Sponsored Search Equilibria for Conservative Bidders

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1 Sponsored Search Equlbra for Conservatve Bdders Renato Paes Leme Department of Computer Scence Cornell Unversty Ithaca, NY Éva Tardos Department of Computer Scence Cornell Unversty Ithaca, NY ABSTRACT Generalzed Second Prce Aucton and ts varants has been the man mechansm used by search companes to aucton postons for sponsored search lns. In ths paper we study the socal welfare of the Nash equlbra of ths game. It s nown that socally optmal Nash equlbra exsts, and ts not hard to see that n the general case there are also very bad equlbra: the gap between a Nash equlbrum and the socally optmal can be arbtrarly large. In ths paper, we consder the case when the bdders are conservatve, n the sense that they do not bd above ther own valuatons. We show that a certan analog of the tremblng hand equlbra are equlbra wth conservatve bdders. Our man result s to show that for conservatve bdders the worse Nash equlbrum and the socal optmum are wthn a factor of the golden rato,.68. Keywords Game Theory, Keyword Auctons. INTRODUCTION Search engnes and other onlne nformaton sources use sponsored search aucton to monetze ther servces. These actons allocate advertsement slots to companes, and companes are charged pay per clc, that s, they are charged a fee for any user that clcs on the ln assocated wth the advertsement. The fee for such a clc s decded by varant of the so-called Generalzed Second Prce Aucton (GSP), a smple generalzaton of the well-nown Vcrey aucton [0] for a sngle tem (or a sngle advertsng slot). The Vcrey aucton [0] for a sngle tem, and ts generalzaton, the Vcrey-Clare-Groves Mechansm (VCG) [, 5], mae truthful behavor (when the advertsers reveal ther true valuaton) domnant strategy, and mae the resultng Supported n part by NSF grants CCF and CCF , ONR grant N , and a Yahoo! Research Allance Grant. outcome maxmze the socal welfare. See also [] about truthful sponsored search auctons. Generalzed Second Prce Aucton, the mechansm adopted by all search companes, s a natural generalzaton of the Vcrey aucton for a sngle slot, but t s nether truthful nor maxmzes socal welfare. In ths paper we wll consder the socal welfare of the GSP aucton outcomes. Our goal n ths paper s to show that the ntuton based on the smlarty of GSP to the truthful Vcrey aucton s not so far from truth: we prove that the socal welfare s wthn a factor of.68 of the optmal n any Nash equlbrum for conservatve bdders. We consder the full nformaton game, assumng all advertsers now the valuatons of all players. In addton, we wll assume that the players are conservatve, and do not rs bddng above ther valuaton. A bd value b above the valuaton v for a player, opens the player up to the rs of an outcome wth negatve utlty (f another bdder b appears n the range v < b < b ). To formally justfy our conservatve bdder model, we assume that an addtonal random bd wll show up wth a small ǫ probablty, and study the Nash equlbra of the game for the orgnal bdders as ǫ tends to zero. Ths s analogous to the tradtonal noton of tremblng hand equlbrum [8]. We ll show that n the Nash equlbra of the game that survve ths perturbaton all bdders are conservatve. Our results. Our focus n ths wor s to analyze the socal welfare n the Generalzed Second Prce Aucton mechansm. We start by consderng the smple model when clc-though rates depend only on the slots,.e., the probablty of clc for all bdders f assgned to slot s. At the end of the paper, we extend our results to the model wth separable clc-through rate, where f advertser j s assgned to slot the probablty of ths resultng n a clc s γ j. It s nown that there are Nash equlbra that are socally optmal. We show smple examples of Nash equlbra where the socal welfare s arbtrarly smaller than the optmum. However, these equlbra are unnatural, as some bd exceeds the players valuatons, and hence the player taes unnecessary rs by playng above ther own valuaton f a new bdder shows up between ther bd and valuaton. We defne conservatve bdders as bdders who won t bd above ther valuatons. Our man contrbuton s to prove that f all bdder are con-

2 servatve, then the socal welfare n a Nash equlbrum can t be very far from the optmal. To analyze the Nash equlbrum when all advertsers are conservatve, we exhbt a smple property of those equlbra: consder two slots and j, and let v denote the valuaton of advertser for a clc. We show that f n a Nash equlbrum wth conservatve bdders, π() and π(j) are assgned to these slots respectvely, than we must have that + v π() v π(j). We say that as assgnment of bdders to slots s wealy feasble f t satsfes the above nequalty for all and j, and we show that the socal welfare of a wealy feasble assgnment s at least a.68 fracton of the socally optmal assgnment. Although only a necessary condton, wea feasblty s a smple and ntutve property. It s not hard to see that wealy feasble assgnments cannot be too far from the optmal: f two advertsers are assgned to postons not n ther order of bds, then ether () the two advertsers have smlar values for a clc; or () the clc-through rates of the two slots are not very dfferent, and hence n ether case ther relatve order doesn t affect the socal welfare very much. Related wor. Sponsored search has been a very actve area of research n the last several years. For the basc model of Nash equlbra n such auctons see the papers by Edelman et al [3] and Varan [9], for a truthful aucton see Aggarwal et all [], and see the survey of Lahae et al [7] for a general ntroducton. Snce the orgnal models, there has been much wor n the area, explorng more complex models of clc-through rates, tang nto account budgets, analyzng dynamcs, consderng more complex models of ncentves (such as vndctve bddng), etc. A lot of ths wor have been reported n the frst four Worshops on Ad Auctons 005 through 008. Closest to our wor s the paper Lahae [6], that provdes prce of anarchy bounds on effcency of equlbra, provded that the clc-through-rate decays exponentally along the slots wth a factor of δ. Here we consder the smpler models of ether clc-through rates that s a property of slot ndependent of the advertser, or separable clc though rates, where the clc through rate for bdder j n slot can be expressed n a smple product form γ j. For these models Edelman et al [3] and Varan [9] show that there exsts Nash equlbra that are socally optmal. More precsely, they consder a restrcted class of Nash equlbra called Envy-free equlbra or Symmetrc Nash Equlbra, and show that such equlbra exsts, and all such equlbra are socally optmal. In ths class of equlbra, an advertser wouldn t be better off after swtchng hs bds wth the advertser just above hm. Note that ths s a stronger requrement than Nash, as an advertser cannot unlaterally swtch to a poston wth hgher clc-through by smply ncreasng ther bd. Edelman et al [3] clam that f the bds eventually converge, they wll converge to an envy-free equlbrum, otherwse some advertser could ncrease hs bd mang the slot just above more expensve and therefore mang the advertser occupyng t underbd hm. They do not provde a formal game model that selects such equlbra. Vorobeych and Reeves [] use smulaton to study stable equlbra. Lahae [6] also consders the problem of quantfyng the socal effcency of an equlbrum. He proves a prce of anarchy of mn{, } provded that the clc-through-rate decays δ δ exponentally along the slots wth a factor of. Feng et al [4] δ gves expermental evdence that clc-through-rates decay exponentally. To prove the clamed bound, Lahae develops a tool whch s smlar to ours. He proves π s a feasble allocaton f and only f v π() + v π(j) + for n and j +. In ths paper, we consder a dfferent restrcton of Nash, we assume that bdders are conservatve, n the sense that no bdder s bddng above ther own valuaton. We can justfy ths assumpton by assumng that a new random bds can show up wth a vanshngly small probablty ǫ 0. In equlbra that survve ths perturbaton, the bdders are conservatve. Wthout any addtonal requrement Nash equlbra can have socal welfare that s arbtrarly bad compared to the optmal socal welfare. However, we show that Nash equlbra of conservatve bdders s wthn a factor to the optmum. We assume only that the clc-through-rates are separable (the product form) and are monotone.. PRELIMINARIES We consder an aucton wth n advertsers and n slots (f there are less slots than advertsers, consder addtonal vrtual slots wth clc-through-rate zero). Let v be the value that advertser has for one clc and be the clc-throughrate of slot j. We wll extend the results to separable clcthrough-rate at the end of the paper. Assume that advertsers and slots ordered so that v v... v n and α α... α n. Gven those parameters of the model, the mechansm of the Generalzed Second Prce Aucton (GSP) s:. each advertser submts a bd b 0. the advertser are sorted by ther bds (tes are broen arbtrarly) 3. the hghest slot s assgned to the advertser wth hghest bd, the second hghest slot to the one wth second hghest bd and so on. 4. the advertser occupyng slot pays the bd of the advertser occupyng slot +. The advertser occupyng the last slot pays zero. Let S n be the set of permutatons of n elements. We characterze the order of the advertsers n the slots usng a permutaton π so that π() s the advertser occupyng slot, whch s the same of the advertser wth the th hghest bd. We defne the utlty of a user when occupyng slot j as gven by u = (v b π(j+) ). Gven a set of bds b,..., b n we say that they consttute a Nash equlbrum f no advertser can ncrease ts own utlty by changng hs own bd. Suppose advertser s currently bddng b and occupyng slot j. Changng hs bd to somethng between b π(j ) and b π(j+) won t change the permutaton π and therefore won t

3 change the allocaton nor hs payment. So, he could try to ncrease hs valuaton by dong one of two thngs: ncreasng hs bd to get a slot wth a better clcthrough-rate. If he wants to get a slot < j he needs to overbd advertser π(), say by bddng b π() + ǫ. Ths way he would get slot for the prce b π() per clc, gettng utlty α (v b π() ). decreasng hs bd to get a worse but cheaper slot. If he wants to get slot > j he needs to bd below advertser π(). Ths way he would get slot for the prce b π(+) per clc, gettng utlty α (v b π(+) ). Therefore we say that b s a Nash equlbrum f the followng equatons hold: b π() b π()... b π(n) (v π() b π(+) ) (v π() b π(j) ) (v π() b π(+) ) (v π() b π(j+) ) j < j > where π s the permutaton defned by b. We say that π s a feasble permutaton for α, v f there s a b that generates π and s a Nash equlbrum. We measure the total qualty of an equlbrum by the socal welfare, whch s defned as P j αjv π(j). The optmal socal welfare s naturally acheved when π s the dentty permutaton and [3] proves that there s always a Nash equlbrum that acheves that (n partcular, allocaton and payments n ths equlbrum are equal to VCG). However not every Nash equlbrum s optmal, as we wll see shortly. We are nterested n quantfyng the prce of anarchy for ths game, whch s gven by the maxmum over all permutatons that defne Nash equlbra of P j αjvj/ P j αjv π(j).. Equlbra wth Low Socal Welfare Even for two slots the gap between the best and the worse Nash equlbrum can be arbtrarly large. For example, consder two slots wth clc-through-rates α = and α = r and two advertsers wth valuatons v = and v = 0. It s easy to chec that the bds b = 0 and b = r are a Nash equlbrum where advertser gets the second slot and advertser gets the frst slot. The socal welfare n ths equlbrum s r whle the optmal s. The prce of anarchy s therefore /r. Snce r can be any number from 0 to, the gap between the optmal and the worse Nash can be arbtrarly large. Notce however that ths Nash equlbrum seems very artfcal: advertser s exposed to the rs of negatve utlty: f advertser (or another advertser) adds a bd somewhere between 0 and r ths mposes a negatve utlty on advertser. Bddng r whle havng valuaton 0 s acceptng a lot of rs. We clam that f bdders are not wllng to accept such rs (or accepts only a lmted amount of such rs) then the prce of anarchy s bounded. 3. CONSERVATIVE BIDDER EQUILIBRIA We say an advertser s γ-conservatve f b v. So, γ generc advertsers are 0-conservatve. We call conservatve bdders the -conservatve advertsers. () Note that f bdder has to pay a prce above v she has negatve utlty, and hence ths cannot happen n a Nash equlbrum. A non-conservatve bd b > v can only be part of a Nash equlbrum f the resultng prce p (the next smallest bd) s small enough v p. In ths case all bds b n the range (p, b ] of user result n the same outcome, and same payments, hence same utlty. Now consder how the outcome and utlty s effected f a new bd b s added to the system. If v < b < b then user remans to be assgned to the same slot, but wll pay a rate b resultng n negatve utlty. In contrast, by bddng b = v bdder does not effect ts utlty n the orgnal game, and avods the danger of negatve utlty when the bd b s added. Gven the parameters α, v, we say that b s a conservatve bdder equlbrum f t s a Nash equlbrum and b v for all bdders. Theorem. A Nash equlbrum that remans an equlbrum n the game when a random bd s added wth a small probablty ǫ > 0 s a conservatve bdder equlbrum, and conservatve bdder equlbra exsts. Proof. We argued above that Nash equlbra that survve a small enough perturbaton are conservatve bdder equlbra. To see that conservatve bdder equlbra exst we use the equlbra of Edelman et al [3], where b = v and b = Pj (αj αj+)vj+ for > s clearly conservatve. For the remander of the paper we consder conservatve equlbra. Theorem. For slots, f all advertsers are γ-conservatve, then the prce of anarchy s bounded by +γr( r), where γ+r( γ) r = α α In partcular, tang γ = 0 we recover the /r bound for the general case and for γ = we have a quadratc functon wth maxmum equal to.5. It s not hard to see that ths bound s lmted for any γ > 0. Proof. We can suppose wthout loss of generalty that α =, α = r and α v +α v =, snce what we are tryng to prove s nvarant under rescalng α or v. In any nonoptmal Nash equlbrum b b and by the Nash condton r(v 0) (v b ) and by the conservatve condton b γ v. Substtutng v = rv n those two expressons and combnng them to elmnate the b term we get: r v r(r ) () γ Therefore the socal welfare n any non-optmal Nash s α v + α v = v + r( rv ) +γr( r) γ+r( γ).

4 3. Wealy Feasble Assgnments Next we show that equlbra wth conservatve bdders satsfes the smple property mentoned n the ntroducton. We wll call the assgnments satsfyng ths property wealy feasble. In the next secton we analyze the welfare propertes of wealy feasble equlbra. We start by showng that an assgnment when no bdder can ncrease ts utlty unless he bds above hs valuaton s n fact a Nash equlbrum n the usual sense (equatons ) n whch b v. For ths equlbrum we stll have the relatons for j > as n equaton but for j <, now we have: (v π() b π(j) ) > (v π() b π(+) ) b π(j) > v π() that s equvalent to: v π() b π(j) α (v π() b π(+) ) or v π() b π(j) < 0 and we can rewrte t as: v π() b π(j) max j ff α (v π() b π(+) ),0 = α (v π() b π(+) ) snce v π() b π() b π(+). So t s a Nash equlbrum n the standard sense wth the addtonal constrants that b v. The equatons are not very easy to wor wth, snce they are not very symmetrc and they depend on b. We propose a cleaner form of representng an equlbrum that just uses α, v and the permutaton π. Although t s a weaer property t stll captures most of the trade-offs:. f values v are very close then the order of the bdders doesn t nfluence the socal welfare that much. f values v are very well separated, then permutatons that would produce a bad socal welfare are not feasble because they volate Nash constrants Theorem 3. Gven v, α and a Nash permutaton π, f < j and π() > π(j) then: n partcular, or v π() v π(j). + v π() v π(j) (3) Proof. Snce t s a Nash equlbrum bdder n slot j s happy wth hs condton and don t want to ncrease hs bd to tae slot, so: (v π(j) b π(j+) ) (v π(j) b π() ) snce b π(j+) 0 and b π() v π() then: v π(j) (v π(j) v π() ) Inspred by the last theorem, gven parameters α, v we say that permutaton π s wealy feasble f equaton 3 holds for each < j, π() > π(j). From Theorem we now that: Corollary 4. Gven α, v, any permutaton correspondng to a Nash equlbrum wth conservatve bds s wealy feasble. Our man results follow from analyzng the prce of anarchy rato P j αjvj/ P j αjv π(j) over all wealy feasble permutatons π. Before proceedng to the man result. we re-prove the bound n [6] for the conservatve case. α Theorem 5. If + δ > for all, then f π s a wealy feasble permutaton, then the prce of anarchy s bounded by,.e.: δ v π() ( δ ) v Proof. If π() > then there s some j > such that π(j) (by the pgeonhole prncple, snce there are only slots wth ndex <, so at least one of the frst bdders must occupy one slot after ). So, as π(j) < π() and j > we can apply our relaton: v π() αj «v π(j) αj «v ( δ )v where the frst nequalty s that of Theorem 3. The theorem follows almost drectly: v π() = v π() + v π() π() π() v + ( δ ) π()> π()> v v ( δ ) 3. The Man Results Here we present the bound on the prce of anarchy for wealy feasble permutatons, and hence for GSP for conservatve bdders. Our man result s that t s bounded by.68. As a warm-up we wll prove that t s bounded by, snce the proof s easer and captures the man deas. We wll prove ths bound for wealy feasble permutatons and t wll automatcally be deduced to a bound for feasble permutatons. Notce that the wealy feasble permutaton ncely capture the fact that f advertsers and j are n the wrong relatve poston (.e. dfferent to the one n the optmal) then ether ther values are close (wthn a factor of ) or ther clc-through-rates are close (wthn a factor of ). Theorem 6. For conservatve bdders, the prce of anarchy for GSP s bounded by. Proof. We wll prove t by nducton on n that all wealy feasble permutatons result n socal welfare at most of factor of less than the maxmum possble. For advertsers and slots we now that the worst possble socal welfare for a wealy feasble permutaton s at most a.5 fracton of

5 γ bdders slots j γ Fgure : Allocaton of slots n the proof of Theorems 5 and 6 the optmum. So, now we need to prove the nductve step. Consder parameters v, α and a wealy feasble permutaton π. Let = π () be the slot occuped by the advertser of hgher value and j = π() be the advertser occupyng the frst slot (as shown n Fgure ). If = j = then we can apply the nductve hypothess rght away. If not, equaton α 3 tells us that: α or v j v. Suppose α and consder an nput wth slot and advertser deleted. Ths nput has n advertsers and n slots and the permutaton π restrcted to those s stll wealy feasble, so by the nductve hypothess: α v π() (αv α v + α+v αnvn) (αv αv + α+v αnvn) therefore: α v π() = v + α v π() αv + > If v j v we just do the same but deletng slot and advertser j from the nput. Now, we prove the tghter result. Theorem 7. For conservatve bdders, the prce of anarchy s bounded by Proof. As before, we prove the concluson for all wealy feasble permutatons. We use here a dynamcal systems argument: we defne a sequence of values r so that we can prove that for slots socal welfare s at least an r fracton of the optmum, and prove that r converges to the desred bound. Let r =.5 and suppose we have r, r 3,..., and that ths property holds for them. Let s calculate some small value of r n so that the property stll holds. Agan, consder parameter α, v, a wealy feasble permutaton π and let s assume = π () and j = π() (as shown n Fgure ). If = j =, ths s an easy case and t s straghtforward to see that n ths case the prce of anarchy can be bounded by. If not, assume wthout loss of generalty that j (snce equaton 3 s symmetrc n α and v we can just nterchange the roles of them n the proof f > j). Let = α and γ = v v j. We now that +. γ Followng the lnes of the proof of the last theorem we have: α v π() = v + α v π() αv + = αv + α v + = " n =+! # α )v + =(α > αv + (α )v + > Now, we can use j to say: v v j = γ v α v π() " + «# α v + v. > So, we would le to fnd some r n such that we can say that P α v π() r n P α v for all, so we would le to have: ( mn r n, + «) for any. But notce some other bound we can get s: α v π() γ αv + > «α v + > by followng the lnes of the proof of last theorem, but removng slot and advertser j n the nductve step. So another alternatve s to get: j mn, ff r n for every. So f we can get /r n bounded by the maxmum of those two quanttes, we are done. Summarzng that, we need: r n max ( for all., " max (, + «)# ) Now we need to evaluate for whch value of (0,] the j ff expresson max, + has ts mn- mum. The mnmum can be n two ponts: the mnmum of the quadratc functon or the ntersecton between those two functons. They ntersect at = r + + r r (where r stands for ) and the quadratc mnmum s at r. So, for r 4, the mnmum occurs n the ntersecton and 3

6 3.3 Extenson to separable clc-through-rates So far, we have consdered that the clc-through-rates of advertser placed on slot j depends only on the slot n whch he s placed. A more general model called separable clc-through-rates assumes t depends on a product of two factors: one dependng on the bdder and other dependng on the slot. Let s say that f advertser s placed on slot j, t wll get clc-through-rate γ where γ s some qualty factor attrbuted to each advertser. The generalzaton of Second Prce Aucton for ths settng rans the advertsers n order of γ b and charges an advertser the mnmum value t needed to bd to conserve hs poston. For example, f π s the permutaton defned by sortng γ b (.e, π() s the advertser wth the th hghest value of γ b ) then we charge advertser π(j) the amount of: b π(j+) γ π(j+) /γ π(j). Fgure : Sequence of values r that are an upper bound of the prce of anarchy for slots for r < 4, t occurs n the quadratc mnmum. So: 3 8 >< rn, < r n = q «>: rn rn, 4 3 snce we want the smallest possble rato. Ths allows to defne r recursvely from r =.5 and t s easy to see that the sequence monotoncally converges to the fxed pont of that functon whch s the golden raton ϕ = , as shown n Fgure. Ths happens because the functon that maps to r n s non-decreasng and has a fxed pont n ϕ, so f ϕ then r n ϕ. To llustrate how symmetrc and easy to wor wth ths new formulaton s, we also add the followng result: Theorem 8. The worse possble prce of anarchy among all possble parameters n, α, v and all possble wealy feasble permutatons π occurs when π s a smple cycle,.e, exst {x,..., x n} = {,..., n} such that π(x ) = x + for < n and π(x n) = x. Proof. If π s wealy feasble but s not a smple cycle, then we can decompose ths permutaton as a product of two dsjont permutatons π = π π wth supports N and N,.e, π moves the bdders and slots wth ndces n N. So, we have: P α P v π() N P α = α v π() + P N α v π() v P N + P N max ( P N α v π() P N, P α ) N v P π() N and π s wealy feasble over N (.e., the restrcted nput of slots wth ndces n N and advertsers wth ndces n N ). In ths settng the utlty of bdder assgned to slot j s u = γ v b π(j+) γ π(j+) γ and the socal welfare s gven by P α γ π() v π(). Consder that α... α n and that γ v... γ nv n. The defnton of Nash equlbrum s analogous. Notce we can obtan a result very smlar wth Theorem 3 just by repeatng the same calculatons for ths model: Theorem 9. Gven v, α, γ and a feasble permutaton π (a permutaton from a Nash equlbrum) n the separable clc-through-rate model, f < j and π() > π(j) then: + γ π()v π() γ π(j) v π(j) (4) Proof. Snce advertser π(j) can t ncrease hs utlty by tang slot, we have that: γ π(j) v π(j) b «π(j+)γ π(j+) γ π(j) v π(j) b «π()γ π() γ π(j) γ π(j) usng that b π(j+) 0 and b π() v π() we get the desred result. And all other results follow wth almost no change. Acnowledgements We than the anonymous revewer for pontng us to the paper by Lahae [6]. 4. REFERENCES [] G. Aggarwal, A. Goel, and R. Motwan. Truthful auctons for prcng search eywords. In EC 06: Proceedngs of the 7th ACM conference on Electronc commerce, pages 7, New Yor, NY, USA, 006. ACM. [] E. H. Clare. Multpart prcng of publc goods. Publc Choce, (), September 97. [3] Edelman, Benjamn, Ostrovsy, Mchael, Schwarz, and Mchael. Internet advertsng and the generalzed second-prce aucton: Sellng bllons of dollars worth of eywords. The Amercan Economc Revew, 97():4 59, March 007. [4] J. Feng, H. K. Bhargava, and D. M. Pennoc. Implementng sponsored search n web search engnes:

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