Revenue Guarantees in Sponsored Search Auctions

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1 Revenue Guarantees n Sponsored Search Auctons Ioanns Caraganns, Chrstos Kaklamans, Panagots Kanellopoulos, and Mara Kyropoulou Computer Technology Insttute and Press Dophantus & Department of Computer Engneerng and Informatcs, Unversty of Patras, Ro, Greece Abstract. Sponsored search auctons are the man source of revenue for search engnes. In such an aucton, a set of utlty-maxmzng advertsers compete for a set of ad slots. The assgnment of advertsers to slots depends on bds they submt; these bds may be dfferent than the true valuatons of the advertsers for the slots. Varants of the celebrated VCG aucton mechansm guarantee that advertsers act truthfully and, under mld assumptons, lead to revenue or socal welfare maxmzaton. Stll, the sponsored search ndustry mostly uses generalzed second prce (GSP) auctons; these auctons are known to be non-truthful and suboptmal n terms of socal welfare and revenue. In an attempt to explan ths tradton, we study a Bayesan settng where the valuatons of advertsers are drawn ndependently from a regular probablty dstrbuton. In ths settng, t s well known by the work of Myerson (1981) that the optmal revenue s obtaned by the VCG mechansm wth a partcular reserve prce that depends on the probablty dstrbuton. We show that by approprately settng the reserve prce, the revenue over any Bayes- Nash equlbrum of the game nduced by the GSP aucton s at most a small constant fracton of the optmal revenue, mprovng recent results of Lucer, Paes Leme, and Tardos (2012). Our analyss s based on the Bayes-Nash equlbrum condtons and on the propertes of regular probablty dstrbutons. 1 Introducton The sale of advertsng space s the man source of ncome for nformaton provders on the Internet. For example, a query to a search engne creates advertsng space that s sold to potental advertsers through auctons that are known as sponsored search auctons (or ad auctons). In ther nfluental papers, Edelman et al. [6] and Varan [18] have proposed a (now standard) model for ths process. Accordng to ths model, a set of utlty-maxmzng advertsers compete for a set of ad slots wth non-ncreasng clck-through rates. The auctoneer collects bds from the advertsers and assgns them to slots (usually, n Ths work s co-fnanced by the European Socal Fund and Greek natonal funds through the research fundng program Thales on Algorthmc Game Theory.

2 non-ncreasng order of ther bds). In addton, t assgns a payment per clck to each advertser. Dependng on the way the payments are computed, dfferent auctons can be defned. Typcal examples are the Vckrey-Clark-Groves (VCG), the generalzed second prce (GSP), and the generalzed frst prce (GFP) aucton. Naturally, the advertsers are engaged as players n a strategc game defned by the aucton; the bd submtted by each player s such that t maxmzes her utlty (.e., the total dfference of her valuaton mnus her payment over all clcks) gven the bds of the other players. Ths behavor leads to equlbra,.e., states of the nduced game from whch no player has an ncentve to unlaterally devate. Tradtonally, truthfulness has been recognzed as an mportant desderatum n the Economcs lterature on auctons [11]. In truthful auctons, truth-tellng s an equlbrum accordng to specfc equlbrum notons (e.g., domnant strategy, Nash, or Bayes-Nash equlbrum). Such a mechansm guarantees that the socal welfare (.e., the total value of the players) s maxmzed. VCG s a typcal example of a truthful aucton [5, 8, 19]. In contrast, GSP auctons are not truthful [6, 18]; stll, they are the man aucton mechansms used n the sponsored search ndustry adopted by leaders such as Google and Yahoo! In an attempt to explan ths prevalence, several papers have provded bounds on the socal welfare of GSP auctons [2, 12, 13, 17] over dfferent classes of equlbra (pure Nash, coarse-correlated, Bayes-Nash). The man message from these studes s that the socal welfare s always a constant fracton of the optmal one. However, one would expect that revenue (as opposed to socal welfare) maxmzaton s the major concern from the pont of vew of the sponsored search ndustry. In ths paper, followng the recent paper by Lucer et al. [14], we am to provde a theoretcal justfcaton for the wde adopton of GSP by focusng on the revenue generated by these auctons. In order to model the nherent uncertanty n advertsers belefs, we consder a Bayesan settng where the advertsers have random valuatons drawn ndependently from a common probablty dstrbuton. Ths s the classcal settng that has been studed extensvely snce the semnal work of Myerson [15] for sngletem auctons (whch s a specal case of ad auctons). The results of [15] carry over to our model as follows. Under mld assumptons, the revenue generated by a player n a Bayes-Nash equlbrum depends only on the dstrbuton of the clck-through rate of the ad slot the player s assgned to for her dfferent valuatons. Hence, two Bayes-Nash equlbra that correspond to the same allocaton yeld the same revenue even f they are nduced by dfferent aucton mechansms; ths statement s known as revenue equvalence. The allocaton that optmzes the expected revenue s one n whch low-bddng advertsers are excluded and the remanng ones are assgned to ad slots n non-ncreasng order of ther valuatons. Such an allocaton s a Bayes-Nash equlbrum of the varaton of the VCG mechansm where an approprate reserve prce (the Myerson reserve) s set n order to exclude the low-bddng advertsers. GSP auctons may lead to dfferent Bayes-Nash equlbra [7] n whch a player wth a hgher valuaton s assgned wth postve probablty to a slot

3 wth lower clck-through rate than another player wth lower valuaton. Ths mples that the revenue s suboptmal. Our purpose s to quantfy the loss of revenue over all Bayes-Nash equlbra of GSP auctons by provng worst-case revenue guarantees. A revenue guarantee of ρ for an aucton mechansm mples that, at any Bayes-Nash equlbrum, the revenue generated s at most ρ tmes smaller than the optmal one. Note that, t s not even clear whether Myerson reserve s the choce that mnmzes the revenue guarantee n GSP auctons. Ths ssue s the subject of exstng expermental work (see [16]). Recently, Lucer et al. [14] proved theoretcal revenue guarantees for GSP auctons. Among other results for full nformaton settngs, they consder two dfferent Bayesan models. When the advertsers valuatons are drawn ndependently from a common probablty dstrbuton wth monotone hazard rate (MHR), GSP auctons wth Myerson reserve have a revenue guarantee of 6. Ths bound s obtaned by comparng the utlty of players at the Bayes-Nash equlbrum wth the utlty they would have by devatng to a sngle alternatve bd (and by explotng the specal propertes of MHR dstrbutons). The class of MHR dstrbutons s wde enough and ncludes many common dstrbutons (such as unform, normal, and exponental). In the more general case where the valuatons are regular, the same bound s obtaned usng a dfferent reserve prce. Ths reserve s computed usng a prophet nequalty [10]. Prophet nequaltes have been proved useful n several Bayesan aucton settngs n the past [4, 9]. In ths work, we consder the same Bayesan settngs wth [14] and mprove ther results. We show that when the players have..d. valuatons drawn from a regular dstrbuton, there s a reserve prce so that the revenue guarantee s at most For MHR valuatons, we present a bound of In both cases, the reserve prce s ether Myerson s or another one that maxmzes the revenue obtaned by the player allocated to the frst slot. The latter s computed by developng new prophet-lke nequaltes that explot the partcular characterstcs of the valuatons. Furthermore, we show that the revenue guarantee of GSP auctons wth Myerson reserve s at most 3.90 for MHR valuatons. In order to analyze GSP auctons wth Myerson reserve, we extend the technques recently developed n [2, 13] (see also [3]). The Bayes-Nash equlbrum condton mples that the utlty of each player does not mprove when she devates to any other bd. Ths yelds a seres of nequaltes whch we take nto account wth dfferent weghts. These weghts are gven by famles of functons that are defned n such a way that a relaton between the revenue at a Bayes-Nash equlbrum and the optmal revenue s revealed; we refer to them as devaton weght functon famles. The rest of the paper s structured as follows. We begn wth prelmnary defntons n Secton 2. Our prophet-type bounds are presented n Secton 3. The role of devaton weght functon famles n the analyss s explored n Secton 4. Then, Secton 5 s devoted to the proofs of our man statements for GSP auctons. We conclude wth open problems n Secton 6. Due to lack of space several proofs have been omtted.

4 2 Prelmnares 1 F (x) f(x) We consder a Bayesan settng wth n players and n slots 1 where slot j [n] has a clck-through rate α j that corresponds to the frequency of clckng an ad n slot j. We add an artfcal (n+1)-th slot wth clck-through rate 0 and ndex the slots so that α 1 α 2 α n α n+1 = 0. Each player s valuaton (per clck) s non-negatve and s drawn from a publcly known probablty dstrbuton. The aucton mechansms we consder use a reserve prce t and assgn slots to players accordng to the bds they submt. Player submts a bd b (v ) that depends on her valuaton v ; the bddng functon b s the strategy of player. Gven a realzaton of valuatons, let b = (b 1,..., b n ) denote a bd vector and defne the random permutaton π so that π(j) s the player wth the j-th hghest bd (breakng tes arbtrarly). The mechansm assgns slot j to player π(j) whenever b π(j) t; f b π(j) < t, the player s not allocated any slot. In such an allocaton, let σ() denote the slot that s allocated to player. Ths s well-defned when player s assgned a slot; f ths s not the case, we follow the conventon that σ() = n + 1. Gven b, the mechansm also defnes a payment p t for each player that s allocated a slot. Then, the utlty of player s u (b) = α σ() (v p ). A set of players strateges s a Bayes-Nash equlbrum f no player has an ncentve to devate from her strategy n order to ncrease her expected utlty. Ths means that for every player and every possble valuaton x, E[u (b) v = x] E[u (b, b ) v = x] for every alternatve bd b. Note that the expectaton s taken over the randomness of the valuatons of the other players and the notaton (b, b ) s used for the bd vector where player has devated to b and the remanng players bd as n b. The socal welfare at a Bayes-Nash equlbrum b s W t (b) = E[ α σ()v ], whle the revenue generated by the mechansm s R t (b) = E[ α σ()p ]. We focus on the case where the valuatons of players are drawn ndependently from a common probablty dstrbuton D wth probablty densty functon f and cumulatve dstrbuton functon F. Gven a dstrbuton D over players valuatons, the vrtual valuaton functon s φ(x) = x. We consder regular probablty dstrbutons where φ(x) s non-decreasng. The work of Myerson [15] mples that the expected revenue from player at a Bayes-Nash equlbrum b of any aucton mechansm s E[α σ() φ(v )],.e., t depends only on the allocaton of player and her vrtual valuaton. Hence, the total expected revenue s maxmzed when the players wth non-negatve vrtual valuatons are assgned to slots n non-ncreasng order of ther vrtual valuatons and players wth negatve vrtual valuatons are not assgned any slot. A mechansm that mposes ths allocaton as a Bayes-Nash equlbrum (and, hence, s revenue-maxmzng) s the celebrated VCG mechansm wth reserve prce t such that φ(t) = 0. We refer to ths as Myerson reserve and denote t by r n the followng. We use the notaton µ to denote such an allocaton. Note that, n µ, players wth zero vrtual 1 Our model can smulate cases where the number of slots s smaller than the number of players by addng fcttous slots wth zero clck-through rate.

5 valuaton can be ether allocated slots or not; such players do not contrbute to the optmal revenue. A partcular subclass of regular probablty dstrbutons are those wth monotone hazard rate (MHR). A regular dstrbuton D s MHR f ts hazard rate functon h(x) = f(x)/(1 F (x)) s non-decreasng. These dstrbutons have some nce propertes (see [1]). For example, F (r) 1 1/e and φ(x) x r for every x r. In ths paper, we focus on the GSP mechansm. For each player that s allocated a slot (.e., wth bd at least t), GSP computes her payment as the maxmum between the reserve prce t and the next hghest bd b π(+1) (assumng that b π(n+1) = 0). As t has been observed n [7], GSP may not admt the allocaton µ as a Bayes-Nash equlbrum. Ths mmedately mples that the revenue over Bayes-Nash equlbra would be suboptmal. In order to capture the revenue loss due to the selfsh behavor of the players, we use the noton of revenue guarantee. Defnton 1. The revenue guarantee of an aucton game wth reserve prce t R s max OP T b R t(b), where b runs over all Bayes-Nash equlbra of the game. In our proofs, we use the notaton σ to refer to the random allocaton that corresponds to a Bayes-Nash equlbrum. Note that, a player wth valuaton strctly hgher than the reserve has always an ncentve to bd at least the reserve and be allocated a slot. When her valuaton equals the reserve, she s ndfferent between bddng the reserve or not partcpatng n the aucton. For auctons wth Myerson reserve, when comparng a Bayes-Nash equlbrum to the revenue-maxmzng allocaton µ, we assume that a player wth valuaton equal to the reserve has the same behavor n both σ and µ (ths mples that E[ α σ()] = E[ α µ()]). Ths assumpton s wthout loss of generalty snce such a player contrbutes zero to the optmal revenue anyway. In our proofs, we also use the random varable o(j) to denote the player wth the j-th hghest valuaton (breakng tes arbtrarly). Hence, µ() = o 1 () f the vrtual valuaton of player s postve and µ() = n + 1 f t s negatve. When the vrtual valuaton of player s zero, t can be ether µ() = o 1 () or µ() = n + 1. When consderng GSP auctons, we make the assumpton that players are conservatve: whenever the valuaton of player s v, she only selects a bd b (v ) [0, v ] at Bayes-Nash equlbra. Ths s a rather natural assumpton snce any bd b (v ) > v s weakly domnated by bddng b (v ) = v [17]. In the followng, we use the notaton x + to denote max{x, 0} whle the expresson x1{e} equals x when the event E s true and 0 otherwse. 3 Achevng Mnmum Revenue Guarantees Our purpose n ths secton s to show that by approprately settng the reserve prce, we can guarantee a hgh revenue from the advertser that occupes the frst slot at any Bayes-Nash equlbrum. Even though ths approach wll not gve us a standalone result, t wll be very useful later when we wll combne t wth

6 the analyss of GSP auctons wth Myerson reserve. These bounds are smlar n sprt to prophet nequaltes n optmal stoppng theory [10]. We begn wth a smple lemma. Lemma 1. Consder n random valuatons v 1,..., v n that are drawn..d. from a regular dstrbuton D. Then, for every t r, t holds that E[max φ(v ) + ] φ(t) + n(1 F (t))2. f(t) We can use Lemma 1 n order to bound the revenue n the case of regular valuatons. Lemma 2. Let b be a Bayes-Nash equlbrum for a GSP aucton game wth n players wth random valuatons v 1,..., v n drawn..d. from a regular dstrbuton D. Then, there exsts r r such that R r (b) (1 1/e)α 1 E[max φ(v ) + ]. For MHR valuatons, we show an mproved bound. Lemma 3. Let b be a Bayes-Nash equlbrum for a GSP aucton game wth n players wth random valuatons v 1,..., v n drawn..d. from an MHR dstrbuton D. Then, there exsts r r such that R r (b) (1 e 2 )α 1 E[max φ(v ) + ] (1 e 2 )α 1 r(1 F n (r)). Proof. We wll assume that E[max φ(v ) + ] r(1 F n (r)) snce the lemma holds trvally otherwse. Let t be such that F (t ) = 1 η/n where η = 2 (1 1/e) n. We wll dstngush between two cases dependng on whether t r or not. We frst consder the case t r. We wll use the defnton of the vrtual valuaton, the fact that the hazard rate functon satsfes h(t ) h(r) = 1/r, the defnton of t, Lemma 1 (wth t = t ), and the fact that F (r) 1 1/e whch mples that 1 F n (r) η 1. We have t (1 F n (t )) = φ(t )(1 F n (t )) + 1 h(t ) (1 F n (t )) = φ(t )(1 F n (t )) + η h(t ) (1 F n (t )) η 1 h(t ) (1 F n (t )) φ(t )(1 F n (t )) + n(1 F (t )) 2 f(t η(1 F n (t )) ) n(1 F (t )) (η 1)r(1 F n (t )) = (1 F n (t )) (φ(t ) + n(1 F (t )) 2 ) f(t (η 1)r ) ( ) n ) 2 (1 1/e)n ( ) 1 (1 E[max φ(v ) + ] r(1 F n (r)). n Note that the left sde of the above equalty multpled wth) α 1 s a lower bound n on the revenue of GSP wth reserve t. Also, (1 2 (1 1/e)n n s non-decreasng

7 n n and approaches e 2 from below as n tends to nfnty. Furthermore, the rght-hand sde of the above nequalty n non-negatve. Hence, R t (b) (1 e 2 )α 1 E[max φ(v ) + ] (1 e 2 )α 1 r(1 F n (r)) as desred. We now consder the case t < r. We have 1 η/n = F (t ) F (r) 1 1/e whch mples that n 5. Tedous calculatons yeld 1 F n (r) n(1 F (r)) = 1 + F (r) F n 1 (r) 1 e 2 n 2 e 2 for n {2, 3, 4, 5} snce F (r) 1 η/n. Hence, R r (b) α 1 r(1 F n (r)) (1 e 2 )α 1 nr(1 F (r)) (1 e 2 )α 1 r(1 F n (r)) (1 e 2 )α 1 E[max φ(v ) + ] (1 e 2 )α 1 r(1 F n (r)), where the last nequalty follows by applyng Lemma 1 wth t = r. 4 Devaton Weght Functon Famles The man dea we use for the analyss of Bayes-Nash equlbra of aucton games wth reserve prce t s that the utlty of player wth valuaton v = x t does not ncrease when ths player devates to any other bd n [t, x]. Ths provdes us wth nfntely many nequaltes on the utlty of player that are expressed n terms of her valuaton, the bds of the other players, and the reserve prce. Our technque combnes these nfnte lower bounds by consderng ther weghted average. The specfc weghts wth whch we consder the dfferent nequaltes are gven by famles of functons wth partcular propertes that we call devaton weght functon famles. Defnton 2. Let β, γ, δ 0 and consder the famly of functons G = {g ξ : ξ [0, 1)} where g ξ s a non-negatve functon defned n [ξ, 1]. G s a (β, γ, δ)-dwff (devaton weght functon famly) f the followng two propertes hold for every ξ [0, 1): ) ) 1 ξ 1 z g ξ (y) dy = 1, (1 y)g ξ (y) dy β γz + δξ, z [ξ, 1]. The next lemma s used n order to prove most of our bounds together wth the devaton weght functon famly presented n Lemma 5.

8 Lemma 4. Consder a Bayes-Nash equlbrum b for a GSP aucton game wth n players and reserve prce t. Then, the followng two nequaltes hold for every player. E[u (b)] E[α σ() φ(v )] n E[α j (βv γb π(j) + δt)1{µ() = j}], (1) j=c n E[α j (βφ(v ) γb π(j) )1{µ() = j}], (2) j=c where c s any nteger n [n], β, γ, and δ are such that a (β, γ, δ)-dwff exsts, and µ s any revenue-maxmzng allocaton. Lemma 5. Consder the famly of functons G 1 consstng of the functons g ξ : [ξ, 1] R + defned as follows for every ξ [0, 1): { κ g ξ (y) = 1 y, y [ξ, ξ + (1 ξ)λ), 0, otherwse, where λ (0, 1) and κ = 1 ln(1 λ). Then, G 1 s a (κλ, κ, κ(1 λ))-dwff. We remark that the bound for GSP auctons wth Myerson reserve (and players wth MHR valuatons) follows by a slghtly more nvolved devaton weght functon famly. Due to lack of space, we omt t from ths extended abstract; t wll appear n the fnal verson of the paper. 5 Revenue Guarantees n GSP Auctons We wll now explot the technques developed n the prevous sectons n order to prove our bounds for GSP auctons. Throughout ths secton, we denote by O j the event that slot j s occuped n the revenue-maxmzng allocaton consdered. The next lemma provdes a lower bound on the revenue of GSP auctons. Lemma 6. Consder a Bayes-Nash equlbrum b for a GSP aucton game wth Myerson reserve prce r and n players. It holds that E[α j b π(j) 1{O j }] R r (b) α 1 r Pr[O 1 ]. Proof. Consder a Bayes-Nash equlbrum b for a GSP aucton game wth Myerson reserve prce r. Defne Pr[O n+1 ] = 0. Consder some player whose valuaton exceeds r and s thus allocated some slot. Note that the player s payment per clck s determned by the bd of the player allocated just below her, f there s one, otherwse, the player s (per clck) payment s set to r. It holds that R r (b) = j α j r(pr[o j ] Pr[O j+1 ]) + j E[α j b π(j+1) 1{O j+1 }]

9 = E[α j b π(j) 1{O j }] + j α j r(pr[o j ] Pr[O j+1 ]) + j E[(α j α j+1 )b π(j+1) 1{O j+1 }] E[α j b π(j) 1{O j }] + j α j r(pr[o j ] Pr[O j+1 ]) + j (α j α j+1 )r Pr[O j+1 ] = E[α j b π(j) 1{O j }] + j α j r Pr[O j ] j α j+1 r Pr[O j+1 ] = E[α j b π(j) 1{O j }] + α 1 r Pr[O 1 ]. The proof follows by rearrangng the terms n the last nequalty. The next statement follows by Lemmas 2 and 4 usng the DWFF defned n Lemma 5. Theorem 1. Consder a regular dstrbuton D. There exsts some r, such that the revenue guarantee over Bayes-Nash equlbra of GSP aucton games wth reserve prce r s 4.72, when valuatons are drawn..d. from D. Proof. By Lemma 2, we have that there exsts r r such that the expected revenue over any Bayes-Nash equlbrum b of the GSP aucton game wth reserve prce r satsfes R r (b ) (1 1/e)E[α 1 φ(v o(1) ) + ]. (3) Now, let b be any Bayes-Nash equlbrum of the GSP aucton game wth Myerson reserve and let β, γ, and δ be parameters so that a (β, γ, δ)-dwff exsts. Usng nequalty (2) from Lemma 4 wth c = 2 and Lemma 6 we obtan R r (b ) = = β E[α σ() φ(v )] E[α j (βφ(v ) γb π(j) )1{µ() = j}] E[α j φ(v o(j) ) + ] γ E[α j b π(j) 1{O j }] β E[α j φ(v o(j) ) + ] γr r (b ). In other words, (1 + γ)r r (b ) β E[α j φ(v o(j) ) + ].

10 Usng ths last nequalty together wth nequalty (3), we obtan ( 1 + γ + eβ ) max{r r (b ), R r (b )} (1 + γ)r r (b ) + eβ e 1 e 1 R r (b ) β j = βr OP T. E[α j φ(v o(j) ) + ] We conclude that there exsts some reserve prce r (ether r or r ) such that for any Bayes-Nash equlbrum b t holds that R OP T R r (b) 1 + γ + e β e 1. By Lemma 5, the famly G 1 s a (β, γ, 0)-DWFF wth β = κλ and γ = κ, where λ (0, 1) and κ = 1 ln (1 λ). By substtutng β and γ wth these values and usng λ 0.682, the rght-hand sde of our last nequalty s upper-bounded by The next statement apples to MHR valuatons. It follows by Lemmas 3 and 4 usng the DWFF defned n Lemma 5. Theorem 2. Consder an MHR dstrbuton D. There exsts some r, such that the revenue guarantee over Bayes-Nash equlbra of GSP aucton games wth reserve prce r s 3.46, when valuatons are drawn..d. from D. Proof. Let b be any Bayes-Nash equlbrum of the GSP aucton game wth Myerson reserve and let β, γ, and δ be parameters so that a (β, γ, δ)-dwff exsts. Snce D s an MHR probablty dstrbuton, we have E[α σ() r] E[α σ() (v φ(v ))] = E[u (b )] for every player. By summng over all players and usng nequalty (1) from Lemma 4 wth c = 2, we obtan E[α σ() r] E[u (b )] n E[α j (βv γb π(j) + δr)1{µ() = j}] j=2 E[α j (βφ(v o(j) ) + γb π(j) + δr)1{o j }] β E[α j φ(v o(j) ) + ] γ E[α j b π(j) 1{O j }] + δ E[α j r1{o j }] β E[α j φ(v o(j) ) + ] γr r (b ) + (γ δ)e[α 1 r1{o 1 }] + δ j = β E[α j φ(v o(j) ) + ] γr r (b ) + (γ δ)e[α 1 r1{o 1 }] + δ E[α j r1{o j }] E[α µ() r].

11 The last nequalty follows by Lemma 6. Snce E[α µ()r] = E[α σ()r], we obtan that γr r (b ) β E[α j φ(v o(j) ) + ] + (γ δ)α 1 r Pr[O 1 ] + (δ 1) E[α σ() r]. By Lemma 3, we have that there exsts r r such that the expected revenue over any Bayes-Nash equlbrum b of the GSP aucton game wth reserve prce r satsfes R r (b ) (1 e 2 )E[α 1 φ(v o(1) ) + ] (1 e 2 )E[α 1 r1{o 1 }]. Usng ths last nequalty together wth nequalty (4), we obtan ( ) γ + e2 β e 2 max{r r (b ), R r (b )} 1 γr r (b ) + β j e2 β e 2 1 R r (b ) E[α j φ(v o(j) ) + ] + (γ δ β)e[α 1 r1{o 1 }] + (δ 1) E[α σ() r] (4) βr OP T + (γ δ β)e[α 1 r1{o 1 }] + (δ 1) E[α σ() r]. By Lemma 5, the famly G 1 s a (β, γ, δ)-dwff wth β = γ δ = κλ, γ = κ, and δ = κ(1 λ), where λ (0, 1) and κ = 1 ln(1 λ). By settng λ so that δ = κ(1 λ) = 1, the above nequalty mples that there exsts some reserve prce r (ether r or r ) such that for any Bayes-Nash equlbrum b of the correspondng GSP aucton game, t holds that as desred. R OP T R r (b) 1 λ + e2 e , For GSP auctons wth Myerson reserve, our revenue bound follows usng a slghtly more nvolved devaton weght functon famly. Theorem 3. Consder an MHR dstrbuton D. The revenue guarantee over Bayes-Nash equlbra of GSP aucton games wth Myerson reserve prce r s 3.90, when valuatons are drawn..d. from D. 6 Conclusons Even though we have sgnfcantly mproved the results of [14], we conjecture that our revenue guarantees could be further mproved. The work of Gomes and Sweeney [7] mples that the revenue guarantee of GSP auctons wth Myerson reserve s n general hgher than 1; however, no explct lower bound s known. Due to the dffculty n computng Bayes-Nash equlbra analytcally, comng up wth a concrete lower bound constructon s nterestng and would reveal the gap of our revenue guarantees.

12 References 1. R. Barlow and R. Marshall. Bounds for dstrbutons wth monotone hazard rate. Annals of Mathematcals Statstcs, 35(3): , I. Caraganns, C. Kaklamans, P. Kanellopoulos, and M. Kyropoulou. On the effcency of equlbra n generalzed second prce auctons. In Proceedngs of the 12th ACM Conference on Electronc Commerce (EC), pp , I. Caraganns, C. Kaklamans, P. Kanellopoulos, M. Kyropoulou, B. Lucer, R. Paes Leme, and É. Tardos. On the effcency of equlbra n generalzed second prce auctons. arxv: , S. Chawla, J. Hartlne, D. Malec, and B. Svan. Mult-parameter mechansm desgn and sequental posted prcng. In Proceedngs of the 41th ACM Symposum on Theory of Computng (STOC), pp , E.H. Clarke. Multpart prcng of publc goods. Publc Choce, 11: 17 33, B. Edelman, M. Ostrovsky, and M. Schwarz. Internet advertzng and the generalzed second-prce aucton: sellng bllons of dollars worth of keywords. The Amercan Economc Revew, 97(1): , R. Gomes and K. Sweeney. Bayes-Nash equlbra of the generalzed second prce aucton. Workng paper, Prelmnary verson n Proceedngs of the 10th ACM Conference on Electronc Commerce (EC), pp , T. Groves. Incentves n teams. Econometrca, 41(4): , M. Hajaghay, R. Klenberg, and T. Sandholm. Automated mechansm desgn and prophet nequaltes. In Proceedngs of the 22nd AAAI Conference on Artfcal Intellgence (AAAI), pp , U. Krengel and L. Sucheston. Semamarts and fnte values. Bulletn of the Amercan Mathematcal Socety, 83(4): , V. Krshna Aucton Theory. Academc Press, S. Lahae. An analyss of alternatve slot aucton desgns for sponsored search. In Proceedngs of the 7th ACM Conference on Electronc Commerce (EC), pp , B. Lucer and R. Paes Leme. GSP auctons wth correlated types. In Proceedngs of the 12th ACM Conference on Electronc Commerce (EC), pp , B. Lucer, R. Paes Leme, and É. Tardos. On revenue n generalzed second prce auctons. In Proceedngs of the 21st World Wde Web Conference (WWW), pp , R. Myerson. Optmal aucton desgn. Mathematcs of Operatons Research, 6(1): 58 73, M. Ostrovsky and M. Schwarz. Reserve prces n Internet advertsng auctons: a feld experment. In Proceedngs of the 12th ACM Conference on Electronc Commerce (EC), pp , R. Paes Leme and É. Tardos. Pure and Bayes-Nash prce of anarchy for generalzed second prce aucton. In Proceedngs of the 51st Annual IEEE Symposum on Foundatons of Computer Scence (FOCS), pp , H. Varan. Poston auctons. Internatonal Journal of Industral Organzaton, 25: , W. Vckrey. Counterspeculaton, auctons, and compettve sealed tenders. The Journal of Fnance, 16(1): 8 37, 1961.