Bayesian Incentive Compatibility via Fractional Assignments

Transcription

1 Bayesan Incentve Compatblty va Fractonal Assgnments Xaohu Be Zhy Huang Abstract Very recently, Hartlne and Lucer [14] studed sngleparameter mechansm desgn problems n the Bayesan settng. They proposed a blac-box reducton that converted Bayesan approxmaton algorthms nto Bayesan-Incentve- Compatble (BIC) mechansms whle preservng socal welfare. It remans a major open queston f one can fnd smlar reducton n the more mportant mult-parameter settng. In ths paper, we gve postve answer to ths queston when the pror dstrbuton has fnte and small support. We propose a blac-box reducton for desgnng BIC mult-parameter mechansms. The reducton converts any algorthm nto an ɛ-bic mechansm wth only margnal loss n socal welfare. As a result, for combnatoral auctons wth sub-addtve agents we get an ɛ-bic mechansm that acheves constant approxmaton. 1 Introducton In ths paper, we consder the problem of desgnng computatonally effcent and truthful mechansm for mult-parameter mechansm desgn problems n the Bayesan settng. Suppose a major Internet search servce provder wants to sell multple advertsement slots to a number of companes. From the hstory of prevous transactons, we can estmate a pror dstrbuton of each company s valuaton of the advertsement slots. What mechansm shall the search servce provder use to obtan good socal welfare, or good revenue? Ths s a typcal mult-parameter mechansm desgn problem. In general, we consder the scenaro n whch a prncpal wants to sell a number of dfferent servces to multple heterogeneous strategc agents subject to some feasblty constrants (e.g. total cost of provdng these servces must not exceed the budget), so that some desred objectve (e.g. socal welfare, revenue, resdual surplus) s acheved. If we nterpret ths as smply a combnatoral optmzaton problem, then there exsts Insttute for Theoretcal Computer Scence, Tsnghua Unversty. Emal: bxh08@mals.tsnghua.edu.cn. Supported n part by the Natonal Natural Scence Foundaton of Chna Grant , the Natonal Basc Research Program of Chna Grant 2007CB807900, 2007CB Computer and Informaton Scence, Unversty of Pennsylvana. Emal: hzhy@cs.upenn.edu. approxmaton algorthms for many of these problems. And the approxmaton ratos of many of these algorthms are tght subject to certan computatonal complexty assumptons. However, f we wants to desgn protocols of allocatons and settng prces n order to acheve the desred objectve n the equlbrum strategc behavor of the agents, we usually have much worse approxmaton rato. Therefore, t s natural to as the followng queston: Can we convert any algorthm nto a truthful mechansm whle preservng the performance, say, socal welfare? Unfortunately, from prevous wor we learn that ths s mpossble for some problems. Papadmtrou et al. [18] showed the frst sgnfcant gap between the performance of determnstc algorthms and determnstc truthful mechansms va the Combnatoral Publc Project problem. Bayesan settng. The standard game theoretc model for ncomplete nformaton s the Bayesan settng, n whch the agent valuatons are drawn from a publcly nown dstrbuton. The standard soluton concept n ths settng s Bayesan-Nash Equlbrum. In a Bayesan-Nash equlbrum, each player maxmzes ts expected payoff by followng the strategy profle gven the pror dstrbuton of the agent valuatons. In ths paper, we wll consder mult-parameter welfare-preservng algorthm/mechansm reductons n the Bayesan settng, and weaen truthfulness constrant from Incentve Compatblty (IC) to Bayesan Incentve Compatblty (BIC), whch means truth tellng s the equlbrum strategy over random choce of the mechansm as well as the random realzaton of the other agent valuatons. In many real world applcatons such as onlne auctons, AdWords auctons, spectrum auctons etc., the avalablty of data of past transactons mae t possble to obtan good estmaton of the pror dstrbuton of the agent valuatons. Thus, revstng the algorthm/mechansm reducton problem n the Bayesan settng s of both theoretcal and practcal mportance. Hartlne and Lucer [14] studed ths problem n the sngle-parameter settng. They showed a brllant

2 blac-box reducton from any approxmaton algorthm to BIC mechansm that preserves the performance wth respect to socal welfare maxmzaton. In ths paper, we prove that smlar reducton also exsts for the realm of mult-parameter mechansm desgn for socal welfare! Moreover, we can also obtan BIC mechansm for revenue or resdual surplus va some varants of our blac-box reducton. Our results and technque. Our man result s a blac-box reducton that converts algorthms nto BIC mechansms wth essentally the same socal welfare for arbtrary mult-parameter mechansm desgn problem n the Bayesan settng. More concretely, gven an algorthm A that provdes SW A socal welfare, the reducton provdes a mechansm that gves SW A ɛ socal welfare and s ɛ-bic. The runnng tme s polynomal n the nput sze and 1/ɛ. Ths resolves an open problem n [14]. The ey dea s to decouple the reported valuatons and the nput valuatons for the algorthm A. When the reported valuatons are v 1, v 2,..., v n, we wll manpulate the valuatons va some carefully desgned ntermedate algorthms B 1,..., B n, and use allocaton A(B 1 (v 1 ),..., B n (v n )). We prove that there exst ntermedate algorthms B 1,..., B n so that there are prces that acheve BIC. Under certan condtons, the margnal loss factor n socal welfare can be made multplcatve. As an applcaton of ths reducton, we get a ( 1 2 ɛ)-approxmate and ɛv max -BIC mechansm for socal welfare maxmzaton n combnatoral auctons wth sub-addtve agents. For the more restrcted case of fractonally sub-addtve agents, we obtan (1 1 e ɛ)- approxmate mechansm. Related wor. The problem of maxmzng socal welfare aganst strategc agents s one of the oldest and most famous problems n the area of mechansm desgn. It has been extensvely studed by the economsts n both Bayesan and pror-free settng wthout consderng computatonal power constrant. The celebrated VCG mechansm [3, 10, 20] whch guarantees optmal socal welfare and ncentve compatblty s one of the most exctng results n ths doman. However, mplementng the VCG mechansm s NP-hard n general. Ths s one of the reasons that VCG mechansm s rarely used n practce despte of ts lovely theoretcal features. In the past decade, computer scentsts ntroduced many novel technques n the pror-free settng to desgn computatonally effcent mechansms that provde ncentve compatblty and/or good approxmaton to optmal socal welfare for varous famles of valuaton functons. On the one hand, Dobzns, Nsan and Schapra [6] proposed poly-tme mechansms whch acheved Ω(1/ n)-approxmaton for general agents and Ω(1/ log 2 n)-approxmaton for sub-modular agnets. Dobzns [4] later proposed a truthful mechansm whch acheved an mproved Ω(1/ log n)-approxmaton for a strctly broader class of sub-addtve agents. On the other hand, f we focus on the algorthmc problem of maxmzng socal welfare assumng all valuatons are truthfully revealed, then the algorthm by Dobzns, Nsan and Schapra [5] gave Ω(1/ n)-approxmaton for general case and Ω(1/ log n)-approxmaton for sub-addtve agents. The latter approxmaton rato s later mproved to 1 2 for subaddtve agents [8] and (1 1 e ) for the more restrcted class of fractonally sub-addtve agents [4, 9]. The above results suggest that there exsts a gap between the performance of the best poly-tme algorthms and that of the best poly-tme and ncentve compatble mechansm. As an effort to study the relaton between desgnng algorthms and desgnng truthful mechansms wth good approxmaton rato, Lav and Swamy [16] proposed a meta-mechansm that converted strong roundng algorthms for the standard LP of socal welfare maxmzaton nto IC mechansms. However, ther approach requred the roundng algorthm to wor for arbtrary valuaton functons. Ths requrement prevents ther technque to get good approxmaton beyond cases of general valuatons and addtve valuatons (va a dfferent lnear program). But the more nterestng classes of valuatons (e.g. sub-addtve valuatons and sub-modular valuatons) les between these two extremes. Another notable attempt on reducng IC mechansm desgn to algorthm desgn s the very recent wor by Dughm and Roughgarden [7]. They proved that for any pacng problem that admtted an FPTAS, there was an IC mechansm that was also an FPTAS. Most of the prevous effort from computer scentsts has focused on the pror-free settng. Untl very recently, there has been a few wor that brought more and more Bayesan analyss nto the feld of algorthmc mechansm desgn. Hartlne and Lucer [14] gave a blac-box reducton that converted any Bayesan approxmaton algorthm nto a Bayesan ncentve compatble mechansm that preserved socal welfare n the sngle parameter doman. Bhattacharya et al. [1] studed the revenue maxmzaton problem for auctonng heterogeneous tems when the valuatons of the agents were addtve. Ther result gave constant approxmaton n the Bayesan settng even when the agents had publc nown budget constrants. Chawla et al. [2] consdered the revenue maxmzaton problem n the mult-dmensonal mult-unt auctons. They ntroduced mechansm that gave constant approxmaton n varous settngs va se-

3 quental posted prcng. Fnally, n concurrent and ndependent wor, Hartlne et al. [13] study the relaton of algorthm and mechansm n Bayesan settng and propose smlar reducton. In the dscrete support settng that s consdered n ths paper, they use essentally the same reducton. However, ther wor acheves perfectly BIC nstead of ɛ-bic. They also extend the reducton to the more general contnuous support settng. 2 Prelmnares 2.1 Notatons. We use {x } 1 n to denote an array of sze n. We also use the natural extenson of ths notaton for mult-dmensonal arrays. We wll use bold font x to denote a vector (x 1,..., x n ). We let (S) denote the set of dstrbutons over the elements n a set S. For a random varable x, we let E [x] denote ts expectaton and let σ [x] denote ts standard devaton. We use subscrpts to represent the random choces over whch we consder the expectaton and varance. For nstance, E y F [x] s the expectaton of x when y s drawn from dstrbuton F. We sometmes use E y [x] for short when the dstrbuton F s clear from the context. 2.2 Model and defntons. In ths secton, we wll formally ntroduce the model n ths paper. We study the general mult-parameter mechansm desgn problems. In a mult-parameter mechansm desgn problem, a prncpal wants to sell a set of servces to multple heterogeneous agents n order to optmze the desred objectve (e.g. socal welfare, revenue, resdual surplus, etc.). A Bayesan mult-parameter mechansm desgn problem wth n agents s defned by a tuple I, J, V, F. I (I 1,..., I n ): The set of servces that the prncpal wants to sell to the agents. Snce we can mpose arbtrary feasblty constrants on the allocatons, we can assume wthout loss of generalty that the servces are parttoned nto n dsjont sets I 1,..., I n such that the servces n I only am for agent, and each agent s nterested n any one of the servces n I. J I 1 I n : The set of feasble allocatons. V V 1 V n : The space of agent valuatons. We let V R I denote the set of possble valuatons of agent. We let v max max,v V,S I v(s) denote the maxmal valuaton. F F 1 F 2 F n : The jont pror dstrbuton of the agent valuatons. We assume the pror dstrbuton s a product dstrbuton. We let F (V ) denote the pror dstrbuton of the valuaton of agent. In ths paper, we only consder dstrbutons wth fnte and polynomally large support. We wll assume wthout loss of generalty that the support of each dstrbuton F s {v 1,..., vl }. Suppose v F, We wll let f (t) denote the probablty that v v t. For example, n the combnatoral aucton problem wth n agents and m tems, we let [m] {1, 2,..., m} denote the set of tems. The set of servces for each agent s the set of all subsets of tems, that s, I 2 [m], 1 n. The set of feasble allocatons s J {(S 1,..., S n ) : S I, S S j }. The set of valuatons, V, s the set of mappngs from subset of tems I to R + that are monotone (v (S) v (T ) for S T ) and normalzed (v ( ) 0). We usually assume that the valuatons n V satsfes certan propertes, e.g. sub-addtvty, sub-modularty, etc. Algorthm. An algorthm for a mult-parameter mechansm desgn problem I, J, V, F s a protocol (may or may not be randomzed) that taes a realzaton of agent valuatons v V as nput, and outputs a feasble allocaton S J. Mechansm. A mechansm s an nteractve protocol (may or may not be randomzed) between the prncpal and the agents so that the prncpal can retreve nformaton from the agents (presumably va ther bds), and determne an allocaton of servces S J and a collecton of prces p (p 1,..., p n ). The extra challenge for mechansm desgn, compared to algorthm desgn, s to retreve genune valuatons from the agents and handle ther strategc behavor. For each 1 n, we wll assume the pror dstrbuton F s publc nown. But the actual realzaton v F s prvate nformaton of agent. Each agent ams to maxmzes the quas-lnear utlty v (S ) p, where S s the servce t gets and p s the prce. Thus, the agents may not reveal ther genune valuatons f manpulatng ther bds strategcally can ncrease ther utlty. Objectves. We wll consder three dfferent objectves: socal welfare, revenue, and resdual surplus. The expected socal welfare of a mechansm M s [ n ] SW M E v F,(S,p) M(v) v (S ). 1 Smlarly, we wll let SW A denote the expected socal welfare of an algorthm A.

4 Defnton 2.1. An algorthm A s α-approxmate n socal welfare for a mult-parameter mechansm desgn problem I, J, V, F, f SW A α OPT. The expected revenue of a mechansm s [ n ] R M E v F,(S,p) M(v) p. 1 The last objectve, resdual surplus, was recently proposed by Hartlne and Roughgarden [15] as an alternatve objectve n the flavour of socal welfare. In the resdual surplus maxmzaton problem, the prncpal ams to maxmze the sum of the agents utltes nstead of the sum of ther valuatons. The expected resdual surplus s [ n ] RS M E v F,(S,p) M(v) (v (S) p ). 1 We wll let OPT denote the optmal socal welfare, that s, OPT max M SW M. Snce both revenue and resdual surplus are upper-bounded by socal welfare. We wll use OPT as our benchmar for all three objectves. Soluton concepts. Ideally, a mechansm shall provde ncentve for the agents to reveal ther valuatons truthfully. In ths secton, we wll formalze ths requrement by ntroducng the game-theoretcal soluton concepts that we use n ths paper. Defnton 2.2. A mechansm M s Bayesan ncentve compatble (BIC) f for each agent and any two valuatons v, ṽ V, we have E v,(s,p) M(v,v ) [v (S ) p ] E v,(s,p) M(ṽ,v ) [v (S ) p ]. Defnton 2.3. A mechansm M s ɛ-bayesan Incentve Compatble (ɛ-bic) f for any agent and any two valuatons v, ṽ V, E v,(s,p) M(v,v ) [v (S ) p ] E v,(s,p) M(ṽ,v ) [v (S ) p ] ɛ. Other than the above constrants of ncentve compatblty, the mechansm shall also guarantee that the agents always get non-negatve utlty. Otherwse, the agents may choose not to partcpate n the mechansm. Ths s nown as the ndvdual ratonalty constrant. Defnton 2.4. A mechansm M s ndvdually ratonal (IR) f for any realzaton v of agent valuatons, and any allocaton S and prces p by the mechansm, we always have that for any agent, v (S ) p 0. 3 Characterzaton of BIC mechansms In ths secton, we wll ntroduce a non-trval characterzaton of BIC mult-parameter mechansms va a novel connecton between BIC mechansms and envyfree prces. Ths characterzaton nspres our reducton n the next secton. 3.1 Fractonal assgnment problem. We wll frst ntroduce the fractonal assgnment problems, whch wll play a crtcal role n the results of ths paper, and a useful lemma about envy-free prces n fractonal assgnment problems. In order to dstngush the notatons for fractonal assgnment problems and those for the mechansm desgn problems, we wll use superscrpts nstead of subscrpts to specfy dfferent entres of a vector for the fractonal assgnment problems. For nstance, we wll use x s to denote the s th entry of a vector x. Let us consder a maret wth l buyers and m nfntely dvsble products. Each buyer s has a non-negatve demand α s, whch denotes the maxmal amount of products the buyer wll buy. Each product t has a non-negatve supply β t, whch denotes the avalable amount of ths product n the maret. For each buyer s and each product t, we let w st denote the nonnegatve value of buyer s of product t. The goal s to set prces for the products and to assgn the products to the buyers subject to the demand and supply constrants. Thus, a soluton (x, p) to the fractonal assgnment problem conssts of a collecton of prces p (p 1,..., p l ) and a feasble allocaton x { } 1 s l,1 t m n the polytope: { x : s, m α s ; t, } l β t ; x 0 where denotes the amount of product t that s assgned to buyer s. Defnton 3.1. A soluton (x, p) s envy-free f for any > 0, then t s a product that maxmzes the quas-lnear utlty of agent s, and the utlty for agent s s non-negatve. That s, (3.1) s, t : > 0 w st p t max {ws p } 0. Defnton 3.2. An allocaton x s maret-clearng f all demand constrants and supply constrants hold wth equalty, that s, 1 s l : m α s, 1 t m : l β t.,

5 The socal welfare maxmzaton problem for a fractonal assgnment problem s characterzed by the followng lnear program (P) and ts dual (D). (P) Maxmze Σ l Σ m w st Σ m α s Σ l β t 0 (D) Mnmze Σ l α s u s + Σ m β t p t u s + p t w st u s 0 p t 0 s.t. s t s, t s.t. s, t We wll ntroduce two useful lemmas about the connecton between envy-free prces and socal welfare maxmzaton for fractonal assgnment problems. These lemmas were nown n dfferent forms n the economcs lterature [11]. Lemma 3.1. If there exst envy-free prces p for a maret-clearng allocaton x, then x maxmzes the socal welfare, that s, x w max z z w. Proof. Suppose there exst envy-free prces p for an allocaton x. Let u s max t {w st p t }. We have that u s + p t w st for all s, t. So (u, p) s a feasble soluton for the dual LP. Moreover, by defnton of envy-freeness, we have s, t : > 0 u s w st p t. Therefore, we get that l m w st l l m (u s + p t ) α s u s + t s t β t p t. The last equalty holds because x s maret clearng. Notce that x s a feasble soluton to the prmal LP. By dualty theorem, we get that the allocaton x maxmzes the socal welfare for the fractonal assgnment problem. Lemma 3.2. If an allocaton x maxmzes the socal welfare, then there exst envy-free prces p for the fractonal assgnment problem. Proof. Suppose the allocaton x maxmzes the socal welfare. Let (u, p) be an optmal soluton to the dual LP. By complementary slacness we get that > 0 only f the correspondng dual constrant s tght, that s, u s + p t w st. Therefore, > 0 mples that w st p t u s w s p for all. Thus p s a collecton of envy-free prces for the allocaton x n ths fractonal assgnment problem. Note that the above proof also gves a poly-tme algorthm for fndng the welfare maxmzng allocaton x and the correspondng envy-free prces p by solvng the prmal and dual LPs. Moreover, we also get that the envy-free prces p satsfy a wea unqueness n the sense that t must be part of an optmal soluton for the dual LP. Corollary 3.1. There exsts a poly-tme algorthm that computes the welfare-maxmzng maret-clearng allocaton and the envy-free prces. 3.2 Characterzng BIC va envy-free prces. We frst ntroduce some notatons that wll smplfy our dscusson. Gven a mechansm M for a multparameter mechansm desgn problem I, J, V, F, we wll consder the expected values and expected prces for each agent when t choose a specfc strategy (each strategy corresponds to reportng a specfc valuaton). Assumng the other agents report ther valuatons truthfully, agent s expected value of the servce t gets, when the genune valuaton s v s and the reported valuaton s v t, s w st E v,(s,p) M(v t,v ) [v s (S )]. Smlarly, we let p t denote the expected prce the mechansm would charge to agent f ts reported valuaton s v t, that s, p t E v,(s,p) M(v t,v ) [p ]. By the defnton of BIC and IR, the mechansm M s BIC and IR f and only f for any 1 n and 1 s l, (3.2) w ss p s max{w st p t } 0. t The above equaton (3.2) s smlar to equaton (3.1) n the defnton of envy-freeness n fractonal assgnment problem. In fact, the ey observaton s that the above BIC condton s equvalent to the envyfree condton for a set of properly chosen fractonal assgnment problems. Induced assgnment problems. For each agent, we wll consder the followng nduced assgnment problem. We consder l vrtual buyers wth demands f (1),..., f (l) respectvely, and l vrtual products wth supples f (1),..., f (l) respectvely. For each vrtual buyer s and each vrtual product t, let vrtual buyer s

6 has value w st on vrtual product t. We wll refer to ths fractonal assgnment problem the nduced assgnment problem of agent. Let us consder the dentty allocaton x defned as follows: { f (s), f s t, 0, otherwse. We can easly verfy that a collecton of prces p (p 1,..., pl ) satsfes constrant (3.2) f and only f p satsfes the envy-free condton (3.1) of the nduced assgnment problem of agent wth respect to the above dentty allocaton. Hence, we have the followng connecton between BIC mechansm and the envy-free prces of the nduced assgnment problems. Lemma 3.3. (Characterzaton Lemma [19]) A mechansm M s BIC f and only f n the nduced assgnment problem of each agent the dentty allocaton x { } 1 s,t l maxmzes the socal welfare, and p (p 1,..., pl ) are chosen to be the correspondng envy-free prces. Comparng wth Myerson s characterzaton. Suppose the problem falls nto the sngle-parameter doman. Each valuaton v s s represented by a sngle non-negatve real number. Wth a lttle abuse of notaton, we let v s denote ths value. Wthout loss of generalty, we assume that v 1 > > v l. We let y t denote the probablty that agent would be served f the reported value was v t. The values w n the fractonal assgnment problems of agent are w st v syt for 1 s, t l. Myerson s famous characterzaton [17] of truthfulness n sngle-parameter doman mpled that the mechansm s BIC f and only f y 1 yl. Indeed, due to rearrangement nequalty, the dentty allocaton x maxmzes the socal welfare f and only f y 1 yl. Thus, the characterzaton lemma mples Myerson s characterzaton n the sngleparameter doman. 4 Reducton for socal welfare Lemma 3.3 suggests an nterestng connecton between BIC and envy-free prces for the nduced assgnment problems. Hence, gven an algorthm A, we wll manpulate the allocaton by A based on ths connecton n order to mae t satsfy the condton n Lemma Man deas. Let us frst brefly convey two ey deas n the constructon of the welfare-preservng reducton. The frst dea s to decouple the reported agent valuatons and the nput agent valuatons for algorthm v 1 v 2 v m B 1 B 2 B m ṽ 1 ṽ 2 ṽ m A S Fgure 1: Hgh-level pcture of the reducton for socal welfare. B s are ntermedate algorthms for manpulatng the nput of algorthm A. ṽ s are the reported valuatons. v s are the manpulated nput valuatons for algorthm A. S s the fnal allocaton. A. More concretely, we wll ntroduce n ntermedate algorthm B 1,..., B n. Each B wll tae the reported valuaton v as nput, then output a valuaton ṽ that may or may not equals v. Then, we wll use algorthm A to compute the allocaton S for agent valuatons ṽ 1,..., ṽ n, and allocate servces accordng to S. If we revst the values w n the nduced assgnment problem of agent after ths manpulaton, we wll get that for any 1 s, t l, w st E v,ṽ B(v t,v ),S A(ṽ) [v s (S )]. By Lemma 3.3, we need to choose B s carefully, so that the dentty allocatons n the manpulated assgnment problems are welfare-maxmzng allocatons. However, from the above equaton we can see that by usng B to manpulate the th valuaton, we may change not only the structure of the nduced assgnment problem of agent, but the structure of the nduced assgnment problems of other agents as well. Hence, we need to handle such correlaton among the nduced assgnment problems when we choose ntermedate algorthms B 1,..., B n. The dea that handles ths correlaton s to mpose an extra constrant on each ntermedate algorthm B : f the nput valuaton v s drawn from the dstrbuton F, then the output valuaton ṽ also follows the same dstrbuton, that s, for all 1 n and 1 t l, (4.3) Pr v F,ṽ B (v ) [ṽ v t ] f (t). Wth ths extra constrant, the values w after the manpulaton n the nduced assgnment problem of agent becomes w st E v F,ṽ B(v t,v ),S A(ṽ) [v s (S )] Eṽ F,ṽ B (v t),s A(ṽ) [v s (S )] E v F,ṽ B (v t ),S A(ṽ,v ) [v s (S )]. Thus, from the Bayesan vewpont of agent, the ntermedate algorthms B of other agents are

7 transparent. Ths property enables us to manpulate the structure of each assgnment problem separately. 4.2 Blac-box reducton. Gven an algorthm A, the blac-box reducton for socal welfare wll convert algorthm A nto the followng mechansm M A : 1. For each agent, 1 n (Pre-computaton) (a) Estmate the values w {w st} 1 s,t l of the nduced assgnment problem of wth respect to algorthm A. Let ŵ {ŵ st} 1 s,t l denote the estmated values. (b) Fnd the socal welfare maxmzng allocaton x { } 1 s,t l and the correspondng envy-free prces p (p 1,..., pl ) for the nduced assgnment problem of agent wth estmated values. 2. Manpulate the valuatons wth ntermedate algorthms B, 1 n, as follows: (Decouplng) Suppose the reported valuaton of agent s v vs, 1 n. Let ṽ B (v ) vt wth probablty /f (s) for 1 t l. 3. Allocate servces accordng to A(ṽ). (Allocaton) (a) Let S (S 1,..., S n ) denote the allocaton by algorthm A wth nput ṽ. (b) For each agent, suppose the reported valuaton s v vs and the manpulated valuaton s ṽ v t, charge agent wth prce p t v s (S ) ŵ st The followng theorem states that ths reducton produces BIC whle preservng the performance wth respect to socal welfare. Theorem 4.1. Suppose A s an algorthm for a multparameter mechansm desgn problem I, J, V, F. 1. If the estmated values ŵ are accurate, then mechansm M A s BIC, IR, and guarantees at least SW A of socal welfare. 2. If the estmated values ŵ satsfy that for any 1 s, t l, ŵ st [(1 ɛ)w st, (1 + ɛ)wst ], then mechansm M A s 4ɛv max -BIC, IR, and guarantees at least (1 2ɛ) SW A of socal welfare. 3. If the estmated values ŵ satsfy that for any 1 s, t l, ŵ st [w st ɛ, w st + ɛ], then mechansm M A s 4ɛ-BIC, IR, and guarantees at least SW A 2nɛ of socal welfare.. Let us llustrate the proof of part 1. The proofs of the other two parts are tedous and smple calculatons along the same lne. We wll omt these proofs n ths extended abstract. Proof. We consder the case when the estmated values ŵ are accurate, that s, ŵ st w st for all 1 n and 1 s, t l. Indvdual ratonalty. By our choce of envy-free prces, we have that p t wst for all 1 n and 1 s, t l. Thus, we always guarantee p t v s (S ) w st v s (S ). So the mechansm M A that we get from the reducton always provdes non-negatve utltes for the agents. Essentally the same proof also shows IR for part 2 and 3. Bayesan ncentve compatblty. We wll frst show that the ntermedate algorthms n the decouplng step of the reducton satsfy constrant (4.3). Let x denote the socal welfare maxmzng allocaton that the reducton fnds for the nduced assgnment problem of agent for 1 n. Note that these socal welfare maxmzng allocatons are maret-clearng. We have that f the reported valuaton v follows the dstrbuton F, then the dstrbuton of the manpulated valuaton ṽ satsfes that Pr [ ṽ v t ] l Pr [v v s ] Pr [ ṽ v t : v v s ] l f (s) xst f (s) l f (t). Indeed, the ntermedate algorthms satsfy constrant (4.3). Thus, for each 1 n the ntermedate algorthm B only changes the structure of nduced assgnment problem of agent and leaves the nduced assgnment problems of other agents untouched. Next, we wll verfy that n each of the manpulated assgnment problem, the dentty allocaton maxmzes the socal welfare and the prces are the correspondng envy-free prces. For each agent, we let w { w st} 1 s,t l and p ( p 1,..., pl ) denote the values and the expected prces of the vrtual products respectvely n the manpulated assgnment problem of agent. We have that for any

8 1 r, s l, w rs p s l l l l Pr [ ṽ v t ] Ev,S A(v t,v ) [v r (S )] f (s) wrt ; Pr [ ṽ v] t Ev,S A(v [p t,v ) t v s(s ] ) w rs f (s) pt. Thus, n the manpulated assgnment problem of agent, the utlty of the vrtual buyer r of the vrtual product s, 1 r, s l, s have that x rt w rs p s l l f (s) (wrt p t ) f (s) max {wr max {wr p }. p } Snce p are chosen to be the envy-free prces, we > 0 only f w rt p t max {w r p }. Hence, when agent reports ts valuaton truthfully, that s, r s, the above nequalty holds wth equalty. So the p are envy-free prces wth respect to the dentty allocaton x of the manpulated assgnment problem of agent. By Lemma 3.1 we now the allocaton x maxmzes the socal welfare. Thus, mechansm M A s BIC accordng to Lemma 3.3. Socal welfare. The expected socal welfare for ths mechansm s n l l 1 xst wst. Snce for any 1 n the allocaton x maxmzes the socal welfare for the nduced assgnment problem of agent, the socal welfare of x s at least as large as that of the dentty allocaton, that s, : l l w st Thus, we have that SW M A n l 1 s, l f (s)w ss E v F,S A(v) [v (S )]. w st n E v F,S A(v) [v (S )] 1 [ n ] E v F,S A(v) v (S ) SW A Estmatng values by samplng. There s stll one techncal ssue that we need to settle n the reducton. In ths secton, we wll brefly dscuss how to use the standard samplng technque to obtan good estmated values of w {w st} 1 s,t l for the nduced assgnment problem of agent for 1 n. By defnton, w st s the expectaton of a random varable v s(s ), where S s the allocated servce gven by A over random realzaton of the valuatons v of other agents and random con flps of the algorthm. Hence, f the standard devaton of v s(s ) s not too large compared to ts mean (no more than a polynomal factor), then we can draw polynomally many ndependent samples and tae the average value as our estmated value. More concretely, the samplng algorthm proceeds as follows. 1. Draw N 4 c 2 log(nl 2 /ɛ)/ɛ 2 ndependent samples of v F condtoned on that the valuaton of agent s v t, where c σ v,s A(v t,v ) [v s (S )] E v,s A(v t,v ) [v s (S )] Let v 1,..., v N denote these N sample. 2. Use algorthm A to compute an allocaton S A(v ) for each sample v, 1 N. 3. Let ŵ st be the average of v s(s ), 1 N. Lemma 4.1. The estmated values ŵ, 1 n, by the above samplng procedure satsfy for any 1 n and 1 s, t l, ŵ st wth probablty at least 1 ɛ. [ (1 ɛ)w st, (1 + ɛ)w st ] Proof. We shall have that E [ ŵ st ] Ev,S A(v t,v ) [v s (S ))] w st σ [ ŵ st ] 1 σ v,s A(v N t,v ) [v s (S )] c E [ ŵ st ] c w st N N By Chernoff bound we get Pr [ ŵst w st ] > ɛw st [ ŵst Pr [ ŵst Pr E [ ] ŵ st ɛ N > c E [ ŵ st e log (nl2 /ɛ) ɛ nl 2... σ [ ŵ st ] ], ] > 2 log (nl2 /ɛ) σ [ ŵ st ] ]

9 Snce we only need to estmate nl 2 values, by unon bound we get that wth probablty at least 1 ɛ the estmated value ŵ st s wthn ɛ relatve error compared to w st for all 1 n and 1 s, t l. Thus, f the allocaton algorthm A admts SW A socal welfare and the rato c s only polynomally large, then by part 2 of Theorem 4.1 we get that mechansm M A gves (1 3ɛ) SW A socal welfare and s 4ɛv max - BIC. The runnng tme s polynomal n the nput sze and 1/ɛ, assumng a blac-box call to algorthm A can be done n a sngle step. In other words, we get a FPTAS reducton. The next lemma gves an alternatve bound of the samplng error wth respect to absolute error. Lemma 4.2. If we draw N 4 log(nl 2 /ɛ)/ɛ 2 ndependent samples, then wth probablty at least 1 ɛ the estmated values ŵ st [w st ɛv max, w st + ɛv max ] for all 1 n and 1 s, t l. Proof. In ths case, we have E [ ŵ st ] Ev,S A(v t,v ) [v s (S ))] w st σ [ ŵ st ] 1 σ N v,s A(v t,v ) [v s (S )], 1 max v N s (S ) 1 v max. S N By Chernoff bound we get that Pr [ ŵst w st ] > ɛvmax [ ŵst Pr E [ ] ŵ st ɛ > σ [ ŵ st ] ] N [ Pr ŵ st E [ ] ŵ st > 2 log (nl 2 /ɛ) σ [ ŵ st ] ] e log (nl2/ɛ) ɛ nl 2. By unon bound, we have ŵ st [w st ɛv max, w st + ɛv max ] for all 1 n and 1 s, t l. Suppose the rato v max /SW A s upper bounded by a polynomal of the nput sze. Then, f we choose ɛ δ SW A /2nv max n the above lemma, we wll get that w st < δ SW A /2n. ŵst By part 3 of Theorem 4.1 we obtan that mechansm M A provdes at least (1 δ)sw A of socal welfare and s 4ɛ-BIC and IR. The runnng tme s polynomal n the nput sze and 1/δ. 5 Reductons for revenue and resdual surplus In the reducton for socal welfare n the prevous secton, we only consder maret-clearng allocatons n the nduced assgnment problems. Ths s because for any agent, we want to mae sure that the ntermedate algorthm B s transparent to all agents except agent. If we restrct ourselves to maret-clearng allocatons, we do not now any way to get reasonable bounds on revenue and resdual surplus. However, f we focus on an mportant sub-class of mult-parameter mechansm desgn problems that ncludes the combnatoral aucton problem and ts specal cases, then we have some flexblty n choosng the allocatons for the nduced assgnment problem and obtan theoretcal bounds on revenue and resdual surplus. More concretely, we wll consder mechansm desgn problems that are downward-closed. We let φ denote the null servce so that allocatng φ to an agent mples that agent s not served, that s, v (φ) 0 for all 1 n. Defnton 5.1. A mult-parameter mechansm desgn problem I, J, V, F s downward-closed f for any feasble allocaton S (S 1,..., S n ) J and any 1 n, the allocaton (S 1,..., S 1, φ, S +1,..., S n ) s also feasble. We let δ mn{f (s) : 1 n, 1 s l, f (s) > 0} denote the granularty of the pror dstrbutons. We wll prove the followng result. Theorem 5.1. For any algorthm A, there s a mechansm that s IR, BIC, and provdes at least Ω(SW A / log(1/δ)) of revenue (resdual surplus). 5.1 Meta-reducton. We wll frst ntroduce a meta-reducton scheme based on algorthms that compute envy-free solutons for fractonal assgnment problems. Suppose C s an algorthm that computes envyfree solutons (x, p) for any gven fractonal assgnment problem. Let A be an algorthm for a mult-parameter mechansm desgn problem I, J, V, F. We wll convert algorthm A nto to a mechansm M C A : 1. For each agent (Pre-computaton) (a) Estmate the values w {w st} 1 s,t l for the nduced assgnment problem of agent wth respect to A. Let ŵ {ŵ st} 1 s,t l denote the estmated values. (b) Use C to solve the nduced assgnment problems wth estmated values. Let (x, p ) denote the soluton by C for the nduce assgnment problem of agent.

10 (c) Let y t f (t) l xst denote the unallocated supply of vrtual product t n soluton (x, p ) for all 1 n and 1 t l. (d) Let y l yt denote the total amount of unallocated vrtual products n (x, p ) for all 1 n. 2. Manpulate the valuatons wth ntermedate algorthm B, 1 n, as follows: (Decouplng) (a) Suppose the reported valuaton of agent s v vs. (b) Let ṽ B (v ) vt wth probablty /f (s) for 1 t l. (c) Wth probablty 1 t xst /f (s), the manpulated valuaton ṽ s unspecfed n the prevous step. In ths case, let ṽ v t wth probablty y t/y for 1 t l. 3. Allocate servces as follows: (Allocaton) (a) Compute a tentatve allocaton S ( S 1,..., S n ) A(ṽ). (b) For each agent, let S S f the manpulated valuaton ṽ s specfed n step 2b). Let S φ otherwse. Allocate servces accordng to S. (c) For each agent, suppose the reported valuaton s v vs and the manpulated valuaton s ṽ v t, charge agent wth prce p t v s (S ) ŵ st The followng theorem states the above metareducton scheme converts algorthms nto IR and BIC mechansms. Theorem 5.2. Suppose the algorthm C always provdes envy-free solutons. 1. If the estmated values ŵ are accurate, then mechansm M C A s IR and BIC. 2. If the estmated values ŵ satsfy that for any 1 s, t l, ŵ st [(1 ɛ)w st, (1 + ɛ)wst ], then MC A s IR and 4ɛv max -BIC. 3. If the estmated values ŵ satsfy that for any 1 s, t l, ŵ st [w st ɛ, w st + ɛ], then M C A s IR and 4ɛ-BIC.. Proof. Let us outlne the proof for part 1. Proofs of the other two parts are calculatons along the same lne. Note that p t wst for all 1 n and 1 s, t l. The mechansm s IR because for any 1 n and 1 s l the utlty for an agent wth valuaton v s n any realzaton s v s (S ) p t v s(s ) w st 0. Next, we wll show that mechansm M C A s BIC. We frst verfy that the ntermedate algorthms B, 1 n, satsfy the constrant (4.3). For any agent, f ts valuaton v s drawn from dstrbuton F, then the probablty that the manpulated valuaton ṽ B (v ) v t s [ ( ) ] l l f (s) f (s) + x sr y t 1 f r1 (s) y ( l l ) l l + f (s) x sr y t y r1 ( l l ) l l + f (r) x sr y t y r1 r1 ( ) l l l + f (r) x y t sr y l + r1 l r1 y r y t y l + y t f (t). Thus, we get that for each agent, the ntermedate algorthms B j, 1 j n and j, are transparent to t. So the expected value of agent of the servce t gets, when ts genune valuaton s v v s and the manpulate valuaton, s ṽ v t s exactly w st E v,s A(v t,v ) [v s (S )]. Hence, the expected value of agent of the serve t gets, when ts genune valuaton s v v s and the reported valuaton s v vt, s w st l r1 x tr f (t) wsr. And the expected prce for agent when the reported valuaton s v vt s p t l r1 l r1 x tr [ f (t) E v,s A(v r,v ) p r x tr f (t) pr. v t(s ] ) w tr

11 Thus, the the expected utlty of agent, when ts genune valuaton s v v s and ts reported valuaton s v vt, s have that x sr w st p t l r1 l r1 x tr f (t) (wsr p r ) x tr f (t) max {ws max {ws p }. p } Snce p are chosen to be the envy-free prces, we > 0 only f w sr p r max {w s p }. Hence, when agent reports ts valuaton truthfully, that s, s t, the above nequalty f tght. Moreover, the above utlty s always non-negatve. So mechansm M C A s BIC. Moreover, the revenue and resdual surplus of mechansm M C A s related to the socal welfare and revenue of the nduced assgnment problems as stated n followng proposton. Proposton 5.1. The expected revenue (resdual surplus) of the mechansm M C A equals the sum of the revenue (resdual surplus) of the manpulated assgnment problems. By choosng proper allocaton algorthm C, we can obtan theoretcal bounds for the revenue or resdual surplus n the manpulated nduced assgnment problems and thus theoretcal bounds for mechansm M C A. 5.2 Assgnment algorthms. In ths secton, we wll ntroduce two algorthms for computng envy-free solutons for the nduced assgnment problems. These two algorthms provdes theoretcal bounds for revenue and resdual surplus. Revenue. The frst algorthm provdes revenue that s a Ω(1/ log(1/δ)) fracton of SW A, the socal welfare by algorthm A. The dea s to ntroduce proper reserve prces to the nduced assgnment problems by redundant vrtual buyers. Ths s nspred by the technques by Guruswam et al. [12]. For the nduced assgnment problem of agent, 1 n, the assgnment algorthm C R for revenue maxmzaton proceeds as follows: 1. Fnd the socal welfare maxmzng allocaton x { } 1 s,t l. 2. Suppose u max s the maxmal valuaton among the vrtual buyer-product par (s, t) wth non-zero, that s, u max max{w st : 1 s, t l, > 0}. 3. Recall that δ mn{f (t) : 1 n, 1 t l, f (t) > 0} denotes the granularty of the pror dstrbuton. For 1 log(1/δ): (a) Consder the followng varant of the nduced assgnment nstance of agent : For each vrtual product 1 t l, add a dummy vrtual buyer wth demand 1 + δ and value u u max /2 for vrtual product t and value 0 for other vrtual products. (b) Fnd socal welfare maxmzng allocaton x and envy-free prces p for ths varant. (c) Let (ˆx, ˆp ) be the projecton of (x, p ) on the orgnal nduced assgnment problem of agent, that s, for any 1 s, t l, ˆ, ˆp t p t. 4. Return the (ˆx, ˆp ), 1 log(1/δ), wth best revenue. Lemma 5.1. Assgnment algorthm C R always return an envy-free soluton (x, p). The revenue s at least a Ω(1/ log(1/δ)) fracton of the optmal socal welfare of the assgnment problem. Proof. The envy-freeness follows from the fact that (ˆx, ˆp ), 1 log(1/δ), are projectons of envyfree solutons and thus are also envy-free. Now we consder the revenue by C R. We let r denote the revenue by soluton (ˆx, ˆp ). Note that n (ˆx, ˆp ), all prces are at least u and the amount of vrtual products that are sold s at least s,t:w st u xst. Hence, we have r w. s,t:w st u Note that f we extend the defnton of u and let u u max /2 for all non-negatve nteger, then we have (5.4) 1 u s,t:w st u (u 1 u ) 1 l l l l s,t w st. :w st u s,t:w st u (u 1 u ) max {u 1 : w st u }

12 On the other hand, the contrbuton of the tal s small compared to the socal welfare. (5.5) log(1/δ)+1 log(1/δ)+1 u s,t:w st u w δw max 2 s,t xst wst 2 The last nequalty holds because allocatng the most valuable vrtual product the one of the vrtual buyer s a feasble allocaton. Hence, consder the dfference of the above formulas, (5.4) (5.5), and we get that log(1/δ) 1 r log(1/δ) 1 u s,t:w st u. s,t xst wst 2 Thus, by pgeon-hole-prncple at least one of the assgnment (ˆx, ˆp ) provdes revenue that s at least a 1/2 log(1/δ) fracton of the socal welfare. The above lemma leads to the followng results for revenue maxmzaton. Proposton 5.2. Suppose the socal welfare gven by allocaton algorthm A s SW A, the mechansm M C R A guarantees at least Ω(SW A / log(1/δ)) of revenue. Complementary lower bound. The approxmaton rato wth respect to SW A s tght due to the followng example. Consder the aucton problem wth only one agent and one tem. Suppose wth probablty 1/2 the agent has value 2 for the tem for 1, 2,..., log(1/δ). And wth probablty δ, the agent has value 0 for the tem. In ths example, the granularty of the pror dstrbuton s δ. The optmal socal welfare s log(1/δ) log(1/δ). But no BIC mechansm 2 can acheve revenue better than 1. Resdual surplus. We turn to the resdual surplus maxmzaton problem. Note that revenue and resdual surplus are symmetrc n the nduced assgnment problems. We wll use the followng assgnment algorthm C RS based on the same dea we use for the revenue maxmzaton algorthm. The resdual surplus maxmzng envy-free algorthm C RS s as follows: 1. Fnd the socal welfare maxmzng allocaton x { } 1 s,t l. 2. Suppose u max s the maxmal valuaton among the vrtual buyer-product par (s, t) wth non-zero, that s, u max max{w st : 1 s, t l, > 0}.. 3. Recall that δ mn{f (t) : 1 n, 1 t l, f (t) > 0} denotes the granularty of the pror dstrbuton. For 1 log(1/δ): (a) Consder the followng varant of the nduced assgnment nstance of agent : For each vrtual buyer 1 t l, add a dummy vrtual product wth demand 1 + δ and value u u max /2 for vrtual buyer t and value 0 for other vrtual buyer. (b) Fnd socal welfare maxmzng allocaton x and envy-free prces p for ths varant. (c) Let (ˆx, ˆp ) be the projecton of (x, p ) on the orgnal nduced assgnment problem of agent, that s, for any 1 s, t l, ˆ, ˆp t p t. 4. Return the (ˆx, ˆp ), 1 log(1/δ), wth best revenue. The proofs of the followng lemma and theorem s almost dentcal to the revenue maxmzaton part so we omt the proofs here. Lemma 5.2. Assgnment algorthm C RS always return an envy-free soluton (x, p). The resdual surplus s at least a Ω(1/ log(1/δ)) fracton of the optmal socal welfare of the assgnment problem. Proposton 5.3. Suppose the socal welfare gven by allocaton algorthm A s SW A, the mechansm M C RS A guarantees at least Ω(SW A / log(1/δ)) of resdual surplus. 6 Applcaton n combnatoral auctons In ths secton we wll brefly llustrates how to use the reducton for socal welfare n ths paper to derve a combnatoral aucton mechansm that matches the best algorthmc result. Combnatoral auctons. In the combnatoral auctons, we consder a prncpal wth m tems (exactly one copy of each tem) and n agents. Each agent has a prvate valuaton v F for subsets of tems. The goal s to desgn a protocol to allocate the tems and to charge prces to the agents. We wll show the followng corollares of our reducton for socal welfare. Corollary 6.1. For combnatoral auctons wth subaddtve (or fractonally sub-addtve) agents where the pror dstrbutons have fnte and poly-sze supports, there s a ( 1 2 ɛ) -approxmate (or ( 1 1 e ɛ) - approxmate respectvely), ɛv max -BIC, and IR mechansm for socal welfare maxmzaton. The runnng tme s polynomal n the nput sze and 1/ɛ.

13 Algorthm. We wll consder a varant of the LPbased algorthms by Fege [8] and use the reducton for socal welfare to convert t nto an IR and ɛv max - BIC mechansm. More concretely, we wll consder the Bayesan verson of the standard socal welfare maxmzaton lnear program (CA): Maxmze f (t) v(s) t x,t,s s.t. t S:j S t S f (t) x,t,s 1 j x,t,s 1, t S x,t,s 0, t, S In ths LP, x,t,s denote the probablty that agent s allocated wth a subset of tems S condtoned on ts valuaton s v t. Ths LP can be solved n polynomal tme by the standard prmal dual technque va demand queres. See [5] for more detals. We let LP denote the optmal value of ths LP. Moreover, for any basc feasble optmal soluton of the above LP, there are at most nml non-zero entres snce there are only nml nontrval constrants. Hence, we have the followng lemma: Lemma 6.1. In poly-tme we can fnd an optmal soluton x to (CA) wth at most nml non-zero entres. Next, we wll flter ths soluton x by removng nsgnfcant entres. We let ˆx,t,S x,t,s < ɛ/nml. Note that LP f (t)v t (S) for any, t, and S snce always allocatng subset S to agent s a feasble allocaton. We get that ˆx s a feasble soluton to (CA) wth value at least (1 ɛ)lp. Then, we wll use the roundng algorthms by Fege ([8] to ) get a 1 2-roundng for sub-addtve agents and a 1 1 e -roundng for fractonally sub-addtve agents: 1. Allocate a tentatve subset of tems S to each agent, 1 n, accordng to the reported valuaton v vt and ˆx,t, S. 2. Resolve conflcts properly by choosng S S so that S (S 1,..., S n ) s a feasble allocaton. By extendng Fege s orgnal proof, we can show that there s a randomzed algorthm for choosng S such that for sub-addtve agents, we have: (6.6) E v,s [v (S )] 1 2 v ( S ). And for fractonally sub-addtve agents, we have: ( (6.7) E v,s [v (S )] 1 1 ) v ( e S ). We wll omt the proof n ths extended abstract. We denote the above algorthm as A. Then, A provdes ( 1 ( 2 ɛ) -approxmaton for sub-addtve agents and 1 1 e ɛ) -approxmaton for fractonally sub-addtve agents. Estmatng values. By Theorem 4.1 and 5.1, we only need to show how to estmate the values w, 1 n, for the nduced assgnment problem of agent effcently. Further, by Lemma 4.1, we can effcently estmate the values w {w st} 1 s,t l, 1 n, f the followng lemma holds. Lemma 6.2. For any 1 n, and any 1 s, t l, σ v,s A(v t,v ) [v s(s )] 4nml E v,s A(v t,v ) [v s(s )]. ɛ Proof. By nequaltes (6.6) and (6.7), we get that condtoned on S beng chosen as the tentatve set, [ E v,s A(v t,v ) v s (S ) : S ] 1 ( ) 2 vs S We also have that [ σ v,s A(v t,v ) v s (S ) : S ] max {v s (S ) : S } Hence, σ v,s A(v t,v ) [v s (S )] 2 S S v s ( S ). ˆx,t, S σ v,s A(v t,v ) [ v s (S ) : S ] 2 ˆx,t, S v s ( S ) 2 1 { } 2 ˆx,t, v S s ( S ) mn ˆx,t, > 0 S,t, S nml [ ˆx ɛ,t, E S v,s A(v t,v ) v s (S ) : S ] 4nml ɛ S References E v,s A(v t,v ) [v s (S )] 2. [1] S. Bhattacharya, G. Goel, S. Gollapud, and K. Munagala. Budget constraned auctons wth heterogeneous tems. In ACM 42nd Annual ACM Symposum on Theory of Computng (STOC),

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