiser assigns the reaches the level at which he is indifferent be- choice of dropping out or not for any history of tween getting position i and paying biϩ1 per the game, given that the advertiser has not pre- click and getting position i Ϫ 1 and paying p per viously dropped out. In other words, the strat- click. Next, if for some set of types it is not egy can be represented as a function pk(i, h, sk), optimal to drop out at this “borderline” price where sk is the value per click of advertiser k, pk level, we can consider the lowest such type, and

21 This version of the English auction is also known as 23 Without this restriction, multiple equilibria exist, even the “Japanese” or “button” auction. in the simplest English auction with two bidders and one 22 Of course, the generalized English auction is very object. For example, suppose there is one object for sale and different from the salary adjustment process of Crawford two bidders with independent private values for this object and Knoer (1981) and its application to multi-unit auctions distributed exponentially on [0, ϱ). Consider the following (Demange et al. 1986). In Demange et al. (1986), bidding pair of strategies. If a bidder’s value is in the interval [0, 1] proceeds simultaneously for all items and the auctioneer or in the interval [2, ϱ), he drops out when the clock reaches keeps track of a vector of item-specific prices, while in the his value. If bidder 1’s value is in the interval (1, 2), he generalized English auction bidding proceeds, in essence, drops out at 1, and if bidder 2’s value is in the interval (1, sequentially, and the auctioneer keeps track of only one 2), he drops out at 2. This pair of strategies, together with price. appropriate beliefs, forms a perfect Bayesian equilibrium. VOL. 97 NO. 1 EDELMAN ET AL.: INTERNET ADVERTISING AND THE GSP AUCTION 253 then once the clock reaches this price level, a auction is a particularly interesting example, player of this type will know that he has the because it can be viewed as a model of a mech- lowest per-click value of the remaining players. anism that has “emerged in the wild.” But then he will also know that the other re- maining players will drop out only at price levels at which he will find it unprofitable to V. Conclusion compete with them for the higher positions. The result of Theorem 2 resembles the classic We investigate a new mechanism that we call result on the equivalence of the English auction the generalized second-price auction. GSP is and the second-price sealed-bid auction under tailored to the unique features of the market for private values (Vickrey 1961). Note, however, Internet advertisements. As far as we know, this that the intuition is very different: Vickrey’s mechanism was first used in 2002. As of May result follows simply from the existence of 2006, the annual revenues from GSP auctions equilibria in dominant strategies, whereas in our were on the order of $10 billion. case such strategies do not exist, and bids do GSP looks similar to the VCG mechanism, depend on other player’s bids. Also, our result is because just like in the standard second-price very different from the revenue equivalence the- auction, the payment of a bidder does not di- orem: payoffs in the generalized English auc- rectly depend on his bid. Although GSP looks tion coincide with VCG payments for all similar to VCG, its properties are very different, realizations of values, not only in expectation, and equilibrium behavior is far from straight- and the result does not hinge on the assumptions forward. In particular, unlike the VCG mecha- of symmetric bidders or common priors. nism, GSP generally does not have an The equilibrium described in Theorem 2 is an equilibrium in dominant strategies, and truth- ex post equilibrium. As long as all advertisers telling is not an equilibrium of GSP. We show other than advertiser k follow the equilibrium that the generalized English auction that corre- strategy described in Theorem 2, it is a best sponds to the generalized second-price auction response for advertiser k to follow his equilibrium has a unique equilibrium. strategy, for any realization of other advertisers’ This equilibrium has some notable properties. values. Thus, the outcome implemented by this The bid functions have explicit analytic formu- mechanism depends only on the realization of las, which, combined with equilibrium unique- advertisers’ values and does not depend on adver- ness, make our results a useful starting point for tisers’ beliefs about each other’s types. empirical analysis. Moreover, these functions Clearly, any dominant strategy solvable game do not depend on bidders’ beliefs about each has an ex post equilibrium. However, the gen- other’s types: the outcome of the auction de- eralized English auction is not dominant strat- pends only on the realizations of bidders’ val- egy solvable. This combination of properties is ues. This is one of the very few mechanisms quite striking: the equilibrium is unique and encountered in practice that are not dominant efficient, and the strategy of each advertiser strategy solvable and nevertheless have this does not depend on the distribution of other property. It is particularly interesting that a advertisers’ values, yet advertisers do not have mechanism with such notable features in theory dominant strategies.24 The generalized English and such enormous popularity in practice devel- oped as a result of evolution of inefficient mar- ket institutions, which were gradually replaced 24 Dirk Bergemann and Stephen Morris (2005) show that by increasingly superior designs. an outcome implementable by robust mechanisms must be implementable in dominant strategies. Indeed, the outcome implemented by the generalized English auction can be implemented in dominant strategies by the VCG mecha- Of course, in our model, values are private, and, crucially, nism; however, VCG is not the mechanism that is used in signals are single-dimensional, even though multiple differ- practice. Philippe Jehiel and Benny Moldovanu (2001) and ent objects are for sale. This makes efficient ex post imple- Jehiel et al. (2006) show that, generically, any efficient mentation feasible. For other examples of mechanisms that choice function is not Bayes-Nash implementable and any allocate multiple different objects to bidders with single- nontrivial choice function is not ex post implementable, if dimensional types, see Moldovanu and Aner Sela (2001) values are interdependent and signals are multidimensional. and Thomas Kittsteiner and Moldovanu (2005). 254 THE AMERICAN ECONOMIC REVIEW MARCH 2007

APPENDIX:PROOFS him. Suppose the advertiser assigned to posi- tion i is considering rematching with position PROOF OF LEMMA 1: m Ͻ i Ϫ 1. Since the equilibrium is locally By definition, in any locally envy-free envy-free, we have equilibrium outcome, no advertiser can prof- itably rematch with the position assigned to ␣ Ϫ ͑i͒ Ն ␣ Ϫ ͑i Ϫ 1͒ i sg͑i͒ p i Ϫ 1 sg͑i͒ p , the advertiser right above him. Also, no adver- tiser (a) can profitably rematch with a position ͑i Ϫ 1͒ ͑i Ϫ 2͒ ␣ Ϫ s ͑ Ϫ ͒ Ϫ p Ն ␣ Ϫ s ͑ Ϫ ͒ Ϫ p , assigned to an advertiser below him (b)—if i 1 g i 1 i 2 g i 1 such a profitable rematching existed, advertiser Ӈ a would find it profitable to slightly undercut advertiser b in game ⌫ and get b’s position and ␣ Ϫ ͑mϩ1͒ Ն ␣ Ϫ ͑m͒ payment. But this would contradict the assump- mϩ1sg͑mϩ1͒ p msg͑mϩ1͒ p . tion that we are in equilibrium.25 ␣ Ͼ ␣ Ͼ Hence, we need only show that no advertiser Since j jϩ1 for any j, and sg(i) sg(j) for can profitably rematch with the position as- any i Ͻ j, the inequalities above remain valid signed to an advertiser more than one spot after replacing sg(i) with sg(j). Doing that, then above him. First, note that in any locally envy- adding all inequalities up, and canceling out the ␣ Ϫ (i) Ն free equilibrium, the resulting matching must be redundant elements, we get isg(i) p ␣ Ϫ (m) assortative, i.e., for any i, the advertiser as- msg(i) p . But that implies that the adver- signed to position i has a higher per-click val- tiser assigned to position i cannot rematch prof- uation than the advertiser assigned to position itably with position m, and we are done. i ϩ 1 and, therefore, the advertiser with the highest per-click value must be assigned to the PROOF OF LEMMA 2: top position, the advertiser with the second- Take a stable assignment. By a result of highest per-click value to the second-highest Shapley and Shubik (1972), this assignment position, and so on. must be efficient, and hence assortative, and so Indeed, suppose sg(i) and sg(i ϩ 1) are the without loss of generality we can assume that values of advertisers assigned to positions i advertisers are labeled in decreasing order of ϩ Ͼ Ͻ and i 1. Equilibrium restrictions imply that their bids (i.e., sj sk whenever j k) and that ␣ Ϫ (i) Ն ␣ Ϫ (i ϩ 1) i sg(i) p i ϩ 1sg(i) p (nobody advertiser i is matched with position i, with wants to move one position down), and local associated payment pi. ␣ Ϫ envy-freeness implies that i ϩ 1sg(i ϩ 1) Let us construct a locally envy-free equilib- (i ϩ 1) Ն ␣ Ϫ (i) ϭ p i sg(i ϩ 1) p (nobody wants to rium with the corresponding outcome. Let b1 ϭ ␣ Ͼ move one position up). Manipulating the in- s1 and bi piϪ1/ iϪ1 for i 1. Let us show ␣ Ϫ ␣ ϩ equalities above yields isg(i) isg(iϩ1) that this set of strategies is a locally envy-free ␣ Ն ␣ ␣ Ϫ ␣ Ն Ͼ iϩ1sg(iϩ1) iϩ1sg(i), thus ( i iϩ1)sg(i) equilibrium. First, note that for any i, bi biϩ1 ␣ Ϫ ␣ ␣ Ͼ ␣ ( i iϩ1)sg(iϩ1). Since i iϩ1, we have (because otherwise we would have, for some i, Ն ␣ Յ ␣ f Ϫ ␣ Ն Ϫ sg(i) sg(iϩ1), and hence the locally envy-free piϪ1/ iϪ1 pi/ i si piϪ1/ iϪ1 si ␣ f ␣ Ϫ Ͼ ␣ Ϫ equilibrium outcome must be an assortative pi/ i iϪ1si piϪ1 isi pi, which match. would imply that player i could rematch profit- Now, let us show that no advertiser can ably). Therefore, position allocations and pay- profitably rematch with the position assigned ments resulting from this strategy profile will to an advertiser more than one spot above coincide with those in the original stable assign- ment. To see that this strategy profile is an equilibrium, note that deviating and moving to a 25 This argument relies on the fact that in equilibrium, different position in this strategy profile is at no two (or more) advertisers bid the same amount, which most as profitable for any player as rematching is straightforward to prove: since all advertisers’ per- with the corresponding position in the assign- click values are different, and all ties are broken ran- domly with equal probabilities, at least one such ment game. To see that this equilibrium is lo- advertisers would find it profitable to bid slightly higher cally envy-free, note that the payoff from or slightly lower. swapping with the bidder above is exactly equal VOL. 97 NO. 1 EDELMAN ET AL.: INTERNET ADVERTISING AND THE GSP AUCTION 255 to the payoff from rematching with that player’s truthful equilibrium of VCG. Therefore, by position in the assignment game. construction, payments are also the same. Next, to see that no bidder j can benefit by PROOF OF THEOREM 1: bidding less than b*j, suppose that he bids an ЈϽ ЈϾ First, we need to check that the order of the amount b b*j that puts him in position j j. Ͼ Ͻ bids is preserved, i.e., b*j b*jϩ1 for any j Then, by construction, his payment will be min{N, K}. For j Ն 2, this is equivalent to equal to the amount that he would need to pay to be in position jЈ under VCG, provided that pV,͑ j Ϫ 1͒ pV,͑ j͒ other players bid truthfully. But truthful bidding Ͼ (3) ␣ ␣ is an equilibrium under VCG, and so such de- j Ϫ 1 j viation cannot be profitable there—hence, it
cannot be profitable in strategy profile B*of game ⌫ either. To see that no bidder j can benefit by bidding ͑␣ Ϫ ␣ ͒ ϩ V,͑ j͒ V,͑ j͒ j Ϫ 1 j sj p p * Ͼ more than b j , suppose that he bids an amount ␣ ␣ ЈϾ ЈϽ j Ϫ 1 j b b*j that puts him in position j j. Then the net payoff from this deviation is equal to ␣ Ϫ ␣ Ϫ ␣ Ϫ ␣ Ͻ ␣ Ϫ
( jЈ j)sj ( jЈb*jЈ j bj ϩ 1) ( jЈ ␣ Ϫ ␣ Ϫ ␣ ϭ ¥jϪ1 ␣ Ϫ j)sj ( jЈb*jЈϩ1 j bj ϩ 1) iϭjЈ ( i jϪ1 V,(i) ϩ jϪ1 ͑ ͒ ␣ Ϫ ¥ p Ϫ V,(i 1) ϭ ¥ ␣ ͑␣ Ϫ ␣ ͒ Ͼ ͑␣ Ϫ ␣ ͒ V, j i ϩ 1)sj iϭjЈ ( p ) iϭjЈ j j Ϫ 1 j sj j Ϫ 1 j p ␣ Ϫ ␣ Ϫ ¥jϪ1 ␣ Ϫ ␣ ( i i ϩ 1)sj iϭjЈ ( i i ϩ 1)si ϩ 1.But Յ Ͻ
since sj si ϩ 1 for any i j, the last expres- sion is less than or equal to zero, and hence the deviation is not profitable. ␣ Ͼ V,͑ j͒ j sj p . To check that this equilibrium is locally envy-free, note that if bidder j swapped his bids with bidder j Ϫ 1, his payoff would change by ϭ Ͼ ␣ Ϫ ␣ Ϫ ␣ Ϫ ␣ ϭ ␣ Ϫ For j 1, b*j b*j ϩ 1 is equivalent to ( jϪ1 j)sj ( jϪ1b*j jbjϩ1) ( jϪ1 V,(jϪ1) V,(j) ␣j)sj Ϫ (p Ϫ p ) ϭ (␣jϪ1 Ϫ ␣ )s Ϫ Ϫ j j V,͑1͒ V,(j 1) Ϫ V,(j) ϭ ␣ Ϫ ␣ Ϫ ␣ Ϫ p (p p ) ( jϪ1 j)sj ( jϪ1 (4) s Ͼ ␣ ϭ 1 ␣ j)sj 0. In other words, each bidder is indif- 1 ferent between his actual payoff and his payoff
after swapping bids with the bidder above, and hence the equilibrium is locally envy-free. Let us now show that B* is the best locally ␣ Ͼ V,͑1͒ 1 s1 p . envy-free equilibrium for the bidders and the worst locally envy-free equilibrium for the ␣ Ͼ V,(j) To see that for any j, j sj p , note first search engine. The core-elongation property of that in the game induced by VCG, each player the assignment game (Shapley and Shubik can guarantee himself the payoff of at least 1972; Crawford and Knoer 1981) implies that zero (by bidding zero), and hence in any there exists an assignment that is the best stable equilibrium his payoff from clicks is at least assignment for all advertisers and the worst as high as his payment. To prove that the stable assignment for all positions. Suppose this inequality is strict, note that if player j’s value assignment is characterized by a vector of Ϫ⌬ ϭ V ϭ V per click were slightly lower, e.g., sj payments p (p1, ... , pK). Let p (p1, ... , ⌬Ͻ Ϫ V instead of sj , sj sj ϩ 1, then his pay- pK ) be the set of dominant-strategy VCG pay- ment in the truth-telling equilibrium would ments, i.e., the set of payments in equilibrium still be the same (because it does not depend B* of game ⌫. on his own bid, given the allocation of posi- In any stable assignment, pK must be at least V,(j) Յ ␣ Ϫ⌬ Ͻ ␣ ␣ tions), and so p j(sj ) j sj. as high as KsKϩ1, since otherwise advertiser Ͼ ϩ Thus, for any j, b*j b*j ϩ 1, and therefore K 1 would find it profitable to match with V ϭ ␣ each bidder’s position is the same as in the position K. On the other hand, pK KsKϩ1, 256 THE AMERICAN ECONOMIC REVIEW MARCH 2007 and hence in the advertiser-optimal stable as- received position i ϩ 1. If history h is empty, ϭ V ϭ signment, pK pK. we set bi ϩ 1 0.) We will now show that for ϭ Next, in any stable assignment, it must be the any i, k, h, and sk, pk(i, h, sk) q(i, h, sk). Ϫ Ն ␣ Ϫ ␣ case that pKϪ1 pK ( KϪ1 K)sK — Suppose that is not the case, and take the otherwise, advertiser K would find it profitable largest i for which there exist such history h Ϫ Ն to rematch with position K 1. Hence, pKϪ1 (with the last player dropping out at biϩ1), ␣ Ϫ ␣ ϩ Ն ␣ Ϫ ␣ ϩ V ϭ ( KϪ1 K)sK pK ( KϪ1 K)sK pK player k, and type sk (surviving with positive V pKϪ1, and so in the advertiser-optimal stable as- probability on the equilibrium path) that pk(i, h, ϭ V signment, pKϪ1 pKϪ1. sk) q(i, h, sk). Since by assumption, all strat- ϭ V ⅐ ⅐ ⅐ ϭ ⅐ Proceeding by induction, we get pj pj for egies up to this stage were pk( , , ) q( , Յ ⅐ ⅐ Ն any j K in the advertiser-optimal stable as- , ), we know that there exists a value smin signment, and so in any locally envy-free equi- biϩ1, such that all players with values less than ⌫ librium of game , the total revenue of the seller smin have dropped out, and all players with ¥K V is at least as high as jϭ1 pj . values greater than smin are still in the auction. Ն Step 1: Suppose for some type s smin , ϵ Ͼ PROOF OF THEOREM 2: pmax(i, h, s) maxk pk(i, h, s) q(i, h, s). Let First, note that in equilibrium, for any player s0 be the smallest type, and let k be the corre- ϭ k, any history h, and any number of remaining sponding player, such that pk(i, h, s0) pmax(i, Յ players i, the drop-out price pk(i, h, sk) tends to h, s); clearly, s0 s. Without loss of generality, infinity as sk tends to infinity. (Otherwise, there we can assume that there is a positive mass of would exist a player for whom it was optimal to types of other players dropping out at or before 26 deviate from his strategy and stay longer, for a pk(i, h, s0). sufficiently high value s.) Next, take any equi- Step 1(a). Suppose first that there is a positive librium of the generalized English auction. Note mass of types of other players dropping out at Ͼ ϭ that if in this equilibrium pk(i, h, sk) pk(i, h, pk(i, h, s0) pmax(i, h, s). That implies that with Ј Ͻ Ј sk) for some k, h, i, and types sk sk, then it has positive probability, player k of type s0 will Ј to be the case that both types sk and sk are remain in the subgame following the drop-out indifferent between dropping out at pk(i, h, sk) of some other player at pk(i, h, s0) (since ties are Ј and pk(i, h, sk). (Otherwise, one of them would broken randomly). Let us show that in this be able to increase his payoff by mimicking the subgame, player k of type s0 will be the first other.) Consequently, we can “swap” such play- player to drop out with probability 1. Suppose ers’ strategies, and therefore there exists an “ob- that is not the case, and let l Ͻ i Ϫ 1bethe servationally equivalent” equilibrium in which smallest number such that he gets position l strategies are nondecreasing in types; also, they with positive probability. are still continuous in own values. Consider this Consider any continuation of history h, hlϩ2, equilibrium profile of strategies pk(i, h, sk). such that the last player to drop out in that ϩ Let q(i, biϩ1, s) be such a price that a player history gets position l 2 and drops out at price with value s is indifferent between getting po- blϩ2, player k of type s0 is one of the remaining Ϫ ϩ sition i at price biϩ1 and position i 1 at price l 1 players, there is a positive probability that q(i, biϩ1, s). That is, player k gets position l in the continuation sub- game following history hlϩ2, and there is zero ␣ ͑ Ϫ ͑ ͒͒ ϭ ␣ ͑ Ϫ ͒ (5) i Ϫ 1 s q i, bi ϩ 1 , s i s bi ϩ 1

26 @ Յ Յ Otherwise, we have j k, pj(i, h, s0) pj(i, h, s) ϭ f @ ϭ pmax(i, h, s) pk(i, h, s0) j k, pj(i, h, s0) pk(i, h, s0) @ ЈϾ Ј Ͼ ␣ and s s0, pj(i, h, s ) pk(i, h, s0). But we also have pk(i, i ϭ Ͼ Ն ͑ ͒ ϭ Ϫ ͑ Ϫ ͒ h, s0) pmax(i, h, s) q(i, h, s) q(i, h, s0), and so for q i, bi ϩ 1 , s s ␣ s bi ϩ 1 . ЈϾ Ј Ͼ Ј i Ϫ 1 some s s0 we have pmax(i, h, s ) q(i, h, s ) and pmax(i, Ј Ͼ Ј h, s ) pmax(i, h, s). We can then consider s0 in place of s0, Ј Ј ϭ where s0 is the smallest type, and k is the corresponding Slightly abusing notation, let q(i, h, s) q(i, Ј ϭ Ј player, such that pkЈ(i, h, s0) pmax(i, h, s ). There is a bi ϩ 1, s), where bi ϩ 1 is the last bid at which positive mass of types of other players dropping out before Ј a player dropped out in history h. (This player pkЈ(i, h, s0). VOL. 97 NO. 1 EDELMAN ET AL.: INTERNET ADVERTISING AND THE GSP AUCTION 257

␣ Ϫ probability that player k gets position m for any iϪ1(s0 pk(i, h, s0)). Now suppose player k Ͻ Ն Ϫ ␧ m l. Note that s0 blϩ2—otherwise, it dropped out at a price pk(i, h, s0) instead of would have been optimal for player k to drop waiting until pk(i, h, s0). If somebody else drops ϩ Ϫ ␧ out earlier. Consider pk(l 1, hlϩ2, s0). Since out before pk(i, h, s0) or after pk(i, h, s0), or player k of type s0 gets position l with positive drops out at pk(i, h, s0) but player k is chosen to probability in this subgame, there must be a drop out first, then these two strategies result in positive mass of types of other players who drop identical payoffs. The probability that some- ϩ Ϫ ␧ out no later than pk(l 1, hlϩ2, s0). Take the body drops out in the interval (pk(i, h, s0) , Ј ␧ highest such type, s , and the corresponding pk(i, h, s0)) goes to zero as goes to zero, and ЈϾ Ն player j. It has to be the case that s s0 the possible difference in the payoffs is finite, so ϩ blϩ2. It also has to be the case that q(l 1, the difference in payoffs due to this contingency Ј ϩ ␧ hlϩ2, s ) is less than or equal to pk(l 1, hlϩ2, goes to zero as goes to zero. Finally, there is Ј s0). (Otherwise, player j with value s would be a positive probability that somebody else drops playing a strategy weakly dominated by drop- out at pk(i, h, s0) and is chosen to drop out first. ϩ Ј Ϫ ping out at q(l 1, hlϩ2, s ), with a positive If player k drops out before that, at pk(i, h, s0) ␧ ␣ Ϫ probability of earning strictly less than he would , his payoff is i(s0 biϩ1). If he waits until have earned if he waited until that price level.) pk(i, h, s0), we know that in the subsequent ϩ Ն ϩ ␣ Ϫ Ͻ Therefore, pk(l 1, hlϩ2, s0) q(l 1, hlϩ2, subgame his payoff is iϪ1(s0 pk(i, h, s0)) Ј Ͼ ϩ Ն ␣ Ϫ ϭ ␣ Ϫ s ) q(l 1, hlϩ2, s0) blϩ2. Let us show that iϪ1(s0 q(i, h, s0)) i(s0 biϩ1). There- it would be strictly better for player k with type fore, for a sufficiently small ␧, it is strictly better ϩ s0 to drop out at q(l 1, hlϩ2, s0) instead of for player k with value s0 to drop out at pk(i, h, ϩ Ϫ ␧ waiting until pk(l 1, hlϩ2, s0). Indeed, if s0) instead of waiting until pk(i, h, s0), ⅐ ⅐ nobody else drops out in between, or someone which contradicts the assumption that {pj( , , ϩ ⅐ drops out before q(l 1, hlϩ2, s0), these strat- )} is an equilibrium. egies would result in identical payoffs. Other- Step 1(b). Now, suppose there is mass zero of wise, payoffs are different, and this happens types of other players dropping out at pk(i, h, ϭ with positive probability. Under the former s0) pmax(i, h, s), but there is a positive mass strategy, player k earns dropping out before pk(i, h, s0). Consider a ␧ sequence of small positive numbers ( n) con- ␣ ͑ Ϫ ͒ 3 ϱ ␲1 (6) l ϩ 1 s0 bl ϩ 2 . verging to 0 as n and sequences ( n), ␲ 2 ␲ iϪ1 ␲ l ( n), ... , ( n ), where n is the probability that player k with value s0 will end up in posi- Under the latter strategy, he earns tion l if another player drops out at price (pk(i, Ϫ ␧ ϭ ␣ h, s0) n). Let B 1s0, i.e., the maximum ␣ ͑ Ϫ ͒ (7) l s0 bl ϩ 1 , payoff that a player with value s0 can possibly ␲ iϪ1 get in the auction. Now, if ( n ) converges to ␲ l Ͻ Ϫ 1 and ( n) converges to zero for all l i 1, where bl ϩ 1 is the price at which somebody then, by an argument similar to the one at the else dropped out. (The probability of getting a end of Step 1(a), it is better for player k of type Ͻ Ϫ ␧ 27 spot m l is zero by construction.) With s0 to drop out at some time pk(i, h, s0) . If Ն ϩ ␲ iϪ1 probability 1, bl ϩ 1 q(l 1, hl ϩ 2, s0), and ( n ) does not converge to 1, take the smallest Ͼ ϩ ␲ l with positive probability, bl ϩ 1 q(l 1, (i.e., best) l for which ( n) does not converge to ␧ hl ϩ 2, s0), so the expected payoff from wait- zero, and take a subsequence of n along which ϩ ␲ l ␳ ing until pk(l 1, hl ϩ 2, s0) is strictly less ( n) converges to some positive number . Let than the expected payoff from dropping out at s1 be the value such that for a random draw of ϩ ␣ Ϫ Ͻ q(l 1, hl ϩ 2, s0): E[ l(s0 bl ϩ 1)] types of remaining players other than k, condi- ␣ Ϫ ϩ ϭ ␣ Ϫ Ϫ l(s0 q(l 1, hl ϩ 2, s0)) l(s0 (s0 ␣ ␣ Ϫ ϭ ␣ Ϫ ( l ϩ 1/ l)(s0 bl ϩ 2))) l ϩ 1(s0 bl ϩ 2). 27 Ϫ If some other player drops out between pk(i, h, s0) Therefore, in the subgame following the ␧ and pk(i, h, s0), the benefit of staying longer tends to zero drop-out of some other player at pk(i, h, s0), Ϫ ␲ iϪ1 Ϫ (it is at most B(1 )), while the cost converges to a player k of type s0 gets position i 1 with positive number (the difference between getting position i at Ϫ probability 1, and therefore his payoff is price biϩ1 and position i 1 at price pk(i, h, s0)). 258 THE AMERICAN ECONOMIC REVIEW MARCH 2007

tional on each draw being greater than s0, the Search Engine Auctions.” Paper presented at probability that at least one type is less than s1 the Second Workshop on Sponsored Search ␳ ͟ Ϫ Ϫ is equal to /2 (i.e., j k [(1 Fj(s1))/(1 Auctions, Ann Arbor, MI. ϭ Ϫ ␳ Ͼ Fj(s0))] 1 /2). Clearly, s1 s0. Take a Bergemann, Dirk, and Stephen Morris. 2005. ␧ small n, and consider a subgame following “Robust Mechanism Design.” Econometrica, some continuation of history h, hlϩ2, where the 73(6): 1771–1813. ϩ nd ϩ (l 2) player drops out at blϩ2, l 1 players, Binmore, Ken, and Paul Klemperer. 2002. “The including player k, remain, and player k gets Biggest Auction Ever: The Sale of the British position l with probability close to ␳ (and any 3G Telecom Licenses.” Economic Journal, position better than l with probability close to 112(478): C74–-96. ϩ zero). Consider pk(l 1, hlϩ2, s0). There must Brooks, Nico. 2004. The Atlas Rank Report— ϩ Յ exist a player, j, such that pj(l 1, hlϩ2, s1) Part II: How Search Engine Rank Impacts ϩ pk(l 1, hlϩ2, s0). (Otherwise, the probability Conversion. Seattle: Atlas Institute. of player k surviving until position l is less than Clarke, Edward H. 1971. “Multipart Pricing of or close to ␳/2, and thus cannot be close to ␳.) Public Goods.” Public Choice, 11(0): 17–33. But then, by an argument similar to the one in Crawford, Vincent P., and Elsie M. Knoer. 1981. Step 1(a), in this subgame it is strictly better for “Job Matching with Heterogeneous Firms ϩ player k to drop out slightly earlier than pk(l and Workers.” Econometrica, 49(2): 437–50. 1, hlϩ2, s0): conditional on somebody else drop- Demange, Gabrielle, David Gale, and Marilda So- ping out in between, the benefit is close to zero tomayor. 1986. “Multi-Item Auctions.” Jour- (the probability of getting a position better than nal of Political Economy, 94(4): 863–72. l times the highest possible benefit B), while the Edelman, Benjamin, and Michael Ostrovsky. cost is close to a positive number (the payoff from Forthcoming. “Strategic Bidder Behavior in ϩ being in position l 1 at price blϩ2 versus the Sponsored Search Auctions.” Decision Sup- payoff from being in position l at price at least port Systems. Ϫ ␣ ␣ Ϫ [s1 ( lϩ1/ l)(s1 blϩ2)])—contradiction. Groves, Theodore. 1973. “Incentives in Teams.” Step 2: In Step 1, we showed that pmax(i, h, Econometrica, 41(4): 617–31. ϵ s) maxk pk(i, h, s) cannot be greater than q(i, Hatfield, John William, and Paul R. Milgrom. h, s), and therefore for any player k and type s Ն 2005. “Matching with Contracts.” American Յ Ͼ smin , pk(i, h, s) q(i, h, s). Take some value s Economic Review, 95(4): 913–35. Ͻ smin and player k. Suppose pk(i, h, s) q(i, h, s). Jehiel, Philippe, and Benny Moldovanu. 2001. Take some other player j. From Step 1, we have “Efficient Design with Interdependent Valu- Յ Ͻ pj(i, h, smin) q(i, h, smin) q(i, h, s), and ations.” Econometrica, 69(5): 1237–59. therefore if player k waited until q(i, h, s) in- Jehiel, Philippe, Moritz Meyer-ter-Vehn, Benny stead of dropping out at pk(i, h, s), the proba- Moldovanu, and William R. Zame. 2006. “The bility that someone dropped out in between Limits of Ex Post Implementation.” Econo- would be positive, and hence the payoff would metrica, 74(3): 585–610. be strictly greater (by the definition of function Kagel, John H., Ronald M. Harstad, and Dan -player k with value s strictly prefers being Levin. 1987. “Information Impact and Al ,q٩ in position i Ϫ 1 or higher at any price less than location Rules in Auctions with Affiliated q(i, h, s) to being in position i at price biϩ1), Private Values: A Laboratory Study.” which is impossible in equilibrium. Hence, pk(i, Econometrica, 55(6): 1275–1304. ϭ Ͼ h, s) q(i, h, s) for all s smin. By continuity, Kelso, Alexander S., Jr., and Vincent P. Craw- ϭ we also have pk(i, h, smin) q(i, h, smin). ford. 1982. “Job Matching, Coalition Forma- tion, and Gross Substitutes.” Econometrica, REFERENCES 50(6): 1483–1504. Kittsteiner, Thomas, and Benny Moldovanu. Aggarwal, Gagan, and Jason D. Hartline. 2005. 2005. Management Science, 51(2): 236–48. “Knapsack Auctions.” Paper presented at the Leonard, Herman B. 1983. “Elicitation of Hon- First Workshop on Sponsored Search Auc- est Preferences for the Assignment of Indi- tions, Vancouver, BC. viduals to Positions.” Journal of Political Asdemir, Kursad. 2006. “Bidding Patterns in Economy, 91(3): 461–79. VOL. 97 NO. 1 EDELMAN ET AL.: INTERNET ADVERTISING AND THE GSP AUCTION 259

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