Accuracy for Sale: Aggregating Data with a Variance Constraint

Transcription

1 Accuracy for Sale: Aggregatng Data wth a Varance Constrant Rachel Cummngs Katrna Lgett Aaron Roth Zhwe Steven Wu Juba Zan August 8, 204 Abstract We consder the problem of a data analyst who may purchase an unbased estmate of some statstc from multple data provders. From each provder, the analyst has a choce: she may purchase an estmate from that provder that has varance chosen from a fnte menu of optons. Each level of varance has a cost assocated wth t, reported (possbly strategcally) by the data provder. The analyst wants to choose the mnmum cost set of varance levels, one from each provder, that wll let hm combne hs purchased estmators nto an aggregate estmator that has varance at most some fxed desred level. Moreover, he wants to do so n such a way that ncentvzes the data provders to truthfully report ther costs to the mechansm. We gve a domnant strategy truthful soluton to ths problem that yelds an estmator that has optmal expected cost, and volates the varance constrant by at most an addtve term that tends to zero as the number of data provders grows large. Computng and Mathematcal Scences, Calforna Insttute of Technology; {rachelc, katrna, zan}@caltech.edu Computer and Informaton Scence, Unversty of Pennsylvana; {aaroth,wuzhwe}@cs.upenn.edu

2 Introducton We consder a data analyst who wshes to compute an unbased estmate of some underlyng populaton statstc, by buyng and aggregatng data from multple strategc data provders. Each data provder may experence dfferent costs for dfferent levels of data accuracy (varance), and may strategcally prce access to hs data f dong so would beneft hm. The analyst must desgn a mechansm for choosng whch level of accuracy to purchase from each provder, and for combnng the purchased data nto a sngle aggregate quantty that forms an unbased estmator of the statstc of nterest. Her goal s to do so at mnmum cost, gven some target level of overall accuracy. Ths model captures a number of nterestng scenaros. For example: Each data provder mght n fact be a sngle ndvdual, who s sellng a (possbly perturbed) bt sgnfyng some property of nterest to the data analyst (e.g., the cancer status of the ndvdual). Here, the varance of each of these estmates comes from two sources: each ndvdual s bt s the realzaton of an ndependent sample from some underlyng populaton dstrbuton, wth an nherent varance. A data provder may also add hs own perturbaton (e.g., nose from a Gaussan or Laplace dstrbuton) n order to guarantee a certan level of (dfferental) prvacy. He can therefore potentally offer the data analyst access to hs data at a menu of dfferent varance levels (and costs), correspondng to dfferng levels of prvacy protecton. Intutvely, the dfferent costs the ndvdual experences at dfferent levels of accuracy may correspond to hs (potentally arbtrary) preferences for prvacy. Snce we model data provders as strategc agents, they wll report the cost that maxmzes ther utlty, and ther reported costs need not necessarly match ther true costs. We allow each ndvdual to report an arbtrary cost separately for each varance level, so ths approach does not requre assumng that agent preferences for prvacy respect any fxed functonal form. Each data provder mght be an organzaton (such as a unversty) that has the ablty to collect a random sample of varyng sze from a sub-populaton that t controls (e.g. students, professors, etc). Under the assumpton that the ndvduals n the data provder s populatons are sampled..d. from some underlyng dstrbuton, the varance of the estmate that they offer s nversely proportonal to the number of ndvduals that they sample. Here, the costs for dfferent levels of varance correspond to the costs requred to recrut dfferent numbers of partcpants to a study. These costs may dffer between organzatons, and behave n complcated ways: for example, the margnal cost for each addtonal sample mght be decreasng (f there are economes of scale for example by advertsng on a campus TV staton), or mght be ncreasng (for example, after exhaustng the undergraduate populaton at a unversty, obtanng addtonal samples may requre recrutng faculty, whch s more dffcult). Agan, because we allow data provders to report arbtrary cost schedules correspondng to dfferent varance levels, we need make no assumptons about the form that these costs take.. Our Results and Technques We model the data analyst s problem as a combnatoral optmzaton problem: From each of the data provders, the analyst buys an unbased estmator of the populaton statstc of nterest, for whch she must choose a varance from a fxed, fnte menu of optons. Gven these purchased estmators, the data analyst may then take any convex combnaton to obtan her fnal unbased estmator of the underlyng populaton statstc. The choces made by the data analyst affect both

3 the varance of the fnal estmator that she derves, as well as the total payment that she must make. We consder the problem of fndng the cheapest way of constructng an estmator that has varance below some fxed desred level, specfed n advance by the data analyst. Our man tool n solvng ths problem s lnear programmng. However, the soluton s not straghtforward. Frst, our problem actually conssts of two nested optmzaton problems: we must choose a varance level for each of the estmators, and then we must fnd the optmal weghted lnear combnaton of these estmators. Rather than solvng these problems separately, we use the KKT condtons to derve a closed form for the optmal weghts to use n the lnear combnaton of each of the estmators as a functon of ther varance. Ths allows us to express the problem as a one-shot optmzaton problem, wth decson varables only for the choces of varance for each estmator. Unfortunately, the natural fractonal relaxaton of ths optmzaton problem (n whch the data analyst may fractonally choose dfferent varance levels) s non-convex. Instead, we consder a further (lnear) relaxaton of the constrant n our problem, whch matches the orgnal constrant only for nteger solutons. We show that all optmal extreme ponts of the lnear program that result from ths relaxaton do n fact yeld nteger choces for all but at most one data provder, and then show that f the number of data provders s suffcently large, then roundng the one fractonal assgnment to an nteger assgnment only margnally volates our target varance constrant. We note that our algorthm chooses the mnmum expected cost lottery over purchase decsons from among a pre-specfed feasble set of lotteres, and hence s maxmum-n-dstrbutonal-range. Ths means t can be pared wth payments such that truthful reportng of costs s a domnant strategy for each of the data provders. (We recall that although we allow data provders to msreport ther costs, they cannot le about ther data or ts accuracy.) In summary, we show the followng theorem: Theorem (Informal). Gven any fnte menu of varance levels, and any feasble aggregate varance level for the data analyst, there exsts a domnant strategy truthful mechansm that selects the mnmum cost assgnment of varance levels to provders, and generates an unbased lnear estmator that satsfes the analyst s varance constrant up to an addtve term that tends to 0 as the number of data provders grow large. Fnally, we observe that VCG payments (although always truthful) do not guarantee ndvdual ratonalty n our settng. We prove an upper bound on the degree to whch ndvdual ratonalty can be volated for any player, and hence can add a fxed amount to the payment gven to each player, to guarantee ndvdual ratonalty for all provders wth suffcently low mnmum costs..2 Related Work There s a growng body of work [0, 5, 9, 7, 8, 7, 9] related to our frst motvatng example: buyng senstve data from ndvduals. Ths lne of work consders the problem of ncentvzng ndvduals to provde ther data to an analyst, when they experence a cost usually due to prvacy loss from sharng ther data. These papers have used dfferental prvacy, defned n [6], to combat ths prvacy loss, but have generally offered only a sngle prvacy level to partcpants, or have made assumptons about the functonal form of ths prvacy loss n terms of the dfferental prvacy parameter. Our constructon, on the other hand, allows the analyst to offer each data provder a menu of dfferent varance levels, correspondng to dfferent levels of dfferental prvacy, and allows the agents to express arbtrary costs for each level ndependently. Ths requres no assumptons at all about the functonal form of agent costs. As n our settng, these papers all 2

4 consder the ndvdual data provders to be selfsh agents, and thus allow agents to strategcally msreport ther costs to secure them a hgher payment. Recently, [9] consdered a settng where ndvduals have unverfable data, and can also msreport ther data. We restrct to a settng where data s verfable (as the other papers n ths lterature do), but allow ndvduals to le about ther costs for provdng data. In ths settng, we are makng the mplct assumpton that the senstve data held by ndvduals s ndependent of ther prvacy costs. Ths s motvated by an mpossblty result of [0], later strengthened by [7], that when prvacy costs are correlated wth data, no mechansm can satsfy ndvdual ratonalty and estmate the statstc of nterest wth non-trval accuracy, whle makng fnte payments. Our paper s also related to the vast body of work on optmal experment desgn, n whch an analyst wshes to learn parameters of an underlyng dstrbuton by optmzng a mult-set of samples to observe from the populaton, each at some cost. (For a survey of results see [8] or [2]; for a textbook treatment see Secton 7.5 of [3].) Each data sample s an experment wth observable attrbutes. The analyst assumes a lnear relatonshp between experment attrbutes and outcomes, and wshes to accurately learn the lnear parameter by performng a collecton of experments subect to a budget constrant. Although the problem we consder seems to be a specal case of experment desgn (.e. wth attrbute vector of dmenson 0), these two problems dffer n a few key aspects. Frst, optmal experment desgn allows the analyst to repeat experments, and performng the same experment multple tmes may result n dfferent outcomes. We do not allow ths, and we further constran the problem to requre the analyst to buy exactly one experment (.e. observaton of data) from each data provder. The technques used n ths lne of work do not genercally extend to the more constraned settng that we consder. Addtonally, optmal experment desgn s a problem n the full nformaton settng, so the cost of each experment s a pror known by the analyst. That s, data provders cannot msreport ther costs. The expermental desgn lterature generally gves approxmaton algorthms whch are not maxmal n range, and hence do not yeld truthful mechansms. We consder a settng n whch data provders are strategc agents, and we must addtonally ensure that our optmzaton process s ncentve compatble. Recent work of [3] consdered expermental desgn for strategc agents, but ther work consdered a dfferent accuracy obectve (other than mnmzng varance), and ther technques fal under the addtonal constrant that the analyst can buy at most one estmate from each data provder. The truthfulness of our mechansm depends on a property called maxmal-n-dstrbutonalrange (MIDR), defned n [4]. MIDR mechansms are guaranteed to select a dstrbuton over outputs that maxmzes expected welfare. Smlar propertes were also used n [], [5] and [4]. [4] showed that MIDR mechansms are truthful-n-expectaton when pared wth VCG payments. That s, wth an MIDR mechansm, no player can ncrease her expected utlty by lyng about her prvate nformaton. We frst show that our proposed mechansm s MIDR, and then use ths result to show that data provders do not have an ncentve to msreport ther costs. 2 Prelmnares and Model We consder an analyst who wshes to estmate the expected value µ of some statstc on the underlyng populaton. She has access to a set of n data provders, each of whch s capable of provdng some unbased estmate µ of the statstc of nterest wth dfferent levels of varance E [ (µ µ) 2]. The provder may also experence some cost for computng the estmate at each 3

5 varance level. The analyst s goal s to obtan an accurate unbased estmate for µ, usng the estmates from the provders, whle mnmzng the socal cost for computng such data. We equp the analyst wth a mechansm that offers a menu specfyng a dscrete range of possble varance levels 0 < v < v 2 <... < v m <, and asks each provder report back a set of costs {c } m = for computng the estmates at all levels. The mechansm then selects a varance level to purchase from each provder, and generates an estmate for µ that s a weghted sum of the provders reported estmates µ s: µ = w µ. Note that the expectaton E [ µ] = w E [µ ] = w µ, so µ wll be an unbased estmate as long as w =. The followng proposton, often called the Benaymé formula, allows us to express the varance of µ as a lnear combnaton of the varances of µ. Proposton. Let X,..., X n be uncorrelated real-valued random varables, and w,..., w n be any real numbers, then ( ) Var w X = w 2 Var(X ). The goal of the analyst s to mnmze the total cost among all provders, whle mantanng a guarantee that the varance of µ s below some threshold α. Ths can be expressed n the followng program, where each x ndcates whether we assgn provder to varance level : mn, x c () subect to w 2 x v α (2) x = for all (3) x {0, } for all (, ) (4) w = and for all, w 0 (5) 2. Mechansm Desgn Bascs We study our optmzaton problem n the settng of mechansm desgn, wth n players, and a set Ω of possble outcomes. In partcular, ths set Ω corresponds to the set of possble assgnments of players to varance levels. Each player also has a cost functon c : Ω R, where c (ω) s player s cost for outcome ω. Let c = (c,..., c n ) denote the profle of cost functons for all players. We want to mnmze total cost, so our obectve s n c (ω). A (drect-revelaton) mechansm M conssts of an allocaton rule A, a functon mappng reported cost profles to outcomes, and a payment rule p, a functon mappng cost profles to a payments to each player. Such a mechansm takes as nput reported cost functons from the players, and outputs (possbly randomly) an allocaton ω and payments to all the players. Two mportant desderata n mechansm desgn are truthfulness and ndvdual ratonalty. Defnton (Truthful-n-Expectaton). A mechansm M = (A, p) on n players s (domnant strategy) truthful-n-expectaton f for any reported cost profle c and for all [n]: E [p (c) c (A(c))] E [p ((c, c )) c (A(c, c ))]. M M 4

6 Defnton 2 (Indvdually Ratonal). A mechansm M = (A, p) s ndvdually ratonal (IR) f for any reported cost profle c and for all [n]: E [p (c) c (A(c))] 0. M We wll use VCG-based mechansms to mnmze total cost whle achevng truthfulness. A VCG mechansm s defned by the allocaton rule that selects the cost-mnmzng outcome ω arg mn ω Ω c (ω) for any reported cost functons, and the payment rule p that rewards each player hs externalty : p (c) = mn c (ω) c (ω ). (6) ω Ω Let dst(ω) be the set of all probablty dstrbutons over the set of outcomes Ω, and let R dst(ω) be a compact subset. Then a maxmal-n-dstrbutonal-range (MIDR) allocaton rule s defned as samplng an outcome ω from dstrbuton D R, where D mnmzes the expected total cost E ω D [ c (ω)] over all dstrbutons n R. A VCG payment rule can be defned accordngly: p (c) = mn E D dst(ω ) ω D c (ω) E ω D c (ω). It s known from [4] that when an MIDR allocaton rule s pared wth a VCG payment rule, the resultng mechansm s truthful-n-expectaton. To guarantee ndvdual ratonalty, we pay each player some entrance reward R before runnng the MIDR mechansm so that R + E [p (c) c (A(c))] 0 for all player. It suffces to set R max E [p (c) c (A(c))], and n Secton 4.2 we derve a more refned bound for R to get ndvdual ratonalty. 3 Rewrtng the Program The optmzaton problem ntroduced n Secton 2 s non-convex because the varance constrant (2) contans the product of decson varables x and w. To acheve convexty, we wll transform the program n three steps:. Frst, we wll elmnate the decson varables w by dervng a closed form soluton for the weghts w that mnmze varance, once the varables x are fxed. However, ths wll stll leave us wth a non-convex optmzaton problem. 2. Next, we wll replace the non-convex constrant derved above wth a lnear constrant, that s dentcal whenever the x varables take on ntegral values. 3. Fnally n Secton 4, we relax the ntegralty constrant. Because our lnear varance constrant s no longer dentcal to the orgnal correct non-convex varance constrant, we must n the end argue that a rounded soluton does not substantally volate the orgnal constrant. Frst, to smplfy notaton, for any assgnment {x }, let v denote the varance level assgned to provder. We want to wrte w as a functon of v s. In partcular, gven the varance assgnments, we want to choose the weghts w so that the varance of the aggregate statstc µ s mnmzed. 5

7 Lemma. Gven a varance level assgnment { v }, the weght vector w that mnmzes the varance of µ = w µ satsfes w = / v / v for all. Proof. The problem can be wrtten as a convex program mn w 2 v subect to w = and w 0 for all (7) We know that strong dualty holds because the program satsfes Slater s condton, and the Lagrangan s gven by L(w, λ) = ( v w 2 λ ) w = w T V w λ( T w), where V = dag(v,..., v n ). Note that w L(w, λ) T = 2V w+λ. By KKT condtons, w L(w, λ) T = 0, and so w = λ 2 V, whch gves mn w L(w, λ) T = λ2 4 / v λ. Now the dual problem can be wrtten as [ ] max mn L(w, λ) = max λ2 / v λ λ w 0 λ 4 [ ( ) ( ) ] 2 = max / v λ/2 + λ / v + / v. It s easy to see that the maxmum s reached at λ = 2 / v. It follows that w = λ 2 V = V / v, and so, w = / v / v for all. Lemma shows that we can rewrte the varance constrant of µ as ( ) 2 / v / v v = / v ( / v ) 2 = / v α. Pluggng n v = x v and takng the nverse on both sdes, constrant (2) becomes /α x v (8) 6

8 Note that the constrants are not lnear, but snce each x {0, }, and only one x = for each, we have / x v = x /v. Thus, we can wrte our whole program as the followng ILP. subect to mn x x c (9), /α x /v (0) x = for all () x {0, } for all (, ) (2) Remark. Note that our problem s only nterestng f the target varance α s n the range of [v /n, v m /n]. Ths s due to the followng observaton based on constrant (0): f /α < n/v m, then the problem s trval snce the varance constrant s satsfed by any assgnment; f /α > n/v, then the problem s nfeasble,.e. even f we assgn the lowest varance level to all provders, the varance constrant s stll volated. 4 An MIDR Mechansm va a Lnear Programmng Relaxaton In order to obtan a computatonally effcent mechansm, we consder the LP relaxaton of the nteger lnear program we derved n the prevous secton, by replacng constrant (2) by x 0 for all (, ). We nterpret a fractonal soluton x = (x,..., x m ) as a lottery over assgnments for player,.e. the probabltes of gettng assgned to dfferent varance levels. Snce the obectve s to mnmze the total cost, the LP gves a maxmum n dstrbutonal range allocaton rule, where the restrcted dstrbutonal range s, S α = {x 0 x = for all, and x /v /α}. Gven a collecton of reported costs, our mechansm frst computes a dstrbuton x over assgnments, based on the MIDR allocaton rule defned by the LP. We then pay each provder based on the VCG payment rule, n addton to some entrance reward R. Gven the realzed varance assgnment sampled from x, we ask the provders to compute ther estmates µ at the correspondng varance levels. Fnally, we re-weght the estmates to obtan the lnear combnaton estmator µ wth the mnmum varance based on the optmal re-weghtng rule n Lemma. The formal descrpton of our mechansm s presented n Algorthm. Theorem 2. Gven n data provders wth reported costs {c } for varance levels {v } and a feasble target varance level α, Algorthm selects an mnmum expected cost assgnment wth a truthfuln-expectaton mechansm, and,. for any ε > 0, computes an estmate µ wth varance Var( µ) ( + ε)α as long as ( ) ( ) vm n v ε +, 7

9 Algorthm MIDR Mechansm for Buyng Estmates Input: Data provders reported costs {c } for dfferent varance levels {v,..., v m }, target varance α, ntal payment R Compute assgnment and payment based on MIDR allocaton rule and VCG payment rule: x arg mn x S α c x p = mn x S α c (x) c (x ) + R Let v = ( v,..., v n ) be the realzed varance assgnments sampled from x and w = / v / v for all. Collect the estmates from provders {µ } based on v Output: w µ as our estmate µ 2. the mechansm s ndvdually ratonal f the entrance reward R max mn c. The propertes of cost mnmzaton and truthfulness follow from the MIDR allocaton rule and VCG payments. We show the other two propertes n the followng subsectons. Remark 2. To acheve a 2-approxmaton for the varance (.e. ε = ), t wll suffce to have n = 2v m /v provders. Pluggng n the bound n Remark, the meanngful range of target varance should be v 2/2v m α v /2. Note that v /v m <, so ths range s always non-empty. 4. Varance Volaton The fractonal soluton we obtan could volate the varance constrant (8), and so could the fnal assgnment sampled from the fractonal soluton. Let x be an optmal soluton to the LP, then x volates the varance constrant (8) by at most (x) = x /v / x v = ( x /v / x v ). The quantty (x) represents the dstance between the real desred varance constrant and our lnear relaxaton. Note that for any agent who happens to receve an ntegral allocaton, the correspondng terms n the two constrants are equal, but they may dverge for agents who have fractonal allocatons. To smplfy and bound ths quantty, we show that at any optmal fractonal soluton, all but at most one agent receves an ntegral allocaton: Lemma 2. At any extreme pont x of the feasble regon for the LP, there are at least n ndces such that x {0, } for all. Proof. Suppose not. Then let x be a pont n the feasble set S α such that at least two players (say players and 2) are assgned to lotteres. In other words, each of these two players are assgned 8

10 nonzero weght on at least two dfferent varance levels. Let a < b, k < l be the ndces such that x a, x b, x 2k, x 2l {0, }. Let ε > 0 be a small enough number such that x a ± ε, x b ± ε, x 2k ± ε, x 2l ± ε [0, ] and x a ± ε, x b ± ε, x 2k ± ε, x 2l ± ε [0, ], where ε = ε ( /va /v b /v k /v l ). Now consder the followng two ponts that dffer from x only n four coordnates: y : y a = x a + ε, y b = x b ε, y 2k = x 2k ε, and y 2l = x 2l + ε z : z a = x a ε, z b = x b + ε, z 2k = x 2k + ε, and z 2l = x 2l ε Note that x = /2(y + z), and recall that /α x /v because x S α. Furthermore, x /v + ε(/v a /v b ) + ε (/v l /v k ) y /v = = = [ x /v + ε /v a /v b + (/v l /v k ) /v ] a /v b /v k /v l x /v /α. Smlarly,, z /v =, x /v /α, so both y and z are n the feasble regon S α. Snce x s a convex combnaton of y and z that are both n S α, we know that x cannot be an extreme pont of the feasble regon. Lemma 2 says that at any extreme pont x, at least n players have an ntegral assgnment n x. To use ths property, we wll compute the soluton usng an (ellpsod-based) polynomal-tme LP solver from [6] that always returns an optmal extreme pont soluton. Now we can bound the varance of our aggregate estmate µ. Lemma 3. For any ε > 0, the varance of our estmate Var( µ) ( + ε)α, as long as ( ) ( ) vm n v ε +. Proof. Suppose that n satsfes the bound above. If the soluton x s fully ntegral, then the varance s no more than α. Otherwse let a be the data provder recevng a lottery n x. Snce for every player wth an ntegral assgnment x /v = / x v, we can further smplfy, (x) = x a /v / x a v. The algorthm conssts of two steps: frst compute a suffcently near optmal soluton x usng the ellpsod algorthm; then round the soluton x to an optmal extreme pont soluton x usng the method of contnued fractons. For more detals, see [6]. 9

11 Then we can bound the volaton of (8) by the fnal assgnment v: x a /v /v m /v /v m. In other words, the resultng varance Var( µ) satsfes Var( µ) α ( ). v v m Snce we assume n > v m /v, we have n/v m (/v /v m ) > 0. As stated earler n Remark, the only nterestng range of α s v /n α v m /n. (Recall that f α < v /n, then the problem s nfeasble; f α > v m /n, then the problem s trval.) For the remander of the proof, we assume α [v /n, v m /n]. By ths assumpton, /α (/v /v m ) > 0, and so, ( ) Var( µ) α v + = α α v m v v m (v m v ) ( ) n α ( + ε)α. n ( vm v ) We also gve an example n Appendx A showng that ths analyss cannot be mproved, and we do need n = Ω(v m /v ) to approxmately satsfy the target varance constrant. 4.2 Indvdual Ratonalty In order to ensure ndvdual ratonalty, we need to set the entrance reward R large enough, so that for each player, R + p c 0, where c denotes the cost for player to provde ts assgned estmate. To reason about the payment player gets, we need to compute the followng two costs C and C 2, for all players except. Let x be the optmal (fractonal) soluton for our LP, and v be the expected varance level assgned to player : x v. Let OPT denote the optmal mn-cost value n the LP, and C denote the total cost for all players except n x : C = mn a, x a c a (3) subect to x a /v /α / v (4) a, x a = for all a (5) x a 0 for all (a, ) (6) 0

12 Let C 2 be the mnmum cost had we removed agent from the nput: C 2 = mn a, x a c a (7) subect to a, x a /v /α (8) x a = for all a (9) x a 0 for all (a, ) (20) The VCG payment gven to player n Algorthm s p = C 2 C. Note that snce the second LP s more constraned than the frst, we know C 2 C and the payment s always non-negatve. We can wrte down the expected utlty of player : R + p c = R + C 2 C c = R + C 2 OPT. Lemma 4. The mechansm n Algorthm s ndvdually ratonal f the entrance reward satsfes R max mn c. Proof. Let x be the optmal assgnment for the second program (wth optmal obectve value at C 2 ). Now let s add back player to the problem, and construct an assgnment x such that x = (x, x ), where x assgns player to the assgnment wth mnmum cost (mn c ). Note that x s a feasble soluton to our orgnal problem snce x already satsfes the varance constrant. It follows that the cost gven by x s at least as large as OPT, the optmal soluton: OPT C 2 + mn c. Therefore, as long as R mn c for each player, we have ndvdual ratonalty. We gve an example n Appendx A to show that ths bound s tght. In partcular, our example shows that wthout the entrance reward, ndvdual ratonalty constrant could be volated by up to mn c for each player. Remark 3. Let c mn = max mn c. If costs are drawn from a known dstrbuton, the analyst can set R to ensure that wth hgh probablty, all players have c mn R. If c mn s unbounded, t s clear that no Groves mechansm 2 can be ndvdually ratonal for all players n ths settng. The Green-Laffont-Holmström theorem [, 2] shows that under certan techncal condtons, any mechansm whch s domnant strategy ncentve compatble and maxmzes welfare must be a Groves mechansm. Thus wthout addtonal nformaton on the players costs, we should not hope to satsfy ndvdual ratonalty for all players whle stll achevng our other desderata. 2 A Groves mechansm s one whch selects the welfare maxmzng outcome, and each player s payment s hs externalty plus an amount that s ndependent of hs report. In partcular, the payments nduced by any Groves mechansm to a player are shfts of the payments nduced by our mechansm, by an amount that s ndependent of player s report. Hence by reportng a large enough value of c mn, ndvdual ratonalty can always be volated by a Groves mechansm.

13 References [] Aaron Archer, Chrstos Papadmtrou, Kunal Talwar, and Éva Tardos. An approxmate truthful mechansm for combnatoral auctons wth sngle parameter agents. In Proceedngs of the 4th Annual ACM-SIAM Symposum on Dscrete Algorthms, SODA 03, pages , [2] Anthony Atknson, Alexander Donev, and Randall Tobas. Optmum Expermental Desgns, wth SAS. Oxford Statstcal Scence, [3] Stephen Boyd and Leven Vandenberghe. Convex Optmzaton. Cambrdge Unversty Press, [4] Shahar Dobznsk and Shaddn Dughm. On the power of randomzaton n algorthmc mechansm desgn. In Proceedngs of the 50th Annual IEEE Symposum on Foundatons of Computer Scence, FOCS 09, pages , [5] Shahar Dobznsk and Noam Nsan. Lmtatons of VCG-based mechansms. In Proceedngs of the 39th Annual ACM Symposum on Theory of Computng, STOC 07, pages , [6] Cyntha Dwork, Frank McSherry, Kobb Nssm, and Adam Smth. Calbratng nose to senstvty n prvate data analyss. In Proceedngs of the 3rd Conference on Theory of Cryptography, TCC 06, pages , [7] Lsa K. Flescher and Yu-Han Lyu. Approxmately optmal auctons for sellng prvacy when costs are correlated wth data. In Proceedngs of the 3th ACM Conference on Electronc Commerce, EC 2, pages , 202. [8] Arpta Ghosh and Katrna Lgett. Prvacy and coordnaton: computng on databases wth endogenous partcpaton. In Proceedngs of the 4th ACM conference on Electronc Commerce, pages ACM, 203. [9] Arpta Ghosh, Katrna Lgett, Aaron Roth, and Grant Schoenebeck. Buyng prvate data wthout verfcaton. In Proceedngs of the 5th ACM Conference on Economcs and Computaton, EC 4, pages , 204. [0] Arpta Ghosh and Aaron Roth. Sellng prvacy at aucton. In Proceedngs of the 2th ACM Conference on Electronc Commerce, EC, pages , 20. [] Jerry R. Green and Jean-Jacques Laffont. Characterzaton of satsfactory mechansms for the revelaton of preferences for publc goods. Econometrca, 45(2): , 977. [2] Bengt Holmström. Groves scheme on restrcted domans. Econometrca, 47(5):37 44, 979. [3] Thbaut Horel, Strats Ioannds, and S. Muthukrshnan. Budget feasble mechansms for expermental desgn. In Alberto Pardo and Alfredo Vola, edtors, LATIN 204: Theoretcal Informatcs, Lecture Notes n Computer Scence, pages [4] Ron Lav and Chatanya Swamy. Truthful and near-optmal mechansm desgn va lnear programmng. J. ACM, 58(6): 24, December 20. 2

14 [5] Katrna Lgett and Aaron Roth. Take t or leave t: Runnng a survey when prvacy comes at a cost. In Paul W. Goldberg, edtor, Internet and Network Economcs, volume 7695 of Lecture Notes n Computer Scence, pages [6] George L. Nemhauser and Laurence A. Wolsey. Integer and combnatoral optmzaton. Wley nterscence seres n dscrete mathematcs and optmzaton. Wley, 988. [7] Kobb Nssm, Sall Vadhan, and Davd Xao. Redrawng the boundares on purchasng data from prvacy-senstve ndvduals. In Proceedngs of the 5th Conference on Innovatons n Theoretcal Computer Scence, ITCS 4, pages 4 422, 204. [8] Fredrch Pukelshem. Optmal Desgn of Experments, volume 50. Socety for Industral and Appled Mathematcs, [9] Aaron Roth and Grant Schoenebeck. Conductng truthful surveys, cheaply. In Proceedngs of the 3th ACM Conference on Electronc Commerce, EC 2, pages , 202. A Tghtness of Our Bounds A. Example for Varance Volaton Bound Consder an example where there are only two optons of varance levels, v and v 2, and we set the v target varance α = v 2 nv +δ(v 2 v ). Suppose the reported costs c = t and c 2 = t 2 for each player [n ], and c n < t and c n2 = t 2 for player n. We also assume that t 2 < t. Let x denote the assgnment such that x = 0 and x 2 = for each [n ], and x n = δ (0, ) and x n2 = δ. That s, the assgnment gves v 2 to the frst (n ) players, and gve a lottery between the two levels to player n. Note that α = n δ + δ. v 2 v We know that the fractonal soluton x exactly satsfes the varance constrant 8, and s also the optmal mn-cost soluton. Therefore, wth probablty ( δ), the realzed varance s Var( µ) = n = n δ + δ + δ δ = v 2 v 2 v v 2 v α δ( ) > 0. v v 2 It follows that Var( µ) = If we want δ(v 2/v ) n α αδ( v v 2 ) = α ( δ ) v v ( 2 n +δ v 2 v v 2 ) = + δ ( v2 v ε, we would need to have the number of provders n ) α n ( ) v2 δ v ε. For δ close to and constant ε, the number of provders we need does scale wth v 2 /v, whch shows that the Ω(v m /v ) for n s essentally tght. 3

15 A.2 Example for Entrance Reward Bound Consder an example wth two provders, two possble varance levels v, v 2 such that v 2 = 2v, and target varance α = v. Suppose the costs satsfy c = c 2 = t and c 2 = c 22 = t ε for some ε > 0. Snce we need to an estmate from each provder, the optmal soluton s to assgn v 2 to both players, whch yelds cost OPT = 2t 2ε. Now suppose we remove any provder from the mechansm. Then we would assgn the remanng provder to v, whch yeld cost C 2 = t. Therefore, the utlty for each provder s R + C 2 OPT = R + t 2(t + ε) = R + 2ε t = R + ε t. In order to ensure non-negatve utlty, we need R t ε. Note that the rght hand sde tends to max mn c when ε tends to 0. Therefore, the bound n Lemma 4 s tght. 4