On the Efficiency of Equilibria in Generalized Second Price Auctions

Transcription

1 On the Effcency of Equlba n Genealzed Second Pce Auctons Ioanns Caaganns Panagots Kanellopoulos Chstos Kaklamans Maa Kyopoulou Depatment of Compute Engneeng and Infomatcs Unvesty of Patas and RACTI, Geece {caagan,kakl,kanellop,kyopoul}@ced.upatas.g ABSTRACT In sponsoed seach auctons, advetses compete fo a numbe of avalable advetsement slots of dffeent qualty. The auctonee decdes the allocaton of advetses to slots usng bds povded by them. Snce the advetses may act stategcally and submt the bds n ode to maxmze the ndvdual objectves, such an aucton natually defnes a stategc game among the advetses. In ode to quantfy the effcency of outcomes n genealzed second pce auctons, we study the coespondng games and pesent new bounds on the pce of anachy, mpovng the ecent esults of Paes Leme and Tados [16] and Luce and Paes Leme [13]. Fo the full nfomaton settng, we pove a supsngly low uppe bound of on the pce of anachy ove pue Nash equlba. Gven the exstng lowe bounds, ths bound denotes that the numbe of advetses has almost no mpact on the pce of anachy. The poof explots the equlbum condtons developed n [16] and follows by a detaled easonng about the stuctue of equlba and a novel elaton of the pce of anachy to the objectve value of a compact mathematcal pogam. Fo moe geneal equlbum classes.e., mxed Nash, coelated, and coase coelated equlba, we pesent an uppe bound of 2.31 on the pce of anachy. We also consde the settng whee advetses have ncomplete nfomaton about the compettos and pove a pce of anachy uppe bound of 3.37 ove Bayes-Nash equlba. In ode to obtan the last two bounds, we adapt technques of Luce and Paes Leme [13] and sgnfcantly extend them wth new aguments. Categoes and Subject Descptos F.2 [Theoy of Computaton]: Analyss of Algothms and Poblem Complexty Mscellaneous; J.4 [Compute Applcatons]: Socal and Behavoal Scences Economcs Pemsson to make dgtal o had copes of all o pat of ths wok fo pesonal o classoom use s ganted wthout fee povded that copes ae not made o dstbuted fo poft o commecal advantage and that copes bea ths notce and the full ctaton on the fst page. To copy othewse, to epublsh, to post on seves o to edstbute to lsts, eques po specfc pemsson and/o a fee. EC 11, June 5 9, 211, San Jose, Calfona, USA. Copyght 211 ACM /11/6...$1.. Geneal Tems Algothms, Economcs, Theoy Keywods Auctons, equlba, genealzed second pce, pce of anachy 1. INTRODUCTION Sponsoed seach auctons [11] have become extemely popula dung the last decade as the man tool used by seach engnes and othe nfomaton sevces n ode to ceate ncome. A numbe of advetsement slots ae avalable whch can be thought of as anked accodng to the sgnfcance e.g., accodng to the numbe of clcks by vewes an advetsement assgned to a slot s expected to have. A sponsoed seach aucton ams to allocate advetsement slots to advetses. Advetses cast the bds fo the avalable slots and the auctonee uses the bds n ode to compute the allocaton to slots and the fee each advetse should pay fo ths sevce. The patcula ule used n ode to compute both the allocaton and the payments defnes a dstnct aucton. We consde genealzed second pce auctons whose vaatons ae wdely used by leades n the sponsoed seach ndusty such as Google and Yahoo! In genealzed second pce auctons, the advetses ae assgned the slots n deceasng ode of the bds.e., the advetse wth the hghest bd s assgned to the most sgnfcant slot, and so on and each of them s equed to pay the next hghest bd.e., the advetse assgned to the most sgnfcant slot wll pay pe clck an amount equal to the bd of the advetse assgned to the second sgnfcant slot, and so on. Tadtonally, Aucton Theoy see [9] fo an ntoducton has gven a cental ole to the equement that an aucton should mpose tuthful behavo by the potentally stategcally actng patcpants. Ths equement has seveal mplcatons to the maxmzaton of the socal welfae,.e., the quantty that ncludes both the auctonee s evenue and the poft of the patcpants. Even though genealzed second pce auctons genealze the famous tuthful Vckey aucton [21], they ae known nethe to be tuthful [1] no to guaantee socal welfae maxmzaton [11, 2] and ths seems to come n shap contast to the ecent success. In an attempt to povde a patal justfcaton of ths success, followng pevous wok, we consde the followng queston:

2 how much can the stategc behavo of the advetses affect the socal welfae? We addess ths queston by consdeng the natual stategc games among the advetses that genealzed second pce auctons defne. Each advetse hencefoth also called bdde has he own valuaton fo each clck and he utlty fom an allocaton depends on the total valuaton fom the clcks on the slot she s assgned to mnus he payment to the auctonee. Hence, actng stategcally means that each bdde ams to maxmze he utlty gven the stateges bds of the othe bddes. Such a behavo natually leads to an equlbum,.e., a set of stateges fom whch no bdde has an ncentve to devate. We consde both the full nfomaton and the moe ealstc ncomplete nfomaton o Bayesan settng. The coespondng equlbum concepts of nteest ae the pue Nash, mxed Nash, coelated, and coase coelated equlba n the fome and Bayes-Nash equlba n the latte settng. In both cases, we make the easonable assumpton that bddes always cast consevatve bds that do not exceed the valuatons. We povde new socal welfae guaantees fo genealzed second pce auctons that ae expessed as bounds on the pce of anachy of the coespondng games ove equlba of patcula classes, mpovng pevous esults n the lteatue. In a nutshell, ou esults ndcate that, despte the stategc behavo of the advetses, the socal welfae s always hgh. Related wok. The game-theoetc model we adopt n the cuent pape was poposed by [5] and [2] and was futhe used n the sequence of papes [6, 1, 13, 16, 19] see also the suveys [11, 15]. Edelman et al. [5] and Vaan [2] pove that Nash equlba wth optmal socal welfae always exst fo genealzed second pce aucton games n the full nfomaton settng. In contast, ths s not the case fo games n the Bayesan settng as poved n [6]. Lahae [1] povdes bounds on the socal welfae of equlba unde stong assumptons fo the clck-though ates of the slots. Thompson and Leyton-Bown [19] study the effcency of equlba though expementaton. Ou wok s closely elated to, and mpoves the ecent esults of Paes Leme and Tados [16] and Luce and Paes Leme [13]. They consde consevatve bddes and justfy ths assumpton snce bddes stateges ae domnated othewse. Ths s a natual assumpton that s usually made n smla contexts as well, such as n combnatoal auctons e.g., see [3, 4, 12]. The authos of [16] consde pue and mxed Nash equlba n the full-nfomaton settng, fo whch they uppe-bound the pce of anachy by and 4, espectvely. The esult fo mxed Nash equlba s vald fo moe geneal equlbum classes as well. Futhemoe, they pesent a tght lowe and uppe bound of 5/4 fo pue Nash equlba and two bddes. Fo the ncomplete nfomaton settng and Bayes-Nash equlba, they show an uppe bound of 8. The authos of [13] mpove ths last bound to 3.162, whle n the pelmnay veson of the pape they pesent a tght bound of on the pce of anachy ove pue Nash equlba fo the case of thee bddes the lowe bound has also been clamed n [16] wthout pesentng the explct constucton. Roughgaden [18] pesents a suffcent condton fo games, temed smoothness, so that the pce of anachy of a smooth game ove pue Nash equlba mmedately extends also to mxed Nash, coelated, and coase coelated equlba as well. Smoothness aguments have been mplctly o explctly used see [18] and the efeences theen n ode to povde such obust bounds on the pce of anachy of seveal games. As obseved n [16], genealzed second pce auctons do not coespond to smooth games. Ou esults. We fst consde pue Nash equlba of genealzed second pce aucton games unde the full nfomaton settng Secton 3. We wam up by consdeng the case of thee bddes n Secton 3.1, fo whch we pesent the uppe bound of appoxmately by povdng an altenatve poof to the one n the pelmnay veson of [13]. Ths uppe bound togethe wth the bound of 5/4 fo two bddes fom [16] seves as the base of the nductve poof of ou moe geneal esult pesented n Secton 3.2: an extemely low uppe bound of on the pce of anachy n games wth abtaly many bddes. Ths esult mples that the numbe of bddes nvolved n the aucton has almost no effect on the pce of anachy. The poof elaboates on technques developed n [16] and follows by a detaled easonng about the stuctue of equlba and a novel elaton of the pce of anachy to the objectve value of a compact mathematcal pogam. Then, n Secton 4, we consde the boad class of coase coelated equlba n the full nfomaton settng and pove an uppe bound of on the pce of anachy, mpovng the uppe bound of 4 fom [16]. Clealy, ths bound holds fo moe estcted equlba classes, namely coelated and mxed Nash equlba. In the ncomplete nfomaton settng, we pove an uppe bound of 3.37 on the pce of anachy ove Bayes-Nash equlba Secton 5, mpovng the bound of fom [13]. The poofs of these esults explctly take nto account the bds of the bddes and bound the utlty n dffeent ways by consdeng seveal possble devatons. In ode to obtan ou bounds, we adapt technques fom [13] and sgnfcantly extend them wth new aguments. All ou bounds hold wth appopate adaptatons n the poofs n the moe geneal model of sepaable clck-though ates [11] n whch the clck-though ate of a slot depends on the bdde allocated to that slot as well. In ode to keep the exposton smple, we do not consde ths extenson n the cuent text. We begn wth pelmnay defntons n Secton 2 and conclude wth open poblems n Secton 6. Due to lack of space, some poofs have been omtted. 2. PRELIMINARIES Befoe poceedng wth the pesentaton of ou esults, we gve some fomal defntons. Thoughout the pape, we consde genealzed second pce auctons wth n bddes and n slots. In such an aucton, each bdde has a valuaton v that denotes how much the bdde values a clck on he ad. Each slot has a non-negatve clck-though ate a that denotes the ate by whch ths slot s clcked by the vewes. Wthout loss of genealty, we assume that bddes and slots ae soted accodng to the valuatons and the clck-though ates, so that v 1 v 2... v n and a 1 a 2... a n. Gven a vecto of bds b = b 1, b 2,..., b n wth one bd pe bdde, the genealzed second pce aucton defnes an assgnment π accodng to whch the bdde wth the j-th hghest bd s assgned to slot j tes ae boken abtaly. We denote by πj the bdde assgned to slot j and by π 1 the slot to whch bdde s

3 allocated. Then, each bdde s equed to pay the bd of the bdde that s assgned to the slot below hes.e., slot π 1 1 f any. We study the stategc game among the bddes that s nduced by such an aucton and we efe to t as a GSP aucton game. In ths game, each bdde acts selfshly and ams to maxmze he utlty gven the bds of the othe bddes. Gven a bd vecto b and the coespondng assgnment π t nduces, the utlty of bdde s u b = a π 1 v b ππ 1 1, assumng that b πn1 =. We assume that bddes ae consevatve,.e., each bdde selects as he stategy a bd that does not exceed he valuaton. A pue stategy fo bdde conssts of a sngle bd b [, v ], whle a mxed stategy s a pobablty dstbuton ove pue stateges. We say that a bd vecto b s a mxed Nash equlbum fo a GSP aucton game f no bdde has an ncentve to unlateally devate fom he stategy n ode to stctly ncease he expected utlty. I.e., fo each bdde and each altenatve bd b [, v ], t holds that E[u b] E[u b, b ], whee b denotes the bds of all bddes apat fom and the expectaton s taken accodng to the andomness of the bds. If all bddes play pue stateges, then b s a pue Nash equlbum. Coelated and coase coelated equlba can be vewed as genealzatons of mxed Nash equlba, whee the bddes have a jont pobablty dstbuton nstead of ndependent ones. Infomally, n both settngs, a medato daws a bd vecto fom a publcly known dstbuton and secetly nfoms each bdde of he suggested stategy. If no bdde has an ncentve to devate fom the suggested stategy, then ths s a coelated equlbum [2], whle f no bdde has a pue stategy that she can always follow, espectve of the outcome, and mpove he expected utlty, then ths s a coase coelated equlbum [14] see also [22]. Moe fomally, thee exsts a jont pobablty dstbuton that daws the bd vecto b fom the unvese B of all bd vectos y 1, y 2,..., y n wth y [, v ]. The expected utlty of a bdde s then E[u b] = y B P[b = y] u y. In a coase coelated equlbum, fo any bdde, thee exsts no altenatve bd b [, v ] such that E[u b] < E[u b, b ]. A coelated equlbum s also a coase coelated one. The socal welfae of an assgnment π nduced by a bd vecto b s then defned as Wb = E[ a v π ], whee the expectaton s taken accodng to the andomness of the bds. Equvalently, we have Wb = E[ a π 1 v ] = E[ The optmal socal welfae s OPT = u b] E[ a v. a b π1 ]. We also consde the Bayesan settng [7] whee bddes valuatons ae andom. In ths settng, each bdde has a pobablty dstbuton on he valuaton and he stategy depends on the actual valuaton,.e., a bd s now a functon on the valuaton. The optmal socal welfae s defned as OPT = E[ a o v ], whee o s the andom vaable denotng the slot that bdde occupes n the optmal assgnment and the expectaton s taken accodng to the andomness n the bddes valuatons. In a Bayes-Nash equlbum, fo any bdde, any possble value x fo he valuaton, and any altenatve bd b x, t holds that E[u b x, b v v = x] E[u b x, b v v = x]. The socal welfae s agan Wb = E[ a v π ], whee the expectaton s taken accodng to the andomness n the valuatons and the bds. The pce of anachy ntoduced n [8]; see also [17] of a game ove a gven class of equlba s defned as the wost ato of the optmal socal welfae ove the socal welfae of an equlbum ove all equlba of the class,.e., max b OPT/Wb, whee b s estcted to equlba. The pce of anachy fo a class of games ove equlba of a patcula type s the wost pce of anachy among the games n the class ove equlba of the same type. 3. THE PRICE OF ANARCHY OVER PURE NASH EQUILIBRIA In ths secton we pesent ou esults fo pue Nash equlba. We consde GSP aucton games wth n bddes wth valuatons v 1... v n and n slots wth clck-though ates a 1... a n. We assume that nethe all slots have the same clck-though ate no all bddes have the same valuaton n both cases, the pce of anachy s 1. We elaboate on the appoach taken n [16] and use the noton of weakly feasble assgnments defned theen. Defnton 1. An assgnment π s weakly feasble f fo each pa of bddes, j, t holds that a π 1 v a π 1 jv v j. We efe to the nequaltes n ths defnton as weak feasblty condtons. Wth some abuse of notaton, we denote by Wπ = a v π the socal welfae of assgnment π and use the tem effcency of assgnment π to efe to the ato OPT/Wπ. As poved n [16], evey pue Nash equlbum coesponds to a weakly feasble assgnment. Hence, the pce of anachy of a GSP aucton game ove pue Nash equlba s uppe-bounded by the wost-case effcency among weakly feasble assgnments. Defnton 2. An assgnment π s called pope f fo any two slots < j wth equal clck-though ates, t holds π < πj. Clealy, fo any non-pope weakly feasble assgnment, we can constuct a pope weakly feasble one wth equal socal welfae. Hence, n ode to pove ou uppe bounds, we

4 essentally uppe-bound the wost-case effcency ove pope weakly feasble assgnments. Gven an assgnment π, consde the dected gaph Gπ that has one node fo each slot, and a dected edge fo each bdde that connects the node coespondng to slot to the node coespondng to slot π 1. In geneal, Gπ conssts of a set of dsjont cycles and may contan self-loops. Defnton 3. An assgnment π s called educble f ts dected gaph Gπ has moe than one cycles. Othewse, t s called educble. Gven a educble assgnment π such that Gπ has c 2 cycles, we can constuct c GSP aucton subgames by consdeng the slots and the bddes that coespond to the nodes and edges of each cycle. Smlaly, fo l = 1,..., c, the estcton π l of π to the slots and bddes of the l-th subgame s an assgnment fo ths game. The next fact s mplct n [16]. Fact 4. If assgnment π s weakly feasble fo the ognal GSP aucton game, then π l s weakly feasble fo the l-th subgame as well, fo l = 1,..., c. Then, the effcency of π s at most the maxmum effcency among the assgnments π l fo l = 1,..., c. When consdeng educble weakly feasble assgnments, we futhe assume that the ndex of the slot bdde 1 occupes s smalle than the ndex of the bdde that s assgned to slot 1. Ths s wthout loss of genealty due to the followng agument. Consde an educble weakly feasble assgnment π fo a GSP aucton game wth n bddes such that π 1 1 > π1. We constuct a new game wth clckthough ate a = v fo slot and valuaton v = a fo bdde, fo = 1,..., n and the assgnment π = π 1. Obseve that π 1 1 = π1 < π 1 1 = π 1. Clealy, the optmal socal welfae s the same n both games whle the socal welfae of π fo the new game s Wπ = a v π = v a π = a π 1 v = Wπ. We can also pove the weak feasblty condtons fo π n the new game fo each, j. In ode to do so, consde the weak feasblty condton fo π n the ognal game fo bddes πj, π. It s a π 1 πjv πj a π 1 πv πj v π and, equvalently, v π a π 1 π v πj a π 1 π a π 1 πj. By the defnton of the clck-though ates and the valuatons n the new game and the defnton of π, we obtan that v a π 1 j v v j as desed. a π 1 We futhemoe note that when v n =, any pope weakly feasble assgnment s educble. Ths s obvously the case f all bddes wth zeo valuaton use the last slots. Othewse, consde a bdde wth non-zeo valuaton that s assgned a slot π 1 > π 1 j whee j s a bdde wth zeo valuaton. Snce the assgnment s pope, t holds that a π 1 < a π 1 j. Then, we obtan a contadcton by the weak feasblty condton a π 1 v a π 1 jv v j fo bddes, j. 3.1 GSP aucton games wth thee bddes We ae eady to pesent ou fst esult Theoem 6. In the poof, we use the followng techncal lemma. Lemma 5. Let ζ = Fo any λ [, 1], t holds that λ ζλ λ2 2. Poof. Snce both pats of the nequalty ae non-negatve fo λ [, 1], t suffces to show that the functon fλ = 2 λ ζλ λ2 2 s non-negatve fo λ [, 1]. Let gλ = λ3 4 1 ζλ2 1 ζ 2 λ 2ζ and obseve that fλ = λ gλ. The poof wll follow by povng that gλ when λ [, 1]. Obseve that the devatve of g s stctly negatve fo λ = and stctly postve fo λ = 1. Hence, the mnmum of g n [, 1] s acheved at the pont λ = 44ζ 2 ζ 2 8ζ1 3 whee the devatve of g becomes zeo. Staghtfowad calculatons yeld that gλ > and the lemma follows. Theoem 6. The pce of anachy ove pue Nash equlba of GSP aucton games wth thee consevatve bddes s at most Poof. Consde a GSP aucton game wth thee slots wth clck-though ates a 1 a 2 a 3 and thee bddes wth valuatons v 1 v 2 v 3 and a pope weakly feasble assgnment π of slots to bddes. We wll pove the theoem by uppe-boundng the effcency of π by If π s educble, then the effcency s bounded by the effcency fo games wth two bddes and the theoem follows by the uppe bound of 5/4 poved n [16] fo ths case. So, n the followng, we assume that π s educble; by the obsevaton above, ths mples that v 3 >. Thee ae only two such assgnments whch ae n fact symmetc: n the fst, slots 1, 2, 3 ae allocated to bddes 3, 1, 2, espectvely, and n the second, slots 1, 2, 3 ae allocated to bddes 2, 3, 1, espectvely. Wthout loss of genealty see the dscusson above, we assume that π s the fome assgnment. Let β, γ, λ, and µ be such that a 2 = βa 1, a 3 = γa 1, v 2 = λv 1, and v 3 = µv 1. Clealy, t holds that 1 β γ and 1 λ µ >. The socal welfae of assgnment π s Wπ = a 1v 1µ β γλ wheeas the optmal socal welfae s OPT = a 1 v 1 1 βλ γµ. Futhemoe, snce π s weakly feasble, the weak feasblty condtons fo bddes 1 and 3 and bddes 2 and 3 ae a 2 v 1 a 1 v 1 v 3 and a 3 v 2 a 1 v 2 v 3, espectvely,.e., β 1 µ and γ 1 µ. λ We ae now eady to bound the effcency of π. Let δ, ϵ be such that β = 1 µ δ and γ = 1 µ ϵ. We have λ OPT Wπ = 1 βλ γµ µ β γλ = 1 λ µλ µ µ 2 δλ ϵµ λ 1 λ µ δ ϵλ 1 λ µλ µ µ 2 λ. 1 λ µ The nequalty follows snce 1 λ µ > mples that 1 λ µλ µ µ2 = 1 λ µ µ1 λ µ1 µ/λ λ 1 λ µ 1 and δ ϵλ δλ ϵµ. Fo µ [, 1], ths last expesson s maxmzed fo the value of µ that makes ts devatve wth espect to µ equal to zeo,.e., µ = λ 3 1 λ 1. By substtutng µ, we obtan that OPT Wπ λ2 λ 2 2 λ 3 1 λ 1 2ζ = , whee ζ = and the second nequalty follows by Lemma 5.

5 3.2 GSP aucton games wth many bddes In ths secton, we pove ou man esult fo pue Nash equlba. Theoem 7. The pce of anachy ove pue Nash equlba of GSP aucton games wth consevatve bddes s at most In ou poof, we wll need the followng techncal lemma. Lemma 8. Let = and fβ, γ, λ, µ = 128 µ β 1 λ γλ µ 1. Then, the objectve value of the mathematcal pogam s non-negatve. mnmze fβ, γ, λ, µ subject to β 1 µ γ 1 µ/λ 1 λ µ > 1 β, γ Poof. Snce µ λ 1, we have that fβ, γ, λ, µ s non-deceasng n β and γ. Usng the fst two constants, we have that the objectve value of the mathematcal pogam s at least f 1 µ, 1 µ λ, λ, µ = 1 1 λ λ µ 2 λ µ2 λ, whch s mnmzed fo µ = λ λ2 2 f 1 λ λ2 2, λ λ2, λ, λ 2 2 to = 1 1 λ λ2 λ In ode to complete the poof t suffces to show that the functon gλ = 1 1 λ λ2 λ3 s non-negatve fo 4 λ [, 1]. Obseve that gλ s a polynomal 2 of degee 3 and, hence, t has at most one local mnmum. Also obseve that the devatve of gλ s 1 2λ 3λ2 whch s stctly 4 negatve fo λ = and stctly postve fo 2 λ = 1. Hence, ts mnmum n [, 1] s acheved at the pont λ = whee the devatve becomes zeo. Staghtfowad calculatons yeld that gλ = and the lemma follows. Poof of Theoem 7. In ode to pove the theoem, we wll pove that the wost-case effcency among weakly feasble assgnments of any GSP aucton game s at most = We use nducton. As the base 128 of ou nducton, we use the fact that GSP aucton games wth one, two, o thee bddes have wost-case effcency among weakly feasble assgnments at most Fo a sngle bdde, the clam s tval. Fo two bddes, t follows by [16], and fo thee bddes, t follows by the poof of Theoem 6. Let n 4 be an ntege. Usng the nductve hypothess that the wost-case effcency among weakly feasble assgnments of any GSP aucton game wth at most n 1 bddes s at most, we wll show that ths s also the case fo any GSP aucton game wth n bddes. Consde a GSP aucton game wth n bddes wth valuatons v 1 v 2... v n and n slots wth clck-though ates a 1 a 2... a n and let π be a pope weakly feasble assgnment. If π s educble, the clam follows by Fact 4 and the nductve hypothess. So, n the followng, we assume that π s educble; ths mples that v n >. Let j be the bdde that s assgned slot 1 and 1 be the slot assgned to bdde 1. Wthout loss of genealty, we assume that 1 < j snce the othe case s symmetc; see the dscusson at the begnnng of Secton 3. Also, let 2 be the slot assgned to bdde 1. By ou assumptons, the nteges j, 1, 1, and 2 ae dffeent. We wll show that Wπ a 1v j a 1 v 1 v 1 a 2v 1 v j a1v1 OPT. 1 Assumng that 1 holds, we apply Lemma 8 wth β = a 1 /a 1, γ = a 2 /a 1, λ = v 1 /v 1, and µ = v j /v 1. Clealy, the last two constants of the mathematcal pogam n Lemma 8 ae satsfed. Also, obseve that the weak feasblty condtons fo bddes 1 and j and bddes 1 and j n assgnment π ae a 1 v 1 a 1 v 1 v j and a 2 v 1 a 1 v 1 v j, espectvely,.e., β 1 µ and γ 1 µ/λ and the fst two constants of the mathematcal pogam n Lemma 8 ae satsfed as well. Now, usng nequalty 1 and Lemma 8, we have that Wπ f a1, a 2, v 1, v j a 1 v 1 OPT OPT a 1 a 1 v 1 v 1 and the poof follows. It emans to pove nequalty 1. We dstngush between thee cases dependng on the elatve ode of j, 1, and 2; n each of these cases, we futhe dstngush between two subcases. Case I.1: 1 < 1 < j < 2 and a j a 2. Consde the estcton of the ognal game that conssts of the bddes dffeent than j, 1, and 1 and the slots dffeent than 1, 1, and 2. Let π be the estcton of π to the bddes and slots of the new game. Clealy, ths assgnment s weakly feasble fo the new game snce the weak feasblty condtons fo π ae just a subset of the coespondng condtons fo π fo the ognal game. Also, note that the optmal assgnment fo the estcted game assgns bdde k to slot k fo k = 2,..., 1 1, 1 1,..., j 1, 2 1,..., n and bdde k 1 to slot k fo k = j,..., 2 1. By the nductve hypothess, we know that the effcency of π s at most. Hence, we can bound the socal welfae of π as Wπ = a 1 v j a 1 v 1 a 2 v 1 k {1, 1, 2 } = a 1 v j a 1 v 1 a 2 v 1 Wπ a 1v j a 1 v 1 a 2 v j 1 k= 1 1 k= k=j a k v πk n k= 2 1

6 a 1 v j a 1 v 1 a 2 v k=j1 n k= 2 1 k=2 j 1 k= 1 1 = a 1v j a 1 v 1 a 2 v 1 1 n a 1v 1 a 1 v 1 a j v j = a 1v j a 1 v 1 a 2 v 1 1 a1v1 a 1v 1 a jv j OPT a 1v j a 1 v 1 v 1 a 2v 1 v j a 1v 1 OPT and nequalty 1 follows. The fst nequalty follows by the nductve hypothess and the defnton of the optmal assgnment fo the estcted game. The second nequalty follows snce a k a k1 fo k = j,..., 2 1. The last nequalty follows snce a j a 2. Case I.2: 1 < 1 < j < 2 and a j > a 2. Consde the estcton of the ognal game that conssts of the bddes dffeent than j and 1 and the slots dffeent than 1 and 1. Let π be the estcton of π to the bddes and slots of the new game. Agan, ths assgnment s weakly feasble fo the new game snce the weak feasblty condtons fo π ae just a subset of the ones fo π fo the ognal game. Also, note that the optmal assgnment fo the estcted game assgns bdde k to slot k fo k = 2,..., 1 1, j 1,..., n and bdde k 1 to slot k fo k = 1 1,..., j. By the nductve hypothess, we know that the effcency of π s at most. Hence, we can bound the socal welfae of π as Wπ = a 1 v j a 1 v 1 k {1, 1 } a k v πk = a 1v j a 1 v 1 Wπ a 1 v j a 1 v n k=j1 k=2 j k= = a 1 v j a 1 v 1 1 n a 1 v 1 a 1 v 1 j 1 v k k= 1 1 a 1v j a 1 v 1 1 n a 1v 1 a 1 v 1 j a j v k 1 v k k= 1 1 = a 1v j a 1 v 1 1 a1v1 a 1v 1 a jv j a jv 1 OPT > a 1v j a 1 v 1 v 1 a 2v 1 v j a 1v 1 OPT and nequalty 1 follows. The fst nequalty follows by the nductve hypothess and the defnton of the optmal assgnment fo the estcted game. The second nequalty follows snce a k a j and v k 1 v k fo k = 1 1,..., j. The last nequalty follows snce a j > a 2. Case II.1: 1 < 1 < 2 < j and v 2 v j. Consde the estcton of the ognal game that conssts of the bddes dffeent than j, 1, and 1 and the slots dffeent than 1, 1, and 2. Let π be the estcton of π to the bddes and slots of the new game. Clealy, ths assgnment s weakly feasble fo the new game snce the weak feasblty condtons fo π ae just a subset of the ones fo Π fo the ognal game. Also, note that the optmal assgnment fo the estcted game assgns bdde k to slot k fo k = 2,..., 1 1, 1 1,..., 2 1, j 1,..., n and bdde k 1 to slot k fo k = 2 1,..., j. By the nductve hypothess, we know that the effcency of π s at most. Hence, we can bound the socal welfae of π as Wπ = a 1 v j a 1 v 1 a 2 v 1 k {1, 1, 2 } = a 1 v j a 1 v 1 a 2 v 1 Wπ a 1 v j a 1 v 1 a 2 v k= 1 1 j k= 2 1 a 1 v j a 1 v 1 a 2 v k= 1 1 j k= 2 1 k= k=2 a k v πk n k=j1 n k=j1 = a 1v j a 1 v 1 a 2 v 1 1 n a 1v 1 a 1 v 1 a 2 v 2 = a 1 v j a 1 v 1 a 2 v 1 1 a 1v 1 a 1 v 1 a 2 v 2 OPT a 1v j a 1 v 1 v 1 a 2v 1 v j a1v1 OPT and nequalty 1 follows. The fst nequalty follows by the nductve hypothess and the defnton of the optmal assgnment fo the estcted game. The second nequalty follows snce v k 1 v k fo k = 2 1,..., j. The last nequalty follows snce v 2 v j. Case II.2: 1 < 1 < 2 < j and v 2 > v j. Consde the estcton of the ognal game that conssts of the bddes

7 dffeent than 1 and 1 and the slots dffeent than 1 and 2. Let π be the estcton of π to the bddes and slots of the new game. Agan, ths assgnment s weakly feasble fo the new game snce the weak feasblty condtons fo π ae just a subset of the ones fo π fo the ognal game. Also, note that the optmal assgnment fo the estcted game assgns bdde k to slot k fo k = 2 1,..., n, bdde 1 1 to slot 1 1, and bdde k 1 to slot k fo k = 1,..., 1 2, 1 1,..., 2 1. By the nductve hypothess, we know that the effcency of π s at most. Hence, we can bound the socal welfae of π as Wπ = a 1 v 1 a 2 v 1 k { 1, 2 } a k v πk = a 1 v 1 a 2 v 1 Wπ a 1 v 1 a 2 v a 1 1v n k= 1 1 = a 1 v 1 a 2 v 1 1 k= 2 1 n a 1 1 a 1 1v k= a k a k1 v k1 a k a k1 v k1 a 1 v 1 a 1 v 1 a 1 v 1 a 2 v 1 1 a 1 1 a 1 1v k= 1 1 n 1 2 a k a k1 v 2 a k a k1 v 2 a 1 v 1 a 1 v 1 = a 1 v 1 a 2 v 1 1 a 1v 1 a 1 v 1 a 2 v 2 a 1 v 2 OPT > a 1v j a 1 v 1 v 1 a 2v 1 v j a1v1 OPT and nequalty 1 follows. The fst nequalty follows by the nductve hypothess and the defnton of the optmal assgnment fo the estcted game. The second nequalty follows snce a k a k1 and v k1 v 2 fo k = 1,..., 1 2, 1 1,..., 2 1 and a 1 1 a 1 1 and v 1 1 v 2. The last nequalty follows snce v 2 > v j, and a 1 > a 2. Due to lack of space, the thd case whee 1 < 2 < 1 < j s omtted. 4. COARSE CORRELATED EQUILIBRIA In ths secton, we pove ou uppe bound on the pce of anachy ove coase coelated equlba. Unlke the poof of pevous uppe bounds e.g., n [16], ou poof s based on takng nto account the bds n the analyss. We consde a GSP aucton game wth n slots wth clck-though ates a 1 a 2... a n and n bddes wth valuatons v 1 v 2... v n. Let b denote the andom bd vecto dawn accodng to a pobablty dstbuton that coesponds to a coase coelated equlbum. We denote by π the andom assgnment nduced by b. Also, let b π be the andom vaable denotng the -th hghest bd among all bddes. We can extend the man agument n the poof of Lemma 6 n [13] adapted to coase coelated equlba n ode to obtan the followng lowe bound on the expected utlty of each bdde at a coase coelated equlbum. Lemma 9. Consde a coase coelated equlbum and bdde. Fo any β >, t holds that E[u b] β1 e 1/β a v βe[a b π ]. We ae now eady to pove the man esult of ths secton. In ou poof, we combne the above lemma togethe wth a stonge popety concenng the bdde wth the hghest valuaton. Theoem 1. The pce of anachy ove coase coelated equlba of GSP aucton games wth consevatve bddes s at most Poof. We pove the theoem by lowe-boundng the expected utlty of each bdde at a coase coelated equlbum. Let β 1 be a paamete to be fxed late. Consde bdde 1 and he devaton to the bd v 1. Then, bdde 1 would always be allocated slot 1 and pay a 1 tmes the hghest bd among the emanng bddes whch s uppe bounded by b π1. Hence, E[u 1b] E[u 1v 1, b 1] a 1v 1 E[a 1b π1 ] β1 e 1/β a 1 v 1 E[a 1 b π1 ], 2 whee the last nequalty follows snce v 1 b π1 and snce β 1 mples that β1 e 1/β 1. Now, we wll use the fact that the socal welfae s the sum of the expected utltes of the bddes plus the total bds pad. We have βwb = Wb β 1Wb = E[u 1 b] E[ 2 β 1E[ u b] E[ a v π ] a b π1 ] β1 e 1/β a 1 v 1 β1 e 1/β E[a 1 b π1 ] β1 e 1/β 2 a v β 2 E[a b π ] 2 E[a b π ] β 1E[a 1 v π1 ] β 1 2 E[a v π ] β1 e 1/β OPT βe 1/β 1E[a 1b π1 ], whee the fst nequalty follows by the lowe bounds on the bddes utltes n Lemma 9 and nequalty 2 and the fact that a a 1 fo = 1,..., n 1, and the second nequalty follows by the defnton of OPT and snce β 1 and v π b π fo any.

8 By settng β such that βe 1/β = 1, we obtan that the pce of anachy OPT/Wb s at most β/ β 1. Ths occus fo β , whee the ato becomes THE BAYESIAN SETTING In ths secton, we pove ou uppe bound on the pce of anachy n the Bayesan settng. In ou poof, we consde a GSP aucton game wth n slots wth clck-though ates a 1 a 2... a n and n bddes wth andom valuatons v 1, v 2,..., v n. Let b denote the bd functons of the bddes at a Bayes-Nash equlbum. We denote by π the andom assgnment nduced by b and by o the andom vaable that ndcates the slot bdde occupes n the optmal assgnment; theefoe, o 1 j stands fo the bdde that occupes slot j n the optmal assgnment. Also, let b π be the andom vaable denotng the -th hghest bd among all bddes. We wll pove ou man esult by appopately combnng two lowe bounds fo the expected utltes of the bddes. In the poof of the fst lowe bound Lemma 12, we wll need the followng techncal lemma. Lemma 11. Fo any ξ and any non-negatve X, Y, t holds that X Y 2 ξ ξ2 X 2 ξxy. 4 Poof. Let fx, Y = X Y 2 ξ ξ 2 /4X 2 ξxy. It suffces to pove that fx, Y fo any ξ. Then, fx, Y = 1 ξ ξ 2 /4X 2 Y 2 2 ξxy = 1 ξ/2x Y 2. Lemma 12. Consde a Bayes-Nash equlbum. fo any γ and δ, t holds that E[u b] γ/2 γ 2 /8 OPT γ/2 Then, E[a b π ] δ δ 2 /4 γ/2 γ 2 /8 E[a 1 v o 1 1] δ γ/2 E[a 1 b π1 ]. Poof. By the defnton of the Bayes-Nash equlbum, t holds that any bdde can not ncease he expected utlty when he valuaton s v = x, by devatng to any bd b < x. We wll ague about all possble slots that bdde can be assgned to, when havng valuaton x and bddng b. Obseve that fo any bd b < x such that b > b π1, bdde s allocated to the fst slot when bddng b and pays at most a 1b π1. Smlaly, fo any bd b < x such that b > b πj, bdde s allocated to slot j o a hghe one and pays at most b pe clck. Let A j x denote the event that v = x and o = j and Bx j denote the event that o = j gven that v = x. Snce the bd b maxmzes the expected utlty of bdde, we have that E[u b v = x] E[u b, b v = x] n = E[u b, b A j P[Bx j ] j=1 a 1 x E[b π1 A 1 n P[b π1 < b A 1 P[B 1 a j x b P[bπj < b A j P[B j = a 1 x E[b π1 A 1 P[Bx 1 ] n a j x b Usng ths nequalty we have = xe[u b v = x] x a 1 x E[u b v = x] db 1 P[b π1 b A 1 1 P[b πj b A j P[Bx j ]. x E[b π1 A 1 1 P[b π1 b A 1 db P[Bx 1 ] n x a j x b 1 P[b πj b A j db P[Bx j ] = a 1 x 2 2xE[b π1 A 1 E 2 [b π1 A 1 n x 2 a j 2 xe[b πj A j 1 2 E[b2 πj A j a 1 x E[b π1 A n 2 P[B 1 2 a j x E[b πj A j P[B j, P[B 1 P[B j whee the second equalty holds snce A 1 x mples b π1 x and, hence, E[b π1 A 1 = x P[b π1 b A 1 db, and, futhemoe, snce z P[Z z] dz = 1 2 E[Z2 ], fo any andom vaable Z that takes non-negatve values. The second nequalty holds snce E[X 2 ] E 2 [X] fo any non-negatve andom vaable X. We wll now use the above nequalty and wll apply Lemma 11 to ts ght-hand sde wth X = x, Y = E[b πj A j and ξ equal to γ and δ n ode to lowe-bound E[u b v = x]. We wll also explot the fact that x = E[v A j, fo any j = 1,..., n, when P[Bx j ] >. We have E[u b v = x] δ δ 2 /4 E[a 1 v A 1 δe[a 1 b π1 A 1 P[Bx 1 ] 1 n γ γ 2 /4 E[a j v A j γe[a j b πj A j 2 P[B j. We can now bound the uncondtonal expected utlty of bdde usng ths last nequalty. We have = E[u b] E[u b v = x] P[v = x] dx

9 δ δ 2 /4 E[a 1 v A 1 P[B 1 P[v = x] dx δ E[a 1 b π1 A 1 P[Bx 1 ] P[v = x] dx γ/2 γ 2 /8 n γ/2 n E[a jv A j P[B j P[v = x] dx E[a jb πj A j P[B j P[v = x] dx = δ δ 2 /4 E[a 1 v o = 1] P[o = 1] δe[a 1 b π1 o = 1] P[o = 1] γ/2 γ 2 /8 n E[a jv o = j] P[o = j] γ/2 n E[a j b πj o = j] P[o = j] = γ/2 γ 2 /8 E[a o v ] γ/2 E[a o b πo ] δ δ 2 /4 γ/2 γ 2 /8 E[a 1v o = 1] P[o = 1] δ γ/2 E[a 1b π1 o = 1] P[o = 1], whee the second equalty holds snce E[Z A j P[B j P[v = x] dx = E[Z o = j] P[o = j], fo any non-negatve andom vaable Z. By summng ove all bddes, we have E[u b] γ/2 γ 2 /8 E[a o v ] γ/2 δ δ 2 /4 γ/2 γ 2 /8 E[a 1 v o = 1] P[o = 1] δ γ/2 = γ/2 γ 2 /8 OPT γ/2 E[a o b πo ] E[a 1 b π1 o = 1] P[o = 1] E[a b π ] δ δ 2 /4 γ/2 γ 2 /8 E[a 1 v o 1 1] δ γ/2 E[a 1 b π1 ], and the lemma follows. The next lemma povdes a second lowe bound on the sum of expected utltes. Its poof omtted due to space constants extends the aguments n [13]. Besdes the use of a techncal lemma n ode to elate the quanttes to paamete β, a subtle and athe supsng techncal pont s that the bound s obtaned by gnong possble gans the bddes may have when allocated to the slot wth the hghest clck-though ate. Ths pont s cucal n ode to obtan ou mpoved bound. Lemma 13. Consde a Bayes-Nash equlbum. Then, fo any β >, t holds that E[u b] β1 e 1/β OPT β E[a b π ] β1 e 1/β E[a 1 v o 1 1] βe[a 1 b π1 ]. We ae now eady to pove ou man esult concenng the pce of anachy fo Bayes-Nash equlba. The poof follows by appopately takng nto account the lowe bounds on the sum of expected utltes that ae poven n Lemmas 12 and 13. Theoem 14. The pce of anachy ove Bayes-Nash equlba of GSP aucton games wth consevatve bddes s at most Poof. Consde a Bayes-Nash equlbum. We wll use the fact that the socal welfae s the sum of the expected utltes of the bddes plus the total bds pad. Let β >, µ [, 1], and non-negatve γ, δ and λ be paametes to be fxed late. We have 1 λwb = µe[ µ = λe[ µe[ u b] 1 µe[ a v π ] u b] 1 µe[ u b] E[ u b] 1 λe[ 2 a b π ] λe[a 1 b π1 ] β1 e 1/β OPT β E[a b π ] β1 e 1/β E[a 1v o 1 1] βe[a 1b π1 ] 1 µ δ δ2 δ γ 2 γ 2 γ2 8 OPT γ 2 E[a 1v o 1 1] E[a b π ] 4 γ 2 γ2 8 E[a 1 b π1 ] 1 λe[ a b π ] 2 a b π1 ] λe[a 1 b π1 ] γ µβ1 e 1/β 1 µ 2 γ2 OPT 8 λ 1 µβ 1 µ γ E[a b π ] µ δ δ2 4 γ 2 γ2 µβ1 e 1/β 8 E[a 1v o 1 1] λ 1 µ δ E[a 1b π1 ]. The fst nequalty follows snce a a 1 fo = 1,..., n 1 and v π b π, and the second one follows by applyng Lemmas 12 and 13. By settng λ =.76, µ =.2, β = 5.5, γ = 1.86, and δ =.95, we have that the second, thd and fouth tem n the ght-hand sde of the above nequalty ae non-negatve and the ato OPT/Wb s then bounded by 3.37 as desed.

10 6. OPEN PROBLEMS Ou wok leaves seveal open poblems. Stll, thee s a small gap between the uppe and lowe bounds on the pce of anachy ove pue Nash equlba. Even though we have found seveal weakly feasble assgnments wth effcency hghe than n games wth moe than thee bddes, none of them coesponds to a pue Nash equlbum. So, t s nteestng to pove o dspove whethe the wost pce of anachy ove pue Nash equlba s obtaned n GSP aucton games wth just thee bddes. A poof of such a statement would eque an explct accountng of the bds and should not be based on weakly feasble assgnments. The case of moe geneal equlba s even moe challengng. Hee, thee s no known lowe bound besdes the one fo pue Nash equlba. Computng tght bounds fo mxed Nash, coelated, o coase coelated equlba ae challengng open poblems. Is the pce of anachy ove equlba n some of these classes wose than the pce of anachy ove pue Nash equlba? The same queston apples to the Bayesan settng as well. The fact obseved n [16] that GSP aucton games ae not smooth games accodng to the defnton n [18] does not peclude a negatve answe. On the postve sde, ou new uppe bound fo pue Nash equlba mght make the seach fo a game wth a stctly wose non-pue Nash equlbum ease. 7. REFERENCES [1] G. Aggawal, A. Goel, and R. Motwan. Tuthful auctons fo pcng seach keywods. In Poceedngs of the 7th ACM Confeence on Electonc Commece EC, pages 1 7, 26. [2] R. J. Aumann. Subjectvty and coelaton n andomzed stateges. Jounal of Mathematcal Economcs, 11:67 96, [3] K. Bhawalka and T. Roughgaden. Welfae guaantees fo combnatoal auctons wth tem bddng. In Poceedngs of the 22nd Annual ACM-SIAM Symposum on Dscete Algothms SODA, 211. [4] G. Chstodoulou, A. Kovács, and M. Shapa. Bayesan combnatoal auctons. In Poceedngs of the 35th Intenatonal Colloquum on Automata, Languages and Pogammng ICALP, pages , 28. [5] B. Edelman, M. Ostovsky, and M. Schwaz. Intenet advetzng and the genealzed second-pce aucton: sellng bllons of dollas woth of keywods. The Amecan Economc Revew, 971: , 27. [6] R. Gomes and K. Sweeney. Bayes-Nash equlba of the genealzed second pce aucton. In Poceedngs of the 1th ACM Confeence on Electonc Commece EC, page 17, 29. [7] J. C. Hasany. Games wth ncomplete nfomaton played by Bayesan playes, pats -. Management Scence, 14: , , , [8] E. Koutsoupas and C. H. Papadmtou. Wost-case equlba. In Poceedngs of the 16th Annual Symposum on Theoetcal Aspects of Compute Scence STACS, pages , [9] V. Kshna. Aucton Theoy. Elseve Scence, 22. [1] S. Lahae. An analyss of altenatve slot aucton desgns fo sponsoed seach. In Poceedngs of the 7th ACM Confeence on Electonc Commece EC, pages , 26. [11] S. Lahae, D. Pennock, A. Sabe, and R. Voha. Sponsoed seach auctons. In N. Nsan, T. Roughgaden, E. Tados, and V. V. Vazan, edtos, Algothmc Game Theoy, pages [12] B. Luce and A. Boodn. Pce of anachy fo geedy auctons. In Poceedngs of the 21th Annual ACM-SIAM Symposum on Dscete Algothms SODA, pages , 21. [13] B. Luce and R. Paes Leme. GSP auctons wth coelated types. In Poceedngs of the 12th Annual ACM Confeence on Electonc Commece EC, 211. Pelmnay veson: Impoved socal welfae bounds fo GSP at equlbum. axv: , 21. [14] H. Mouln and J. P. Val. Stategcally zeo-sum games: the class of games whose completely mxed equlba cannot be mpoved upon. Intenatonal Jounal of Game Theoy, 7:21 221, [15] S. Muthukshnan. Intenet ad auctons: nsghts and dectons. In Poceedngs of the 35th Intenatonal Colloquum on Automata, Languages and Pogammng ICALP, pages 14 23, 28. [16] R. Paes Leme and E. Tados. Pue and Bayes-Nash pce of anachy fo genealzed second pce auctons. In Poceedngs of the 51st Annual IEEE Symposum on Foundatons of Compute Scence FOCS, pages , 21. [17] C. H. Papadmtou. Algothms, games and the Intenet. In Poceedngs of the 33d Annual ACM Symposum on Theoy of Computng STOC, pages , 21. [18] T. Roughgaden. Intnsc obustness of the pce of anachy. In Poceedngs of the 41st Annual ACM Symposum on Theoy of Computng STOC, pages , 29. [19] D. R. M. Thompson and K. Leyton-Bown. Computatonal analyss of pefect-nfomaton poston auctons. In Poceedngs of the 1th ACM Confeence on Electonc Commece EC, pages 51 6, 29. [2] H. Vaan. Poston auctons. Intenatonal Jounal of Industal Oganzaton, 25: , 27. [21] W. Vckey. Countespeculaton, auctons, and compettve sealed tendes. The Jounal of Fnance, 161:8 37, [22] H. P. Young. Stategc Leanng and ts Lmts. Oxfod Unvesty Pess.