Hybrid Keyword Search Auctions

Hybrid Keyw ord Searc h A uctions ∗ Ashish Go el † Stanford Univ ersit y Kamesh Munagala ‡ Duk e Univ ersit y Octob er 29, 2018 Abstract Search auctions have b ecome a dominant source of revenu e generation on the Internet. Such auctions ha ve t ypica lly used per -clic k bidding and pr icing. W e prop ose the use o f hybrid auctio ns where an advertiser can make a p er-impression as well as a per-c lick bid, and the auctioneer then c ho oses one of the tw o as the pricing mechanism. W e assume that the advertiser and the auctioneer b oth have separ ate b eliefs (called priors ) on the click-probability of an advertisemen t. W e first prove tha t the hybrid auction is truthful, assuming that the advertisers are risk-neutral. W e then show that this a uction is sup erior to the existing p er-click auction in multiple wa ys: 1. W e s ho w that r isk-seeking advertisers will choose only a p er-impression bid whereas ris k- av erse advertisers will choose only a p er-clic k bid, a nd ar gue tha t b oth kind of advertisers arise natura lly . Hence, the ability to bid in a hybrid fas hion is imp ortant to a ccoun t for the r isk characteristics of the advertisers. 2. F or obscure keyw ords, the auctioneer is unlikely to have a very sha rp prior o n the click- probabilities. In s uch s itua tions, we show that having the extra information from the advertisers in the form of a per -impression bid can result in significantly higher reven ue. 3. An a dv ertiser who b elieves that its c lick-proba bilit y is muc h hig her than the auctioneer’s estimate ca n use p er-impression bids to co rrect the a uctioneer’s prior without incur ring any ex t r a c ost. 4. The hybrid auctio n can allow the advertiser and auctioneer to implement complex dynamic progra mming strategies to deal with the uncertaint y in the click-probability using the same bas ic auction. The p er-click and pe r-impression bidding schemes can o nly b e use d to implement tw o extre me case s of these str ategies. As In ternet co mmerce ma tures, w e ne e d more sophistica ted pricing mo dels to exploit a ll the information held by each of the participants. W e believe that h ybrid auctions could be an impo rtan t step in this direction. The h ybrid auction easily extends to m ultiple slots, and is also applicable to scenario s where the hybrid bidding is p er-impression a nd p er-actio n (i.e. CPM and CP A), or p er-click and pe r -action (i.e. CP C and CP A). ∗ T o app ear in the proceedings of ACM WWW ’09 † Departments of Managmen t Science and Engineering and (by courtesy) Computer Science, Stanford U niv ersity . Researc h supp orted in part by NSF ITR grant 0428868, by gifts from Google, Microsoft, and Cisco, and by the Stanford-KAUST alliance. Email: ashishg@stanford.edu . ‡ Department of Computer Science, Duke Universit y . Researc h supp orted by NSF v ia a CAR EER aw ard and gran t CNS-0540347. Email: k amesh@cs.duke.edu . 1 1 In tro duction While searc h engines had a trans formatio nal effect on Internet use and ind eed, on human interac- tion, it wa s only with the adv ent of k eyw ord auctions that these searc h engines b ecame commercially viable. Most of the ma jor searc h en gines d ispla y adv ertisemen ts along with searc h results; the rev- en ue from these adve rtisements d riv es m uc h of the innov ation that o ccurs in searc h in particular, and Internet applications in general. Cost-p er-clic k (CPC ) auctions ha v e ev olv ed to b e the dom- inan t means by whic h suc h advertisemen ts are sold [11]. An advertiser places a bid on a s p ecific k eyw ord or keyw ord group. The auctioneer (i.e . the searc h en gine) main tains an estimate of the clic k-through probabilit y (CTR) of eac h adv ertiser for eac h keyw ord. When a u ser searc hes for a k eyw ord, the fi rst advertising sp ot is sold to th e advertiser which has the highest p rod uct of the bid and the CTR; in the even t that this adv ertisemen t is clic ked up on b y th e user , this adve rtiser is c harged the min imum b id it would hav e to mak e to r eta in its p osition. The same pro cess is rep eated for the next slot, and so on. A fu ll d esc rip tio n of the tremendous amount of work r el ated to keyw ord auctions is b ey ond the scop e of th is pap er; the reader is referred to the excellen t survey b y Lahaie et al [6]. Two other alternativ es to CPC auctions are widely used: 1. CPM, or Cost P er (thousand) Imp ressions: The p ublisher c harges the adve rtiser for ev ery instance of an adv ertisement sho wn to a user, r ega rdless of the clic k. This is widely used to sell banner adve rtisements. 2. CP A, or Cost Pe r Action (also known as Cost P er Acquisition): The p u blisher c harges the adv ertiser when an actual sale happ ens . This is w idely u sed in associate p rograms such as the ones run b y Am azon, and by lead generation in termediaries. The three mo dels are equiv alen t when precise estimates of the clic k-through-probabilit y and clic k-to-sal e-con ve rsion p robabilit y are kno wn. In the absence of such estimates, CPC has emerged as a goo d wa y of in formally d ividing the risk b et wee n the auctioneer and the adv ertiser: the auctioneer is vuln er ab le only to uncertaint y in its o wn estimates of CT R , whereas the adve rtiser is vulnerable only to uncertain t y in its own estimates of th e clic k-to-sale-c onv ersion probabilit y , assuming its adve rtisement gets d isp la yed in a fa v orable sp ot. A great deal of effort has gone in to obtaining go o d p redictio ns of the CT R. The pr oblem is m ade harder b y the fact that man y k eyw ords are searc hed for only a f ew times, and t ypical C TRs are low. A dvertisers often w an t to also bid by customer d emo graphics, which fu rther exacerbates the sparsity of the d at a. Hence, there has to b e a great reliance on predictiv e mo dels of user b ehavio r and n ew ads (e.g. see [10]). Arguably , an other approac h wo uld to devise pricing mo dels th at explicitly tak e the uncertaint y of the CTR estimates in to accoun t, an d allo w advertisers and auctioneers to j ointly optimize o v er this uncertain t y . In general, w e b eliev e that as Inte rn et commerce matur es, we need n ot just b etter estimation and learning algorithms but also m ore sophisticate d pricing mo dels to exploit all the information held by eac h of the participan ts. In th is pap er, we prop ose th e use of hybrid auctions for searc h k eywo rds where an adve rtiser can mak e a p er-impression as w ell as a p er-clic k bid, and the auctioneer then c ho ose s one of the t w o as the pr icing mec hanism. Informally , the p er-impression bids can b e though t of as an additional signal w hic h ind ica tes the adve rtiser’s b elief of the CT R. This signal ma y b e quite v aluable wh en the k eyword is obscure, when the advertiser is aggregating d ata from m ultiple pu blishers or has a goo d pr edicti ve mo del based on domain kn owledge, and when the adv ertiser is willing to pay a higher amoun t in order to p erform inte rn al exp eriment s/ke yword-selec tion. W e assume that the 2 adv ertiser and the auctioneer b oth ha v e s eparate b eliefs (called priors) on the clic k-probabilit y of an adv ertisemen t. 1.1 Our results W e d escribe the h ybrid auction in section 2, where w e also ou tline the strategic mo del (that of discoun ted rewards) used by the auctioneer and the advertiser. W e in tro duce the m ulti-armed bandit problem as it o ccurs naturally in th is co ntext. Our results, which w e hav e already describ ed at a high lev el in the abs tr ac t, are sp lit in to tw o p arts. My opic adv ertisers. W e fi rst study (section 3) the case of m yo pic advertise rs, whic h on ly optimize the exp ected profit at th e cur ren t step. When these adv ertisers are risk-neutral, we sh o w that truth- telling is a str ongly dominant strategy: the adv ertiser bids the exp ected p rofit from an impression as a p er-impression b id , and the v alue it exp ects f r om a clic k as the p er-clic k bid, r eg ardless of the auctioneer’s prior or optimization strategy . F u rther, if the adv ertiser is certain ab out its CTR, and if this CTR is dra wn from the auctioneer’s prior whic h follo ws the natural Beta distribution (defined later), then the w orst case loss in rev en ue of the auctioneer o v er pure p er-clic k b idding is at most 1 /e ≈ 37%. In con trast, the reve nue-loss for the auctioneer when h e uses the p er-clic k sc heme as opp osed to the hybrid auction c an appr o ach 100% for a fairly natur al sc enario, one that c orr esp onds to obscur e keywor ds . W e finally consider risk taking b eha vior of the adv ertisers when they are not certain ab out their CT R. W e s ho w that p er-clic k bidding is d ominan t w h en the adv ertisers are risk-a ve rse, but p er-impression bidding is d esir ab le when they are risk-seeking. Th us , the hybrid auction 1. Naturally extends the truthfu lness of the single-slot p er-clic k bidding auctions cur ren tly in use, for the case of m yopic, risk-neutr al advertisers (which is the situatio n und er whic h the prop erties of the p er-clic k auction are t ypically analyzed [11, 4, 1, 8]). 2. Results in b ounded p ossible rev en ue loss but unb ounded p ossible rev enue gain for the auc- tioneer in the natural setting of risk-neutral, m y opic adv ertisers, and where the auctioneer uses the Gittins ind ex. T he r ev en ue gain o ccurs in the common setting of obs cure keyw ords. 3. Naturally tak es the risk p osture of the adv ertiser into accoun t, whic h neither p er-clic k nor p er- impression bidding could h a ve d one on its own (b oth r isk a v erse and risk seeking adv ertisers o ccur naturally). The result b oundin g the p ossible reven ue loss of the auctioneer un d er the h ybrid auction is for an arb itrary discoun t factor u sed b y the auctioneer; the results ab out the p ossible rev en ue gain and the risk p osture assu me a m y opic auctioneer. W e b eliev e these are th e most appropriate assumptions, since we w ant to pro vide b ounds on the rev en ue loss using hybrid auctions un der the most general s cenario, and w ant to illustrate the b enefits of u sing the h ybrid auction under natural, non-pathological scenarios. Semi-m y opic adv ertisers. I n section 4, w e remo v e the assump tio n that th e advertisers are only optimizing some function of th e profit at the current step. W e generalize to th e case where the adv ertisers op timize rev en ue o v er a time-horizon. W e deve lop a tractable mo del for the adve rtisers, and sh o w a simple dominan t strategy for the adv ertisers, based on wh at we call th e bidding index . Though this strategy do es not ha v e a closed f orm in general, we sho w that in many natural cases (detailed later), it redu ce s to a natural pu re p er-clic k or pur e p er-impr essio n strategy that is so ciall y 3 optimal. Thus, our hybrid auctions are flexible en ough to allo w th e auctioneer and the adv ertiser to implemen t complex d ynamic programming strategies co llab orativ ely , under a wide range of scenarios. Neither p er-impr essio n nor p er-clic k bidding can exhaustiv ely mimic the biddin g index in these natural scenarios. Finally , we sho w a simple bidding strategy for a certain (i.e. well -informed ) advertiser to m ak e the auctioneer’s prior con v erge to the tru e CTR, w hile in curring no extra cost for the adv ertiser; p er-clic k bidd ing would ha v e resulted in th e advertiser incur r ing a large cost. This is our final argumen t in su pp ort of hybrid auctions, an d ma y b e the most con vincing from an adv ertiser’s p oin t of view. W e explain thr oughout the p aper why the s ce narios w e consider are not arbitrarily c hosen, but are quite n at ur al (indeed, we b eliev e the most n atural ones) to analyze. In th e pr ocess, we obtain man y inte resting prop erties of the h ybrid auction, which are describ ed in the tec hnical sections once w e hav e the b enefit of additional notation. Multiple Slots. Th e main fo cus of the pap er is analyzing the pr operties of the hybrid sc heme on a single ad slot. Ho w ev er, the auction itself can b e easil y generalized to m ultiple s lo ts in t wo differen t w a ys; b efore describin g these, w e n ee d to note that the hybrid auction assigns an “effectiv e b id” to eac h adv ertiser based on the p er-impression bid, the p er-clic k bid, and the exp ected CTR or qualit y measure. The fi rst generaliza tion is akin to th e widely used generalized s ec ond price auction [4 , 11] (also referred to as the “n ext-pr ic e” auction [1]) for CPC-only bidding: the advertisers are rank ed in decreasing order of effectiv e bids, and the “effectiv e c harge” made to eac h adv ertiser is the effectiv e bid of the next adv ertiser. W e do not discus s this v arian t in the rest of this p aper, sin ce the computation method ology is no different from single slot auctions. Note that w e can n ot exp ect this multi-slot generalization of th e hybrid auction to b e tru thful b ec ause eve n the CPC-only n ext- price auction is not tr uthful [11, 4, 1]. Ho w ev er, given th e immens e p opularity of the next-price auction, w e b eliev e th at this generalization of the h ybrid auction is the most lik ely to b e u sed in real-life settings. The second generalizatio n mirrors V CG [4] (or equiv alen tly , a laddered CPC auction [1]). Th is generalizat ion assumes that the CT R is multiplica tiv ely s eparable int o a p osition d epend en t term and an adv ertiser d epend en t term, and und er this assump tio n, guaran tees tru th fulness (on b oth the p er-impr essio n and p er-clic k b ids) for m y opic, r isk-neutral adv ertisers. Details of th is auction are in section 6. Th e p roof follo ws by extending th e pro of of theorem 3.1 exac tly along th e lines of [1] and is omitted. The hybrid auction is also applicable to scenarios where the h ybr id b id ding is p er-impr essio n and p er-action (i.e. CPM and CP A), or p er-clic k and p er-action (i.e. CPC and CP A). 2 The Hybrid Auc tio n Sc heme As mentio ned b efore, we will assum e that there is a s in gle slot that is b eing auctioned. Th ere are n advertisers inte rested in a single keyw ord. When an adv ertiser j arr ives at time t = 0, it submits a b id ( m j t , c j t ) to th e adve rtiser at time-slot t ≥ 1. The in terpretation of this bid is that the adv ertiser is willing to pa y at most m j t p er imp ression or at most c j t p er clic k. These v alues are p ossibly conditioned on the outcomes at the previous time slots. The auctioneer c ho oses a pub lic ly kno wn v alue 1 q j t whic h w e term the auctione er index , which can p ossibly dep end on the outcomes 1 It is conceiv able th at an auctioneer may strategically decide to n ot reveal its tru e prior; this would b e an in teresting direction to consider in future w ork. 4 for this adve rtiser at the previous time slots when its advertise ment was sho wn, b ut is indep endent of all the bids. The Hyb rid auction sc heme mimics V CG on the quantit y R j t = max( m j t , c j t q j t ). W e w ill call R j t the effectiv e bid of user j at time t . Let j ∗ denote the adv ertiser with highest R j v alue, and R − j ∗ denote the second h ighest R j . There are t w o cases. First, s upp ose m j ∗ > c j ∗ q j ∗ , then j ∗ gets the s lo t at p er-impression pr ice R − j ∗ . In the other case, j ∗ gets the slot at p er-clic k p rice R − j ∗ q j ∗ . It is clear that the Hybrid sc heme is feasible, since the p er-impression price charged to j is at most m j , and the p er-clic k price is at most c j . The auction generalizes in a natural w a y to multiple slots, but w e will fo cus on the sin gl e slot case in this pap er. I f the auctioneer c ho oses q j,t to b e an estimate of the clic k-through-rate (CTR) and the advertiser sub m its only a p er-clic k bid, then th is reduces to the traditional next-price auction cur ren tly in u s e. In order to analyze prop erties of the Hy brid au ction, w e need to mak e mo deling assump tions ab out the advertiser, ab out the auctioneer index q , and ab out time durations. 2.1 Time Hor izon and the Discount F actor T o model th e time scale o v er whic h the auction is run, w e assume there is a global discount factor γ . Inf orm al ly , this corresp onds to the presen t v alue of reve nue/profit/cost that w ill b e realized in the n ext step, and is an essentia l p aramete r in d et ermin in g the correct tradeoff b et w een maximizing present exp ected reward (exploitat ion) vs. obtaining more in formatio n w ith a view to w ards impro ving future r ew ards (exploration). The exp ected reven ue at time step t gets m ultiplied b y a factor of γ t . Note that γ = 0 corresp onds to optimizing for th e cur ren t step (the myopic case). In the discussion b elo w, w e assu me the auctioneer and adv ertiser b eha v e strategically in optimizing their o wn reve nues, and can compute parameters and bids based on their o wn discoun t factors whic h could b e differen t from the global discount factor γ u sed for discussing so cial optimalit y . 2.2 Auctioneer Mo del and the Gittins Index F or th e pu rp ose of designing an auctioneer ind ex q j t , w e assume the auctioneer starts with a p rior distribution Q j on the CTR of adv ertiser j . W e assume furth er that h e announces this pub licly , so that the advertiser is aw are of th is distribu tio n. Therefore, the a priori exp ected v alue of the C TR of advertiser i from the p oin t of view of th e auctioneer is E [ Q j ]. Supp ose at some time instan t t , T j t impressions ha ve b een offered to adve rtiser j , and n j t clic ks hav e b een observe d. The natural p osterior distrib ution Q j t for the advertiser is giv en by: Pr[ Q j t = x ] ∝ x n j t (1 − x ) T j t − n j t · Pr[ Q j = x ]; this corresp ond s to Ba y esian up dates, and when in itia lized with the u niform con tin uous prior, corresp onds to the natural Beta distribution, defined later. The auctioneer chooses a function f ( Q j t ) that maps a p osterior distrib ution Q j t to a q v alue. The fun ction f is c hosen b ased on the rev en ue gu arantees th e auctioneer d esires. W e will us e the follo wing idealized scenario to il lustr ate a concrete c hoice of f ; our resu lts apply broadly and are not limited to the scenario we describ e. Supp ose th e auctioneer wishes to optimize ov er a time horizon giv en by discoun t factor γ a . Then, if the auctioneer were to ignore the p er impression bid s, and us es a first price auction on c j q j t assuming that c j is the true p er-clic k v aluation, then h is c hoice of q j t should maximize his exp ected discounted rev enue. Let v j denote the true p er-clic k v aluatio n 5 of the adv ertiser. B y assumption, v j = c j . A t time t , the auctioneer offers the slot to the bidder j ∗ with highest v j q j t at p er-clic k price v j ∗ , earn in g v j ∗ E [ Q j t ] in exp ectation. It is well-kno w n that the d iscounted rewa rd of this scheme is maximized wh en q j t is set to the Gittins index (describ ed next) of Q j t with d isco unt factor γ a , and hence setting f ( Q j t ) = the Gittins index of Q j,t w ould b e a natural choic e. The Gittins index [5, 12] of a distribu tio n Q for discoun t fact or γ a is d efined as f ol lo ws: Cons ider a coin with th is prior distribution on probabilit y of h ea ds Q , and that yields r ew ard 1 on heads. The Gittins in dex is (1 − γ ) M , w here M is the largest n umb er satisfying the follo wing condition: Some op timal d isco unted reward tossing p olicy that is allo wed to retire at any time p oin t and collect a retiremen t r eward of M will toss the coin at least once. It is w ell-kno wn that the Gittins index is at least the mean E [ Q ] of the prior, and for a given mean, the Gittins index increases w ith the v ariance of the p rior, taking the lo we st v alue equal to th e mean only when the prior h as zero v ariance. F urther, the Gittins index also increases with th e d isco unt factor γ a , b eing equal to the mean when γ a = 0. An equiv ale nt definition will b e usefu l: Consid er a coin with the prior distribu tio n on p robabilit y of heads Q j,t that yields rewa rd 1 on h ea ds . Sup pose the coin is c harged G amount w henev er it is tossed, but is allo wed to retire anytime. Th e Gittins ind ex is the largest G for whic h the exp ected discoun ted difference b et ween the reward from tossing minus the amoun t c harged in the optimal tossing p olicy is n on -n eg ativ e 2 . T ypically , the distribution Q is set to b e the conjugate of the Bern oulli distribution, called the Beta distribu tio n [5]. T he distr ib ution Beta( α, β ) corresp onds to starting with a uniform distribution ov er the C TR and observing α − 1 clic ks in α + β − 2 impr essio ns. Therefore, if the initial p rior is Beta( α, β ), and n clic ks are then observed in T imp ressions, the p oste rior distribution is Beta( α + n, β + T − n ). Beta(1 , 1) corr esp ond s to the unif orm d istr ibution. Beta distributions are wid el y used mainly b ecause they are easy to up date. How ev er, unless otherwise stated, our results will not dep end on the distribution Q b eing a Beta d istribution. 2.3 Adv ertiser Mo del The v alue bid b y the adv ertisers will dep end on their optimizati on criteria. The true p er-cli ck v alue of adv ertiser j is v j . The adve rtiser j main tains a time-indexed distribution P j t o v er th e p ossible v alues of the actual CTR p that is u pd ate d wh enev er h e receiv es an impression. W e assume adv ertiser j ’s p rior is up dated based on the observ ed clic ks in a fashion similar to the auctioneer’s prior, but again, this is not essenti al to our results except where sp ecifically m en tioned. The advertiser’s bid will d epend on its optimization criteria. In the n ext section, we consider the case wher e the advertise rs only optimize their r even ue at the cur ren t step, and could p ossibly tak e risk. In later sections, w e consider the case wher e the advertisers att empt to optimize long-te rm rev en ue b y biddin g strategicall y ov er time. In eac h case, the advertiser could b e well-informed (or certai n) ab out its CTR, so that P j t is a p oint distribution, or unin formed ab ou t its CTR, so that it trusts the auctioneer’s p r ior, i.e. , P j t = Q j t , or somewhere in b et w een. Dep ending on the optimization criterion of the adv ertiser, these cases lead to different r ev enue pr operties for the auctioneer and advertiser, and show the adv antag es of the Hybrid scheme o v er pure p er-clic k bid ding as well as o ver p er-impression b id ding. 2 The Gittins index is usually defined as M (i.e., G/ (1 − γ )) but the alternate definition (1 − γ ) M (i.e., G ) is more conv enient for this pap er. 6 3 My opic Adv ertisers In this section, we analyze single time-step prop erties of the auction. Sp ecificall y , w e assum e that the adv ertisers are my op ic, meaning that they optimize some function of the reven ue at the curr en t time step. Since we consider m y opic pr operties, w e drop the subs cr ip t t denoting the time step f rom this section. T he auctioneer’s pr ior is therefore Q j , and the adv ertiser’s pr ior is P j . Let p j = E [ P j ]. W e fi rst s ho w that w hen the advertisers are risk-neutral, then bidding ( v j p j , v j ) is the dominan t strategy , i ndep endent of the auctione er’s prior or the choic e of f . F ur ther, if the advertiser is certain ab out its CTR, and if this CTR is d ra wn from the auctioneer’s pr io r whic h follo ws a Beta distribution, then the worst case loss in rev en ue of the auctioneer ov er p ure p er-clic k bidd ing is at most 1 /e ≈ 37%. In con trast, the rev en ue-loss for the auctioneer when he u ses the p er-clic k sc heme as opp osed to the hybrid auction c an appr o ach 100% for a fairly natur al sc enario, one that c orr esp onds to obscur e keywor ds . W e finally consider risk taking b eha vior of the adv ertisers when they are not certain ab out their CT R. W e s ho w that p er-clic k bidding is d ominan t w h en the adv ertisers are risk-a ve rse, but p er-impression bidding is d esir ab le when they are risk-seeking. Th us , the h ybrid auction naturally extends the truthfuln ess of the single-slot p er-clic k b idding auctions currently in use, results in b ounded p ossible reve nue loss but unb ounded p ossible reve nue gain, and naturally tak es the risk p osture of the advertiser into accoun t; the precise q u alit ativ e conclusions are detailed in the introdu cti on and the formal statemen ts are prov ed b elo w. The result b oun ding the p ossible rev enue loss of the auctioneer under the hybrid auction is for an arbitrary discoun t factor used by th e auctioneer; the resu lts ab out the p ossible rev enue gain and the r isk p osture assume a my opic auctioneer. W e b eliev e th ese are the most app ropriate assumptions, since we w ant to pro vide b ounds on the rev en ue loss using hybrid auctions un der the most general s cenario, and w ant to illustrate the b enefits of u sing the h ybrid auction under natural, non-pathological scenarios. 3.1 T ruthfulness W e first show that th e dominan t strategy in v olv es truthfully rev ealing the expected CT R, p j . Recall that the advertiser bids ( m j , c j ). F urth er , th e auctioneer computes an index q j based on the distribution Q j , and do es VCG on the quan tit y R j = max( m j , c j q j ). Theorem 3.1 If p j = E [ P j ] and the advertiser is myopic and risk-neutr al, then r e gar d less of the choic e of q j , the (str ongly) dominant str ate gy is to bid ( v j p j , v j ) . Pro of: First, consider th e case where q j ≤ p j . Supp ose the advertise r b ids ( m j , c j ) and wins the auction. Then , the exp ected profit of th is advertiser is at most p j v j − min { R − j ∗ , R − j ∗ · ( p j /q j ) } which is at m ost p j v j − R − j ∗ . Thus, the maximum profit of the advertiser can b e at most max { 0 , p j v j − R − j ∗ } wh ic h is obtained by bidd in g ( p j v j , v j ). Next, consider the case where q j > p j . Sup p ose the advertiser bids ( m j , c j ) and wins the auction. Th en, the exp ected profit of this adve rtiser is at most p j v j − min { R − j ∗ , R − j ∗ · ( p j /q j ) } whic h is at most p j v j − R − j ∗ · ( p j /q j ). Thus, the maxim um profit of the adv ertiser can b e at most max { 0 , p j v j − R − j ∗ · ( p j /q j ) } whic h is again obtained by bid ding ( p j v j , v j ). Th us , it is n ev er su b optimal to bid truthfu lly . Let R ∗ denote the s econd h ig hest v alue of max( m j , c j q j ). In ord er to s h o w that ( p j v j , v j ) is a (strongly) dominant s tr at egy , we need to show 7 that for an y other b id-pair ( m j , c j ), there exist v alues of q j , R ∗ suc h that the profit obtained b y bidding ( m j , c j ) is strictly less than that obtained by truthfu l bids. Supp ose ǫ is an arbitrary small but p ositiv e n um b er. First consider the scenario where q j = p j , i.e., the auctioneer has a p erfect prior. In this scenario, bid d ing ( m j , c j ) with either m j > p j v j + ǫ or c j > v j + ǫ/p j results in a negativ e profit w h en p j v j < R ∗ < p j v j + ǫ ; tru thful biddin g w ould h a ve resu lte d in zero pr ofit. F ur ther, if m j < p j v j − ǫ and c j < v j − ǫ/p j , then the adv ertiser obtains zero profit in the case where p j v j > R ∗ > p j v j − ǫ ; tru thful biddin g would hav e obtained p ositiv e profit. This lea ve s the cases wh ere m j = p j v j , c j < v j / (1 + ǫ ) or where m j < p j v j / (1 + ǫ ) , c j = v j . In the form er case, the advertiser obtains zero p rofit in the s itu ation wh ere q j > p j > q j / (1 + ǫ ) and R ∗ = p j v j ; truthful bidding would ha ve obtained p ositiv e profit. In the latter case, th e adve rtiser obtains zero p rofit wh en q j < p j / (1 + ǫ ) and R ∗ = p j v j / (1 + ǫ ); truthfu l bidd ing would h av e obtained p ositiv e profit. 3.2 W ell-Informed Adv ertisers: Loss in Auctioneer’s R ev en ue W e n o w consid er the case where the advertisers are certain ab out th eir CTR p j and risk-neutral; b y the r esults of the pr evious section, w e will assume that they bid truthfully . More formally , w e assum e the prior P j is the p oin t distr ib ution at p j . W e su pp ose that th e p j are drawn from the auctio neer’s pr io r that is of the form Q j = Beta( α j , β j ). W e no w sho w that for q j b eing the Gittins index of Beta( α j , β j ) for any discoun t factor γ a , the exp ected rev enue of the auctioneer at the current step is at least 1 − 1 /e times th e reven ue had he ignored the p er-impression b ids. Theorem 3.2 In the ab ove mentione d sc enario, the exp e cte d r evenue of the auctione er at the cur- r ent step is at le ast 1 − 1 /e ≈ 63% of the c orr esp onding auction that ignor es the p er-impr ession bid. Pro of: Let q j denote the Gitti ns index of Q j =Beta( α j , β j ). Let adv ertiser 1 ha v e the highest v j q j , and adve rtiser 2 the next highest. Let R ∗ = v 2 q 2 . If the p er-impression bids are ignored, adv ertiser 1 gets the impression at a p er-clic k price of v 2 q 2 /q 1 , s o that the exp ected reven ue is R ∗ E [ Q 1 ] q 1 . In th e Hybrid sc heme, v 1 q 1 and v 2 q 2 are b oth at least as large as R ∗ . Hence, if the auctioneer mak es a p er-impression c harge, then this charge must b e at least R ∗ p er impression. If the advertise r mak es a p er-clic k charge (w h ic h must b e to advertiser 1), then the exp ected rev enue is at least R ∗ Q 1 /q 1 . Hence the exp ected rev enue of the Hybrid sc heme is at least R ∗ E h min  1 , Q 1 q 1 i and the ratio of the rev enue of the Hy brid s c h eme to the p er-clic k sc heme is at least E [min( q 1 , Q 1 )] E [ Q 1 ] . F or p dra wn from the distribu tio n Q 1 , w e no w need to sho w that E [min( q 1 , Q 1 )] E [ Q 1 ] ≥ 1 − 1 /e . T o sho w this, observ e that for a fixed Q 1 , this ratio is smallest wh en q 1 is as small as p ossible. Th is implies w e should c ho ose q 1 = E [ Q 1 ] = α α + β , whic h corresp onds to a discount factor of 0. Denote µ = E [ Q 1 ]. Th en, the go al is to min imize the ratio 1 µ E [min( µ, Q 1 )] as a function of α, β . Lemma 3.3 sho ws that this ratio is 1 − 1 /e , completing the pr oof. Lemma 3.3 If w is dr awn fr om the Beta distribution with p ar ameters α, β ≥ 1 , and µ = α/ ( α + β ) is the me an of w , then E [min( µ, w )] ≥ µ (1 − 1 /e ) . Pro of: W e will allo w α, β to tak e on f ract ional v alues as long as they are b oth at least 1. Sup p ose α, β are b oth strictly bigger than 1. Let z denote the r andom v ariable dra wn from the Beta distribution with p arameters α ′ = α − µθ , β ′ = β − (1 − µ ) θ , where θ > 0 is chosen su c h that 8 α ′ , β ′ ≥ 1 and at least one of α ′ , β ′ is exactly 1. The mean of z is α − µθ α + β − θ = µ , w h ic h is the same as the mean of w . Let f w , f z denote the p r obabilit y density fun cti ons of w , z resp ectiv ely , and let F w ( x ) (resp. F z ( x )) denote Pr[ w ≥ x ] (resp. Pr[ z ≥ x ]). Consider the ratio r ( x ) = f w ( x ) /f z ( x ) = φx µθ (1 − x ) (1 − µ ) θ , where φ is the ratio of the corresp ondin g normalizing terms and h ence do es not dep end on x . Observe that r ( x ) is uni-mo dal (since th e deriv ativ e of r is 0 exactly once in the interv al [0 , 1]); and that r ( x ) → 0 as x → 0 + and as x → 1 − . Since b oth F w ( x ) and F z ( x ) are monotonically decreasing curv es connecting (0 , 1) and (1 , 0) , the ab o v e p rop ertie s of r ( x ) = F ′ w ( x ) F ′ z ( x ) easily imp ly that for some s ∈ (0 , 1), o v er the int erv al x ∈ [0 , s ], F w ( x ) ≥ F z ( x ), and o ve r x ∈ [ s, 1], F w ( x ) ≤ F z ( x ). This com bined with the fact th at E [ w ] = E [ z ] = µ implies w Lorenz-dominates z , so that f or all conca v e functions g , w e hav e E [ g ( w )] ≥ E [ g ( z )] [7]. Since g ( w ) = min( w, y ) is conca ve in w for fixed y , w e ha ve E [min( w , µ )] ≥ E [min( z , µ )]. Therefore, it is sufficien t to analyze E [min( µ, z )] /µ , i.e. the case wh ere either α or β is exactly 1, and the other is at least 1. It is easy to explicitly v erify b oth these cases, and show that the worst case is when α = 1 and β → ∞ when E [m in( µ, z )] = (1 − 1 /e ) µ . A Typical Case. Though the Hybrid sc heme is not r ev en ue dominan t o v er the pu re p er-clic k sc heme in pathological cases, the k ey adv an tage is in the follo wing t ypical situation. There are n advertisers whose CTR s p 1 ≥ p 2 ≥ · · · ≥ p n are d ra wn from a common pr ior Q = Beta ( 1 , K ) , whose mean is µ = O  1 K  . Assu me fur th er th at n = 4 K or K = log n 2 . W e hav e: Pr[ p 2 ≥ 1 / 2] ≥    1 −   1 − 2 log n  1 2  log n 2   n 2    2 = 1 −  1 − 2 √ n log n  n 2 ! 2 = 1 − o (1) Recall that the adv ertisers are aw are of their CTR, bu t the auctioneer is only a wa re of the p rior. Supp ose the p er-clic k v alue f or all th e adv ertisers is v , and these are truthfu lly reve aled. In the p er-clic k sc heme, the auctioneer sells the impression to an arbitrary advertise r at p er-clic k p rice v , and in exp ectation earns µv . If the auctioneer is m yo pic ( γ a = 0), then q = E [ Q ] < p 2 w.h.p, and the Hybr id sc heme c harges p er-impression. Here, the auctioneer sells to adv ertiser 1 at p er- impression price v p 2 . F rom the ab o v e, E [ p 2 ] = Ω(1), so that E [ p 2 ] /µ = Ω(log n ). Therefore, for n advertisers with d iffuse pr iors of the form Beta  1 , 1 log n  , th e auctioneer gains a facto r Ω(log n ) in reven ue. Th is is particular relev ant for obscure keyw ords, where the auctioneer will hav e very diffuse priors. 3.3 Uninformed Advertisers and R isk So far, we ha v e assumed that the adve rtiser is risk neutral and certain ab out th e CTR, so that he is optimizing his exp ected pr ofit. W e now supp ose that the adv ertiser is uncertain and trying to maximize a utilit y function U on his p rofit. T he function U ( x ) is monotone w ith monotone deriv ativ e, a nd U (0) = 0. If U is conv ex, the adve rtiser is said to b e risk-seeking, and if it is 9 conca v e, the advertise r is said to b e risk-a v erse. W e sh o w that for risk-a v erse adv ertisers, pu re p er- clic k b idding is dominant, whereas pure p er-impression biddin g is dominan t when the advertisers are risk-seeking. It is n atural to assume that some adv ertisers ma y b e either risk-a v erse or risk -seeking. Risk- a v ersion mo dels adv ertisers with tigh t b udget constrain ts. Risk-seeking adve rtisers also o ccur naturally in m an y settings; one example is when advertisers are conducting exp erimen ts to identify high p erformance adv ertising c hannels and keyw ords. Finding a high reward keyw ord may r esult in a higher bu dget allo ca ted to this keyw ord and more reven ue in the future, making the pr esen t utilit y fu nction of winning this ad slot app ear conv ex at the pr esent time. W e assume the adv ertisers are uninformed, wh ic h is equiv alen t to assuming the adve rtiser and the auctioneer share a common prior, so that P j = Q j . Essentia lly , the advertise r has n o information and simply tru sts the auctioneer’s prior 3 . In this s ec tion, w e f ocus on a single advertiser, and dr op the sub script corresp onding to it. Let p = E [ P ] = E [ Q ]. As mentioned earlier, w e assum e that th e auctioneer is m yo pic as w ell ( γ a = 0), so that q = E [ Q ] = p . Let ( m, c ) denote the advertiser’s bid, and let v d en ot e the true p er-clic k v aluation. Let I R b e the indicator corresp onding to whether the bidder gets the imp ression if R − j = R . The bidding strategy of the adv ertiser w ill attempt to m aximize: I R · max  E [ U ( v P − R )] , E  U  v P − R P p  In th e ab o ve exp ression, the first term is the exp ected p rofit if the impression is obtained b ased on the p er-impression bid; an d the latter term is the exp ected pr ofit if the impression is obtained based on the p er-clic k bid. O ur next lemma captures the str u cture of the dominant strategy . Lemma 3.4 If U is c onc ave, bidding (0 , v ) is a dominant str ate gy. If U is c onvex, the dominant str ate gy is of the form ( m, 0) for a suitably chosen m . Pro of: First consid er the case when v p < R . In this case, r eg ardless of U , E h U  v P − R P p i ≤ 0. Therefore, to obtain p ositiv e profit, the bidder has to obtain th e impression based on his p er- impression bid. In this case, the exp ected p r ofit is E [ U ( v P − R )]. Note that E [ v P − R ] < 0. Therefore if g is conca ve : E [ U ( v P − R )] ≤ U ( E [ v P − R ]) ≤ U (0) = 0 Therefore, if U is conca v e and v p < R , then biddin g (0 , v ) is a dominan t strategy . Since obtaining the impression based on the p er-clic k bid do es n ot yield p ositiv e profit, if U is con vex, b idding ( m, 0) with appropr iat ely c hosen m is a dominan t strategy . The next and m ost inte resting case is when v p ≥ R . Define random v ariable X = v P − R and Y = v P − R P p . First n ot e that E [ X ] = E [ Y ] = v p − R ≥ 0. F urther, the cumulat ive distribution functions (CDFs) of X and Y cross exactly once, w ith the CDF of X b eing initially larger than the CDF of Y . This is sufficient to sho w that Y Lorenz-dominates X . This imp lies th at for U b eing conca v e, E [ U ( Y )] ≥ E [ U ( X )] [7], so th at the adve rtiser only bids p er clic k. F urther, if U is con ve x, E [ U ( X )] ≥ E [ U ( Y )], so that the adv ertiser only bids p er imp ression. Our main result in this sub-section is the follo wing p roperty w h ic h giv es a single natural c harac- terizatio n of the optimum hybrid bid for b oth risk-a v erse and risk-seeking adv ertisers. W e will then 3 F or the other extreme case of well-info rmed advertis ers, there is no uncertaint y , and hence th e risk-ave rse an d risk-seeking cases collapse to risk-neutral. 10 sho w that for risk-seeking adve rtisers ( U is strictly increasing an d conv ex), the exp ected m y opic rev en ue of the auctioneer is larger in th e Hybrid auction compared to the pu r e p er-clic k auction, and for risk-a v erse adv ertisers, the H ybrid and p er-clic k auctions coincide. Theorem 3.5 L et m ∗ = max { y | E [ U ( v P − y )] ≥ 0 } . Bidding ( m ∗ , v ) is a dominant str ate gy. F urther, the auctione er’s r evenue i n the Hybrid scheme dominates the r eve nue in the pur e p er- click scheme. Pro of: F irst consider the case when U is conca ve . Then, E [ U ( v P − p )] ≤ U ( E [ v P − v p ]) = 0. Therefore, m ∗ ≤ v p , so that bidding ( m ∗ , v ) is equiv alent to bidding (0 , v ), which is a dominan t strategy . Next, when U is con v ex, w e hav e m ∗ ≥ v p , so that bid ding ( m ∗ , v ) is equ iv alen t to bidding ( m ∗ , 0). The p revious lemma shows that the dominant strategy is of th e form ( m, 0). S in ce m ∗ is the largest v alue of R − j for wh ic h the advertise r mak es a non-negativ e p rofit, bidding ( m ∗ , 0) m ust b e the dominant strategy . The auctioneer’s reven ue in the Hybr i d scheme is the second largest v alue of max( m ∗ j , v j p j ) while that in the p er clic k sc heme is th e second largest v alue of v j p j , w hic h cannot b e larger. 4 Semi-My opic Adv ertisers In this section, we remo ve the assump tio n th at the advertisers are optimizing some function of the profit at the curren t step. W e no w generaliz e to the case where the adv ertisers optimize rev en ue o v er a time-horizon. W e d evelo p a tractable mo del for the adv ertisers, and sho w a simple d ominan t strategy for the adv ertisers, based on what w e call the bidding index . Though th is strategy d oes not h a ve a closed form in general, we sho w that in man y natural cases (detailed later) cases, it reduces to a natural pur e p er-clic k or p u re p er-impression strategy that is so cially optimal. Th us, our hybrid auctions are flexible en ough to allo w th e auctio neer and the adv ertiser to implemen t complex d ynamic programming strategies collab orativ ely , un der a wide range of scenarios. Neither p er-impression nor p er-clic k bidding can exhaustive ly mimic the bid ding ind ex in these n at ur al scenarios. Recall that the true p er-clic k v alue of advertise r j is v j , and that the advertiser j main tains a time-indexed d istribution P j t o v er the p ossible v alues of th e actual CTR p that is up dated whenev er he receiv es an impr ession. W e assume advertise r j ’s prior is up dated based on the observe d clic ks in a fashion similar to the auctioneer’s prior, Q j t . W e assume the b idding strategy of the adv ertiser is semi-m yo pic , whic h we defin e as follo ws: The adv ertiser has a discount facto r γ b . T he b id of an adve rtiser j dep ends on its current state h v j , P j t , Q j t i , and on R − j in a fashion describ ed n ext. A t ev ery step, the v alue of R − j is r ev ealed. If the advertise r j got the impression the p revious time step, the v alue of R − j remains the same sin ce the states of the ot her adv ertisers remains the same, else it c hanges adversariall y . T he optimization criterion of the adv ertiser is to maximize its discounte d exp ected gain (using d iscoun t factor γ b ) in the conti guous time that it receiv es impressions (so th at the v alue of R − j remains the same). W e mak e the reasonable assumption that the adv ertiser cannot optimize for a horizon b ey ond that, since the v alue of R − j c hanges in an unknown fash io n. Finally note th at a m y opic advertiser is equiv alent to assuming γ b = 0. Discussion. The semi-m y opic mo del is closely related to the MDP appr oa ch of analyzing r ep eated auctions; see for instance [2, 3]. These wo rks mak e the assumption that the priors P j of the 11 adv ertisers are pu blic kno wledge. Ho wev er, this leads to somewh at p erv erse incent ives in whic h the optimal strategy for an adv ertiser could b e to underbid at the curren t time step in the h ope that the priors of the other ad vertisers resolv e to low v alues, and he then w in s th e auction on the remaining time steps at a lo w er price. Ho wev er, note that if th ere are sufficien tly man y bidders, this s cenario is unlikel y to happ en, and the bid der will attempt to win the auction at the current time slot. W e make th is explicit by making the follo wing assumptions: 1. The bidder j is a ware of the reveal ed R − j v alues of the other bidders, bu t may n ot b e aw are of their prior distribu tio ns, wh ic h are usually priv ate inform at ion. 2. The bid der only optimizes ov er th e con tiguous time horizon in whic h he receiv es the im- pressions. In this horizon, R − j is fixed, and fur ther, this remo v es the p erv erse incen tiv e to under-bid describ ed ab o v e. Note that in our mo del, the bidder is indeed awar e of the cur r en t bids R − j of the other bidders. Ho w ev er, unlik e the mod el in [2, 3], the optimization time-horizo n of the b idder leads to the existence of a n icely sp ecified d omin an t strategy . W e hop e th at our mo deling, th at is only sligh tly more restrictiv e than ones consid ered in literature, but whic h h a ve nice analytic pr op erties, will b e of indep endent interest in this and other con texts. 4.1 The Dominan t Bidding Index Strategy W e first sho w a bid d ing strategy that we term the bidding index str at egy , and show that it is w eakly dominant in the class of semi-myopic strategies. The bidding index B ( v , P , Q ) is defined as follo ws: Supp ose the adv ertiser’s cur r en t prior is P and the auctioneer’s current prior is Q . Denote th e current time instant as t = 0. S ince the adve rtiser computes this index, we assume th e adv ertiser tru sts his o wn prior b u t not the auctioneer’s. F or a parameter W , d efine the follo wing game b et w een the adv ertiser and the auctioneer w ith discount factor γ b : A t step t ≥ 0, supp ose the adv ertiser has prior P t (with mean E [ P t ] = p t ) and the auctioneer, Q t (with q t = f ( Q t ) b eing the auctio neer’s ind ex), the advertiser can either stop the game, or con tin ue. If he con tin ues, he gains v p t in exp ectatio n and pays the auctioneer W min  1 , p t q t  ; the difference is his gain. The adv ertiser’s v alue f or the game is the exp ected discoun ted (acc ording to γ b ) gain for the optimal strategy . Define W ( v , P , Q ) as the largest v alue of W for w h ic h the v alue of the game w ith initial priors P and Q , is p ositiv e. Th is v alue can easily b e compu te d by d ynamic programming, muc h lik e the Gittins ind ex. The bidding index B ( v , P , Q ) is defined as: B ( v , P , Q ) = W ( v , P , Q ) min  1 , p 0 q 0  This is the largest p er impression price at time t = 0 for whic h the v alue of the ab o v e game is p ositiv e. The Strategy: At any time step, when the advertise r j ’s prior is P j t with mean p j t , and the auctioneer’s prior is Q j t , with q j t = f ( Q j t ), let W j t = W ( v j , P j t , Q j t ) and B j t = B ( v j , P j t , Q j t ). The bidding index st rategy inv olves bidding ( B j t , B j t p j t ). It is clear that the bidd in g index strategy is w ell-defined for q j t b eing an arbitrary f unction f ( Q j t ) chosen b y the auctioneer,and not j u st for f b eing the Gittins in d ex of Q j t using d isco unt factor γ a . 12 Theorem 4.1 The bidding index str ate gy is (we akly) dominant in the class of semi-myopic str ate- gies. Pro of: Consider a sequ ence of time steps when ad vertiser j gets the impression; call this a phase. During this time, the v alue R − j used in the V CG sc heme is fixed ; denote this v alue R ∗ . Supp ose at a certain time step, the mean of the adv ertiser’s prior is p j t and the auctionee r computes q j t . If the adve rtiser gets the imp r ession, the p rice he is charge d in the V CG scheme is either R ∗ p er impression or R ∗ /q j t p er clic k. The advertiser optimizes this b y pa ying R ∗ min(1 , p j t /q j t ) in exp ectat ion p er impression. The state ev olution is only conditioned on getting the impr essio n and not on the price paid for it. Since the advertise r’s strategy is semi-my opic, at any time step, th e b id should fetc h him a non-negativ e exp ected profit for the rest of the p hase. This implies that R ∗ ≤ W j t . T here are t wo cases. First, if p j t < q j t , the adv ertiser essenti ally bids R j t = B j t q j t p j t = W j t ≥ R ∗ , and receiv es the impression at a p er-clic k price of R ∗ q j t . Th erefore, th e exp ected p er impression price is R ∗ p j t q j t = R ∗ min  1 , p j t q j t  . Next, if p j t > q j t , the adv ertiser essentia lly bids R j t = B j t = W j t ≥ R ∗ , and receiv es the impression at a p er-impression price of R ∗ = R ∗ min  1 , p j t q j t  . Therefore, the bidding sc heme ensures that the advertiser receiv es the impression and make s the most p ossible p rofit in the rest of the phase. Note fi nally that if W j t < R ∗ , th e maximum p ossible profit in the rest of the p hase is negativ e, and the bidding sc heme ensures th e advertiser do es not r eceiv e the impression. 4.2 So cial Optimality of Bidding Index Supp ose the global d isco unt factor is γ . W e defin e the so cially optimal strategy as follo ws: Sup pose at time t , adv ertiser j with prior P j t receiv es the impression resu lting in v alue v j p j t for him. The so cial ly optimal solution maximizes the infin ite horizon exp ected discount ed v alue with discount factor γ . W e show that the biddin g ind ex strategy implement s the so cially optimal solution in eac h of the follo wing t wo situations: (1) T h e adv ertiser and the auctioneer share th e s ame prior ( P j t = Q j t ), and either (1a) only the advertisers are str ategic ( γ a = 0 and γ b = γ ) or (1b) only the auctioneer is strategic ( γ a = γ and γ b = 0); and (2) Th e advertisers are certain ab out their CTR ( P j = p j ) and (2a) the aucti oneer’s index q j t is alw a ys at most p j . Th e b idding index also h as a particularly simple form when the adv ertisers are certain, and (2b) the auctioneer’s q j,t is monotonically decreasing with t and alwa ys larger than p j,t . In b oth (2a) and (2b), the biddin g index strateg y reduces to bidding ( v j p j , v j ). These scenarios are n ot arbitrarily chosen, and are the most illustrativ e scenarios w e could find. Scenario (1) corresp onds to an adv ertiser and an auctioneer that trust eac h other and hence ha v e a common prior; in case (1a), the auctioneer merely discloses its cur r en t estimate and trusts the adv ertisers to bid in an optimal fashion, whereas in (1b) th e adv ertisers delegate the strategic decision making to the auctioneer. In scenario 2, the advertisers h a ve a defin itiv e mo del of the CTR; in (2a), w e m odel the case where the auctioneer starts with an underestimate of the clic k- through rate and hence the q j,t are alwa ys s mall er than p j to wh ic h they will hop efully con v erge as this adv ertisement is sh o w n more times and the auctioneer’s prior gets sharp ened, and in (2b) 13 w e mo del th e m ir ror situation where the q j,t ’s are alwa y s an o ve r-estimate. It will b e int eresting to find a general theorem ab out th e bidding index that u n ifies all these div erse scenarios. In eac h of these cases, the bidd ing strategy can b e imp lemen ted usin g either p er-impression or p er-clic k bid d ing or b oth, b u t n eit her p er-imp r ession nor p er-clic k biddin g can exhaustiv ely mimic the biddin g ind ex in all scenarios. Shared Priors. W hen the adve rtisers are uncertain and s imply sh are the au ctioneer’s prior, we ha v e P j t = Q j t . Let G j t denote the Gittins index of P j t with discount factor γ . The so ciall y optimal solution alw a ys giv es the impr ession to the ad vertiser with highest v j G j t at time t . Theorem 4.2 F or shar e d priors, the bidding index str ate g y implements the so cial ly optimal solu- tion in the fol lowing two c ases: 1. The advertisers ar e str ate gic, i.e., γ b = γ , and the auctione er is myopic, i.e ., γ a = 0 . 2. The advertisers ar e myopic, i.e., γ b = 0 , and the auctione er is str ate gic, i.e., γ a = γ . Pro of: F or the fir st part, w e ha v e q j t = p j t since γ a = 0. Therefore, min  1 , p j t q j t  = 1, so that the v alue W j t is the largest c harge p er impression so that the advertiser’s discounted rev en ue is non-negativ e. This is precisely the defin ition of the Gittins index w ith discount factor γ b = γ . Therefore, the bidd ing index strategy in vo lve s biddin g ( v j G j t , v j G j t p j t ). T h is can easily b e seen to b e equiv alen t to b idding either ( v j G j t , 0) or (0 , v j G j t p j t ), and hen ce can b e mimiced with either pu re- impression or pu re-cl ic k bidding. Also, we h a ve R j t = v j G j t , so that the bidding index implemen ts the so cially optimal strategy . F or the seco nd part, since the adv ertiser is m y opic, the bidd ing index reduces to bidding ( v j p j t , v j ). Sin ce γ a = γ , w e h a ve q j t = G j t ≥ p j t . Th erefore, R j t = v j G j t , so that the biddin g index implemen ts the so cially optimal solution; this can also b e mimiced using p er-clic k bidding but not p er-impression b id ding. W ell -Informed Adv ertisers. W e next consider th e case where the adv ertisers are certain ab out their CT R p j , so th at P j t = p j . Th e so cia lly optimal solution alwa ys giv es the impression to the adv ertiser with the largest v j p j . W e sh o w th e follo wing theorem: Theorem 4.3 When P j t = p j , then the bidding index str ate gy r e duc es to b i dd ing ( v j p j , v j ) i n the fol lowing two sc enarios: 1. The auctione er’s q j t is always at most p j . In this c ase, the str ate gy is e quivalent to bidding ( v j p j , 0) and is so cial ly optimal. 2. The auctione er’s q j t is at le ast p j , and i s monotonic al ly de cr e asing with t . Pro of: When p j > q j t for all t , we ha v e min  1 , p j q j t  = 1, so that the v alue W j t is th e largest p er-impression price for whic h the adv ertiser’s discount ed r ev enue is non-negativ e. This is precisely v j p j , so th at the bid ding index strategy r educes to bidd ing ( v j p j , v j ). Th is is clearly so cially optimal. Since q j t < p j , this is equiv alent to bidding ( v j p j , 0, but can not b e simulat ed u sing p er-clic k b ids. When p j ≤ q j t for all t and q j t is monotonica lly decreasing with t , the exp ected price W j t p j q j t c harged to the adv ertiser increases with time. At any time t , th e advertiser maximizes W j t b y setting it to v j q j t and stoppin g after the first step. Therefore, W j t = v j q j t , and B j t = v j q j t p j q j t = v j p j . Therefore, the biddin g in dex strategy redu ces to biddin g ( v j p j , v j ); this is also equiv alen t to (0 , v j ), but can not b e simulated u sing p er-impression bids. 14 5 Exploration b y Ad v ertisers W e no w sh o w a simple bidding s trate gy for a certai n (i.e. w ell-informed) adv ertiser to mak e the auctioneer’s p rior con ve rge to the true CTR, while in curring n o extra cost for the advertiser; p er- clic k bidding would hav e resu lte d in the advertise r in cu rring a large cost. Mo re concretely , this mo dels the scenario where th e advertiser has s id e inf orm at ion ab out the adv ertisemen t’s CTR but the auctioneer do es not ha ve a go o d pr ior, for example, b ecause the keyw ord ma y b e obscur e. The adv ertiser has incent ive to help the auctioneer “learn” the true CTR b ecause this impro ves the adv ertiser’s c hance of winning an ad slot in a pure p er-clic k sc heme. T o m ot iv ate why this is imp ortan t, imagine a situation wh ere the advertise r w ould not get the slot if the sc heme w ere pu re p er-clic k, and he were to bid truthf u lly p er-clic k, letting the auctioneer use his o wn estimate q j of th e CTR. Therefore, in the pure p er-clic k sc heme, the adve rtiser has to o v erbid on the p er-clic k v aluation to get the slot enough num b er of times to mak e the C T R used b y the auctio neer conv erge to the true v alue; we sho w that this resu lts in loss in rev en ue for the adv ertiser. Ho w eve r, allo wing f or p er imp ression b id s pr eserv es truthfulness, and furth er m ore, h elps the auctioneer “learn” th e tru e C T R, while in curring no r ev en ue loss to the adv ertiser. This is our final argument in supp ort of hybrid auctions, and may b e the m ost con vincing from an advertiser’s p oin t of view. F ormally , w e consider an adv ertiser th at is certain ab out its C T R p j , where v j p j > R − j so th at the adv ertiser can (and wo uld lik e to) win the auction but where q j t < p j , and wh ere the goal of the advertiser is to make the auctioneer’s pr ior conv erge to the true CTR. W e sh o w that the adv ertiser can ac hieve this goal without an y loss in reven ue, whereas ac hieving th e same ob j ective using p er-clic k bidd in g would ha v e resulted in a large reven ue-loss. W e assume the auctioneer’s prior is a Beta distribution. W e sho w a candidate strategy f or an adv ertiser to mak e the Gittins index of the auctioneer’s distribution, Q j t = Beta( α j t , β j t ) conv er ge close to its true CTR p j while incurr ing no loss in rev en ue. The loss is defi ned as the v alue earned from actual clic ks min us the amount paid to the auctioneer. W e fo cus on a single advertise r and dr op its sub script. F or an y ǫ > 0, su p p ose the advertiser’s strategy is as follo ws: Du r ing an “explore” ph ase, sub mit a bid of ( v p ′ , v ) where p ′ = p (1 − ǫ ), and then sw itch to bidding (0 , v ). During the explore phase, supp ose th e adv ertiser gets T impressions on a pr ice p er impression basis resulting in n clic ks. Then the worst-ca se loss in rev en ue of the adv ertiser dur ing the explore p hase is v ( T p ′ − n ). Th e “explore” phase stops wh en the auctioneer’s p osterior mean of the distribu tio n Beta( α + n , β + T − n ) is at least p (1 − ǫ ). Not e that this also implies that the Gittins index for Beta( α + n, β + T − n ) is at least p (1 − ǫ ) irresp ectiv e of the discoun t factor γ ; this in turn implies that by switching to pur e p er-clic k bid ding, the adv ertiser is assured that q ≥ p (1 − ǫ ), so that biddin g (0 , v ) yields R j ≥ v p (1 − ǫ ). Claim 5.1 Supp ose the advertiser knows its true CTR is p , and the auctione er’s initial prior is Beta ( α, β ) . F or any ǫ > 0 , the explor e phase incu rs no loss in r eve nue for the advertiser. Pro of: Let T den ot e th e r andom stopp ing time of the explore p hase and supp ose it results in N clic ks. Firs t note that if T > 0, then p (1 − ǫ ) > α α + β . The p osterior mean on stopping is α + N α + β + T ≥ p (1 − ǫ ), which implies N/T > p (1 − ǫ ). Therefore, T p (1 − ǫ ) − N < 0, wh ic h sh o w s there is no loss in reven ue (pro vided T is finite w ith probab ility 1, which follo ws from the la w of large n umb ers in this case). 15 Supp ose R − j = v j p j (1 − ǫ ). In a pure p er-c lic k b id ding sc heme, the adv ertiser w ould ha v e to bid at least v j (1 − ǫ ) p j /q j t at time t < T w ith an exp ected loss (i.e. pr ofit − cost) of p j v j ((1 − ǫ ) p j /q j t − 1). F or a my opic auctioneer with initial p rior (1 , β ), the total loss of r ev enue for the adv ertiser till time T is Ω( v j p j β ) wh ich can b e arbitrarily large. 6 Multi-Slot Auction In this section, we generalize the h ybrid auction to multiple slots u n der the standard separable CTR assumption, s uc h that the resulting generaliz ation is truthfu l in a m y opic setting analog ous to Section 3. Assume th ere are K slots, wh ere slot i is asso cia ted with a CTR multiplier θ i ∈ [0 , 1]. Slot 1 is the topm ost slot; since the CTRs decrease with slot num b er, we h a ve 1 = θ 1 ≥ θ 2 ≥ · · · ≥ θ K ≥ 0. W e will also defin e θ K +1 = 0. As b efore, adv ertiser j and the auctioneer main tain p r iors on the CTR v alue for this adve rtiser in ad s lot 1. A s b efore, w e den ot e these priors as P j and Q j resp ectiv ely . Let p j = E [ P j ] b e the exp ected CT R estimated by the adv ertiser, an d let q j = f ( Q j ) denote the Gitti ns index (or for that matter, an y other function) of the auctioneer’s p rior. Let v j denote the true p er-clic k v aluation of adv ertiser j . Note th at the pr iors P j and Q j corresp ond to the estimated CTR for adv ertiser j in the first ad slot , s o that the exp ected CTR for the i th slot based on the adv ertiser’s estimate is θ i p j . Adv ertiser j bids ( m j , c j ), whic h is interpreted as the p er-impression and p er-clic k bid s for obtaining the first slot . The auctio n is mo deled after the laddered auction in [1], whic h is equiv alen t to V CG u nder the separabilit y assumption [4]. First, compute the effectiv e b id R j = max { m j , c j q j } for ev er y adv ertiser as d escribed in section 2. Assume without loss of generalit y that there are K + 1 adv ertisers, and that R 1 ≥ R 2 ≥ . . . ≥ R K +1 . T hen, the auction pro ceeds as follo ws: 1. Advertiser j is p laced in slot j , for 1 ≤ j ≤ K . 2. An “effectiv e c harge”, e j is computed for adv ertiser j as e j = P K i = j  θ i − θ i +1 θ j  R i +1 . 3. If m j > c j q j then the adv ertiser is c harged e j p er impression; else it is c harged e j /q j p er clic k. It is easy to s ee that e j θ j = R j ( θ j − θ j +1 ) + e j +1 θ j +1 . Inform ally , ad vertiser j ’s effectiv e charge is the same as the effectiv e bid of the ( j + 1)-th advertiser f or the additional clic k-rate at th e j -th p osition, and the same as the effectiv e c harge of the ( j + 1)- th adv ertiser for the clic k-r at e that w ould hav e already b een r ea lized at the ( j + 1)-th p osition. Theorem 6.1 If p j = E [ P j ] and the advertiser is myopic and risk-neutr al, then r e gar d less of the choic e of q j , the (str ongly) dominant str ate gy is to bid ( v j p j , v j ) . The p roof of the ab o v e theorem is obtained b y extending the pro of of theorem 3.1 exactly along the line of th e pro of of truthf ulness of the ladd ered auction in [1], and is omitted from th is ve rsion. This pro of can also b e obtained using the analysis of VCG with probabilistic allo ca tions, due to My erson [9]. 7 Conclusion Adv ertising is a ma jor source of reven ue for searc h engi nes and other w eb-sites, and a ma jor driv er of inn ov ation in w eb tec hnology an d services. Adv ertising sp ots are t ypically sold on the 16 w eb using auctions, and th ese auctions hav e typicall y b een either Cost-P er-Clic k (CPC), Cost- P er-Impression (CPM), or Cost-P er-Action (CP A). W e defined a sin gl e-slot hybrid auction, whic h allo ws adv ertisers to ent er p er-imp r ession as w ell as p er-clic k bids. W e sho w ed that this auction is truthful for risk-neutral, my opic ad vertisers, the setting under which such auctions ha v e t ypically b een analyzed. When adv ertisers are risk-seeking, or n on-m y opic, or wh en the adv ertiser has m uch b etter information ab out the Click-Through-Rate (CTR) than the auctioneer, w e sh o w that the h ybr id auction offers stronger rev en ue guaran tees and adv ertiser flexibilit y than either pure CPC or CPM. The hybrid auction generalizes n aturally to multi-slo t scenarios and is equally app licable to (CPM,CP A) or (CPC,CP A) bidding. Finally , the h ybrid auction is fully b ac kwards compatible with a CPC auction, in the sense that advertise rs ent ering (optional) p er-impression bids in addition to p er-clic k bids can seamlessly co-exist with advertisers ent ering only p er-clic k bid s in the same auction. References [1] G. Aggarwa l, A. Go el, and R. Mot w ani. T ruthful auctions for p ricing searc h keyw ords. Pr o- c e e dings of the seventh ACM c onfer enc e on Ele ctr onic Commer c e , pages 1–7, Jun e 2006. [2] S . A they and I. Segal. An efficie nt dynamic mec hanism. 2007 . Av ailable at http://k uznets.fas.har vard. edu / ~athey . [3] A. Bapna and T . W eb er. Efficien t dy n amic allocation with uncertain v aluatio n. Working Pap er, De p artment of Management Scienc e and E ngine ering, Stanfor d University , 2005. [4] B. Edelman, M. Ostrovsky , and M. Sc hw arz. In ternet adv ertising and th e generalized second price auction: Selling b illio ns of dollars w orth of ke ywords. Americ an Ec onomic R eview , 97(1): 242–259, 2007. [5] J . C. Gittins. Bandit pro cesses and dynamic allocation indices. J R oyal Statistic al So ciete Series B , 14:14 8–167, 1979. [6] S . Lahaie, D. P enno c k, A. S aberi, and R. V ohra. Sp onsored s ea rch auctions. In Algor ithmic Game The ory, e dite d by N isan, R oughgar den, T ar dos, and V azir ani , 2007. [7] A.W. Marshall and I. Olkin. Ine qualities: the ory of majorization and its applic ations . Academic Press (V olume 143 of Mathematics in Science and Engineering), 1979. [8] A. Meh ta, A. S aberi, U. V azirani, , and V. V azirani. Adw ords and generalized online matc hing. Pr o c e e dings of the 46th IEEE Symp osium on F oundations of Computer Scienc e , 2005. [9] R. B. My erson. Optimal auction design. Mathematics of Op er ations R ese ar ch , 6(1):58 –73, 1981. [10] M. Ric hardson, E. Domino wsk a, and R. Ragno. Predicting clic ks: estimating the clic k-through rate for new ads. Pr o c e e dings of the 16th international c onfer enc e on World W ide Web , p ag es 521–5 30, 2007. [11] H. V arian. P osition auctions. International Journal of Industrial Or ganization , Octob er 2006. 17 [12] R. W eb er. On the gi ttins index for multiarmed band its. A nnals of app lie d pr ob ability , 2(4):1 024–1033 , 1992. 18