A Truthful Mechanism for Offline Ad Slot Scheduling

A T ruthful Mec hanism for Offline Ad Slot Sc heduling Jon F eldman 1 , S. Muth ukr ishnan 1 , Evdokia Nikolo v a 2 , and Martin P´ al 1 1 Google, Inc. Email : { jonfeld,mu thu,mpal } @google.com 2 Massac husetts Institute of T echnology ⋆ . Email: nikolova@ mit.edu Abstract. W e consider the Offline A d Slot Sche duling problem, where advertise rs must b e scheduled to sp onsor e d se ar ch slots du ring a given p eriod o f time. Adve rtisers specify a budget constrain t, as well as a max- im um cost p er clic k, and may not be assigned to more than one slot for a particular search. W e give a truthful mec hanism under the utilit y mo del where bidd ers try to maximize their clicks, sub ject to th eir p ersonal constraints. In addi- tion, w e show that the reven ue-maximizing mec hanism is not t ru thful, but has a Nash equilibrium whose outcome is identical to our mechanism. As far as w e can t ell, this is the first treatment of sp onsored search that directly incorp orates b oth multiple slots and bu dget constrain ts into an analysis of incentiv es. Our mechanism employs a descend ing-price auction that maintains a solution to a certain machine sc hed uling problem whose job lengths d e- p end on th e p rice, and hence is vari able over the auction. The price stops when th e set of bidders t h at can afford that price pack exactly into a block of ad slots, at which p oint the mec hanism allo cates that b lock and conti nues on the remaining slots. T o pro ve our result on t he equilibrium of the revenue-maximizing mec hanism, w e first show that a greedy algo- rithm suffices to solv e t he reven ue-maximizing linear program; w e then use this insigh t to prove that bidders allocated in the same blo ck of our mec hanism h ave no incentive to d ev iate from bidd in g the fixed price of that blo ck. ⋆ This w ork was done while the author w as visiting Google, Inc., New Y ork, NY. 1 In tro duction Spo nsored s earch is an inc r easingly impo rtant a dvertising medium, attra cting a wide v ariety o f advertisers, lar ge and small. When a user sends a q uery to a search engine, the advertisemen ts are placed in to slots , usually arranged linear ly down the pa ge. These slots have a v arying degree of e xpo sure, often mea sured in terms of the probability that the ad will b e c licked; a common mo del is tha t the higher ads tend to attract more clicks. The problem of allo cating these s lots to bidders has b een addressed in v arious wa ys . The mos t common metho d is to allo cate ads to each sear ch indep endently via a gener alize d se c ond pric e (GSP) auction, where the ads a re ranked b y (so me function o f ) their bid, a nd placed int o the slots in ra nk or der. (See [18 ] for a survey of this area .) There a re several imp ortant asp ects of sp onso red sear ch no t captured by the original mo dels. Most advertisers are interested in g etting many clicks thr ough- out the day on a v ariety of sea rches, not just a sp ecific slot on a particular search quer y . Also, many advertisers hav e budget constra in ts, wher e they do not allow the s e a rch eng ine to sp end mor e than their budge t during the day . Finally , search engines may have so me knowledge ab out the distribution of queries that will o ccur dur ing the day , and so s ho uld b e able to make mor e efficient allo ca tion decisions than just simple rank ing . The Offline A d Slot Sche duling problem is this: g iven a set of bidders w ith bids (per clic k) and budgets (p er day), and a set o f s lots over the entire day where we know the exp ected num b er o f clicks in each s lot, find a schedule that places bidders into slots. The s chedule must not pla ce a bidder into tw o different slots at the same time. In addition, we must find a price for each bidder that doe s not exceed the bidder’s budget constra in t, nor their p er-click bid. (See Section 1.3 for a for mal sta temen t of the problem.) A go o d algorithm for this proble m will hav e high reven ue. Also, we would like the algo rithm to b e tr u thful ; i.e., ea ch bidder will b e incented to rep or t her true bid and budget. In order to prove something like this, we need a ut ility function for the bidder that captures the deg ree to which she is ha pp y with her allo cation. Natural mo dels in this context (with clicks, bids and budgets) are click-maximization —where she wishes to ma x imize her num ber o f c licks sub ject to her p ers o nal bid and budg e t constraints, o r pr ofit-maximization —where she wishes to maximize her pro fit (clicks × pro fit p er clic k). In this pap er w e fo cus on click-maximization. 3 W e present an efficient mechanism for Offline Ad Slot Sche duling and prov e that it is truthful. W e also prov e tha t the re venue-optimal mechanism fo r Offline 3 Our choi ce is in part motiv ated by the p resence of budgets, which have a natural interpretatio n in t h is application: if an o verall advertising campaign allo cates a fi xed p ortion of its bu dget to online media, then the agent responsible for that budget is incented to sp end the entire bud get to max imize exp osure. I n contrast, under the profit-maximizing utility , a w eak motiv ation for budgets is a limit on liquidity . Also, our choice of utilit y function is out of analytical necessity: Borgs et al. [4] show th at under some reasonable assumptions, truthful mechanisms are imp ossible under a profit-maximizing utility . A d S lot Sche duling is no t truthful, but has a Nash equilibrium (under the sa me utilit y model) whose outco me is equiv alent to our mechanism; this result is strong ev idence that our mec hanis m has desirable reven ue pr op erties. Our r esults generalize to a mo del where ea ch bidder ha s a p ersona l click-thr ough-r ate that m ultiplies her click pro bability . As far as we can tell, this is the first treatment of sp onso r ed search that directly incor po rates b oth multiple p ositions and budg et co nstraints into an analysis of incentiv es (see Section 1.2 for a survey of rela ted work). In its full generality , the pr oblem of sp ons o red sea rch is mor e complex than our mo del; e.g., since the q uer y distr ibution is no isy , g o o d allo c a tion str ategies need to b e online and adaptive. Also, o ur mec hanis m is desig ned for a single query type, whereas advertisers ar e interested in enforcing their budge t a cross multiple quer y t yp es. Howev er, the to ols used in this pap er may be v a luable for der iv ing more general mechanisms in the future. 1.1 Metho ds and Results. A natur a l mechanism for Offline A d S lot Sche dul- ing is the following: find a feasible schedule and a set of prices that maximizes reven ue, s ub ject to the bidders’ constra int s. It is s traightforw ar d to der ive a linear progr am for this optimization pr oblem, but unfortunately this is not a truthful mechanism (see E xample 1 in Section 2). How ever, there is a direct truthful mechanism—the pric e-setting mechanism we present in this pap er— that res ults in the s a me outcome a s an equilibrium of the reven ue-maximizing mechanism. W e der ive this mechanism (and pr ov e that it is truthful) b y starting with the single- slot case in Section 2, wher e tw o ex treme cases hav e natural, instruc- tive interpretations. With only bids (and unlimited budgets), a winner-take-all mechanism works; with o nly budgets (and unlimited bids) the clicks are simply divided up in prop or tion to budg e ts. Combining these ideas in the r ight wa y re- sults in a natural descending- pr ice mechanism, where the pric e (pe r click) stops at the po int where the bidders who can afford that price hav e enough budget to purchase all of the clicks. Generalizing to multiple slots requires under s tanding the s tructure of feasi- ble schedules, even in the spe c ia l budgets- only case. In Section 3 we solve the budgets-only case by characterizing the allow able schedules in terms of the so- lution to a cla ssical machine sche duling pr oblem (to b e precise, the problem Q | pmtn | C max [13]). The difficulty that arises is that the le ng ths of the jobs in the scheduling pro ble m actually dep end o n the price charged. Thus, w e in- corp ora te the s cheduling algor ithm int o a descending-pr ice mechanism, where the price stops a t the p oint where the scheduling c o nstraints are tight; at this po int a blo ck of slots is allo cated at a fixed uniform price (dividing the clicks equally by budget) and the mechanism itera tes. W e pre sent the full mechanism in Section 4 by incorp or ating bids analogous ly to the single- slot case: the price descends until the set o f bidders that can afford that price has eno ugh budget to make the scheduling constraints tigh t. A tricky case arises when a new bidder app ears who s e budget vio lates the s cheduling constra int s; in this cas e the bud- get o f this “thresho ld” bidder is re duce d to make them tight again. Finally in Section 4.2 we s how that the revenue-optimal mechanism has a Nash equilibrium whose outcome is identical to o ur mechanism. This follows from the fact that if all the bidders in a blo ck decla re a bid (ro ughly) equal to the price of the blo ck, nob o dy ha s an incentiv e to deviate, since every bidder is c har ged exactly her bid, and the clicks a re div ided up equa lly by budget. 1.2 Related W ork. Ther e ar e s ome pap ers on sp onsor ed sear ch that analyze the gener alize d se c ond-pric e (GSP) auction, whic h is the auction curr ently in use at Go o g le and Y aho o . The equilibr ia of this auctio n ar e character ized and com- pared with VCG [9, 17, 2, 22]. Here the utilit y function is the pr ofit- m aximizing utilit y where each bidder attempts to maximize her c licks × pro fit per click, a nd budget constraints are genera lly not tr eated. Borgs et al. [4 ] consider the pr oblem of budget-co nstrained bidders for mul- tiple items of a s ingle type, with a utilit y function that is profit-maximizing , mo dulo b eing under the budget (b eing ov er the budget gives an unbo unded neg- ative utilit y). They give a truthful mechanism allo cating so me p ortion of the items that is reven ue-optimal, a nd prov e that in their mo del, under rea sonable assumptions, truthful mechanisms that allo cate all the units are imp ossible. Our work is different b oth b ecaus e of the different utilit y function a nd the general- ization to multiple slots with a scheduling constr aint. Using rela ted metho ds, Mahdian et al. [1 9] co nsider an online se tting w her e an unknown num b er of copies of a n item arrive online, and g ive a truthful mechanism with a constant comp etitive ratio g uarantee. There is s ome work o n algorithms for allo cating bidders with budgets to keyw ords that a rrive online, where the bidders place (p ossibly different) bids on particular keyw ords [2 0, 19]. The application of this work is similar to ours, but their concern is purely online optimizatio n; they do no t consider the g ame- theoretic a sp ects o f the allo cation. Abrams et al. [1] der ive a linear progr am for the offline optimization problem o f allo cating bidders to quer ie s, and handle m ultiple p ositions by using v ariables for “slates ” o f bidders. The ir LP is related to our s, but a gain they do not consider g ame-theoretic asp ects of their prop osed allo cations. Bidder stra teg ies for keyword auc tio ns in the pr esence of budget co nstraints hav e also b een considered [11 , 21, 6 , 5 ]. Genera lly these pap er s are not concerned with mechanism des ign, but there could be some interesting relatio ns hips b e- t ween the mo dels in these pap ers and the o ne we study here. In our setting o ne is tempted to apply a Fisher Market mo del: here m divisible go o ds ar e av aila ble to n buyers with money B i , and u ij ( x ) denotes i ’s utility of receiving x amount of go o d j . It is known [3, 10, 7] that under certain conditions a vector o f prices for go o ds exists such that the m arket cle ars , in that there is no surplus of g o ods , and all the money is sp ent . F urthermor e, this price vector can be found efficie n tly [8]. The natura l way to apply a Fisher mo del to a slot auction is to reg ard the slots as commo dities and have the utilities b e in pr o p o rtion to the num b er of clicks. How ever this be c omes problema tic be cause there do e s not seem to be a wa y to enco de the scheduling c o nstraints in the Fisher mo del; this constraint could make an apparently “ market-clearing” e quilibrium infea sible, and indeed plays a central r ole in our inv estiga tions. 1.3 Our Setting . W e define the Offline A d Slot Sche duling problem as fol- lows. W e hav e n > 1 bidders in tere sted in clicks. E ach bidder i has a budget B i and a maximum cost-p er-click (max- cpc ) m i . Given a n umber of clicks c i , and a price per click p , the utility u i of bidder i is c i if bo th the true max-cp c and the true budget are satisfied, and −∞ otherwise. In other words, u i = c i if p ≤ m i and c i p ≤ B i ; a nd u i = −∞ otherwise. W e have n ′ advertising slots where slot i receives D i clicks during the time interv al [0 , 1]. W e as s ume D 1 > D 2 > . . . > D n ′ . In a sche dule , ea ch bidder is assigned to a se t of (slot, time interv al) pa irs ( j, [ α, β )), where j ≤ n ′ and 0 ≤ α < β ≤ 1. A fe asible sche dule is one where no more than one bidder is assigned to a slot at any given time, and no bidder is assigned to more than one slot a t a ny given time. (F or mally , the interv als for a particular slot do not ov er lap, and the in terv als for a particular bidder do no t ov erlap.) A feasible sc hedule can be applied as follows: when a user que r y co mes at some time α ∈ [0 , 1], the s chedule for that time instant is used to po pulate the ad slots. If we assume that clicks co me a t a constant rate throughout the interv al [0 , 1], the num b er of clicks a bidder is exp ected to receive fro m a schedule is the sum of ( β − α ) D j ov er all pairs ( j, [ α, β )) in its schedule. 4 A me chanism for Offline A d Slot Sche duling takes a s input a declared budg et B i and decla red max-cp c (the “bid”) b i , and r eturns a fea s ible schedule, as well as a price p er click p i ≤ b i for ea ch bidder. The schedule gives some num b er c i of clicks to each bidder i that m ust r e s pe c t the budget at the given price; i.e., we ha ve p i c i ≤ B i . The r evenue of a mechanism is P i p i c i . W e say a mechanism is trut hful if it is a weakly dominant stra tegy to declare o ne ’s true budget and max -cp c; i.e., for any pa r ticular bidder i , g iven any set of bids and budgets declar ed by the other bidders, declar ing her true budget B i and max-cp c m i maximizes her utility u i . A (pure s trategy) Nash e quilibrium is a set of declared bids and budgets such that no bidder want s to change he r declar a tion of bid or budget, given tha t all other declaratio ns stay fixed. An ǫ -Nash e quilibrium is a s et o f bids and budge ts where no bidder can incr ease her utility by mor e than ǫ b y changing her bid or budget. Throughout the pap er w e a ssume some arbitr a ry lexicographic or dering on the bidder s, that do es not necess a rily match the subscripts. When we compare t wo bids b i and b i ′ we say that b i ≻ b i ′ iff either b i > b i ′ , or b i = b i ′ but i o cc ur s first lex icographica lly . 2 One Slot Case In this se c tio n we consider the case k = 1, where there is only one advertising slot, with s o me num b er D := D 1 of clicks. W e will der ive a truthful mechanism 4 All our results generalize to the setting where each bidd er i has a “clic k-through rate” γ i and receiv es ( β − α ) γ i D j clic ks (see Section 5). W e leav e this out for clarit y . for this cas e by firs t c onsidering the t wo ex tr eme cases of infinite bids a nd infinite budgets. The pr o ofs of the theorems in this section ar e in App endix A. Suppo se all budgets B i = ∞ . Then, our input amounts to bids b 1 ≻ b 2 ≻ . . . ≻ b n . Our mechanism is simply to give all the clicks to the highest bidder. W e charge bidder 1 her full price p 1 = b 1 . W e claim that rep orting the truth is a w ea kly domina n t strateg y for this mechanism. Clearly all bidders will rep ort b i ≤ m i , since the price is se t to b i if they win. The losing bidders ca nnot ga in from decreasing b i . The winning bidder can low er her price by low ering b i , but this will not g ain her any more clicks, since she is alr eady g etting all D of them. Now suppo s e all bids b i = ∞ . In this case, o ur input is just a set o f budgets B 1 , . . . , B n , and we need to allo ca te D clicks, with no ceiling on the per -click price. Her e we apply a simple r ule r elated to pricing schemes for netw ork band- width (see [16, 15]): Let B = P i B i . Now to e a ch bidder i , a llo cate ( B i / B ) D clicks. Set all pr ices the same: p i = p = B /D . The mec hanis m g uarantees that each bidder exactly sp ends her budget, thus no bidder will rep or t B ′ i > B i . Now suppo se s o me bidder r ep orts B ′ i = B i − ∆ , for ∆ > 0 . Then this bidder is allo- cated D ( B i − ∆ ) / ( B − ∆ ) clicks, whic h is le ss than D ( B i / B ), since n > 1 a nd all B i > 0. 2.1 Greedy First-Price Mecha nis m. A natura l mechanism for the genera l single-slot ca se is to solve the asso cia ted “fractional k napsack” pr o blem, and charge bidders their bid; i.e., s tarting with the highest bidder, g reedily add bid- ders to the allo catio n, charging them their bid, until all the clicks are allo c ated. W e r efer to this as the gr e e dy first-pric e (GFP) mechanism. Though natural (and reven ue-maximizing as a function of bids) this mechanism is easily s een to be not truthful: Example 1. Supp ose there are tw o bidders and D = 120 clicks. Bidder 1 has ( m 1 = $2, B 1 = $100) and bidd er 2 has ( m 2 = $1, B 2 = $50). In the GFP mechanism, if b oth bidders tell the truth, then bidder 1 gets 50 clic ks for $2 each, and 50 of the remaining 70 clic k s go to bidder 2 for $1 each. Ho wev er, if bidd er 1 instead declares b 1 = $1 + ǫ , then she gets (roughly) 100 clic ks, and bidder 2 is left with (roughly) 20 clic ks. The pro blem here is that the high bidders can get awa y with bidding low er, th us g etting a lower price. The differenc e betw een this and the unlimited-budget case ab ove is that a low er price now res ults in more clicks. It turns o ut that in equilibrium, this mechanism will result in an a llo cation where a prefix of the top bidders a re allo cated, but their prices equalize to (roughly) the low est bid in the prefix (a s in the ex a mple ab ove). 2.2 The Price-Setting Mechanism. An equilibrium allo cation of GFP ca n be computed dir e ctly via the following mechanism, w hich we refer to as the pric e-setting (PS) me chanism . Essentially this is a descending price mechanism: the price stops descending when the bidders willing to pay at that price have enough budget to purchase a ll the clicks. W e hav e to be careful at the moment a bidder is added to the po ol of the willing bidder s; if this new bidder has a large e no ugh budget, then suddenly the willing bidders hav e mor e than eno ugh budget to pay for all o f the clicks. T o co mpens a te, the mechanism decrea ses this “threshold” bidder’s e ffective budget until the clicks are paid for exa ctly . W e formalize the mechanism as follows: Price-Setting (PS) Mec hanism (Single Sl ot) • Ass ume wlog that b 1 ≻ b 2 ≻ . . . ≻ b n ≥ 0. • Let k b e the firs t bidder such that b k +1 ≤ P k i =1 B i /D . Compute price p = min { P k i =1 B i /D , b k } . • Alloc a te B i /p clicks to each i ≤ k − 1. Allo ca te ˆ B k /p clicks to bidder k , where ˆ B k = pD − P k − 1 i =1 B i . Example 2. Supp ose there are three bidders with b 1 = $2, b 2 = $1, b 3 = $0 . 25 and B 1 = $100, B 2 = $50, B 3 = $80, and D = 300 clicks. Runnin g th e PS mechanism, we get k = 2 since B 1 /D = 1 / 3 < b 2 = $1, bu t ( B 1 + B 2 ) /D = $0 . 50 ≥ b 3 = $0 . 25. The price is set to min { $0 . 50 , $1 } = $0 . 50, and bidders 1 and 2 get 200 and 100 clicks at that price, resp ectively . There is n o th reshold bidder. Example 3. Supp ose now bidder 2 c han ges her bid to b 2 = $0 . 40 ( everything else remains the same as Example 2). W e still get k = 2 since B 1 /D = 1 / 3 < b 2 = $0 . 40. But now the price is set to min { $0 . 50 , $0 . 40 } = $0 . 40, and bidders 1 and 2 get 250 and 50 clic ks at that price, resp ective ly . Note that bidder 2 is now a threshold bidd er, does not use her entire budget, and gets few er clic ks. Note that this mechanism r educes to the given mec hanisms in the sp e- cial cases of infinite bids or budgets (with the prop er treatment of infinite bids/budgets). Theorem 1. T he pric e-setting me chanism (single slot) is truthful. 2.3 Price-Setting Mecha ni sm Computes Nash Equili brium of GFP . Consider the greedy first-price auction in which the highest bidder receives B 1 /b 1 clicks, the seco nd B 2 /b 2 clicks and so o n, until the supply of D clicks is exhausted. It is immediate that truthfully rep or ting budgets is a domina nt stra teg y in this mechanism, since when a bidder is considered, her r ep orted budg et is exhausted as muc h a s p ossible, at a fixed pr ice. How ever, rep or ting b i = m i is not a dom- inant str a tegy . Nevertheless, it turns out that GFP has an equilibrium whose outcome is (roughly ) the sa me as the PS mechanism. One c a nnot show that there is a plain Nash equilibrium beca use of the wa y ties are r esolved lexicogr aphically; the following example illustrates why . Example 4. Supp ose w e hav e th e same instance as example 1: tw o b id ders, D = 120 clic ks, ( m 1 = $2, B 1 = $100) and ( m 2 = $1, B 2 = $50). But now supp ose that bidder 2 occurs first lexicographicall y . In GFP , if bidder 2 tells th e truth , and bidder 1 d eclares b 1 = $1, then bidder 2 will get chosen first (since she is fi rst lexicographically), and take 50 clic ks. Bidder 2 will en d up with the remaining 70 clicks. How ever, if bidder 1 instead declares b 1 = $1 + ǫ for some ǫ > 0, th en she gets 100 / (1 + ǫ ) clicks. But this is not a b est response, since sh e could bid 1 + ǫ/ 2 and get slightly more clicks. Thu s, we prove ins tead that the bidder s r e a ch an ǫ -Nash equilibrium: Theorem 2. Supp ose the PS me chanism is run on the truthful input, r esulting in pric e p and clicks c 1 , . . . , c n for e ach bidder. Then, for any ǫ > 0 ther e is a pur e-str ate gy ǫ -Nash e quilibrium of the GFP me chanism wher e e ach bidder r e c eives c i ± ǫ clicks. 3 Multiple Slots: Bids or Budgets Only Generalizing to m ultiple slots makes the scheduling constraint nontrivial. Now instead of splitting a p o ol o f D clicks a rbitrarily , we need to assig n clic ks that corres p o nd to a fea s ible schedule of bidders to s lots. The co nditions under which this is p ossible a dd a complexity that we characterize and incorp ora te int o our mechanism in this section. As in the s ing le-slot case it will b e ins tr uctive to consider first the cases of infinite bids or budgets . Supp o se all B i = ∞ . In this cas e , the input consists of bids only b 1 ≻ b 2 ≻ . . . ≻ b n . Naturally , what we do here is rank by bid, a nd allo cate the slo ts to the bidder s in that o rder. Since each budget is infinite, we can always set the prices p i equal to the bids b i . By the sa me logic a s in the single-slot case, this is easily seen to b e truthful. In the other case, when b i = ∞ , there is a lo t more work to do, and we devote the r e mainder of the section to this case. Without los s of generality , we may assume the num be r o f slo ts equals the nu mber of bids (i.e., n ′ = n ); if this is not the case, then we add dummy bidders with B i = b i = 0, or dumm y slots with D i = 0, as appropr iate. W e keep this assumption for the r e ma inder o f the pap er. The pr o ofs of the theorems in this section a re in Appendix B. 3.1 Assigning slots using a classical sc heduling algorithm. First we give an imp ortant le mma that characterizes the conditions under which a set of bidders can b e a llo cated to a set of slots, which turns out to be just a res tatement of a classical result [1 4] fr o m scheduling theor y . Lemma 1. S upp ose we would like to assign an arbitr ary s et { 1 , . . . , k } of bidders to a set of slots { 1 , . . . , k } with D 1 > . . . > D k . Then, a click al lo c ation c 1 ≥ ... ≥ c k is fe asible iff c 1 + . . . + c ℓ ≤ D 1 + . . . + D ℓ for al l ℓ = 1 , ..., k . (1) Pr o of. In scheduling theo ry , we s ay a job with servic e r e quir ement x is a ta s k that needs x/s units of time to complete on a machine with sp e e d s . The question of whether there is a feasible allo ca tion is equiv alent to the following scheduling problem: Given k jobs with serv ice req uirements x i = c i , and k machines with sp eeds s i = D i , is there a schedule of jobs to machines (with preemption allowed) that completes in one unit of time? As shown in [14], the optimal schedule for this problem (a.k.a . Q | pmtn | C max ) can b e fo und efficiently by the level algorithm , 5 and the s chedu le completes in time ma x ℓ ≤ k { P ℓ i =1 x i / P ℓ i =1 s i } . Thus, the conditions of the lemma ar e exa ctly the conditions under which the schedule completes in one unit of time. ⊓ ⊔ 5 In later w ork, Gonzalez and Sahni [12] giv e a faster ( linear-time) algorithm. 3.2 A multiple-slot budge ts -only me c hanism. Our mechanism will roughly be a desce nding-price mechanism where we decrease the price until a prefix o f budgets fits tightly into a prefix of p ositions at that price, whereupo n we allo cate that prefix, and contin ue to decr ease the price for the r emaining bidders . The following subroutine, which will b e us ed in our mechanism (and later in the general mechanism), takes a set o f budgets and determines a prefix of po sitions that can b e pack ed tig htly with the lar gest budg ets at a unifor m pr ice p . The r outine ensures that all the clicks in thos e pos itio ns are s old at price p , and a ll the allo cated bidders sp end their budget exac tly . Routine “Fi nd-Price-Blo ck” Input: Set of n bidders, set of n slots with D 1 > D 2 > . . . > D n . • If a ll D i = 0, assign bidder s to slots arbitrar ily and exit. • So rt the bidders by budget and a ssume wlo g that B 1 ≥ B 2 ≥ ... ≥ B n . • Define r ℓ = P ℓ i =1 B i / P ℓ i =1 D i . Set price p = max ℓ r ℓ . • Let ℓ ∗ be the largest ℓ s uch that r ℓ = p . Allo cate slots { 1 , . . . ℓ ∗ } to bidders { 1 , . . . , ℓ ∗ } a t price p , using all o f their budgets; i.e., c i := B i /p . Note that in the last step the allo ca tion is alwa ys p os sible since for a ll ℓ ≤ ℓ ∗ , we hav e p ≥ r ℓ = P ℓ i =1 B i / P ℓ i =1 D i , which rew r itten is P ℓ i =1 c i ≤ P ℓ i =1 D i , and so we can apply Lemma 1. Now w e ar e ready to g ive the mec hanism in terms o f this subroutine; an e x ample run is shown in Fig ur e 1 . Price-Setting Mec hanism (M ultiple Slots , Budg ets Only) • Run “Find-Price - Blo ck” on bidders 1 , . . . , n , and slots 1 , . . . , n . This gives an allo cation of ℓ ∗ bidders to the first ℓ ∗ slots. • Rep eat o n the remaining bidder s and slots un til a ll slots ar e allo c ated. Let p 1 , p 2 , . . . b e the prices used for e a ch successive blo ck a ssigned by the a l- gorithm. W e cla im that p 1 > p 2 > . . . ; to see this, note then whe n p 1 is s et, we hav e p 1 = r k and p 1 > r ℓ for all ℓ > k , where k is the las t bidder in the blo ck. Thus for all ℓ > k , w e have p 1 P j ≤ ℓ D j > P i ≤ ℓ B j , which gives p 1 P k P k p 2 > . . . . Theorem 3. T he pric e-setting me chanism (multiple slots, budgets only) is truth- ful. 4 Main R esults In this section we g ive our main res ults, pr esenting our price- setting mec hanism in the gener al case, building on the ideas in the previo us tw o sections. W e b egin in Section 4.1 by stating the mechanism and showing some examples, then proving that the mech a nis m is truthful. In Section 4 .2 we analyze the reven ue-optimal P S f r a g r e p l a c e m e n t s Bidder Budget 1 2 3 4 $80 $70 $20 $1 3 / 5 2 / 5 20 / 21 1 / 2 1 D 1 = 100 D 2 = 50 D 3 = 25 D 4 = 0 p 1 = $1 . 00 p 2 = $0 . 84 Fig. 1. An example of t he PS mechanism (multiple slots, bu dgets only). W e ha ve four slots with D 1 , . . . , D 4 clic ks as shown, and four bidd ers with d eclared bud gets as shown. The first application of Fin d -Price-Block computes r 1 = B 1 /D 1 = 80 / 100, r 2 = ( B 1 + B 2 ) / ( D 1 + D 2 ) = 150 / 150, r 3 = ( B 1 + B 2 + B 3 ) / ( D 1 + D 2 + D 3 ) = 170 / 1 75, r 4 = ( B 1 + B 2 + B 3 + B 4 ) / ( D 1 + D 2 + D 3 + D 4 ) = 171 / 175. Since r 2 is largest, the top t wo slots make up th e first price blo ck with a price p 1 = r 2 = $1; bidder 1 gets 80 clic k s and b idder 2 gets 70 clicks, using the schedule as shown. In the second price block, w e get B 3 /D 3 = 20 / 25 and ( B 3 + B 4 ) / ( D 3 + D 4 ) = 21 / 25. Th us p 2 is set to 21 / 25 = $0 . 84, bid d er 3 gets 500 / 21 clic ks and bidder 4 gets 25 / 21 clicks, using the sc hed u le as sho wn. schedule, and show that it ca n be co mputed with a g eneralization of the gr e e dy first-pric e (GFP) mec hanism. W e then show that GFP has an ǫ -Nash equilibrium whose o utcome is identical to the general PS mechanism. The pro ofs of the theorems in this section are in Appendix C. 4.1 The Price-Setti ng Mec hanism (General Case). The genera lization of the P S mechanism combines the ideas from the bids-and-budgets v er sion of the single slot mec hanis m with the budgets- only version of the multiple-slot mechanism. As our price de s cends, we maintain a set of “a ctive” bidder s with bids at or a bove this price, a s in the single-slo t mechanism. These active bidders are kept ranked by budget , and when the price rea ches the po int w he r e a prefix of bidders fits into a prefix of slots (as in the budgets-o nly mec hanism) we allo ca te them and rep eat. As in the single-slot case, we heav e to be careful when a bidder ent er s the active se t and suddenly causes an ov er-fit; in this case w e aga in reduce the budget of this “ threshold” bidder un til it fits. W e for malize this as follows: Price-Setting Mec hanism (Gene ral Case) (i) Assume wlog that b 1 ≻ b 2 ≻ . . . ≻ b n = 0. (ii) Le t k be the fir s t bidder such that running Find-Price-Blo c k on bidders 1 , . . . , k would result in a price p ≥ b k +1 . (iii) Reduce B k un til r unning Find-Price- B lo ck on bidders 1 , . . . , k would result in a price p ≤ b k . Apply this a llo cation, which for some ℓ ∗ ≤ k gives the first ℓ ∗ slots to the ℓ ∗ bidders among 1 . . . k with the la rgest budgets. (iv) Repe a t on the remaining bidders and slots. An exa mple run of this mechanism is shown in Figure 2. Since the PS mechanism sets prices p er slot, it is natura l to a sk if these prices co nstitute some so rt of “market-clearing” equilibrium in the spirit of a Fis her market. The quick a nswer is no: since the price p er click increases for higher s lots, and each bidder v alues clicks a t each slot eq ua lly , then bidder s will always prefer the b ottom slot. Note that b y the same logic a s the budgets-only mechanism, the prices p 1 , p 2 , . . . for each price blo c k strictly decreas e. P S f r a g r e p l a c e m e n t s Bidder Budget Bid 1 2 3 4 $3 $0 . 75 $1 $0 . 50 $80 $70 $20 $1 29 / 45 1 6 / 4 5 D 1 = 100 D 2 = 50 D 3 = 25 D 4 = 0 p 1 = $0 . 80 p 2 = $0 . 75 p 3 = $0 Fig. 2. Consider the same b idders and slots as in Figure 1, but n o w add bids as shown. Running Find-Price-Blo ck on only bidder 1 gives a price of r 1 = 80 / 10 0, which is less than the n ext bid of $1. So, w e run Find-Price-Block on b idders 1 and 3 (the next- highest bid), giving r 1 = 80 / 100 and r 2 = 100 / 150. W e still get a price of $0 . 80, but now th is is more than the next-h ighest bid of $0 . 75, so w e allocate the first bidder to the fi rst slot at a price of $0 . 80. W e are left with b idders 2-4 and slots 2-4. With just bidder 3 (t h e highest bidd er) and slot 2, we get a price p = 20 / 50 whic h is less than the next-highest bid of $0 . 75, so we consider bidders 2 and 3 on slots 2 and 3. This giv es a p rice of max { 70 / 50 , 90 / 7 5 } = $1 . 40, which is more than $0 . 50. Since this is also m ore than $0 . 75, we must low er B 2 until the price is exactly $0 . 75, which makes B ′ 2 = $36 . 25 . With this setting of B ′ 2 , Find-Price-Block allocates bidders 2 and 3 to slots 2 and 3, giving 75(36 . 25 / 56 . 25) an d 75(20 / 56 . 25 ) clicks resp ectivel y , at a price of $0 . 75 p er clic k. Bidder 4 is allo cated to slot 4, receiving zero clic ks. Efficiency . So far we hav e b een lar gely igno ring the efficiency of co mputing the allo ca tion in the PS mechanism. It is immediately clear that the general PS mechanism can b e executed in time p olynomial in n and log(1 /ǫ ) to some precision ǫ using binary search and linear pr ogra mming . In fact, a purely com binato r ial O ( n 2 ) time a lgorithm is p o s sible. As bidder s get added in step (ii), maint a ining a sor ted list of bidder s and budgets can b e done in time O ( n log n ). Th us it remains to show that running Find-P r ice-Blo ck (and computing the r educed budget) can b e done in O ( n ) time given these sorted lists. In Find-Price-Blo ck, computing the r atios r ℓ can b e done in linear time. Finding the allo c a tion fro m Lemma 1 can also b e done in linear time using the Gonzalez-Sahni alg orithm [12] for sc heduling rela ted parallel machines (in fact the total time for scheduling can b e made O ( n ) s ince each s lot is scheduled only once). Finally , computing the re duce d budget is a simple calc ulation on each relev ant ra tio r ℓ , also doable in linear time. W e susp ect that there is a O ( n · p o ly log( n )) a lgorithm using a mo r e elab orate data structure; we leave this op en. Theorem 4. T he pric e-setting me chanism (gener al c ase) is tru thful. 4.2 Greedy First-Price Mec hanis m for Mul tiple Slots. In the genera l case, as in the single-slot case , there is a na tural gr e e dy first- pric e mechanism when the bidding lang uage includes both bids a nd budgets: O rder the bidder s by bid b 1 ≻ b 2 ≻ . . . ≻ b n . Star ting from the hig hest bidder, for ea ch bidder i compute the maximum p oss ible num b er of clic ks c i that one co uld a llo cate to bidder i at pric e b i , given the budget co nstraint B i and the commitments to previous bidders c 1 , . . . , c i − 1 . This reduces to the “fractiona l kna psack” pro blem in the single-slo t ca se, and so one would hop e that it maximizes r even ue for the given bids and budg ets, as in the sing le-slot case. This is no t immediately clear , but do e s turn out to b e true, as we will prove in this s e ction. As in the single- slot case, the gree dy mechanism is no t a truthful mecha- nism. How ever, we show that it do es hav e a pure-stra tegy eq uilibr ium, a nd that equilibrium has prices and allo cation equiv alent to the price setting mec hanis m. Greedy is Rev enue-Maximizing. Consider a r even ue-maximizing schedule that resp ects b oth bids and budgets. In this allo c a tion, w e ca n a ssume wlog that each bidder i is c har ged exactly b i per click, since otherwise the allo cation can increase the price for bidder i , reduce c i and remain feasible. Th us, by Lemma 1 , we can find a reven ue-ma x imizing schedule c ∗ = ( c ∗ 1 , . . . , c ∗ n ) by ma ximizing P i b i c i sub ject to c i ≤ B i /b i and c 1 + . . . + c ℓ ≤ D 1 + . . . + D ℓ for all ℓ = 1 , ..., n. Theorem 5. T he gr e e dy first-pric e auction gives a r evenue-maximizing sche d- ule. Price-Setting Mecha ni sm is a Nash E quilibrium of the Greedy First Price Mec hanism. W e note that truthfully r ep o rting one’s budg et is a weakly dominant strategy in GFP , since when a bidder is considered for a llo cation, their budget is exha usted at a fixed pr ice, s ub ject to a c a p on the num ber of clicks they can get. Rep or ting one’s bid truthfully is not a dominant stra tegy , but we can still show that there is an ǫ - Nash equilibr ium who se o utcome is arbitr arily close to the P S mechanism. Theorem 6. Supp ose the PS me chanism is run on the truthful input, r esulting in clicks c 1 , . . . , c n for e ach bidder. Then, for any ǫ > 0 ther e is a pur e-stra te gy ǫ -Nash e quilibri um of the GFP me chanism wher e e ach bidder r e c eives c i ± ǫ clicks. 5 Conclusions In this pap er we hav e given a truthful mechanism for a ssigning bidders to click- generating slots that resp ects budg e t and per -click price constr a int s. The mech- anism also resp ects a sc heduling c o nstraint on the slots, using a classica l re sult from scheduling theory to characterize (and compute) the p ossible allo c ations. W e hav e also proved that the reven ue-maximizing mechanism has an ǫ -Nas h equilibrium who se outcome is a rbitrarily clos e to o ur mechanism. This fina l re- sult in some w ay sugg ests that o ur mech a nis m is the rig ht o ne fo r this mo del. It would interesting to make this more formal; we conjecture that a genera l truthful mechanism cannot do better in terms of reven ue. 5.1 Extensions. There a re several natural g eneralizatio ns of the Online A d Slot Sche duling pr oblem where it w ould b e interesting to extend our r esults or apply the k nowledge gained in this pa per . W e mention a few her e. Click-thr ough r ates. In sp o ns ored search (e.g. [9]) it is common for each bidder to have a per sonal click-through-r ate γ i ; in our mo del this would mean that a bidder i assigned to s lo t j for a time p er io d of length α would receive αγ i D j clicks. All our r esults ca n b e generalized to this setting by simply scaling the bids using b ′ i = b i γ i . How ever, our mechanism in this ca se do es not necessar ily pre fer more efficient solutions; i.e., ones that g enerate more overall clicks. It would b e int er e sting to a nalyze a p os sible tradeoff b etw een efficiency and r even ue in this setting. Multiple Keywor ds. T o mo del multiple keywords in o ur mo del, we co uld say that each query q had its own set of click totals D q, 1 . . . D q,n , and each bidder is interested in a subset o f queries. The greedy first-pr ice mech a nis m is eas ily generalized to this cas e: ma ximally allo ca te clicks to bidder s in order of their bid b i (at price b i ) while resp ecting the budgets, the query prefer ences, and the click commitment s to previo us bidders. It would not b e surpr ising if there was an equilibrium o f this extension of the greedy mechanism that could b e computed directly w ith a generaliza tio n o f the PS mechanism. Online queries, u nc ertain supply. In sp onsor ed sea r ch, a llo cations must be made online in resp o nse to user q ueries, and some o f the previous literature has fo cus e d on this asp ect of the problem (e.g., [20 , 19]). Perhaps the ideas in this pap er could be used to help make o nline allo cation de c isions using (unre lia ble) estimates of the supply , a setting consider ed in [19], with game-theo retic consider ations. Ac kno wledgm en ts. W e thank Cliff Stein a nd Yishay Mansour for helpful dis - cussions. References 1. Zo¨ e Abrams, O fer Mendelevitch, and John T omlin. Optimal d eliv ery of sp onsored searc h advertisemen ts sub ject to bu d get constraints. In ACM Confer enc e on Ele c- tr onic Commer c e , pages 272–278, 2007. 2. Gagan Aggarw al, Ashish Go el, and Ra jeev Mot wani. T ru thful au ct ions for pricing searc h keyw ords. In EC ’ 06: Pr o c e e dings of the 7th ACM c onfer enc e on Ele ctr onic c ommer c e , pages 1–7, New Y ork , NY, USA, 2006. ACM. 3. K enneth J. Arrow and Gerard D ebreu. Existence of an equilibrium for a comp et- itive econom y . Ec onometric a , 22(3):265–290, July 1954. 4. Christian Borgs, Jennifer Chay es, Nicole Immorlica, Mohammad Mahdian, an d Amin S ab eri. Multi-unit auctions with budget-constrained bidders. In EC ’ 05: Pr o c e e dings of the 6th ACM c onfer enc e on Ele ctr onic c ommer c e , pages 44–51 , New Y ork, NY, USA, 2005. ACM. 5. Christian Borgs, Jennifer T. Chayes , Nicole Immorlica, Kamal Jain, Omid Etesami, and Mohammad Mahdian. Dy n amics of bid opt imization in online advertisemen t auctions. In Carey L. Williams on, Mary Ellen Zu rko, Peter F. Pa tel-Schneider, and Prashan t J. Shen o y , editors, W WW , pages 531–540 . AC M, 2007. 6. Matthew Cary , Aparna Das, Benjamin Edelman, Ioann is Giotis, Kurtis Heimerl, Anna R . K arlin, Claire Mathieu, and Michael Schw arz. Greedy b idding strategies for keyw ord auctions. In ACM Confer enc e on Ele ctr onic Commer c e , pages 262–271, 2007. 7. X iaotie D eng, Christos Papadimitriou, and Shmuel Safra. On the complexity of equilibria. In STOC ’02: Pr o c e e di ngs of the thiry-fourth annual ACM symp osium on The ory of c omputing , pages 67–71, N ew Y ork, N Y, USA, 2002. AC M. 8. N ikhil R. Dev anur, Christos H. Pa padimitriou, Amin Sab eri, and Vija y V . V azirani. Mark et equilibrium v ia a primal-dual-type algorithm. In FOCS ’02: Pr o c e e dings of the 43r d Symp osium on F oundations of Com puter Scienc e , pages 389–395 , W ash- ington, DC, USA, 2002. IEEE Computer So ciet y . 9. Benjamin Edelman, Mic hael Ostrovsky , and Michael Schw arz. Internet advertis- ing and the generalized second-price auction: Selling billions of dollars worth of keyw ords. Amer i c an Ec onomic R eview , 97(1):242–25 9, March 2007. a v ailable at http://idea s.rep ec.org/a/aea/ aecrev/v97y2007i1p242-259.html. 10. Edmund Eisenberg and David Gale. Consensus of su b jective probabilities: The pari-mutuel metho d. Annals Of Mathematic al Statistics , 30(165):165– 168, March 1959. 11. Jon F eldman, S Muthukrishnan, Martin Pal, and Cliff Stein. Budget optimization in searc h-b ased advertising auctions. In EC ’ 07: Pr o c e e dings of the 8th ACM c onfer enc e on El e ctr onic c ommer c e , p ages 40–49, New Y ork, NY, U SA, 2007. ACM. 12. T eofilo Gonzalez and Sarta j Sahni. Preemptive sc heduling of uniform pro cessor systems. J. ACM , 25(1):92–101, 1978. 13. R.L. Graham, E.L. Lawler, J.K. Lenstra, and A.H.G. Rin n ooy Kan. Optimization and appro ximation in deterministic sequencing and scheduling: a survey . Ann . Discr ete Math. , 4:287–326, 1979. 14. Edward C. Horv ath, S hui Lam, and Ravi Sethi. A level algorithm for preempt ive sc hed u ling. J. ACM , 24(1):32–4 3, 1977. 15. Ramesh Johari and John N . Tsitsiklis. Efficiency loss in a netw ork resource allo- cation game. Math. Op er. R es. , 29(3):407–4 35, 2004. 16. F rank P . Kelly . Charging and rate control for elastic traffic. Eur op e an T r ansactions on T ele c ommuni c ations , 8:33–37, January 1997. 17. S´ ebastien Lahaie. An analysis of alternative slot au ct ion designs for sponsored searc h. In EC ’06: Pr o c e e dings of the 7th ACM c onfer enc e on Ele ctr onic c ommer c e , pages 218–22 7, N ew Y ork, NY, USA, 2006. ACM. 18. S´ ebastien Lahie, David P enn ock, Amin Sab eri, and Rakesh V ohra. Sp onsor e d Se ar ch Auct ions, in: Algorithmic Game The ory , p ages 699–716. Cam birdge Un i- versi ty Press, 2007. 19. Mohammad Mahdian, Hamid N azerzadeh, and Amin Sab eri. Allocating online advertise ment space with un reliable estimates. In ACM Confer enc e on Ele ctr onic Commer c e , pages 288–29 4, 2007. 20. Aranyak Mehta, Amin S ab eri, Umesh V azirani, and Vija y V azira ni. Adwo rds and generalized on - line matching. In FOCS ’05: Pr o c e e dings of the 46th Annual IEEE Symp osium on F oundations of Computer Scienc e , pages 264–273, W ashington, DC, USA, 2005. IEEE Computer So ciet y . 21. Paat Rusmevichien tong an d Da v id P . Williamson. An adaptive algorithm for se- lecting p rofitable keyw ords for search-based advertising services. In EC ’06: Pr o- c e e di ngs of the 7th ACM c onfer enc e on El e ctr onic c ommer c e , p ages 260–269 , New Y ork, NY, USA, 2006. ACM. 22. Hal V arian. Pos ition auctions. Inter national Journal of Industrial Or ganization , 25(6):1163 –1178, D ecemb er 2007. A Pro ofs for Section 2 Pr o of of The or em 1: F o r the purp oses of this pro of, let bidders { 1 , . . . , n } be such that b 1 ≻ . . . ≻ b n = 0 , and consider a new bidder (call her Alice) with true max-cp c m a nd true budge t B ∗ . W e first show tha t r ep orting the true budg et is a weakly domina n t str ategy for Alice, for any fixed bid b > 0. Let ℓ b e the first bidder with b ≻ b ℓ , so b 1 ≻ . . . ≻ b ℓ − 1 ≻ b ≻ b ℓ ≻ . . . ≻ b n . Let B = P ℓ − 1 i =1 B i . If B ≥ bD then the mec hanism will not a llo cate a ny clicks to Alice, regar dless of the rep orted budget, since the price will stop b efore reaching b . If B < bD , w e will argue that Alice’s clicks c a re non-increas ing in B . Define ˆ B = b D − B > 0. – If Alice declare s B ∈ [ ˆ B , ∞ ], then the price will sto p at b . She will s p end ˆ B and receive c = ˆ B / b clicks. – If Alice de c lares B ∈ [0 , ˆ B ), then the price will b e low er than b , and she will sp end all o f her budget. Her final num b er of clicks will b e c = ( B / ( B + B + R )) D , where R is the tota l sp end of bidder s { ℓ, . . . , n } . Since R is non-increa sing in B , we ca n conclude tha t c is no n- decreasing in B . Putting together these interv als, we s e e that c is non- decreasing in B overall, and since Alice’s total spend is min { B , ˆ B } , we may conclude that it is weakly dominant to declare B = B ∗ . It remains to show that it is weakly dominant for Alice to decla r e a bid b = m , given that she declares a budget B = B ∗ . Let R ( b ) b e the total sp end o f bidders { 1 , . . . , n } given that Alice dec lares b . Note that R ( b ) is non-increasing in b . Le t p 1 be the price that would r esult if b = ∞ , and let p 2 be the price that would result if b = 0. Note that p 2 ≤ p 1 . – If b ∈ [0 , p 2 ) then the pr ice sto ps at p 2 and Alice receives zero clicks. – If b ∈ ( p 1 , ∞ ], then the price sto ps at p 1 , and Alice r eceives B /p 1 clicks. – If b ∈ [ p 2 , p 1 ], then the price stops at b . T o see this, note that if Alice ha d bid zero, then the pr ice w ould hav e gone down to p 2 , so it cer tainly stops at b or low er. But at price b , the set of bidders that ca n afford this price consists of at lea s t all the bidders that co uld affo r d price p 1 , and so we must hav e B + P i : b i ≻ b B i ≥ B + P i : b i ≥ p 1 B i ≥ p 1 D ≥ bD . Alice thus receives max  0 , D −  X i : b i ≻ b B i /b  (2) clicks, and we may conclude that in this interv a l, clic ks ar e non-dec r easing with b . Note that in the expressio n (2), plugg ing in p 1 for b yields c = B /p 1 . Thus we hav e that in the int er v al [ p 2 , ∞ ], clicks are non-decrea sing with b , and the pr ice is alwa ys min { b, p 1 } . W e co nclude that bidding b = m is a weakly dominant strategy . ⊓ ⊔ Pr o of of The or em 2: W e will show that for sufficiently sma ll ǫ ′ > 0, if each bidder truthfully r epo rts her budget and bids b i = min { m i , p + ǫ ′ } in the GFP mechanism, then the c o nditions in the theo rem ho ld. There a re tw o ways that the PS mechanism (under truthful input) can rea ch its last allo cated bidder k and final price p : if m k > p ≥ m k +1 and then pD = P k i =1 B i (no threshold bidder), or if p = m k ( k is a thresho ld bidder). In the first case, we have that bidder s i ≤ k all have m i > p . Thus in the suppos ed equilibrium o f GFP , all these bidders are bidding p + ǫ ′ , and all bidders i > k are bidding m i ≤ p . T he r efore in GFP , each i ≤ k will receive B i / ( p + ǫ ′ ) clicks, and the total num b er of clicks allo cated by GFP to bidder s 1 . . . k is P i ≤ k B i / ( p + ǫ ′ ) = ( p p + ǫ ′ ) D . The remaining D ′ = (1 − p p + ǫ ) D clicks, ar e allo cated to bidders i > k . Bidders 1 . . . k lose clicks by incr easing their bid, and can g ain at most D ′ clicks b y low ering their bid. Bidders i > k will never raise their bid (since they are bidding m i ), and cannot gain more clicks by lowering their bid. Since D ′ can b e ma de arbitr arily small, we hav e an ǫ -Nash equilibrium. In the s econd case, p = m k . Let k ′ < k be the la st bidder bidding mo re than p . In the supp osed GFP equilibrium, bidders 1 . . . k ′ are bidding p + ǫ ′ , and bidders ( k ′ + 1 , . . . , k ) are bidding m k = p . Th us GFP allo cates B i / ( p + ǫ ′ ) clicks to bidders 1 . . . k ′ , B i /p c licks to bidders ( k ′ + 1 , . . . , k − 1) (if any such bidders exist) and the re maining clicks to bidder k . As in the previous cas e , no bidder can gain from raising her bid, the num b er of clicks that a bidder i ≤ k ′ can gain from low ering her bid can b e made arbitra rily small, and no o ther bidder can gain from lowering her bid. ⊓ ⊔ B Pro ofs for Section 3 Lemma 2. In Find-Pric e-Blo ck, if B i = B i +1 , then i c annot b e the last slot of the c ompute d pric e blo ck. Pr o of. Suppo se the contrary , namely that i is the last slot of the firs t price blo ck and ( i + 1) is the fir st slot in the second price block. Denote B = B 1 + ... + B i − 1 and D = D 1 + ... + D i − 1 . T he n the price of the first price blo ck sa tisfies (1) p 1 = B + B i D + D i ≥ B D and (2) p 1 = B + B i D + D i > B + B i + B i +1 D + D i + D i +1 . The first condition is e quiv alen t to B i D i ≥ B + B i D + D i , a nd the second condition is eq uiv alen t to B + B i D + D i > B i +1 D i +1 . The latter t wo inequa lities imply B i D i > B i +1 D i +1 , which is a contradiction to the fact that B i = B i +1 and D i > D i +1 . ⊓ ⊔ Pr o of Sketch of The or em 3: Supp os e bidders 1 , . . . , n declare budgets B 1 ≥ . . . ≥ B n , and Alice decla res budget B . Let ℓ B be the r ank of Alice by budget (and lexicogra phic order in ca se o f ties) if she bids B . W e will pr ov e tha t the n umber of clicks Alice receives is non- increasing as she lowers her declared budget B , whic h immediately implies that truthful rep orting of budgets is weakly dominant in the PS mechanism. Let r B j be the ratio r j assuming Alice bids B ; so r B k = ( B + P k − 1 i =1 B i ) / P k i =1 D i if ℓ B ≤ k , and r B k = P k i =1 B i / P k i =1 D i otherwise. F or a declared budget B , let k B be the la st slot in the fir s t price blo ck chosen by the mechanism. So , k B = a rg max k r B k (if there are multiple maxima, then k B is the largest lexico- graphically ). F or sufficiently larg e B > B 1 , we g e t that r B 1 > r B k for all k and so k B = 1. F or any such B Alice receives D 1 clicks, the most p os sible. Now as we low er B , t wo significa nt even ts c o uld o ccur; we could drop to a no ther bidder’s budget B i , or w e could hav e a change in k B , th us changing the set o f bidder s in the first blo ck. If neither of these even ts oc c ur, then Alice remains in the first price blo ck, but gets a smaller s hare of the clicks. Th us it r emains to cov er thes e tw o even ts. If B = B i for some i , then note that by Lemma 2, Alice canno t b e the last bidder in the blo ck, so i is in the same blo ck as Alice. Therefore we may exchange the r oles of Alice and bidder i lexico graphica lly (i.e., increas e Alice’s rank by one) and no thing changes. Now supp ose B r eaches a po int where r k changes b eca use arg max k r B k changes from k B to k ′ . W e use k ∗ = k B for the remainder o f the pro o f for ease of nota- tion. At the bid B we hav e r B k ∗ = r B k ′ . W e claim that either k ′ > k ∗ or k ′ < ℓ B . T o see this note that fo r a ny k betw een ℓ B and k ∗ we have that r B k decreases at a r ate o f 1 / ( P k i =1 D i ), w hich is fas ter than the rate of the highes t ratio r B k ∗ . If k ′ > k ∗ then Alice rema ins in the fir s t blo ck, but it expands from ending at k ∗ to ending at k ′ . Both b efore and a fter the ch a ng e in r k , Alice is s p ending her entire budge t a t price r B k ∗ = r B k ′ , so her c licks r emain the s ame. If k ′ < ℓ B then Alice would remain in a blo ck ending at slot k ∗ , since r B k ∗ remains maximum a mo ng r B ℓ B , ..., r B n (b y the same rea soning ab out “ra te” as ab ov e). Since r B k ∗ = r B k ′ we hav e that the price of Alice’s blo c k and the fir st blo ck will b e the same. Since Alice is sp ending her entire budget b efore and after the change in r k at the s a me price, her clicks r emain the s ame. As w e co nt inue to decrease B b eyond this p oint, we simply remove the bidder s and slots from the first price blo ck, a nd imagine that we a re aga in in the first price blo ck o f a reduced ins tance. ⊓ ⊔ C Pro ofs for Section 4 Pr o of Sketch of The or em 4: W e split the pr o of int o tw o lemmas, showing that clicks are non-decreas ing in both bids and budg ets. This immediately implies the theorem. First we need a small observ ation ab out Find-P rice-Blo ck: Lemma 3. S upp ose Find-Pric e-Blo ck is ru n on a s et of budgets B 1 ≥ . . . ≥ B n and pr o duc es a blo ck 1 , . . . , ℓ ∗ with pric e p . Then if a bidder is adde d to the set with budget B , and Find-Pric e-Blo ck s til l pr o duc es pric e p , we must have that B ≤ B ℓ ∗ . Pr o of. Suppo se no t. Then B > B ℓ ∗ and we ha ve that ( B + P ℓ ∗ − 1 i =1 B i ) / P ℓ ∗ i =1 D i ≤ p . This c o nt r adicts p = P ℓ ∗ i =1 B i / P ℓ ∗ i =1 D i , since B > B ℓ ∗ . ⊓ ⊔ Lemma 4. The numb er of clicks a bidder is al lo c ate d is non-de cr e asing in her de clar e d budget. Pr o of sketch: Let bidders { 1 , . . . , n } be such that b 1 ≻ . . . ≻ b n , and consider a new bidder Alice with bid b ℓ − 1 ≻ b ≻ b ℓ . W e will argue that the num b er of clicks that Alice receives is no n-increasing as she r educes her decla red budg et B . Suppo se Alice declares B = ∞ and let ˆ B be the amount she would spend (Alice would alwa ys b e a threshold bidder if s he declared B = ∞ ). An y declared budget B ∈ [ ˆ B , ∞ ] would result in the same num b er o f clicks, b ecaus e B is reduced by the mechanism in step (iii) to ˆ B . Now as B decr eases from ˆ B , t wo different even ts could o ccur: (a) Alice’s price blo ck threshold ℓ ∗ could change (becaus e Find-Pr ice-Blo ck outputs a different ℓ ∗ ) o r (b) the low est bidder k could change (b ecause running Find-Price-Blo ck on 1 , . . . , k gave a pr ice less tha n b k +1 ). F or even t (a), and b etw een these even ts, the arguments from Theorem 3 imply that Alice’s clicks a re non-incr easing. F or even t (b), when the price of the Alice’s blo ck is exactly b k +1 , if bidder k + 1 is added, the resulting price o utput b y Find-Price -Blo ck in step (ii) is still at least b k +1 , since adding a bidder ca nnot reduce the price. Also Lemma s 3 and 2 together imply that Alice is s till in the price blo ck chosen in step (iii). Thu s Alice’s clicks do not incr ease. ⊓ ⊔ Lemma 5. The numb er of clicks a bidder is al lo c ate d is non-de cr e asing in her de clar e d bid. Pr o of sketch: F or the purp oses of this pro of, let bidders { 1 , . . . , n } b e such that b 1 ≻ . . . ≻ b n , and co nsider a new bidder (call her Alice) with declar ed budget B . W e will argue that the num be r of clicks that Alice r eceives in non- increasing with he r decla red bid b . Let p 1 be the price that Alice would pay if b = ∞ , and s uppo s e Alice is in the j th pr ice blo ck when she bids ∞ . Note that for any bid b ∈ ( p 1 , ∞ ], Alice is still in the j th price block and rec eives the same n umber o f clicks ( B /p 1 ). Let p 2 be the minimum bid req uired to keep Alice in the j th price blo ck. W e claim tha t if b ∈ [ p 2 , p 1 ], the price will a lwa ys be exa c tly b : no a llo cation is made until Alice is considered in step (ii), a nd when she’s co nsidered, Find- Price-Blo c k returns a pr ice p ≥ p 1 , since the set of bidders considered contains all the bidder s who pro duced price p 1 . Thus Alice is a thresho ld bidder, and in step ( iii ) Alice’s budg et is reduced so that the price is e x actly b . Let k b be the num b er of bidders with bid b i ≻ b . Let B b i be the i th large s t budget among bidder s with bid b i ≻ b . W e claim that if b ∈ [ p 2 , p 1 ], we hav e P ℓ i =1 B b i / P ℓ i =1 D i < b for all ℓ ≤ k b , since otherwise Alice would not b e in the j th blo ck. Let ˆ B b be Alice’s reduced budget when s he bids b ∈ [ p 2 , p 1 ], and le t c b = ˆ B b /b denote the num b er o f clicks she receives. T o satisfy the pr ice b eing a t most b in step (iii), w e must have that for all ℓ ≤ k b , ˆ B b ≤ B b ℓ + ∆ , where ∆ > 0 satisfies ( ∆ + P ℓ i =1 B b i ) / P ℓ i =1 D i = b . In additio n, we m ust hav e ( B b + P k b i =1 B b i ) / P k b +1 i =1 D i ≤ b . Putting these constra in ts together we get ˆ B b = min ℓ ≤ k b +1 { b P ℓ i =1 D i − P ℓ − 1 i =1 B b i } and so c b = ˆ B b /b = min ℓ ≤ k b +1 ( ℓ X i =1 D i − 1 b ℓ − 1 X i =1 B b i ) . As b decrea s es, if the s et of bidders with bids ≻ b do esn’t change, then the B b i s don’t change, and so this ex pression implies that c b also decre ases. If b decreas es to the po int w he r e b ′ ≻ b for some new bidder b ′ , then w e claim that c b also cannot incr ease. T o se e this note that for all ℓ , the expres sion P ℓ − 1 i =1 B b i can only increas e or stay the s ame if a new bidder is added. W e conclude that c b is non-increas ing in the interv al b ∈ [ p 2 , p 1 ]. When b decreases to p 2 , we transition fro m Alice b eing in the j th price blo ck to the j + 1st price blo ck. As in Theor e m 3 , at the p oint o f tr ansition the j th price blo ck will have the same price as the j + 1st pr ice blo ck, and in b oth scenarios Alice sp ends exactly ˆ B p 2 . Thus her clicks do no t change. W e can iter ate these arguments for the j + 1st price blo ck, and so the theorem is prov en. ⊓ ⊔ Lemmas 4 a nd 5 immediately imply Theo r em 4. ⊓ ⊔ Pr o of of The or em 5: Note that an equiv alent statement of the c o nstraint c 1 + . . . + c ℓ ≤ D 1 + . . . + D ℓ for a ll ℓ = 1 , ..., n. is: X i ∈ S c ′ i ≤ D 1 + ... + D | S | for a ll subsets S ⊆ { 1 , ..., n } . (3) Suppo se bids are b 1 ≻ b 2 ≻ ... ≻ b n and the corr esp onding c licks given to bidders in the greedy allo cation a re c = ( c 1 , ..., c n ). Le t c ∗ = ( c ∗ 1 , ..., c ∗ n ) b e the reven ue-maximizing solution with the closest pr efix to c , meaning that the first i such that c i 6 = c ∗ i is maximized, and mo dulo that, c i − c ∗ i is minimized. W e shall prove that the g reedy c gives a reven ue-maximizing schedule. Sup- po se the c ontrary and let i b e the first index on which c differs from c ∗ . Note that c i > c ∗ i (b y the de finitio n of gr eedy , c i is the maximum p ossible given c 1 , ..., c i − 1 ). Let c ∗ max = max { c ∗ i +1 , ..., c ∗ n } . Let J = { j > i : c ∗ j = c ∗ max } . Consider an a rbitrary tig h t constra in t o n c ∗ of the for m (3), defined b y the set S . W e claim that if i ∈ S , then all j ∈ J ar e als o in S . Pr o of of claim: Suppos e the co nt r a ry , namely that i ∈ S and j / ∈ S for some j ∈ J . Applying (3), we get X ℓ ∈ S c ∗ ℓ = X ℓ ≤| S | D ℓ . (4) One of the bidders in S must hav e index m > i , other wise (3) would b e viola ted for c and S by P ℓ ∈ S ⊆{ 1 ,. ..,i } c ℓ > P ℓ ∈ S ⊆{ 1 ,. ..,i } c ∗ ℓ = P ℓ ≤| S | D ℓ . If m / ∈ J , then we would violate (3 ) for the set S ′ = S ∪ { j }\{ m } : P ℓ ∈ S ′ c ∗ ℓ > P ℓ ∈ S c ∗ ℓ = P ℓ ≤| S | = | S ′ | D ℓ . Therefore m ∈ J . Now by the feas ibility of c ∗ and the fact that j / ∈ S , we a ls o hav e c ∗ j + P ℓ ∈ S c ∗ ℓ ≤ D | S | +1 + P ℓ ≤| S | D ℓ which implies, tog ether with (4), tha t c ∗ j ≤ D | S | +1 . Again by feasibility , w e also have P ℓ ∈ S \ m c ∗ ℓ ≤ P ℓ ≤| S |− 1 D ℓ and this, together with (4), gives c ∗ m ≥ D | S | . Putting these last tw o obser v ations together yields D | S | ≤ c m = c ∗ max = c ∗ j ≤ D | S | +1 . Unless c m = c ∗ max = c ∗ j = 0, this violates the distinctness of the no n-zero D j ’s. But if c ∗ max = 0, it means that all c ℓ for ℓ > i hav e c ℓ = 0, which mea ns that c g ives strictly more clic ks than c ∗ , a co nt r a diction. ⊓ ⊔ Let j b e an arbitrar y member of J . By the claim, ther e is a n ǫ > 0 such that if we set c ′ = c ∗ except c ′ i = c ∗ i + ǫ and c ′ j = c ∗ j − ǫ , we get a fea s ible allo cation c ′ , since j appea rs in ev er y tight co nstraint in whic h i app ears. This allo cation has r even ue at lea st that of c ∗ , s ince b i ≥ b j . But, it has a clo ser prefix to c tha n c ∗ , a c o nt r adiction. ⊓ ⊔ Pr o of Sketch of The or em 6: W e will abuse notation and let ǫ ′ denote an y po sitive quantit y that ca n b e made a rbitrarily clo s e to zero . When the PS mechanism is run on the truthful input, let p 1 > p 2 > . . . deno te the prices of each blo ck. W e will show that if in GFP ea ch bidder i truthfully rep or ts her budget and bids b i = min { m i , p j + ǫ ′ } , wher e j is the price blo ck of i in the PS mechanism, we meet the conditions of the theorem. Suppo se the first pric e blo ck is determined when bidder k is co nsidered, and ends at s lot ℓ ∗ ≤ k . The price p 1 satisfies m k +1 ≤ p 1 ≤ m k . Let P ⊆ [ k ] deno te the bidders in the firs t blo ck (the ones in [ k ] with the ℓ ∗ highest budgets). Also, we hav e that all i ∈ P spend their en tire budget in the PS mechanism, except po ssibly k , who may spend les s than he r budg e t if m k = p 1 . W e now argue that GFP will pr o duce the sa me allo cation as the PS mechanism for this price blo ck. F or all i ∈ P we hav e b i = min { m i , p 1 + ǫ ′ } ≥ min { m k , p 1 } = p 1 . All bidder s i ∈ ([ k ] \ P ) have b i ≤ p 2 + ǫ ′ < p 1 . All bidders i / ∈ [ k ] hav e m i ≺ m k and so since b i ≤ m i we get b i ≺ b i ′ for all i ′ ∈ P . W e conclude that the bidders in P are the fir st to b e c o nsidered by the GFP mechanism. F urther more, if k ∈ P , and B k is r e duced in the PS mechanism (b ecause k is a thresho ld bidder), then we m ust hav e b k = m k = p 1 , and so b k ≺ b i for a ll i ∈ P, i 6 = k . Thus in this case bidder k is the last bidder in P to b e cons ide r ed by GFP . F rom here it is straightforward to show that GFP will assig n the first ℓ ∗ slots to the bidders in P (almost) exactly like the PS mechanism do es, with at least c i − ǫ ′ clicks to each i ∈ P ; the mechanism will hav e ǫ ′ clicks left ov er, which will be a ssigned to bidders not in P . Applying this same ar gument to subsequent price blo cks, we conclude tha t GFP will a ssign c ′ i = c i ± ǫ ′ clicks to a ll bidders i . T o s how this is an eq uilibr ium, consider a bidder Alice (call her “bidder a ”) that was a ssigned to price blo ck j ∗ and received c ′ a = c a ± ǫ ′ clicks. If Alice spe nt within ǫ ′ of her entire budge t, it mea ns s he would no t wan t to r aise her bid, since she could not p ossibly receive more tha n ǫ ′ additional clic ks. If she did no t sp end her budget, then from the obs e rv ations above we know that she is bidding her tr ue max-cp c m a , and ther efore also do es not wan t to raise her bid. It remains to s how that Alice do es not wan t to lower her bid. Let ℓ j denote the last s lot in price blo ck j . Let P j denote the set of bidder s in price blo ck j . Alice’s current bid b a is at least p j , and if she keeps her bid ab ov e p j her clicks will remain c a ± ǫ from the a rguments ab ov e. Let S = ∪ j ≤ j ∗ P j . If Alice low ers her bid to b ′ a < p j , then all bidders i ∈ S bes ides Alice will hav e b i ≻ b ′ a . Thus when Alice is cons idered by the greedy alg orithm, her clicks will b e constr ained b y the commitment s to these bidders. F ur thermore ea ch of these bidders will still receive at least c ′ i clicks. F or all price blo cks j , w e hav e P i ∈ P j c ′ i ≥ P ℓ j i = ℓ j − 1 +1 D i − ǫ ′ . Thu s P i ∈ S,i 6 = a c ′ i ≥ ( P ℓ j ∗ i =1 D i ) − ǫ ′ − c ′ a . Since S has size ℓ j ∗ , this implies tha t the constraint (3) r estricts Alice’s clicks to at mo st c ′ a + ǫ ′ . ⊓ ⊔