Bayesian Budget Feasibility with Posted Pricing

Transcription

1 Bayesan Budget Feasblty wth Posted Prcng Erc Balkansk Harvard Unversty School of Engneerng and Appled Scences Jason D. Hartlne Northwestern Unversty EECS Department Aprl 12, 2016 Abstract We consder the problem of budget feasble mechansm desgn proposed by Snger (2010), but n a Bayesan settng. A prncpal has a publc value for hrng a subset of the agents and a budget, whle the agents have prvate costs for beng hred. We consder both addtve and submodular value functons of the prncpal. We show that there are smple, practcal, ex post budget balanced posted prcng mechansms that approxmate the value obtaned by the Bayesan optmal mechansm that s budget balanced only n expectaton. A man motvatng applcaton for ths work s crowdsourcng, e.g., on Mechancal Turk, where workers are drawn from a large populaton and posted prcng s standard. Our analyss methods relate to contenton resoluton schemes n submodular optmzaton of Vondrák et al. (2011) and the correlaton gap analyss of Yan (2011). 1 Introducton Consder the problem of hrng workers to complete complex tasks on crowdsourcng platforms such as Mechancal Turk. A prncpal must select a set of workers, henceforth agents, whose contrbutons wll be aggregated to complete the task. The prncpal s value for the task s a functon of the set of agents selected and the prncpal s budget lmts the total payments to agents. We assume that the prncpal s value s submodular,.e., t exhbts dmnshng returns. For example, the accuracy obtaned for the task of labelng mages s mproved wth more agents, but there are dmnshng returns to recrutng addtonal agents. The agents have a prvate cost for partcpatng and wll choose to partcpate strategcally to optmze ther payments receved relatve to ths cost. The prncpal seeks a budget feasble mechansm for selectng agents so as to maxmze the value of the completed task. The lterature on budget feasble mechansm desgn ntated by Snger (2010) studes ths problem; however, t prmarly consders sealed-bd mechansms whch do not tend to be seen on crowdsourcng platforms lke Mechancal Turk. Instead, these platforms use posted prcng mechansms. We follow a tradtonal economcs approach to ths problem where agents costs are drawn from a common pror dstrbuton and a mechansm s sought to optmze the prncpal s value functon n expectaton of ths dstrbuton. The Bayesan approach s especally approprate for the problem of crowdsourcng a task as the agents on crowdsourcng platforms are drawn from a large populaton of avalable agents that repeatedly perform tasks. The pror dstrbuton on the costs of the populaton of agents can be learned statstcally from past crowdsourcng tasks. Our model allows 1

2 asymmetry n the dstrbuton whch may arse, for example, from the agents observable skll levels, prevous partcpaton n smlar tasks, or demographc. We show that posted prcng mechansms gve a good approxmaton to the optmal sealed-bd mechansm. Addtonally, we gve effcent algorthms for calculatng the approprate prces. In comparson to other work n optmzaton of prces n crowdsourcng, our work focuses on the use of prces to control partcpaton and not the level of effort of partcpants. Controllng the level of effort of partcpants was studed n onlne behavoral experments by Ho et al. (2015), theoretcally for crowdsourcng contests by Chawla et al. (2012), and for user generated content by Immorlca et al. (2015). Overvew of Approach. Our approach follows smlarly to that of Alae (2014) and Yan (2011). The startng pont for our analyss s an upper bound on the performance of the optmal sealed bd mechansm that relaxes the ex post budget constrant on the mechansm to hold ex ante,.e., n expectaton over the prvate costs of the agents. Va ths ex ante relaxaton and the Myerson (1981) theory of vrtual values, we construct a posted prce mechansm that s budget feasble n expectaton and a 1 1/e approxmaton to the optmal ex ante mechansm when the prncpal s value functon exhbts decreasng returns,.e., s submodular. For the specal case where the prncpal s value functon s addtve, ths posted prcng s optmal (for the ex ante relaxaton). We then consder postng the prces from the soluton to the ex ante relaxaton untl the budget runs out. The resultng mechansm s ex post budget feasble, but suffers a loss n performance because the budget may run out early. The man techncal contrbuton of ths work s to show that the performance of such a prce postng mechansm compares favorably to the optmal sealed-bd mechansm. Prevous work n mechansm desgn gves technques whch are now well understood to satsfy ex post allocaton constrants. Ex post payment constrants requre dfferent technques and our analyses follow two basc approaches that combne optmzaton and mechansm desgn concepts. To analyze the performance of the posted prcng under any arrval order of the agents, we solve the ex ante relaxaton wth a slghtly smaller budget and then, usng results from the Vondrák et al. (2011) analyss of contenton resoluton schemes, show that t s unlkely for the orgnal ex post budget constrant to bnd. Alternatvely, we obtan better bounds for addtve value functons and when the order of agent arrvals can be specfed by the mechansm va the correlaton gap approach of Yan (2011). As a corollary, we obtan new correlaton gap results for ntegral and fractonal knapsack set functons. Moreover, when the envronment s symmetrc (both n dstrbuton of agent costs and the prncpal s value functon), the submodular case can be reduced to the addtve case. The prces dentfed above can be computed or approxmately computed n polynomal tme. In partcular, for submodular value functons, we reduce the problem of fndng the prces to the well-known greedy algorthm for submodular optmzaton. The dentfed prces approxmate the optmal prces wth relatve loss n the value functon that s wthn a factor of 1 1/e. For addtve value functons, the optmzaton problem smplfes to a monopoly prcng problem of classc mcroeconomcs. Smlarly to the Myerson and Satterthwate (1983) treatment of welfare maxmzaton subject to budget balance n a buyer seller exchange, optmzaton n ths context s based on Lagrangan vrtual surplus. These optmal prces can be approxmated arbtrarly precsely by solvng ths problem on a dscretzed nstance. 2

3 Related work. The pror lterature on budget feasblty prmarly consders a worst-case desgn and analyss framework that compares the performance of the desgned mechansm to the frst-best outcome,.e., the one that could be obtaned f the agents costs were publc. See Snger (2010), Be et al. (2012), Badandyuru et al. (2012), and Anar et al. (2014). Our analyss compares the desgned mechansm, n expectaton for the known pror dstrbuton, to the second-best outcome,.e., the one obtaned by the Bayesan optmal mechansm. The followng results are for pror-free mechansms n comparson to the frst-best outcome. Snger (2010) obtaned a randomzed truthful budget feasble mechansm wth a constant factor approxmaton for submodular value functons, Chen et al. (2011) then mproved the analyss of ths mechansm to a 0.13 approxmaton. In the Bayesan settng, Be et al. (2012) obtaned a constant approxmaton for subaddtve functons. More recently, Anar et al. (2014) obtaned better bounds by consderng large markets, whch we also consder n ths paper. Fnally, Badandyuru et al. (2012) also consdered posted prcng mechansms but when the agents arrve onlne. They obtaned a constant approxmaton for the class of symmetrc submodular functons. They also obtaned a O(log n) mechansm for the case of submodular functons. In comparson to ths last paper, we gve much better bounds when the pror dstrbuton on costs s known. The startng pont for our analyss s the soluton to the relaxed problem of budget balance n expectaton,.e., ex ante. In the addtve case, ths problem was recently studed by Ensthaler and Gebe (2014). They show that posted prcng mechansms solve the relaxed problem and remark that the same performance can be obtaned wth ex post budget balance, but at the expense of relaxng ex post ndvdual ratonalty (for the bdders) and not wth a posted prcng. Ths latter observaton follows, for example, by applyng a general constructon of Esö and Futo (1999). Our analyss of the relaxed problem gves a much smpler proof of ther man theorem. Budget feasblty has also been studed n the context of crowdsourcng. Among that lne of work, the model consdered n Anar et al. (2014) s the closest to ours, and wll be compared n detal below. Sngla and Krause (2013) and Snger and Mttal (2013) consder the specal case of our model where the prncpal s value functon s the number of tasks performed. The former studes posted prcng for agents wth..d. costs from an unknown dstrbuton, whle the latter studes sealed bd mechansms wthout a pror. Dscusson about posted prcng mechansms and benchmarks. Followng a lne of lterature n mechansm desgn that was ntated by Chawla et al. (2010), the goal of ths work s to show that there exsts smple posted prcng mechansms that approxmate the optmal sealed-bd mechansm. Two quanttes of nterest therefore need to be separated. The frst s the cost of ncentve compatblty n budget feasble settngs,.e., the gap between the frst-best and secondbest benchmarks. The second s the cost of smplcty,.e., the loss of a posted prcng mechansm compared to the Bayesan optmal mechansm. Pror work wth comparsons to a frst-best benchmark has approxmatons that are a combnaton of both of these quanttes. Our comparson to the second-best outcome solates the loss from a smple decentralzed prcng over the optmal centralzed mechansm as the quantty of nterest. Our results. Our results are summarzed n Fgure 1. We consder two man classes of valuaton functons, addtve and submodular. We use two dfferent methods to satsfy the ex post payment constrant, one s based on contenton resoluton schemes and the other on correlaton gap. Contenton resoluton schemes gve an oblvous posted prce mechansm,.e., one that ob- 3

4 Value Functon Mechansm Famly Ex Post Constrant Approach General Result Large Markets Addtve Sequental Posted Prcng Correlaton Gap (1 1 2πk )(1 1 k ) 1 Symmetrc Submodular Oblvous Posted Prcng Correlaton Gap (1 1 2πk )(1 1 k ) 1 Submodular (computatonal) Oblvous Posted Prcng Contenton Resoluton (1 1 e )2 (1 ɛ)(1 e ɛ2 (1 ɛ)k/12 ) (1 1 e ) Submodular (non-computatonal) Oblvous Posted Prcng Contenton Resoluton (1 1 e )(1 ɛ)(1 e ɛ2 (1 ɛ)k/12 ) 1 1 e 0.63 Fgure 1: Our results are approxmatons to the Bayesan optmal mechansm. Bounds are parameterzed by the market sze k, a lower bound on the number of agents that can be smultaneously selected wth the gven budget (see Defnton 4). In large markets, k grows large. The gven results wth the contenton resoluton approach requre k 4 and ɛ (2/k, 1/2), a result for k < 4 s mentoned n Secton 4. For the symmetrc submodular results, we also assume symmetrc dstrbutons on costs. Our computatonal results also have an addtonal o(1) loss due to dscretzaton. tans ts proven bound under any arrval order of the agents. The correlaton gap approach, for the case where the prncpal has an addtve value functon, gves a sequental posted prce mechansm. Such a mechansm s specfed by an orderng on agents and take-t-or-leave-t prces to offer each agent. As a specal case, we consder symmetrc envronments where both the value functon and the dstrbuton s symmetrc. Our results can most drectly be compared to those of Anar et al. (2014), but wth the followng caveats. Ther results are for sealed bd mechansms whle ours are for posted prcngs; ther mechansm s pror-free whle ours s parameterzed by the pror dstrbuton on agent costs; ther results compare performance to the frst-best outcome,.e., wthout ncentve constrants, whle ours compare to the second-best outcome,.e., that of the Bayesan optmal mechansm (wth ncentve constrants). They obtan approxmaton ratos of 1 1/e, 1/3 and 1/2 n large markets respectvely for addtve, submodular (computatonal), and submodular (non-computatonal) value functons. Moreover, they show that no truthful mechansm can acheve an approxmaton rato better than 1 1/e wth respect to the frst-best outcome for addtve value functons. Paper Organzaton. We start wth prelmnares n Secton 2 to ntroduce the model and dfferent concepts used n ths paper. We then descrbe posted prce mechansms for the ex ante relaxaton, where the budget holds n expectaton, n Secton 3. We explan how to go from an ex ante posted prce mechansm to an ex post posted prce mechansm usng two dfferent methods, one nspred by contenton resoluton schemes n Secton 4 and another based on a correlaton gap analyss n Secton 5. We tackle the computaton ssues of fndng a good ex ante mechansm n Secton 6. In Secton 7, we study symmetrc envronments. Up to Secton 7, cost dstrbutons are assumed to be regular and Secton 8 consders the case where some dstrbutons mght be rregular. Throughout the paper, we assume that the prncpal s valuaton functon s monotone and submodular. 4

5 2 Prelmnares There are n agents N = {1,..., n}. Agent has a prvate cost c for provdng a servce that s drawn from a dstrbuton F (denotng the cumulatve dstrbuton functon) wth densty f. Indcator varable x denotes whether or not provdes servce and p denotes the payment receves. Agent ams to optmze her utlty gven by p c x. The cost profle s denoted c = (c 1,..., c n ); the jont dstrbuton on costs s the product dstrbuton F = F 1 F n ; the payment profle s denoted p = (p 1,..., p n ); and the allocaton profle s denoted x = (x 1,..., x n ). The prncpal has a value functon v : {0, 1} n R +. For allocaton profle x {0, 1} n or set of agents S = { : x = 1} who provde servce, the value to the prncpal s v(x) = v(s). The prncpal has a budget B and requres the payments to the agents not to exceed the budget,.e., p B. The followng mathematcal program captures the prncpal s objectve. max E c[v(x(c))] (1) x,p s.t. p (c) B c, x( ) and p( ) are ncentve compatble. We consder only mechansms that are ncentve compatble. A mechansm s ncentve compatble (IC), f truthful reportng of the agents s a domnant strategy equlbrum. 1 We wll consder the budget constrant both ex ante,.e., n expectaton over realzatons of agents costs and random choces of the mechansm, and ex post,.e., the payments to the agents never exceed the budget. The man goal of ths paper s to approxmate the optmal ex ante budget feasble mechansm wth an ex post budget feasble posted prcng mechansm. Posted prcng mechansms are trvally ncentve compatble. Defnton 1. The posted prcng (ĉ, σ), for prces ĉ and orderng on agents σ, s: 1. The remanng budget s ntally B. 2. The agents arrve n order σ. 3. If agent arrves wth cost c below her offered prce ĉ whch s below the remanng budget, then select ths agent for servce, pay her ĉ, and deduct ĉ from the remanng budget. Otherwse, dscard ths agent. For (mplct) dstrbuton on costs F, we can equvalently specfy a posted prcng (ĉ, σ) as (ˆq, σ) where ˆq = F (ĉ ) s the margnal probablty that agent wth cost c F would accept the prce ĉ. 2 Note that the prces ĉ are non-adaptve,.e., fxed before the agents arrve. We consder posted prcng mechansms under two dfferent models for agent arrval. In the sequental posted prcng model, the orderng σ can be fxed n advance by the mechansm and, wthout computatonal consderatons, our analyss s for the best case orderng of the prces. In the oblvous posted 1 The restrcton to domnant strategy mechansms over Bayesan ncentve compatble mechansms s wthout loss for the budget feasblty objectve. 2 It s common n Bayesan mechansm desgn to consder the agents prvate costs n quantle space where s quantle q = F (c ) s the measure of cost lower than c accordng to F. Agent quantles are always unformly dstrbuted on [0, 1]. From ths perspectve, ˆq s agent s prce n quantle space. 5

6 prcng model, the orderng σ s unconstraned and our analyss s worst case wth respect to ths orderng. An oblvous posted prcng s denoted ĉ. We compare our mechansms to an ex ante posted prcng ĉ where the budget constrant holds n expectaton,.e., ĉ ˆq B. The value of an ex ante posted prcng s E S ˆq [v(s)] where S ˆq adds each agent to S ndependently wth probablty ˆq. The paper focuses on value functons that are monotone and submodular (Defnton 2). An mportant specal case, whch we wll treat separately, s that of addtve value functons where each agent has a value v and the value functon s v(s) = S v. Defnton 2. A set functon v : {0, 1} n R + s monotone submodular f (monotoncty) v(t ) v(s) for all T S, and (submodularty) for all T S the margnal contrbuton of S to T s at least ts margnal contrbuton to S. In other words, v(t {}) v(t ) v(s {}) v(s). Our analyss s based on the relatonshp between a set functon and two standard extensons of a set functons from the doman {0, 1} n to the doman [0, 1] n. For submodular set functons, these extensons were studed by Calnescu et al. (2007) and Agrawal et al. (2010). Defnton 3. Gven a set functon v : {0, 1} n R +, ts concave closure V + ( ) (a.k.a., correlated value) s the smallest concave functon that upper bounds the set functon. Alternatvely, V + (ˆq) = max D E S D [v(s)] wth the maxmzaton taken over all dstrbutons D wth margnal probabltes ˆq = (ˆq 1,..., ˆq n ); and ts multlnear extenson V ( ) (a.k.a., ndependent value) s the expected value of the set functon when each element s drawn ndependently wth margnal probablty ˆq. In other words, V (ˆq) = E S ˆq [v(s)]. For any set functon, the concave closure s clearly an upper bound on the multlnear extenson. For submodular functons the nequalty approxmately holds n the opposte drecton as well. By the nterpretaton of the multlnear extenson as the expected value of the set functon for ndependent dstrbuton and the concave closure as the expected value of the set functon for correlated dstrbutons, ther worst case rato over margnal probabltes ˆq s known as the correlaton gap (Agrawal et al., 2010). Theorem 1 (Calnescu et al., 2007, Agrawal et al., 2010). For monotone submodular set functon v( ), the correlaton gap s V (ˆq) mn ˆq V + (ˆq) 1 1/e. Theorem 2 (Yan, 2011). For a k-hghest-value-elements set functon v( ), whch s addtve wth value v for element up to a capacty of at most k elements, the correlaton gap s mn ˆq V (ˆq) V + (ˆq) 1 1/ 2πk. 6

7 Our analyss s parameterzed by a measure of the sze of the market. Ths noton of market sze s standard n the lterature, e.g., see Be et al. (2012) and Anar et al. (2014). A large market analyss consders the market sze n the lmt. Although large markets are descrbed as an assumpton by Anar et al. (2014), the market sze k s a parameter n our analyss and we obtan results for any market sze. Defnton 4. A market s k-large for prces ĉ and budget B f B/ĉ k for all agents. Note that the market sze depends on prces and therefore on the mechansm, whch s nherent to our analyss. These prces can trvally be upper bounded by the maxmum cost that can be drawn from the dstrbutons. 3 The Ex Ante Budget Feasble and Concave Closure Relaxatons In ths secton we relax the objectve functon and the budget constrant to make the problem more amenable to optmzaton. We frst relax the budget constrant so that t only holds n expectaton, makng t an ex ante feasblty constrant. We then upper bound the value functon by ts concave closure. Wth an ex ante feasblty constrant, the objectve s to optmze the followng ex ante program over allocaton rule x( ) and payment rule p( ) wth c F. max E c[v(x(c))] (2) x,p s.t. E c[p (c)] B, x( ) and p( ) are IC. When payments are part of the prncpal s objectve or constrants, the Bayesan mechansm desgn problem wll typcally rely on the Myerson (1981) theory of vrtual values or, n our case where the agents are sellers, vrtual costs. The vrtual cost of agent wth cost c drawn from dstrbuton F s φ (c ) = c + F (c ) f (c ). The vrtual surplus of an agent wth vrtual cost φ (c ) and allocaton ndcator x s φ (c ) x. Lemma 3 (Myerson and Satterthwate, 1983). In any ncentve compatble mechansm, any agent s expected payment s equal to her expected vrtual surplus,.e., for c F, E c [p (c)] = E c [φ (c) x (c)]. The defnton of vrtual costs and Lemma 3 allows the ex ante program (2) to be rewrtten n terms of the allocaton rule only. To do so, we nvoke the followng characterzaton of ncentve compatble mechansms of Myerson (1981). Lemma 4 (Myerson, 1981). There exsts an ncentve compatble mechansm wth allocaton rule x( ) f and only f x( ) s monotone n the cost of any agent. We now rewrte the optmzaton program (2) by substtutng n vrtual costs for payments to obtan the followng vrtual surplus program, 7

8 max E c[v(x(c))] (3) x s.t. E c[φ (c) x (c)] B, x( ) s monotone n the cost of any agent. For the general case of submodular value functons, the expected value of the set functon v( ) s upper bounded by ts concave closure (Defnton 3) as follows. The allocaton rule x( ) that optmzes ths vrtual surplus program nduces, for c F, a dstrbuton over sets of wnnng agents. Denote ths dstrbuton by D and denote by ˆq the profle of margnal probabltes,.e., wth ˆq = Pr S D [ S]. By the defnton of the concave closure of the set functon v( ), E c [v(x(c))] = E S D [v(s)] V + (ˆq). The payment to an agent s lower bounded by the payment from prce postng. As above, the optmal mechansm selects agent wth probablty ˆq. When vrtual costs are monotoncally ncreasng,.e., n the case of regular dstrbutons, the expected payment to an agent selected wth probablty ˆq s mnmzed f agent s served f and only f c F 1 (ˆq ) by Lemma 3 snce these costs mnmze φ (c). 3 Thus, the mechansm that mnmzes expected payments and serves each agent wth probablty ˆq s the mechansm that posts prce ĉ = F 1 (ˆq ) to each agent. Lemma 5. For any agent wth cost drawn from regular dstrbuton F and any ncentve compatble mechansm that selects agent wth probablty ˆq, the expected payment of agent s at least ˆq ĉ where ĉ = F 1 (ˆq ). Combnng the relaxaton of the value functon and the relaxaton of the payments we obtan the followng concave closure program, max V + (q) (4) q s.t. q F 1 (q ) B. Lemma 6. Let ˆq + be the optmal soluton to the concave closure program (4), then V + (ˆq + ) upper bounds the performance of the optmal ex ante mechansm n the case of regular cost dstrbutons. Posted prce mechansms are trvally ncentve compatble. Snce the dstrbutons of agents costs are ndependent, the set of agents who wll accept ther offer wth a posted prce mechansm s a set whch wll contan each agent wth some probablty q ndependently. Therefore the performance of a posted prce mechansm where agents accept ther offer wth probabltes q s the multlnear extenson V (q). Ths motvates us to rewrte the concave closure program (4) as the followng multlnear extenson program, max V (q) (5) q s.t. q F 1 (q ) B. Maxmzng the multlnear extenson program gves us an ex ante posted prce mechansm that s approxmately optmal. 3 The case of rregular dstrbutons s consdered n Secton 8. 8

9 Theorem 7. In the case of monotone submodular value functons and regular cost dstrbutons, the ex ante mechansm that posts prce ĉ = F 1 (ˆq ) to each agent s an 1 1/e approxmaton to the optmal ex ante mechansm, where ˆq s the optmal soluton to the multlnear extenson program (5). Proof. Let ˆq + be the optmal soluton to the concave closure program (4). By Theorem 1, V (ˆq + ) (1 1/e)V + (ˆq + ). By the optmalty of ˆq, V (ˆq) V (ˆq + ). Snce the performance of postng prce F 1 (ˆq ) to each agent s V (ˆq) and snce V + (ˆq + ) upper bounds the performance of the optmal ex ante mechansm by Lemma 6, postng prce F 1 (ˆq ) to each agent s an 1 1/e approxmaton to the optmal ex ante mechansm. Note that n the addtve case where each agent has value v, V (q) = V + (q) = v q and we get the followng corollary. Corollary 8. In the case of addtve value functons and regular cost dstrbutons, the ex ante mechansm that posts prce ĉ = F 1 (ˆq ) to each agent s an optmal mechansm, where ˆq s the optmal soluton to the multlnear extenson program (5). We dscuss the computatonal ssues of fndng a good soluton q to the multlnear extenson program (5) n Secton 6. For the case of submodular functons, we reduce the problem to submodular functon maxmzaton (wth a cardnalty constrant) for whch the greedy algorthm gves an 1 1/e approxmaton. In the addtve case, we wll show that the optmal ex ante budget feasble mechansm can be found by takng the Lagrangan relaxaton of the vrtual surplus program (3). 4 Submodular Value and Oblvous Posted Prcng In the prevous secton, we obtaned an ex ante mechansm by optmzng the multlnear extenson program (5). In ths secton we analyze the performance of oblvous posted prcng (wth an ex post budget constrant). The approach of ths secton s the followng: lower the budget by some small amount and optmze the multlnear extenson program (5) so that the lowered budget s satsfed ex ante. Wth the budget suffcently lowered, wth hgh probablty the cost (sum of prces) of the set of agents who would accept ther offer s under the orgnal budget (regardless of ther arrval order and ex post). Ths approach s a specal case of that taken by the contenton resoluton schemes of Vondrák et al. (2011) and we frst revew some known bounds. The frst comes from the submodularty of the value functon; the second comes from the Chernoff bound. Theorem 9 (Bansal et al., 2010). Gven a non-negatve monotone submodular functon v( ), a random set R whch contans each agent ndependently wth probablty ˆq, and a (possbly randomzed) procedure π that maps (possbly nfeasble) sets to feasble sets such that, (margnal property) for all, Pr R ˆq;π [ π(r) R] γ, and (monotoncty property) for all T S and T, Pr π [ π(t )] Pr π [ π(s)], then E R ˆq;π [v(π(r))] γ E R ˆq [v(r)]. 9

10 Theorem 10 (Vondrák et al., ). Gven ɛ (0, 1/2), budget B, ndependent varables p that are the payments to each agent such that, (scaled ex ante budget constrant) E[p ] (1 ɛ) B, (k-large market) p s bounded by [0, B/k] for all, and k > 2/ɛ, then the probablty that the sum of costs of selected agents does not exceed the budget less the cost of any agent,.e., Pr[ p (1 1/k)B], s at least 1 e ɛ2 (1 ɛ)k/12. We now connect these two results by relatng the probablty that the sum of costs does not exceed (1 1/k)B of Theorem 10 to γ of Theorem 9 and then show that posted prcngs satsfy the condtons of Theorem 9. Lemma 11. For sequental posted prcng (ĉ, σ) that satsfy the scaled ex ante budget constrant and k-large market condtons, the probablty that an agent s offered her prce s lower bounded by Pr R ˆq [ R ĉ (1 1/k)B ], the probablty that the sum of the prces of agents who would accept ther offered prce s at most (1 1/k)B. Proof. If the total cost of all agents who would accept ther prce s at most (1 1/k)B then ths budget remans at the tme an agent s consdered n the sequence σ. By the defnton of k B/ĉ t s feasble to serve ths agent and so she s offered her prce ĉ by the sequental posted prcng mechansm. Lemma 12. For sequental posted prcng (ˆq, σ), f each agent s offered her prce wth probablty at least γ, then the expected value of the mechansm s at least γv (ˆq). Proof. It suffces to show, for sequental posted prcng (ˆq, σ) wth an ex post budget constrant B, that the margnal and monotoncty propertes of Theorem 9 hold. In our case, R ˆq s the random set of agents who would accept ther offer f the budget never runs out. Gven a set of agents R who accept ther offer, defne π(r) to be the set of agents who accept ther offer and who arrve before the budget runs out. In our case, π s determnstc gven the orderng σ. Note that Pr R ˆq;π [ π(r) R] s equal to the probablty that an agent gets offered her prce, meanng that she arrves before the budget runs out. Thus, by the assumpton of the lemma the margnal property holds. For the monotoncty property, consder two sets T S. When an agent arrves n the posted prce mechansm, the mechansm has spent less f the set of agents who accept ther offer s T than f ths set s S. Therefore π(s) mples that π(t ) and the monotoncty property holds. By combnng the prevous results, we obtan the man theorem for ths secton. Theorem 13. For ɛ (0, 1/2), f the oblvous posted prcng ĉ correspondng to the optmal soluton ˆq to the multlnear extenson program (5) wth budget (1 ɛ)b (.e., wth ĉ = F 1 (ˆq ) for each agent ) satsfes 2/ɛ k B/ max ĉ, then ths posted prcng mechansm s a (1 1/e)(1 ɛ)(1 e ɛ2 (1 ɛ)k/12 ) approxmaton to the optmal mechansm for submodular value functons and (1 ɛ)(1 e ɛ2 (1 ɛ)k/12 ) for addtve value functons n the case of regular cost dstrbutons. 4 The formulaton of ths theorem s slghtly dfferent than n Vondrák et al. (2011) but follows easly from ther analyss. 10

11 Proof. The proof starts wth the ex ante mechansm from the prevous secton and then apples results from ths secton to modfy t nto an ex post mechansm. Let ˆq be the optmal soluton to the multlnear extenson program (5) wth budget (1 ɛ)b, ˆq + (1 ɛ)b be the optmal soluton to the concave closure program (4) wth budget (1 ɛ)b, and ˆq+ B be the optmal soluton to the concave closure program (4) wth budget B. By the optmalty of ˆq and Theorem 1, V (ˆq) V (ˆq + (1 ɛ)b ) (1 1 e )V + (ˆq + (1 ɛ)b ). Note that the soluton (1 ɛ)ˆq + 1 B has cost at most (1 ɛ)b snce F ( ) s ncreasng. So by the optmalty of ˆq + (1 ɛ)b and by the concavty of the concave closure V + ( ), V + (ˆq + (1 ɛ)b ) V + ((1 ɛ)ˆq + B ) (1 ɛ)v + (ˆq + B ). Snce V + (ˆq + B ) s an upper bound on the performance of the optmal ex ante mechansm by Lemma 6, the ex ante posted prcng mechansm defned for each agent by ĉ = F 1 (ˆq ) s a (1 1/e)(1 ɛ) approxmaton to the optmal mechansm. We now consder the posted prcng mechansm defned by ĉ that s no longer ex ante. Snce the budget has been lowered by a factor 1 ɛ, each agent s offered her prce wth probablty at least Pr R ˆq [ R ĉ (1 1/k)B ] by Lemma 11, regardless of the orderng σ of agents. By Theorem 10, ths probablty s at least 1 e ɛ2 (1 ɛ)k/12. Therefore, by Lemma 12, the expected value of ths mechansm s at least (1 e ɛ2 (1 ɛ)k/12 )V (ˆq) and ths mechansm s a (1 ɛ)(1 1/e)(1 e ɛ2 (1 ɛ)k/12 ) approxmaton to the optmal mechansm n the case of submodular value functons. In the case of addtve functons, there s no loss from the multlnear extenson to the concave closure, so the mechansm s a (1 ɛ)(1 e ɛ2 (1 ɛ)k/12 ) approxmaton. Note that as the sze of the market k grows to nfnty, ths approxmaton rato approaches 1 1/e. Also note that ths mechansm requres the market to be at least 4-large. Usng another result from Vondrák et al. (2011) and a smlar analyss to the one from ths secton, a (1 1/e)/8 posted prcng mechansm can easly be obtaned for any market sze. Ths posted prcng attans ts performance guarantee when agents wth cost at least B/4 arrve before all others, but otherwse the order s oblvous. 5 Addtve Value and Sequental Posted Prcng In ths secton we gve mproved bounds for sequental posted prcng,.e., where the mechansm orders the agents, and when the value functon s addtve,.e., v(s) = S v. In partcular, we analyze the sequental posted prcng (ĉ, σ) wth ĉ = F 1 (ˆq ) from the soluton to the multlnear extenson program (5) wth the full budget B and the orderng σ by decreasng bang-per-buck,.e., v /ĉ for agent. Our results n ths secton are based on the analyss of the correlaton gap of fractonal and ntegral-knapsack set functons (to be defned subsequently). The fractonal-knapsack set functon s a submodular functon, so a correlaton gap of 1 1/e can be drectly obtaned (Theorem 1). In ths secton, we mprove ths bound to 1 1/ 2πk for k-large markets,.e., wth k = B/ max ĉ. From ths bound we observe that the correlaton gap for fractonal-knapsack n large market s 11

12 asymptotcally one. We show that the ntegral-knapsack correlaton gap s nearly the same. Followng the approach of Yan (2011), the factor by whch sequental posted prcng approxmates the ex ante relaxaton s equal to the ntegral-knapsack correlaton gap. Defnton 5. The fractonal-knapsack set functon correspondng to addtve set functon v(s) = S v, szes ĉ, and capacty B s denoted v B (S) and equals the maxmum value soluton to the correspondng fractonal-knapsack problem on elements S. 5 The ntegral-knapsack set functon can be defned analogously to the fractonal one, but t cannot add elements fractonally. Most of ths secton analyzes the rato of the ndependent value of fractonal-knapsack to the correlated value of v( ) (see Defnton 3 for the defnton of ndependent and correlated values) n the case where the budget constrant s met ex ante,.e., E S ˆq [v B (S)] /E S D [v(s)] when ĉˆq B. We then show that ths rato s equal to the approxmaton rato of the sequental posted prcng mechansm. Fnally, we use ths rato to bound the ntegral, and fractonal, knapsack correlaton gap. The man dea to derve a bound on ths rato s to show that t s mnmzed when all agents have equal cost B/k, n whch case, when the budget constrant s met ex ante, we can then apply the result from Yan (2011) for the correlaton gap of the k-hghest-value-elements set functon. Lemma 14. For any addtve value functon v( ) and budget B, over margnal probabltes ˆq and prces ĉ that (a) satsfy the ex ante budget constrant,.e., ĉ ˆq B, and (b) satsfy the k-large market condton,.e., ĉ B/k, the rato of the ndependent value of the fractonal-knapsack and the correlated value of v( ) s mnmzed when ĉ = B/k for all. Proof. For the frst part of the proof, we assume that v = ĉ,.e., that the bang-per-buck s one for all elements. The last step of the proof s to generalze ths specal case to any values. Observe that wth ths assumpton, v B (S) = mn(b, j S ĉj). Assume that there s some ĉ such that ĉ < B/k. We show that when v = ĉ, ncreasng ĉ to any ĉ > ĉ and decreasng ˆq to ˆq = ĉ ˆq /ĉ preserves the correlated value whle only lowerng the ndependent value. Let ĉ j = ĉ j and ˆq j = ˆq j for j. The correlated value of v( ) s E S D [v(s)] = j ĉj ˆq j = j ĉ j ˆq j so t s preserved. Smlarly, the ex ante budget constrant s stll satsfed. The argument for the ndependent value decreasng s the followng. Let v B (S) be defned smlarly as v B (S), but where agents have values and costs equal to ĉ. Condton on the subset of other agents S who accept ther prces and consder the margnal contrbuton to the expected value of v B ( ) and v B ( ) from agent. In the case that C = j S ĉj > B, ths contrbuton s zero for both ĉ and ĉ. When C < B, these contrbutons are ˆq mn(b C, ĉ ) and ˆq mn(b C, ĉ ). By the defnton of ˆq = ĉ ˆq /ĉ and concavty of mn(b C, ), the former s greater than the latter. Ths nequalty holds for all sets S, so removng the condtonng on S, t holds n expectaton and the ndependent value of fractonal-knapsack s lowered. It remans to extend ths result to any v. Fx v and assume wthout loss of generalty that v 1 /ĉ 1 v n /ĉ n. Then the fractonal-knapsack set functon can be rewrtten as v B (S) = N(v /ĉ v +1 /ĉ +1 ) mn(b, ĉ j ) j S {1,...,} 5 Ths value s gven by sortng the elements of S by v /ĉ and admttng them greedly untl the frst element that does not ft wth the remanng capacty, that element s admtted fractonally (provdng a fracton of ts value). 12

13 and the addtve set functon as v(s) = N(v /ĉ v +1 /ĉ +1 )( j S {1,...,} snce these sums telescope. So the rato of ndependent value of v B (S) to the correlated value of v(s) s mnmzed when the ratos of the ndependent value of mn(b, j S {1,...,} ĉj) to the correlated value of j S {1,...,} ĉj are mnmzed for all. We conclude by observng that mn(b, j S {1,...,} ĉj) and j S {1,...,} ĉj are the fractonal-knapsack set functon and the addtve set functon when v = ĉ over ground set {1,..., }, and that ther rato s mnmzed when ĉ = B/k for all agents. Next, we use the result from Yan (2011) to bound the rato of the ndependent value of fractonalknapsack to the correlated value of v( ). Lemma 15. For any dstrbuton over sets D wth margnal probabltes ˆq satsfyng the ex ante budget constrant,.e., ĉ ˆq B, the rato of the ndependent value of fractonal-knapsack to the correlated value of v( ) s at least 1 1/ 2πk when the market s k-large. Proof. Consder the case where each agent has cost ĉ = B/k and assume that the ex ante budget constrant s satsfed, so ˆq k. Snce any set of sze at most k s feasble and snce ˆq k, there s a dstrbuton such that the budget constrant s always met ex post. Therefore, the correlated value of v( ) s equal to the correlated value of fractonal-knapasck. The rato of the ndependent value of fractonal-knapsack to the correlated value of v( ) s thus equal to the correlaton gap of fractonal-knapsack. Snce all agents have cost B/k, the fractonal-knapsack set functon s equal to the k-hghest-value-elements set functon. By Theorem 2, the rato of the ndependent value of fractonal-knapsack to the correlated value of v( ) s therefore 1 1/ 2πk. By Lemma 14, the rato of the ndependent value of fractonal-knapsack to the correlated value of v( ) when the ex ante budget constrant s satsfed s mnmzed when all agents have cost B/k, so ths rato s at least 1 1/ 2πk. We now prove the man theorem of ths secton whch relates the approxmaton factor of sequental posted prcng (wth ex post budget feasblty) to the optmal mechansm wth ex ante budget feasblty. Theorem 16. The sequental posted prcng mechansm (ˆq, σ), where ˆq s the soluton to the multlnear extenson program (5) and where the order σ s decreasng n v ĉ, s a (1 1/ 2πk)(1 1/k) approxmaton to the optmal mechansm n the case of regular cost dstrbutons. Proof. Denote ˆq the optmal soluton to the multlnear extenson program (5). For addtve value functons, lnearty of expectaton mples that the multlnear extenson s equal to the concave closure and the optma of the multlnear extenson program (5) and concave closure program (4) are the same. Ther performance upper bounds that of the optmal mechansm that satsfes ex post budget feasblty by Lemma 6. The objectve value of these programs wth optmal soluton ˆq s v ˆq, whch s equal to the correlated value of the addtve set functon v( ) on dstrbutons wth margnals ˆq. So by Lemma 15, the rato of the ndependent value of fractonal-knapsack to the upper bound of the optmal mechansm s at least 1 1/ 2πk The random set of agents who accept ther offer n the sequental posted prcng s equal to the set of agents who are admtted by the fractonal-knapsack set functon on an ndependent random ĉ j ) 13

14 Fgure 2: Comparson of the approxmaton ratos obtaned for addtve value functons by the two dfferent approaches. On the horzontal axs s k, the sze of the market. set of agents wth margnals ˆq, wthout ncludng the fractonal agent. The loss from ths fractonal agent s at most a factor 1 1/k. Ths posted prcng mechansm therefore has an approxmaton rato of (1 1/ 2πk)(1 1/k). As a corollary of Lemma 15, we get new correlaton gap results for the fractonal, and ntegral, knapsack set functons. Theorem 17. The correlaton gaps of fractonal-knapsack and ntegral-knapsack are at least 1 1/ 2πk and (1 1/ 2πk)(1 1/k) respectvely, n a k-large market. Proof. We frst show the correlaton gap of fractonal-knapsack, the correlaton gap of ntegralknapsack wll then follow easly. We start by showng that the correlaton gap s mnmzed when the budget constrant s satsfed. Then, we upper bound the fractonal-knapsack correlated value by the correlated value of v( ). Fnally, we apply Lemma 15. We clam that the correlaton gap of fractonal-knapsack s mnmzed when the budget constrant s satsfed. Observe that f the budget constrant s not satsfed, then t s possble to decrease some ˆq such that the correlated value of fractonal-knapsack remans the same. Snce decreasng some ˆq only decreases the ndependent value of fractonal-knapsack, the rato of the ndependent value to the correlated value also decreases. Clearly, the fractonal-knapsack correlated value s upper bounded by the correlated value of v( ). Therefore, the correlaton gap of fractonal-knapsack s at least the rato of the ndependent value of fractonal-knapsack to the correlated value of v( ) when the budget constrant s satsfed, so at least 1 1/ 2πk by Lemma 15. Fnally, observe that the correlated value of fractonal-knapsack upper bounds the correlated value of ntegral- knapsack and that the ndependent value of ntegral-knapsack s a 1 1/k approxmaton to the ndependent value of fractonal-knapsack. Therefore, the correlaton gap of ntegral-knapsack s at least (1 1/ 2πk)(1 1/k). Comparson of Sequental and Oblvous posted prcng. We now compare the approxmaton rato for addtve value functons acheved usng the sequental posted prcng mechansm wth the bang per buck order, (1 1/ 2πk)(1 1/k), and usng oblvous posted prcng where the budget s lowered, (1 ɛ)(1 e ɛ2 (1 ɛ)k/12 ). Fgure 2 shows that the approxmaton rato wth the sequental orderng approaches 1 much faster than wth the oblvous orderng as the sze of the market ncreases. To obtan these results for oblvous posted prcng, we numercally solved for the best ɛ. We emphasze that we are comparng the theoretcal bounds of these approaches, and not emprcal performances. 14

15 6 Computng Prces In the two prevous sectons, we gave condtons under whch optmal prces from the multlnear extenson program (5) perform well when offered sequentally or oblvously. In ths secton, we consder the computatonal problem of fndng these prces. For submodular value functons, we reduce the problem to the well-known greedy algorthm for submodular optmzaton. For addtve value functons, we use a smple method based on the Lagrangan relaxaton of the budget constrant. 6.1 The Lagrangan Relaxaton for Addtve Value Functons Consder the case of addtve value functons where the prncpal has a value v for each agent and the value functon s v(s) = S v. Recall the vrtual surplus program (2) from Secton 3: whch can be rewrtten for addtve value functons as: max E c[v(x(c))] (2) x s.t. E c[φ (c) x (c)] B, max q s.t. E c[v x (c)] (6) E c[φ (c) x (c)] B. We show that the ex ante optmal mechansm can be found drectly by takng the Lagrangan relaxaton of the budget constrant (wth parameter λ) of the followng Lagrangan program: max x λb + E c[(v λφ (c )) x (c)]. (7) For any Lagrangan parameter λ, ths objectve can be optmzed by pontwse optmzng (v λφ (c )) x (c), a.k.a., the Lagrangan vrtual surplus. Ths pontwse optmzaton pcks all the agents such that v λφ (c ). If the vrtual cost functons are monotone,.e., n the socalled regular case, then ths optmzaton gves a monotone allocaton rule where an agent s pcked whenever c φ 1 (v /λ) Notce that as the Lagrangan parameter ncreases, the payments of the agents, as represented by vrtual costs, become more costly n the objectve of the lagrangan program (7). Thus, the expected payment of the mechansm s monotoncally decreasng n the Lagrangan parameter. Wth λ = 0 the Lagrangan vrtual surplus optmzer smply maxmzes v(x) and pays each agent selected the maxmum cost n the support of her dstrbuton. If ths payment s under budget then t s optmal, otherwse, we can ncrease λ untl the budget constrant s satsfed. For example, wth λ = the empty set of agents s selected and no payments are made. The optmal mechansm s the one that meets the budget constrant wth equalty. In the case that the expected payment s dscontnuous then mxng between the least over-budget and least under-budget mechansm s optmal. For further dscusson of Lagrangan vrtual surplus optmzers, see Devanur et al. (2013). Proposton 18. The Lagrangan vrtual surplus optmzer (or approprate mxture thereof) that meets the budget constrant wth equalty s the Bayesan optmal ex ante budget feasble mechansm. 15

16 Lagrangan vrtual surplus optmzaton suggests selectng an agent when her prvate cost c s below φ 1 (v /λ). The mechansm that acheves ths outcome posts the prce of ĉ = φ 1 (v /λ) to agent. Denote by ˆq = F (ĉ ) the probablty that accepts the prce ĉ. For the prces ĉ, the total expected payments are ĉ ˆq. When the vrtual cost functons are monotone and strctly ncreasng, there s a Lagrangan parameter for whch the budget constrant s met wth equalty,.e., wth ĉ ˆq = B. The optmal ex ante mechansm s therefore the posted prce mechansm that posts ĉ to each agent for the Lagrangan parameter λ that satsfes ĉ ˆq = B. Note that such a Lagrangan parameter λ can be arbtrarly well approxmated snce ĉ ˆq s decreasng as a functon of λ. Example 1. Consder n agents wth costs drawn unformly and..d. from [0, 1] and unform addtve value functon v = 1 for all,.e., the cardnalty functon. The vrtual cost functon s φ(c) = c + F (c) f(c) = 2c. The Lagrangan parameter λ = 1 2 n/b nduces a unform posted prce of ĉ = B/n whch s accepted wth probablty ˆq = B/n for an expected payment of B/n. Summng over all n agents, the budget s balanced ex ante. 6.2 A Reducton to the Greedy Algorthm for Submodular Optmzaton For general submodular value functons we reduce the optmzaton of the multlnear extenson program (5), restated below, to the problem of optmzng a submodular functon subject to a cardnalty constrant. Ths problem of optmzng a submodular functon under cardnalty, knapsack, or matrod constrants s well studed and the greedy algorthm gves a 1 1/e approxmaton for knapsack and cardnalty constrants; see Nemhauser et al. (1978), Khuller et al. (1999), and Svrdenko (2004). max V (q) (5) q s.t. q F 1 (q ) B. Defne the cost curve of agent to be the expected payment to agent,.e., q F 1 (q ) n our case. The man dfference between the multlnear extenson program (5) and the knapsack settng consdered n the lterature s that the cost curves n the knapsack settng are lnear n q. Our reducton to the greedy algorthm s the followng. We dvde each agent, called a bg agent, n cost space nto m dscrete agents j of equal cost, called the small agents. An agent j corresponds to the jth ncrease of q, startng from q = 0, that has cost B/m. We set 1/m as a fracton of the total budget B whch fxes the number of steps n the algorthm to be m. Wth large m, the reducton becomes a fner dscretzaton. Before formally descrbng the reducton, we ntroduce some notaton. For each and j, let δ j be the jth ncrease n q, startng from q = 0, that has cost B/m,.e., δ j satsfyng B/m = F 1 ( k j δ k) ( k j δ k) F 1 ( k<j δ k) ( k<j δ k). Gven a set S of small agents, the contnuous soluton correspondng to S s q(s) wth q (S) = j: j S δ j. The reducton. 1. For each agent, create m small agents j where 1 j m so that the reduced nstance has mn agents. 16

17 2. For each small agent j, ts cost s B/m. 3. For each small agent j, ts margnal contrbuton V S ( j ) n value to a set S s the margnal contrbuton of ncreasng the fracton of agent correspondng to S by δ j,.e., V (q ) V (q(s)) where q = q (S) + δ j and q j = q j(s) for j. We show that the soluton to the reduced problem that we obtaned wth the greedy algorthm for cardnalty constrant corresponds to a soluton for the multlnear extenson program (5) that s a 1 1/e o(1) approxmaton, almost matchng the performance of the greedy algorthm for knapsack constrant wth ntegral agents and lnear cost curves. We start by showng that f a soluton s feasble n the reduced problem, then the contnuous soluton correspondng to t s a feasble soluton to the multlnear extenson program (5). Then, wth access to exact values of the ncreases δ j and of the margnal contrbutons V S ( j ), the approxmaton rato s 1 1/e o(1). Fnally, we show that t s possble to approxmate δ j and V S ( j ) wth estmates that cause an addtonal loss of o(1) to the approxmaton rato. From a set of small agents to a contnuous soluton for the bg agents. Prevously, we defned a dstrbuton to be regular f the vrtual cost functon s monotoncally ncreasng. An alternate defnton s that a dstrbuton F s regular f the cost curve q F 1 (q) s convex. Ths defnton s the analogue to the revenue curve beng concave for regular dstrbutons when the agents are buyers, and not sellers, from Bulow and Roberts (1989). Recall that gven a set S of small agents, the contnuous soluton correspondng to S s q(s) wth q (S) = j: j S δ j and that δ j s the jth ncrease n q that has cost B/m. Therefore, gven a set S of small agents of sze at most m such that for any δ j S, δ k S for all k < j, then q(s) has cost at most B. The condton that f δ j S, then δ k S for all k < j, s equvalent to the condton that greedy always pcks small agents correspondng to lower quantles before small agents correspondng to hgher quantles, whch we show formally. Lemma 19. Gven two small agents k and j such that k < j, the greedy algorthm wth a cardnalty constrant pcks k before j for regular dstrbutons F. Proof. Snce all small agents have equal cost, we need to show that k has a larger margnal contrbuton than j to any set S of small agents such that k, j S. Snce V ( ) s monotone, t suffces to show that δ k > δ j. In quantle space, the cost of ncreasng some quantle q by a fx amount s ncreasng n q snce q F 1 (q ) s convex by defnton of regular dstrbutons. Therefore, n cost space, the ncrease n quantle δ that s obtaned by ncreasng the cost curve by a fx amount s decreasng, so δ k > δ j. The case of rregular dstrbutons s consdered n Secton 8. Wth exact values of δ j and V S ( j ). We consder the case where the exact values of the ncreases n q and margnal contrbutons are gven by an oracle. We show that fndng a good soluton to ths reduced problem wth small agents gves us a good soluton to the problem wth bg agents. Lemma 20. The optmal soluton S to the reduced problem satsfes V (q(s )) (1 o(1))v (ˆq) where ˆq s the optmal soluton to the multlnear extenson program (5). 17

18 Proof. We pck the step sze to be m = n 2. The proof shows that there exsts a set S that s close to a feasble soluton n the reduced problem and such that q(s) s a better soluton than ˆq. Let S be the set of small agents such that q(s) s maxmzed subject to q(s) ˆq. Defne S +1 to be the set contanng all small agents n S and one addtonal small agent for each bg agent. Observe that V (q(s +1 )) V (ˆq) snce V ( ) s non-decreasng. So there s a feasble soluton to the dscretzed problem such that f we add one small agent for each bg agent, then we obtan a better soluton than the optmal soluton to the orgnal problem. Greedly remove agents by mnmal margnal contrbuton from S +1 untl we get a feasble soluton S. The number of small agents who need to be removed s n snce S s feasble. Snce S contans n 2 small agents, by the greedness and the fact V ( ) s concave along any lne of postve drecton, (1 + 1/n)V (q(s)) V (q(s +1 )). Therefore, (1 + o(1))v (q(s )) (1 + o(1))v (q(s)) V (q(s +1 )) V (ˆq) Next, we show that the reduced problem can be optmzed. Lemma 21. Let S be the set returned by the greedy algorthm for submodular functons under a cardnalty constrant on the reduced problem, then V (q(s)) (1 1/e)V (q(s )) where S s the optmal soluton to the reduced problem. Proof. Observe that the objectve functon n the reduced problem s a submodular functon. Ths follows drectly from the concavty of V ( ) along any postve lne of drecton. In addton, snce all small agents have cost B/m, the constrant s a cardnalty constrant. Snce the greedy algorthm for submodular functons under a cardnalty constrant s a 1 1/e approxmaton for submodular functons, we get the desred result. We now have the tools to show that f we had an oracle for the ncreases and margnal contrbutons, the greedy algorthm on the reduced nstance would gve us a 1 1/e o(1) approxmaton. Lemma 22. Let S be the output of the greedy algorthm on the reduced nstance, where exact values of δ j and V S ( j ) are gven by an oracle at each teraton, then V (q(s)) (1 1/e o(1))v (ˆq), where ˆq s the optmal soluton to the multlnear extenson program (5). Proof. We combne the results from the dscretzaton that causes a o(1) loss wth the greedness of the algorthm that s a 1 1/e approxmaton to obtan the desred result. By Lemma 21 and Lemma 20, V (q(s)) (1 1/e)V (q(s )) (1 1/e o(1))v (ˆq) where S s the optmal soluton to the reduced problem. Wth estmates of δ j and V S ( j ). We now show that we can use the greedy algorthm wth estmates of the ncreases and the margnal contrbutons, that we can compute. Let q(s) be defned smlarly to q(s) but wth estmates δ j. The frst lemma shows that the value of the optmal soluton to the reduced problem has almost the same value as when the ncreases δ j are estmated. The second lemma extends Lemma 21 to the case where greedy s run wth estmated margnal contrbutons ṼS( j ) and any δ j. We defer the proofs of these two lemmas to the appendx. 18