Position Auctions and Nonuniform Conversion Rates


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1 Position Auctions and Nonunifor Conversion Rates Liad Blurosen Microsoft Research Mountain View, CA 944 Jason D. Hartline Shuzhen Nong Electrical Engineering and Microsoft AdCenter Coputer Science Redond, WA 9852 Northwestern University Evanston, IL 628 ABSTRACT The generalized secondprice auction (GSP) is used predoinantly for sponsored search by leading search engines like Google, MSNLive Search and Yahoo!. Previous results showed, in a odel where all clicks on an ad gain the advertiser the sae benefit, that GSP axiizes the social welfare in equilibriu. In practice, however, the probability that a click will convert to a sale for the advertiser depends on the position (a.k.a. slot) of the ad on the search results page. We support this observation by epirical results collected in MSNLive adcenter. We then prove that with nonunifor conversion rates GSP does not adit an optial equilibriu; nonetheless, we are still able to bound the incurred loss. Finally, we devise an increental change in the GSP echanis that achieves sociallyoptial results in equilibriu, while aintaining the sae interface for advertisers and the sae payperclick business odel.. INTRODUCTION The generalized second price auction (GSP) sells trillions of advertising ipressions to illions of advertisers generating gross revenues in the billions annually. Its ability as an auction echanis to effectively allocate the available supply of advertising ipressions to advertisers is of crucial iportance if the allocation produced is only twothirds the value of the optial allocation it represents billions of dollars lost! In this paper we cobine an epirical study of the sponsored search auction arket place with a theoretical study of the GSP auction. Our study suggests, contrary to the iplication of initial analyses of GSP s equilibria in a odel intended to represent the auctioning of ad slots for a single search query [8, 6], that GSP is not ideally suited to the slot auction proble. The total value (welfare) of the equilibriu of GSP can be as sall as twothirds of optial. This nonoptiality of GSP coes fro a failure of GSP to account for the reality that a click on an ad in a top slot is not worth the sae as a click on the sae ad in a botto slot. Perission to ake digital or hard copies of all or part of this work for personal or classroo use is granted without fee provided that copies are not ade or distributed for profit or coercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific perission and/or a fee. Workshop on Ad Auctions 8 Chicago, Illinois USA Copyright 28 ACM XXXXXXXXX/XX/XX...$5.. A conclusion can be ade for payperclick online advertising in general (including, for exaple, banner ads); naely, that the value generated by clicks ay vary due to various reasons (e.g., the position on the web page) and that this should taken into account in the design of the advertising echanis. GSP s inadequacies are any and, in response, other studies have proposed extending GSP in nontrivial ways to allow advertisers a ore expressive bidding language in which to specify their preferences. However, in a arket where already advertisers are required to solve daunting probles of specifying bids across a huge and diverse inventory of search ipressions, aking the bidding language ore coplex sees unlikely to significantly iprove the arket s efficiency. We propose a solution that takes a different approach. For any advertisers, the result of a successful advertiseent is a conversion, e.g., the user both clicks on the ad and then subsequently buys one of the advertiser s products, fills out a for, or otherwise copletes soe electronically trackable action. All ajor web search engines allow advertisers to optin to a conversion tracking syste where the search engine tracks and keeps statistics on how well the advertiser s ads are converting. These conversion tracking systes can be eployed to autoatically learn the relative value of the clicks in each slot position. In our work, we show that GSP can be odified, retaining its original bidding interface, to take into account relative differences in conversion rates across advertisers and slots. The resulting echanis, in a odel that is consistent with the results of our epirical study, has optial equilibria. GSP works as follows. GSP assues that the probability that an ad will be clicked on is a product of a slot specific clickthrough rate and an advertiser specific clickthrough rate. It assues that an advertiser s value for a click is the sae for any ad slot. When a search query is entered into a search engine, all advertiser bids that atch the query are entered into an auction. An advertiser s bid is interpreted as a bidperclick. GSP coputes the advertiser s bidperipression by ultiplying the advertiser s bid by their clickthrough rate. The ads are then ranked by decreasing bidperipression and are assigned to the slots in this order (usually on the right hand side of the search results page). If an ad is clicked on, the advertiser is charged the iniu bid that would have been necessary to aintain their position. This is precisely the bidperipression of the next advertiser divided by this advertiser s clickthrough rate. If the advertisers bid their true valueperclick then the ranking rule akes sense: the total value of the ranking is
2 the su valueperipression of each ad scaled by the clickthrough rate of the slot. Under the assuption that lower slots have lower clickthrough rates, this assignent of ads to slots axiizes the total value. The prior analyses of GSP assued a full inforation setting where each advertiser s actual valueperclick is publicly known. This full inforation assuption is reasonable for repeated gaes such as slot auctions. As iplicit in the definition of GSP, these studies also assue that the valueperclick of an advertiser is the sae for all slots. In this odel, Varian [8] and Edelan, Ostrovsky, and Schwarz [6] show that in equilibriu GSP ranks the bidders in order of valueperipression, even though the advertisers are probably not reporting their true valueperclick as their bid. Thus, in equilibriu GSP s assignent of ads to slots axiizes the total value. The first part of our study is to epirically refute the assuption that the advertisers valueperclick is unifor across slots. A large percentage of advertisers in MSN Search have conversion tracking enabled. With conversion tracking data we can calculate an advertiser s relative valueperclick for each slot. We find that these are never unifor. Instead, they often are ascending across slots, eaning that clicks fro lower slots are often ore valuable than clicks fro top slots. For the sae advertiser, the ratio in valueperclick between slots, according to our data, can be as uch as a factor of three. Motivated by this observation we revisit the theoretical odel of Varian [8] and Edelan et al. [6], but in the ore realistic setting where the advertisers values for clicks are nonunifor across slots. Even in the special case where all advertisers have the sae nonunifority in value for clicks across slots, we show that there are price levels where advertisers prefer botto slots to top slots. This induces equilibriu where bidperipressions are not always in the sae order as valueperipressions. Thus, there are scenarios where the rankings produced by GSP in equilibriu do not axiize the total value. We stress that this ipossibility holds even when we assue that the optial total value is obtained by assigning the bidders to slots in the order defined by their values (that is, the product of the clickthrough rate and the conversion rate decreases with the slot nuber). In fact, we give a priceofstability result that constructs settings where all the equilibria of GSP obtain at ost twothirds of the total value possible. We give the copleentary result that the loss in total value cannot be drastically low; there always exist equilibria that capture at least two thirds of the total value. These upper and lower bounds are shown for 2slot auctions, where it already requires a nontrivial analysis. We conjecture that a siilar constant upper and lower bound holds ore generally for k > 2 slots. Note that this priceofstability result is in the sae spirit of the work of [8] and [6] that proved that there always exists at least one welfareaxiizing equilibriu. Our analyses assue that, like the clickthrough rates, the conversion rates are a product of an advertiser specific ter and a slot specific ter. If this is not the case (for either conversion rates or clickthrough rates) then no GSP variant can hope to optially allocate advertisers to slots. GSP s rank The price of stability easures the ratio of the best equilibriu welfare to the optial welfare. A related notion, the price of anarchy, easures the ratio of the worst equilibriu welfare to the optial welfare. ing algorith is inherently greedy; whereas, optially allocating ads to slots when clickthrough rates (or conversion rates) do not factor requires a weighted atching algorith. Fortunately, soe of the relevant clickthrough/conversion rate learning algoriths copute these rates as products. The final part of our study deonstrates a siple fix to GSP to take into account nonunifor conversion rates across slots and results in optial equilibria. This fix is as siple as ultiplying the price an advertiser pays for a click in GSP by the slot specific conversion rate of the slot that the ad was shown in. This effectively siulates a payperconversion odel without having to ake a significant change away fro the predoinant payperclick business odel of the sponsored search industry. Related Work. Our work starts fro the original analyses of GSP under unifor conversion rates by Varian [8] and Edelan et al. [6], and extends their study to the practically relevant case where conversion rates are nonunifor. In this case GSP equilibria are nonoptial, so to quantify the extent to which GSP is nonoptial we adopt the atheatical analysis techniques of the priceofanarchy literature (see [7] for a survey). Our work is very siilar in spirit to the recent work of Abras et al. [] that discusses the cost of conciseness in GSP, i.e., the possible effect of using a single bid to represent an advertiser s value per click when in fact the advertisers value per click is nonunifor across slots. [] considers cases where the valuations are arbitrary coplicated and not given forulaicly fro known conversion rates. They give a general inforation theoretic lower bound of /k on the perforance of any kslot auction that allows only a single bid per advertiser. This result holds even in the special case where advertisers values are twoparaeter: an advertiser i has a constant value v i for the top k i slots and no value for lower slots. We ephasize that their result is inforation theoretic and applies to advertiser preferences that are ore general than our singlevalueperconversion preferences. Further ore, since the conversion rates are known in our setting, there is a singlebidperadvertiser auction that is optial. Thus, we arrive at the opposite conclusion: there is no cost of conciseness, instead there is a cost due to failure of GSP to account for nonunifor conversion rates. A final note: the techniques necessary for the inforation theoretic lower bounds of [] are copletely difference fro those necessary for our priceofanarchy style results. There have been any papers advocating alternative auction forats for slot auctions. For exaple, in the twoparaeter case considered in [], Aggarwal et al. [] give a new echanis with good equilibriu properties. The line of work in studying nongspbased auction forats for the next generation slot auctions is extreely iportant; however, it is also orthogonal to direction proposed in this paper that provides an easily ipleentable odification to GSP that can significantly iprove the equilibriu perforance of the current de facto standard slot auction. In a very recent paper Milgro [5] studies general properties of gaes where (where the set of actions is restricted), and analyses settings where such siplifications eliinate inefficient equilibria.
3 2. SPONSORED SEARCH AUCTIONS: GSP AND CONVERSION RATES Consider a set of n advertisers copeting for k slots. Each player gains a valueperconversion of v i, i.e., v i is their value for the transaction that results fro their ad; what is exactly a conversion varies between advertisers, ranging between filling out an online for and buying an airline ticket for thousands of dollars. A central ingredient of the sponsored search auction odel, and what distinguishes it fro standard ultiite auctions, is the special paraeters that affect the utilities of the bidders. These paraeters describe the behavior of nuerous users that search for inforation on the web. The first iportant set of paraeters are the clickthrough rates, where c j i denotes the probability that advertiser i s ad is clicked on when it is shown in the j th slot. Our work focuses on conversion rates, where a j i denotes the probability that a click on advertiser i s ad will convert to a transaction or acquisition when the ad is shown in slot j. 2. ClickThrough Rates and Conversion Rates We wish to call explicit attention to several coon assuptions that concern clicks on ads:. The clickthrough rates, c j i, are factorable as the product of an advertiser specific ter, c i, and a slot specific ter, c j. So, c j i cicj. 2. The clickthrough rates are onotone nonincreasing in slot nuber. I.e., top (lower nubered) slots generate ore clicks than botto (higher nubered) slots. So, c j c j+. A special case analyzed by prior studies of GSP s equilibriu akes an additional unifor conversion rate assuption: the advertiser s value for a click is the sae regardless of the slot it coes fro. Under this assuption the advertisers valueperclick are equal to their valueperconversion so we slightly abuse notation to let v i represent both of these quantities for advertiser i. We show epirically that this assuption does not hold. In its place we provide the following ore general assuptions:. The conversion rates, a j i, are factorable as the product of an advertiser specific ter, a i, and a slot specific ter, a j. So, a j i aiaj. 4. The values per ipression are onotone nonincreasing in slot nuber. This is iplied by the assuptions on factorability when the product of the slot specific clickthrough rate and the slot specific conversion rate is onotone. I.e., top (lower nubered) slots generate ore conversions per ipression than botto (higher nubered) slots. So, c j a j c j+ a j+. Notice that if the clickthrough rates are not factorable (Assuption ) then GSP is not well defined. If the conversion rates are not factorable (Assuption ) then no greedy ranking rule (like that of GSP) can give an econoically efficient allocation. The assuption of onotone clickthrough rates (Assuption 2) is a consensus in the industry, and we observe it in our data. If the values per ipression are not onotone (Assuption 4) then the ranking order used by GSP is inappropriate. (However, in such a case we could, without loss of generality, renae the slots so that slot j is the one with the jth highest nuber of conversions per ipression.) Therefore, we are soewhat justified in these assuptions. 5. The advertiser specific clickthrough and conversion rates are unifor. So, a j i aj and c j i cj. This assuption is eployed in our theoretical bounds only and is without loss of generality. Clearly for lower bounds it cannot weaken our results to add this assuption; ore generally, equilibriu probles with general advertiser specific coefficients can be reduced to an analogous proble in the unifor case. 2.2 The Generalized Second Price Auction We will now forally define GSP: The Generalized Second Price Auction (GSP). The generalized second price auction, for k slots, is given input bids fro advertisers. Let b i represent the bidperclick of advertiser i.. Let b ic i be the bidperipression of advertiser i. 2. Sort the bidders by bidperipression: Let i j be the index of the bidder with the jth highest bidperipression. For all j, b ij c ij b ij+ c ij+. For all j, let p j be the iniu bid necessary for advertiser i j to aintain their position in slot j: p j bi j+ ci j+ c ij.. For j k, show advertiser i j in slot j. 4. For j k, if ad i j is clicked, charge p j to advertiser i j. Note that GSP does not explicitly take into account the conversion rates or slot specific clickthrough rates. We will ake two iportant assuptions on ipleentations of GSP: 6. Ties are broken uniforly at rando. 7. The bid space is discrete, e.g., restricted to be ultiples of soe inial bid increent, ǫ. The assuption that ties are broken uniforly at rando (Assuption 6) is consistent with soe ipleentations of GSP. It is also consistent with other equilibriu analyses of pricing gaes, e.g., Bertrand gaes. Nonetheless, we do not believe changing this assuption significantly affect our results. The assuption of a discrete bid space (Assuption 7) is necessary for the existence of pure Nash equilibria in our odel. It is well justified as every ipleentation of GSP we are aware of has a iniu bid increent of one penny. We do not require that the agent valuations be discrete.
4 Gaetheoretic Analysis of GSP. A truthful auction is one where it is a doinant strategy for all bidders to subit bids equal to their true values. GSP is a generalization of the second price auction which is truthful [9]; however it is not truthful nor does it belong to the faily of truthful VCG echaniss. This eans that we cannot expect advertiser i to subit a bidperclick, that is equal to their valueperclick. Nonetheless, the ain theoretical result of [8] and [6] is that in equilibriu, under the unifor conversion rate assuption, when the advertisers have valuesperclick of v,..., v n and valuesperipression of v c,..., v nc n, the sorted orders of advertisers by valueperipression and by bidperipression are identical. Thus, we can conclude that in equilibriu the advertiser with the jth highest valueperipression indeed gets the jth highest slot, which has the jth highest clickthrough rate. Such an equilibriu is optial. The equilibriu concept used in [8, 6] and this paper, is Nash equilibriu in the full inforation gae induced on GSP by the clickthrough rates, conversion rates, and advertiser valuations. This concept is well suited for repeated gaes where private inforation will not stay private for long. This is the predoinant odel in the literature on sponsored search and on priceofanarchy questions. In this gae, the actions (pure strategies) of the bidders are their bids, and their payoffs are deterined by their utility fro the GSP outcoe, u j i (p) cj i (aj ivi p) (where bidder i wins slot j and pays p, and v i denotes their value per conversion). Definition. A profile of bids b,..., b n is a (pure Nash) equilibriu of GSP, if no bidder i can gain a better utility by deviating to any other bid. Forally, fixing the bids b i of the other bidders, and assuing that by bidding b i bidder i wins slot j and pays p, then for every other bid b i for which bidder i is assigned to slot j and pays p we have u j i (p) uj i (p ). 2 Throughout the paper we will use the ter equilibriu to denote a pure Nash equilibriu. We will not consider ixed Nash equilibria in this paper, unless explicitly entioned. Social Welfare. Let i,..., i k be the ordering defined by the bidsperipression in GSP, then the social welfare is P k j vi j cj i j a j i j. The axiu su is achieved when the ordering of valuesperipression coincides with the ordering of bidsperipression. Notice that the social welfare includes both the welfare of the advertisers (which is their total value less their total payents) and the payoff of the search engine (which is the total payents). Finally, we forally note the result of [8, 6] on the econoic efficiency of GSP (assuing unifor conversion rates): Theore. [8, 6] For unifor conversion rates, GSP always adits a welfare axiizing equilibriu.. EMPIRICAL RESULTS In this section, we present epirical data collected in Microsoft adcenter during August 27. Soe of the advertisers that advertise in MicrosoftLive Search use a conversiontracking syste to onitor conversions that are done electronically. We present data collected fro all advertisers 2 We can assue that slot k + exists and gains the bidders values of zero to handle losing players. Slot Adv. Adv. 2 Ind. Main Line % % % Main Line 2 9% 84% % Main Line 87% 5% 7% Side Bar 8% 5% 22% Side Bar 2 7% 77% 2% Side Bar 7% 9% 25% Side Bar 4 5% 4% 2% Side Bar 5 9% 95% 52% Figure : Data on the relative conversion rates of two individual advertisers over a period of onth, and aggregate data for all the advertisers fro the sae industry over a period of two onths. Slot CR CTR CR*CTR Main Line % % % Main Line 2 8% 49% 4% Main Line 87% % 28% Side Bar 78% 9% 7.2% Side Bar 2 82% 6% 5.6% Side Bar 96% 5% 5.% Side Bar 4 % 4% 4.6% Side Bar 5 6% 4% 4.% Figure 2: Aggregate data on conversion rates (CR) and on clickthrough rates (CTR) in different slots, relative to those paraeters in the top slot in the ain line. The data was collected in Microsoft adcenter over a period of a onth, and consists of an aggregation of any advertisers fro different industries. that are affiliated to an industry (e.g., travel, finance, insurance, etc.). This study is preliinary and eant only as otivation for our theoretical results in subsequent sections. A ore thorough epirical study will be included with the full paper. The Live search engine currently shows up to 8 ads in one page. Three appear above the true (organic) search results and these slots are referred to as the ain line. Up to five additional ads appear to the right of the organic search result, and we denote these slots as the side bar. We denote the slots in the ain line as slots , and in the side bar as slots 48. We ephasize that we only collected data for search queries where all the 8 ads were presented. Our ain goal in this section is to exhibit aggregate data that describes how conversion rates tend to change over slots. This aggregation is not a straightforward task due to several reasons. First, the concept of conversion rate does not refer to the sae thing for different advertisers and different industries, and we observe fro the data that conversion rates vary considerably over different advertisers. Second, even after noralizing, the slot effect on conversion rates varies fro advertiser to advertiser and industry to industry (Figure ). It is not clear that an aggregate slot conversion rate is eaningful. Third, ost of the data is sparse, representing queries in the longtail of the distribution. A full description of our epirical ethodology is given in Appendix C. The aggregate results in Figure 2 show that the conversion rates tends to increase with the slot nuber. The highest
5 .6.4 Utility.5.4. Advertiser, slot Advertiser, slot 2 Advertiser 2, slot Advertiser 2, slot 2 Advertiser, slot Advertiser, slot 2 Utility Advertiser, slot Advertiser, slot 2 Advertiser 2, slot Advertiser 2, slot 2 Advertiser, slot Advertiser, slot Price Price Figure : The utilities of the bidders as a function of the price with unifor conversion rates in a 2slot auction. The figure depicts the preferences of bidders with two curves per bidder, one per each slot. The figure shows that with unifor conversion rates, bidders prefer having the first slot over the second slot at every price level. Figure 4: The figure describes the utility for three advertisers in GSP as a function of the price they pay. Each advertiser has a different utility curve for each one of the two slot. At high prices, bidders will prefer having the second slot over the first one. It turns out that in this exaple, no efficient Nash equilibriu exists. conversion rate, on average, is achieved in the two lowest slots on the side bar (slots seven and eight). 4. GSP IS SUBOPTIMAL In this section, we show that with nonunifor conversion rates there are profiles of valuations for which no welfareaxiizing equilibria exist in GSP. We prove this clai for the special case of 2slot bidder auctions. We will analyze this 2slot case in detail as it highlights significant differences between the nonunifor conversion rate case and the unifor case studied by previous work of [8, 6]. We will first give an intuition for the atheatical effect of nonunifor conversion rates. Figure shows the utility of bidder i when allocated to slot j (j {, 2}) as a function of his payent in the standard odel with unifor conversion rates. This utility is a linear function of the price with a negative slope, c j (v i p), where v i here represents bidder i s value per click. An iediate property of this odel, as shown in Figure, is that at all price levels, the utility fro slot one is at least that of slot two. With nonunifor conversion rates, however, this property no longer holds. As Figure 4 shows that with nonunifor conversion rates the curves shift; for sufficiently high price levels, bidders will actually prefer the lower slot! Intuitively, when the price of a click is high, the bidder ay prefer to have a saller nuber of clicks in the lower slot when these clicks have higher quality (that is, higher conversion rates). Theore 2. 2slot GSP adits a welfareaxiizing equilibriu if and only if v a c a 2 c 2 v a 2 c a 2 c 2 and the three highest valuations satisfy v > v 2 > v. Proof. Consider two price levels. Let β be the axial price that advertiser three would be willing to pay for slot two, i.e., β v a 2. Let α be the inial price for which advertiser one prefers slot two over slot one, i.e., the payent α for which c (a v α) c 2 (a 2 v α), i.e., α c a c 2 a 2 c c v 2. Notice that β α is the condition of the lea. First, we show that α < β iplies no welfare axiizing equilibria. Since v > v 2 > v, for axiizing the social welfare, we ust have that b > b 2 > b. We will now use the fact that advertiser three would be willing to be assigned to the second slot with any payent less than β and advertiser one prefers slot two over slot one for payents higher than β. Therefore, if b 2 < β, advertiser three is better off by bidding b 2 since his utility for the second slot is nonnegative. If b 2 β, then advertiser one is better off by bidding b 2, since for such prices he prefers a lottery of the second and first slots over receiving the first slot for sure. Thus, neither case could be an equilibriu. Second, if α β then the following set of bids is an equilibriu: b a 2 v, b 2 b +ǫ and b is any bid greater than p, where p is the price for which winning slot one at price p gains advertiser two exactly the sae utility as winning slot two and paying b 2. The proof iplies that GSP will lack welfareaxiizing equilibria whenever the crossing point of the two utility curves of advertiser one (denoted in the proof by α) is saller than β v a 2 (the highest payent for which advertiser three desires slot two). It is this observation that enables our bounds on inefficiency of GSP s equilibria; when GSP is inefficient, it ust be that the advertiser values are not too far apart, so it cannot be too inefficient. 5. BOUNDS ON THE WELFARE LOSS In the previous section, we saw that soeties GSP does not adit an efficient equilibriu with nonunifor conversion rates. How do equilibria look like in GSP in such cases? We present a siple characterization of such equilibria. It turns out that given any equilibriu, there also exists an orderpreserving equilibriu where the advertisers bids are weakly ordered by the advertisers values. Note that these equilibria will not axiize social welfare, since soe of the bids ay be tied and we break ties uniforly. It turns out that such ties are inevitable under nonunifor conversion rates, and they will be central in our analysis. Definition 2. An equilibriu b (b,..., b n) is order
6 preserving, if for every two bidders i, i 2, if v i > v i2 then b i b i2 Lea. Consider a profile of valuations v,..., v n for which an equilibriu exists, denoted by b (b,..., b n). Then, there also exists an order preserving equilibriu b. Moreover, b achieves at least the sae social welfare as b. Proof. Notation: Let u i(v i,b) denote the expected utility of bidder i with value v i when the bidders bid b. That is, the expected utility is (let n i(b) denote the nuber of bidders bidding exactly b i and let the consecutive slots that they are assigned be t,..., t). u i(v X c j a j v i ( n i(b) )b i b i A n i(b) t j t Pjt,...,t+n h (b) cj a j where b i denotes the highest bid in b which is saller than b i. We will now prove the lea. Consider an equilibriu b (b,..., b n). Consider two bidders h and l such that b h > b l but h > l. We will show that the bid profile where bidder l bids b k and bidder k bids b l is also an equilibriu. It follows that we can continue swapping bidders until reaching an orderpreserving equilibriu. Let b be a profile of bids identical to b except bidder h bids b l. Since b is an equilibriu, clearly u h (v h,b ) u h (v h,b ). Note that both u h (v h,b ) and u h (v h,b ) are linear functions of v h. The slope of u h (v h,b ) is n h (b) (the first slot assigned to bidders that bid b h is t) and the Pjt,...,t +s cj a j slope of u h (v h,b ) is n h (where s is the nuber of slots assigned to bidders who bid b l and t is the first (b) such slot), which is clearly saller since we assue that a j c j is decreasing in j. It follows, that if we increase v h to be v l, we will have that u h (v l,b ) u h (v l,b ). That is, for the bid profile b swap which is siilar to b except h bids v l and l bids v h, bidder l will not be willing to bid b l. Since b was an equilibriu, bidder l will not be willing to bid any other bid level in b swap for the sae reason. Siilar arguents show that h will not be willing to change his bid as well. 5. The Social Welfare Loss Might be Substantial We now give our first bound on the welfare loss in GSP with nonunifor conversion rates. For given paraeters (CTR s and conversion rates), we construct a profile of valuation such that all equilibria incur a large welfare loss. We show that as the paraeters of the proble becoe extree (but still agree with all the assuptions in our odel), the welfare loss can be arbitrarily close to of the total welfare. The reason for GSP inefficiency is that advertisers with distinct values bid in pools. To analyze the equilibria we distinguish between the pooling outcoes of the three advertisers with the highest values. Definition. An order preserving equilibriu is nonpooling if the top three advertisers have distinct bids, i.e., b > b 2 > b, Identifying pooling and separating equilibria plays a central role in icroeconoics, ainly in the context of signaling gaes (see [4]). pooling if the top three advertisers bid the sae, i.e., b b 2 b, (, 2)pooling if the top advertiser beats the second two who pool, i.e., b > b 2 b, and (2, )pooling if the top two advertisers pool and beat the third, i.e., b b 2 > b. The above definitions include the case where the fourth and lower valued advertisers pool with the third highest value advertiser. For 2slot auctions, we siplify notation and denote c /c 2 x and a /a 2 y. Notice that by our previous onotonicity assuptions x and y. We also denote to be the nuber of players whose values v i are at least the third highest value b ( n). Theore. For every ǫ >, and given any set of clickthrough rates and conversion rates, there exist valuations profile for which the social welfare achieved by every equilibriu of GSP is at ost a xy+y xy+ xy+y xy+ +ǫ fraction of the optial welfare. When xy and y, this fraction approaches 2 + ǫ. Proof. Consider the following set of bidders in a 2 slot auction. We first denote β c a +c 2 a 2, and note that c +c βv is exactly the price where bidders with 2 value v are indifferent between losing and winning the lottery of slot and 2. Bidder has value v, and all the other n bidders have the sae value v 2, such that a v < βv 2, or equivalently v 2 > xy+y xy+y v. Denote δ v2 v, and note xy+ xy+ that δ can be ade arbitrarily sall and that it is defined independently of. Figure 5 in the appendix describes utility functions for this construction. Assuption: the exact bid level βv 2 is not in the bid space for any discretization. Denote the highest winning bid by p. For the above valuations, every pooling equilibriu will gain social welfare which is arbitrarily close to 2 of the social welfare in the optial slot allocation when xy and y. (a + b)(v + ( )v2) av + bv 2 (xy + ) v + ( ) xy+y xy+ v + ( )δ (xy + y)v xyv + xy+y xy+ v + δ + xy+ v + xyv + xy+y xy+ v + δ (xy + )δ (xy + y)v xyv + xy+y xy+ v + δ < xy + y + O(ax{ xy + xy+y, δ}) xy+ Clais 5, below, establish that the only equilibria in the above setting are pooling. Finally, if the paraeters are chosen such that xy and y approaches zero, then clearly xy+y xy+ xy+y xy+ approaches 2. Clai. For the above valuations, GSP adits no nonpooling equilibriu.
7 This above clai follows iediately fro Theore 2. Clai 2. For the above valuations, GSP adits no (2,) pooling equilibriu. Proof. If p > βv 2, then bidder 2 prefers losing over his current payoff. If p < βv 2, then all the other losing bidders will gain positive utility fro bidding p. Due to the above assuption, p βv 2 cannot be an equilibriu since it is not in the bid space. Clai. For the above valuations, GSP adits no (,2) pooling equilibriu. Proof. First, p a v, otherwise bidder will prefer losing. Since we selected v 2 such that a v < βv 2, it follows that each one of the bidders with the value v 2 would earn soe positive utility by bidding p and sharing Slots and 2 with bidder. Since this positive utility is independent of the nuber of bidders, for sufficiently large bidder 2 will prefer bidding p over sharing Slot 2 with the other 2 bidders, in contradiction to the existence of an equilibriu. (Note that the bidders that win together Slot 2 gain positive utility, thus if such an equilibriu had existed, then all the bidders except bidder would have shared Slot 2.) Clai 4. For the above valuations, GSP adits a pooling equilibriu. Proof. It is easy to see that bidding the highest available bid p such that p a v by all bidders is an equilibriu. Since a v < β v 2, every bidder with value v 2 gains positive utility, and thus bidder also gains positive utility. However, all bidders would gain zero utility by underbidding p and gain negative utility by overbidding. Clai 5. For the above valuations, GSP adits no nonorderpreserving equilibria (for the top three advertisers). Proof. This clai is a direct corollary of Lea. A  pooling equilibria is the only orderpreserving equilibria that exists and it cannot be a result of a swapping process starting fro other equilibria. The other orderpreserving equilibria do not exist and therefore neither to nonorderpreserving equilibria that can be converted to the via Lea. 5.2 Existence of Equilibria with a Bounded Loss In the proof of Theore we constructed a faily of bidder profiles for which all the equilibria achieve social welfare which is arbitrarily close to 2 of the optiu. In this section we show that this bound is alost tight, at least for 2slot auctions. We show that there always exists an equilibriu that achieves in{ 2 a,/2 + } of the optial social welfare; this does not rule out the existence of other equilibria a 2 with worse perforance. This kind of results is coonly referred to as the price of stability [2], and it is siilar in spirit to the work of [8, 6] that showed the existence of at least one welfareaxiizing equilibriu in GSP. We were able to prove this result only for 2slot auctions, and we conjecture that a siilar constant lower bound on the suboptiality of GSP ay be proved for any nuber of slots; however, we leave this proble open. Note that it is reasonable in this odel to study a sall constant nuber of slot, rather than asyptotically growing nuber of slots, therefore the 2slot case can give us good intuition on the ore general case. We will start by proving that pure Nash equilibria always exist, even under the assuption of nonunifor conversion rates. Again, we were only able to prove this for the 2slot case. The ain reason is that our current proof relies on a case analysis that does not extend to a general nuber of slots. A new technique should probably be used for generalizing this lea. Lea 2. With a discrete bid space, GSP for two slots always adits a (pure Nash) equilibriu, even with nonunifor conversion rates. Proof. (sketch) Consider the following iterative ascendingprice process. All bids are initialized to zero. Each bidder in his turn (i,..., n) raises his bid by ǫ (the inial increent allowed) if it iproves his utility. Also, losing players will raise their bid in their turn as long as they do not decrease their utility by doing so. We clai that this process will always end up in an equilibriu. Consider the three bidders with the highest values (bidders,2,), and consider the first iteration where one of the will not increase his bid. If the three bidders stop bidding at the sae iteration, then it is clearly an equilibriu (they do not to bid higher bids since they stopped, but they do earn positive utilities; if they bid a lower bid they will get zero). If player 2 is the first to stop bidding, this is an equilibriu as well, since bidder one preferred the higher bid, and player and 2 preferred not to raise their bid. Finally, if bidder stopped increasing his bid, it is not necessarily an equilibriu since the iterative process ay proceed, but since we require that the losing bidders will continue raising their bids in the algorith, all we have to show is that bidder will not prefer decreasing his bid when he shares the highest bid with bidder 2. This is clear, since bidder preferred oving to the current bid over sharing slot 2 with the others, and thus gain nonnegative value fro slot. He will therefore prefer the lottery of slots,2 over having slot 2 with probability of at ost /2 (since losers will always continue bidding). Corollary. GSP with 2 slots always adits an order preserving (pure) equilibriu. Proof. Iediate fro Leas 2 and. The following theore shows that the equilibriu social welfare is not fatally low. As long as the conversion rates in the slots are not extreely far fro each other (i.e., a 2 a and this is copatible with what we observed fro the data), GSP achieves at least 2 of the optial welfare (with 2 slots). Theore 4. In GSP for 2 slots and any nuber of bidders there always exists an equilibriu that achieves at least a fraction of in{ 2, + a } of the optial social welfare. 2 a 2 The theore is iediately concluded fro the following lea proved in Appendix 4. This is proved by a case analysis that bounds the social welfare loss in each equilibria for each possible pooling. Recall that the loss in social value in orderpreserving equilibria is due to the randoized assignents in case of ties. An iportant part of the analysis is the use of the paraeterized constraints on the values of the bidders that are derived by the equilibriu constraints. We forulate these constraints to show that the values of the bidders that bid
8 the sae bid level cannot have values that are too far fro each other, and therefore the loss in social welfare is liited. One subtle issue is that in the third part of the lea, we require a ore strict requireent than the equilibriu requireents that correspond to pooling equilibria, and we use the fact that we can assue that no (2,)pooling equilibriu exists (if such an equilibriu exists, we have a bound by the first ite of the lea). Lea.. Every (2)pooling equilibriu achieves at least 4 of the optial social welfare. 2. Every (,2)pooling equilibriu achieves at least 2 + y of the optial social welfare. 4. Every pooling equilibriu achieves at least 2 of the optial social welfare (when (2,)pooling equilibria do not exist). Actually, the proof of Lea gives us a better lower bound that depends on all the paraeters of the odel (CTR s, conversion rates, nuber of bidders). The in{ 2, + 2 a } bound is only a siplification. The better bound is a a iniu 2 of several paraeterized functions, and is given in Appendix B. 6. GASP: CONVERSION RATE AWARE GSP AUCTIONS Conversion tracking allows us to estiate the likelihood that a click will generate a conversion when the user visits the advertiser s website. We presented epirical data (in Section ) that showed that the conversion rates on botto slots are nonunifor across slots. Unfortunately, the equilibriu results of [8, 6] which show that GSP axiizes the social welfare (i.e., perfors optially) fails when conversion rates are not unifor across slots. In Section 5, we have shown that for reasonable clickthrough rates, conversion rates, and advertiser valuations; the social welfare of GSP can be as uch as % less than optial. In this section we deonstrate how we can odify the payent rule in GSP using conversion rates that are autoatically calculated by the conversion tracking syste. The equilibriu in the resulting auction axiizes the social welfare. One iediate suggestion ight be to ove fro a payperclick auction to a payperconversion odel. However, this transition ay be too drastic and fast for such large businesses; an increental iproveent of the current syste, which will aintain the sae payperclick business odel and the sae user interface but will take into account the conversion tracking data sees to be desirable. This is exactly what we suggest in the following variant of GSP which we call GASP. The difference fro standard GSP is in how the payents are noralized. Note that that since GASP preserves the payperclick business odel, it 4 We conjecture that a better analysis ay iprove this bound to 2. can concurrently sell ads to bidders with and without conversion rate tracking systes (e.g., for advertiser with offline conversions). The Generalized Acquisitionaware Second Price Auction. The generalized acquisitionaware second price (GASP) auction, for k slots, is given input bids fro advertisers. Let b i represent the bidperconversion of advertiser i.. Let b ic ia i be the bidperipression of advertiser i. 2. Sort the bidders by bidperipression: Let i j be the index of the bidder with the jth highest bidperipression. For all j, b ij c ij a ij b ij+ c ij+ a ij+. For all j, let p j be the iniu bid necessary for advertiser i j to aintain their position in slot j: p j bi j+ ci j+ ai j+ c ij a ij.. For j k, show advertiser i j in slot j. 4. For j k, if ad i j is clicked, charge a j i j p j to advertiser i j. 5 Gaetheoretic Analysis of GASP. A gaetheoretic analysis of GASP follows directly fro the analysis of GSP in [8, 6]. Under the assuption that clickthrough rates and conversion rates are factorable (which was also required in GSP, see Section 2), and that the ipressions in top (lower nubered) slots lead to ore conversions than botto (higher nubered) slots; in equilibriu, GASP axiizes the social welfare. Proposition. GASP always adits a welfareaxiizing equilibriu, even with nonunifor conversion rates. The proof of this clai is based on the following observation about an advertiser s utility. Using the prices in the definition of the GASP echanis, actually the conversion rates can be factored out in the utility functions of the bidders, returning to the sae utility structure discussed for unifor conversion rates by [8, 6]. In other words, we ove fro a utility structure as in Figure 4 back to the structure in Figure. Forally, advertiser i s utility for slot j at price a j ip is now u j i (p) vicj i aj i (vi p), since if there is a conversion (with probability c j i aj i ), the advertiser gets their value v i, but if there is a click (with probability c j i ), the advertiser ust pay aj ip. Notice that our assuption on the onotonicity of values per ipression across the slots allows us to assue that at any p all advertisers have higher utility for top (lower nubered) slots. 5 First a j i j p j siplifies to b ij+ c ij+ a j i j+ /c ij. Second, if wanted a payperconversion auction instead of a payperclick auction we would siply charge p j advertiser i j upon conversion. Constrained to a payperclick odel, we scale this payent by i j s conversion rate for slot j to get the appropriate payent.
9 7. REFERENCES [] Z. Abras, A. Ghosh, and E. Vee. Cost of conciseness in sponsored search auctions. In Proc. rd International Workshop on Internet and Network Econoics, 27. [2] E. Anshelevich, A. Dasgupta, J. Kleinberg, E. Tardos, T. Wexler, and T. Roughgarden. The price of stability for network design with fair cost allocation. In FOCS 4, pages 295 4, 24. [] S. Muthukrishnan Gagan Aggarwal, Jon Feldan. Bidding to the top: Vcg and equilibria of positionbased auctions. In Proceedings of the Fourth Workshop on Approxiation and Online Algoriths, 26. [4] A. MasCollel, W. Whinston, and J. Green. Microeconoic Theory. Oxford university press, 995. [5] Paul Milgro. Siplified echaniss with applications to sponsored search and package auctions, 27. Working paper, Stanford University. [6] Michael Ostrovsky, Benjain Edelan, and Michael Schwarz. Internet advertising and the generalized second price auction: Selling billions of dollars worth of keywords. Forthcoing in Aerican Econoic Review, 26. [7] Ti Roughgarden and Eva Tardos. In Noa Nisan, Ti Roughgarden, Eva Tardos and Vijay Vazirani (Editors), Algorithic Gae Theory. Chapter 7. Introduction to the Inefficiency of Equillibria. Cabridge University Press., 27. [8] Hal Varian. Position auctions. To appear in International Journal of Industrial Organization, 26. [9] W. Vickrey. Counterspeculation, auctions and copetitive sealed tenders. Journal of Finance, pages 8 7, 96. APPENDIX A. PROOF OF LEMMA Notations: recall that denotes be the nuber of bidders that have at least the rdhighest value (i.e., where v i v ) and that a c a, b a 2 c 2, x c and y a, and let p c 2 a be the lowest winning bid. In the following three 2 subsections we prove the ites in Lea. A. A bound on the welfare loss in pooling equilibria This subsection proves the following lea. Lea 4. Every pooling equilibriu achieves at least 2 of the optial social welfare (if (2)pooling equilibria do not exist). Note that the optial welfare is av + bv 2, and the welfare when > 2 bidders bid the highest bid is (a + b) (v + v 2 + ( 2)v ). In this proof, we will denote the ratio between the as WR (for Welfare Ratio), and we will bound it fro below. WR (a + b) (v + v2 + ( 2)v) () av + bv 2 We first observe that the inefficiency ratio is either always increasing or always decreasing in v 2. It is easy to see (by firstorder conditions) that the function is increasing in v 2 if v < ( )b v a b and decreasing otherwise. We will treat these two cases separately. Case : The function in Equation is decreasing in v 2. We can now change v 2 to be v and the ratio will not increase. WR > (a + b)(v + ( )v) (2) av + bv Since this is a pooling equilibriu, we will now write constraints derived by the equilibriu definition. These constraints will help us bound the WR function. If bidding p is an equilibriu bid for the top bidders, then bidder will not benefit fro deviating and bid p + ǫ. Therefore, c (a v p) + c2 (a 2 v p) c (a v p) Since bidder 4 gains non negative utility, c (a v p) + c2 (a 2 v p) () The two above inequalities give an upper bound and a lower bound on p; taken together, we get that v tv where t (( )a b) `c + c 2 (( )c c 2 ) (a + b) (( )xy ) (x + ) (4) (( )x ) (xy + ) Since Equation 2 is increasing in v (assuing b < and > 2), we get that: WR > (a + b) (v + ( )tv) av + btv (a + b) ( + ( )t) a + bt (xy + ) ( + ( )t) xy + t (5) It is easy to see that in this case WR xy+ xy+2 a+b a+2b > 2.6 Case 2: The function in Equation is increasing in v 2. We can now change the v 2 in equation to be v and the ratio will not increase. WR > (a + b)(2v2 + ( 2)v) (7) av 2 + bv 2 6 Fro Equation 4 clearly t ( )xy. Substituting into ( )(xy+) Equation 5 we get: (xy + ) ( + ( )t) xy + t xy( )(xy + ) ( )(x 2 y 2 + xy) + ( )xy xy(xy + ) (xy + ) (x 2 y 2 + xy) + xy xy + 2 (6)
10 In the analysis of this part of the proof we will actually need a stronger constraints than the one that follows fro the pooling equilibriu properties. We will assue that there is no (2,)pooling equilibriu. (We can do this, since part of Lea shows that (2,)pooling equilibria achieve at least of the social welfare.) It follows that 4 bidder 2 prefers sharing the second slot at price p with the bidders with the value v over sharing the highest bid with bidder : 7 `c2 (a 2 v 2 p) 2 c (a v 2 (p + ǫ)) + 2 c2 (a 2 v 2 p) After rearranging the ters and together with Equation we get: v > (a + (a + b) 2 c + b)(c + c 2 ) > 2 )c2 v 2 And we denote v > tv 2, and note that t > a+( 2 )b a+b v 2. Using Equation 7 we get the following bound on the inefficiency: WR > (a + b)(2v2 + ( 2)tv2) av 2 + bv 2 (2 + ( 2)t) > a + ( 2) > 5 6 ( 2)2b ( )(a + b) 2 a + b b A Where the last equality holds since 2( 2) ( ) < for > 2 and since b < a. A.2 A bound on the loss in (,2)pooling equilibria Lea 5. Every (,2)pooling equilibriu achieves at least + a of the optial social welfare. 2 a 2 Proof. First, bidder ust have non zero utility fro bidding p, where he receives Slot 2 with a positive probability and pays p. Thus, p a 2 v. On the other hand, bidder 2 prefers bidding p to bidding p + ǫ. That is, he prefers no slot (zero utility) to Slot at price p+ǫ. Thus, p+ǫ > a v 2. It follows that a 2 v + ǫ > a v 2, v > yv 2 ǫ a 2 (8) 7 If such an equilibriu does not exists, bidder 2 ust want to deviate. The bidders with the value v will not bid above p since p ust be exactly the price for which bidder is willing to get Slot and Slot 2 at rando, but this lottery will gain hi negative utility for price p + ǫ. Also, if bidder will deviate, necessarily bidder 2 will deviate too. WR av + 2 bv2 + bv av + bv 2 av + bv y O(ǫ) (9) av + bv 2 av + av + 2 y O(ǫ) av + av + ( 2)y + O(ǫ) 2 2( ) > 2 + y O(ǫ) () 2 Where the 2nd inequality is due to Equation 8, and the rd inequality is due to the fact that Equation 9 is decreasing in the value of bv 2 (since + 2 y < ), which is bounded by av. A. A bound on the loss in (2,)pooling equilibriu 4 Lea 6. Every (2,)pooling equilibriu achieves at least of the optial social welfare. Bidders and 2 win by bidding p, and bidder bids p ǫ. Again, we will consider constraints on the valuations of the bidders that follow fro the equilibriu properties. Bidder 2 ust gain nonnegative utility by bidding p. 2 c `a v 2 p + 2 c2 `a 2 v 2 (p ǫ) Therefore: p a + b c + c 2 v2 + ǫc2 () c + c 2 Bidder will not increase his utility fro bidding above p and winning Slot alone: 2 c `a v p + 2 c2 `a 2 v (p ǫ) c (a v p) Therefore: p Fro Inequalities and 2 we get: a b c v (2) c2 v 2 (a b)(c + c 2 ) (a + b)(c c 2 v O(ǫ) ) > xy v O(ǫ) xy + Now we can bound the socialwelfare loss. Note that the expected welfare in (2,)pooling equilibriu is 2 (a+b)(v+ v 2), and recall that the optial welfare is given by av +bv 2. We will use the above constraints to prove the following lower bound on the ratio between the equilibriu welfare and the optial welfare.
11 WR > > (a + b)(v + v2) 2 av + bv 2 2 > 4 (a + b)(v + xy xy+ v O(ǫ)) av + b xy v O(ǫ) xy+ xy (a + b)( + O(ǫ)) 2 xy+ a + b xy xy+ xy (xy + )( + ) 2 xy+ xy + xy O(ǫ) xy+ x 2 y 2 + xy O(ǫ) () x 2 y 2 + 2xy O(ǫ) (4) Where Inequalities and 4 follow fro siple algebra. B. BETTER LOWER BOUND ON THE SO CIAL WELFARE LOSS The proof of Theore 4 (and Lea ) actually directly derives the following lower bound on the socialvalue loss. Siplifying this bound proves a in{ 2, 2 + a a 2 } on the equilibriu social welfare. Let f,..., f 4 be the following functions: f (xy + + ( 2) (xy + ) x + (xy + ) f 2 f 2 f 4 )(x + ) ) 2 + ( ) (( )xy )(x+) (( )x )(xy+) xy + (( )xy )(x+) (( )x )(xy+) + ( 2)y + 2( ) (xy )(x+) (xy + )( + ) 2 (xy+)(x ) xy + (xy )(x+) (xy+)(x ) Theore 5. In GSP for 2 slots and any nuber of bidders there always exists an equilibriu that achieves at least a fraction of in{f, f 2, f, f 4} of the optial social welfare. C. EMPIRICAL METHODOLOGY We describe our experiental ethodology for aggregating advertiser conversion rates. Recall that the concept of conversion rate does not refer to the sae thing for different advertisers and different industries, and we observe fro the data that conversion rates vary considerably over different advertisers. Therefore, we noralized the conversion rates of each advertiser using the top slot (Slot in the ain line) as a benchark before aggregating. We then coputed the weighted average of these noralized conversion rates per each slot, reoving advertisers with extreely low nuber of conversion to avoid bias fro statistically insignificant data. Overall, the data aggregation process of the data presented in Figure was as follows: 8 8 For exaple, consider the following three advertisers in a A Utility x y Price Advertiser, slot Advertiser, slot 2 Advertisers 2 n, slot Advertisers 2 n, slot 2 Advertisers 2 n, lottery Figure 5: The figures show the utility structure of settings where the social welfare loss approaches 2. The solid lines describe the utility of bidder, the dashed lines show the utility of the other bidders; the iddle dashed line shows their utility fro being allocation randoly to slot or 2. Following the notations in the proof of theore, x a v and y β v 2.. Restriction to search queries with eight ads. To control for conversion and click phenoena that result of a varying nuber of slots, we only considered search queries for which eight ads were shown. 2. Noralization at the advertiser level. For each advertiser and for each one of the eight slots, we calculated the conversion rate in this slot, relative to the conversion rate at slot one of the ain line. I.e., the relative conversion rate at slot j for advertiser i is âj i /â i (where âj i stands for the epirical conversion rate).. Ignoring data with very sall nuber of saples. For each advertiser we calculate the total epirical conversion rate â i (total nuber of conversions / total nuber of clicks). For advertiser i we ignore the data in slot i if the expected nuber of clicks was too close to zero in ters of standard deviation. 9 Such ad ipressions are be treated as if they received no clicks. 4. Coputing a weighted average of the relative conversion rates. Finally, we coputed a weighted average of the conversion rates separately for each slot (j,..., 8). The relative conversion rate of each advertiser at slot j is weighted by the nuber of clicks this advertiser received in slot j divided by the total nuber of clicks of all advertisers at slot j. three slot auction. Advertiser had, 9, and 5 clicks respectively on slots, 2 and, where 2%, 5% and % of the clicks were converted. Advertiser 2 had 2, 8 and 4 clicks, and conversion rates of 5%, 7% and 2%. The third advertiser had 4, 2 and clicks and conversion rates of 4%, 5% and %. The aggregate conversion rates are (%, 8% and 9%. Note that the data of the slot of the rd advertiser was disregarded since the nuber of clicks is too low. 9 Specifically, we excluded data fro slots where E[conversions] 2 ST DEV < (STDEV is the standard deviation of the nuber of conversions in this slot). Naely, (#clicks) â i 2 p (#clicks) â i ( â i) < (where #clicks is the nuber of clicks for advertiser i in slot j.
Algorithmica 2001 SpringerVerlag New York Inc.
Algorithica 2001) 30: 101 139 DOI: 101007/s0045300100030 Algorithica 2001 SpringerVerlag New York Inc Optial Search and OneWay Trading Online Algoriths R ElYaniv, 1 A Fiat, 2 R M Karp, 3 and G Turpin
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