A Lower Bound on the Competitive Ratio of Truthful Auctions

Transcription

1 A Lower Bound on the Compettve Rato of Truthful Auctons Andrew V Goldberg 1, Jason D Hartlne 1, Anna R Karln 2, and Mchael Saks 3 1 Mcrosoft Research, SVC/5, 1065 La Avenda, Mountan Vew, CA {goldberg,hartlne}@mcrosoftcom 2 Computer Scence Department, Unversty of Washngton karln@cswashngtonedu 3 Department of Mathematcs Hll Center, Rutgers Unversty 110 Frelnghuysen Rd, Pscataway, NJ saks@mathrutgersedu Abstract We study a class of sngle-round, sealed-bd auctons for a set of dentcal tems We adopt the worst case compettve framework defned by [1,2] that compares the proft of an aucton to that of an optmal sngle prce sale to at least two bdders In ths framework, we gve a lower bound of 242 (an mprovement from the bound of 2 gven n [2]) on the compettve rato of any truthful aucton, one where each bdders best strategy s to declare the true maxmum value an tem s worth to them Ths result contrasts wth the 339 compettve rato of the best known truthful aucton [3] 1 Introducton A combnaton of recent economc and computatonal trends, such as the neglgble cost of duplcatng dgtal goods and, most mportantly, the emergence of the Internet as one of the most mportant arenas for resource sharng between partes wth dverse and selfsh nterests, has created a number of new and nterestng dynamc prcng problems It has also cast new lght on more tradtonal problems such as the problem of proft maxmzaton for the seller n an aucton A number of recent papers [1,2,3] have consdered the problem of desgnng auctons, for sellng dentcal unts of an tem, that perform well n worst case under unknown market condtons In these auctons, there s a seller wth l unts for sale, and bdders each nterested n obtanng one of them Each bdder has a valuaton representng how much the tem s worth to them The aucton s performed by solctng a sealed bd from each of the bdders, and decdng on the allocaton of unts to bdders and the prces to be pad by the bdders The bdders are assumed to follow the strategy of bddng so as to maxmze ther personal utlty, the dfference between ther valuaton and the prce they pay To handle the problem of desgnng and analyzng auctons where bdders may Work was done whle second author was at the Unversty of Washngton Supported by NSF grant CCR Part of the work was done whle vstng Mcrosoft Research

2 falsely declare ther valuatons to get a better deal, we wll adopt the soluton concept of truthful mechansm desgn (see, eg, [1,4,5]) In a truthful aucton, truth-tellng, e, revealng ther true valuaton as ther bd, s an optmal strategy for each bdder regardless of the bds of the other bdders In ths paper, we wll restrct our attenton to truthful (aka, ncentve compatble or strategyproof) auctons In research on such auctons, a form of compettve analyss s used to gauge aucton revenue Specfcally, a truthful aucton s performance on a partcular bd vector s evaluated by comparng t aganst the proft that could be acheved by an optmal omnscent aucton, one that knows the true valuatons of the bdders n advance An aucton s β-compettve f t acheves a proft that s wthn a factor of β 1 of optmal on every nput The goal then becomes to desgn the aucton wth the best compettve rato, e, the aucton that s β-compettve wth the smallest possble value of β A partcularly nterestng specal case of the aucton problem s the unlmted supply case In ths case the number of unts for sale s at least the number of bdders n the aucton Ths s natural for the sale of dgtal goods where there s neglgble cost for duplcatng and dstrbutng the good Pay-per-vew televson and downloadable audo fles are examples of such goods For the unlmted supply aucton problem, the compettve framework ntroduced n [1] and further refned n [2] uses the proft of the optmal omnscent sngle prced mechansm that sells at least two unts as the benchmark for compettve analyss The assumpton that two or more unts are sold s necessary because n the worst case t s mpossble to obtan a constant fracton of the proft of the optmal mechansm when t sells only one unt [1] In ths worst case compettve framework, the best known aucton for the unlmted supply has a compettve rato of 339 [3] In ths paper we also consder the case where the number of unts for sale, l, s lmted, e, less than the number of bdders At the opposte extreme from unlmted supply, s the lmted supply case wth l = 2 4 In ths case the Vckrey aucton [4], whch sells to the hghest bdder at the second hghest bd value, obtans the optmal worst case compettve rato of 2 [2] The man result of ths paper s a lower bound on the compettve rato of any randomzed aucton For l = 2, ths lower bound s 2 (ths was orgnally proven n [2], though we gve a much smpler proof of t here) For l = 3, the lower bound s 13/6 217, and as l grows the bound approaches 242 n the lmt We conjecture that ths lower bound s tght Yet, even n the case of three unts, the problem of constructng the aucton matchng our lower bound of 13/6 s open The rest of the paper s organzed as follows In Secton 2 we gve the mathematcal formulaton of the aucton problem that we wll be studyng, and we descrbe the compettve framework that s used to analyze such auctons n worst case In Secton 3 we gve our man result, a bound on how well any auc- 4 Notce that the compettve framework s not well defned for the l = 1 case as the optmal aucton that sells at least two unts cannot sell just one unt

3 ton can perform n worst case In Secton 4 we descrbe attempts to obtan a matchng upper bound 2 Prelmnares and Notaton We consder sngle-round, sealed-bd auctons for a set of l dentcal unts As mentoned n the ntroducton, we adopt the game theoretc soluton concept of truthful mechansm desgn A useful smplfcaton of the problem of desgnng truthful auctons s obtaned through the followng algorthmc characterzaton Related formulatons to the one we gve here have appeared n numerous places n recent lterature (eg, [6,7,2,8]) To the best of our knowledge, the earlest dates back to the 1970s [9] Defnton 1 Gven a bd vector of n bds, b = (b 1,,b n ), let b denote the vector of wth b replaced wth a?, e, b = (b 1,, b 1,?, b +1,, b n ) Defnton 2 (Bd-ndependent Aucton, BI f ) Let f be a functon from bd vectors (wth a? ) to prces (non-negatve real numbers) The determnstc bd-ndependent aucton defned by f, BI f, works as follows For each bdder : 1 Set t = f(b ) 2 If t < b, bdder wns at prce t 3 If t > b, bdder loses 4 Otherwse, (t = b ) the aucton can ether accept the bd at prce t or reject t A randomzed bd-ndependent aucton s a dstrbuton over determnstc bdndependent auctons The proof of the followng theorem can be found, for example, n [2] Theorem 1 An aucton s truthful f and only f t s equvalent to a bdndependent aucton Gven ths equvalence, we wll use the the termnology bd-ndependent and truthful nterchangeably We denote the proft of a truthful aucton A on nput b as A(b) Ths proft s gven by the sum of the prces charged bdders that are not rejected For a randomzed bd-ndependent aucton, A(b) and f(b ) are random varables It s natural to consder a worst case compettve analyss of truthful auctons In the compettve framework of [2] and subsequent papers, the performance of a truthful aucton s gauged n comparson to the optmal aucton that sells at least two unts There are a number reasons to choose ths metrc for comparson, nterested readers should see [2] or [10] for a more detaled dscusson

4 Defnton 3 The optmal sngle prce omnscent aucton that sells at least two unts (and at most l unts), F (2,l), s defned as follows: Let b be a bd vector of n bds, and let v be the -th largest bd n the vector b Aucton F (2,l) on b chooses k {2,,l} to maxmze kv k The k hghest bdders are each sold a unt at prce v k (tes broken arbtrarly); all remanng bdders lose Its proft s: F (2,l) (b) = max 2 k l kv k In the unlmted supply case, e, when l = n, we defne F (2) = F (2,n) Defnton 4 We say that aucton A s β-compettve f for all bd vectors b, the expected proft of A on b satsfes E[A(b)] F(2,l) (b) β The compettve rato of the aucton A s the nfmum of β for whch the aucton s β-compettve 21 Lmted Supply Versus Unlmted Supply Throughout the remander of ths paper we wll be makng the assumpton that n = l, e, the number of bdders s equal to the number of tems for sale The justfcaton for ths s that any lower bound that apples to the n = l case also extends to the case where n l To see ths, note that an l tem aucton A that s β-compettve for any n > l bdder nput must also be β-compettve on the subset of all n bdder bd vectors that have n l bds at value zero Thus, we can smply construct an A that takes l bdder nput b, augments t wth n l zeros to get b, and smulates the outcome of A on b Snce F (2) (b ) = F (2,l) (b), A obtans at least the compettve rato of A In the other drecton, a reducton from the unlmted supply aucton problem to the lmted supply aucton problem gven n [10] shows how to take an unlmted supply aucton that s β-compettve wth F (2) and construct a lmted supply aucton parameterzed by l that s β-compettve wth F (2,l) Henceforth, we wll assume that we are n the unlmted supply case, and we wll examne lower bounds for lmted supply problems by placng a restrcton on the number of bdders n the aucton 22 Symmetrc Auctons In the remander of ths paper, we restrct attenton to symmetrc auctons An aucton s symmetrc f ts output s not a functon of the order of the bds n the nput vector, b We note that there s no loss of generalty n ths assumpton, as the followng result shows Lemma 1 For any β-compettve asymmetrc truthful aucton there s a symmetrc randomzed truthful aucton wth compettve rato at least β

5 Proof Gven a β-compettve asymmetrc truthful aucton, A, we construct a symmetrc truthful aucton A that frst permutes the nput bds b at random to get π(b) and then runs A on π(b) Note, F (2) (b) = F (2) (π(b)) and snce A s β-compettve on π(b) for any choce of π, A s β-compettve on b 23 Example: The Vckrey Aucton The classcal truthful aucton s the 1-tem Vckrey aucton (aka the second prce aucton) Ths aucton sells to the hghest bdder at the second hghest bd value To see how ths fts nto the bd-ndependent framework, note that the aucton BI max (the bd-ndependent aucton wth f = max) does exactly ths (assumng that the largest bd s unque) As an example we consder the compettve rato of the Vckrey aucton n the case where there are only two bdders Gven two bds, b = {b 1, b 2 }, the optmal sngle prce sale of two unts just sells both unts for the smaller of the two bd values, e, the optmal proft s F (2) (b) = 2 mn(b 1, b 2 ) Of course, the 1-tem Vckrey aucton sells to the hghest bdder at the second hghest prce and thus has a proft of mn(b 1, b 2 ) Therefore, we have: Observaton 1 The Vckrey aucton on two bdders s 2-compettve It turns out that ths s optmal for two bdders Along wth the general lower bound of 242, n the next secton we gve a smplfed proof of the result, orgnally from [2], that no two bdder truthful aucton s better than 2-compettve 3 A Lower Bound on the Compettve Rato In ths secton we prove a lower bound on the compettve rato of any truthful aucton n comparson to F (2) ; we show that for any randomzed truthful aucton, A, there exsts an nput bd vector, b, on whch E[A(b)] F(2) (b) 242 In our lower bound proof we wll be consderng randomzed dstrbutons over bd vectors To avod confuson, we wll adopt the followng notaton A real valued random varable wll be gven n uppercase, eg, X and T In accordance wth ths notaton, we wll use B as the random varable for bdder s bd value A vector of real valued random varables wll be a bold uppercase letter, eg, B s a vector of random bds To prove the lower bound, we analyze the behavor of A on a bd vector chosen from a probablty dstrbuton over bd vectors The outcome of the aucton s then a random varable dependng on both the randomness n A and the randomness n B We wll gve a dstrbuton on bdder bds and show that t satsfes E B [E A [A(B)]] EB[F(2) (B)] 242 We then use the followng fact to clam that there must exst a fxed choce of bds, b (dependng on A), for whch E[A(b)] F(2) (b) 242

6 Fact 1 Gven random varable X and two functons f and g, E[f(X)] E[g(X)] mples that there exsts x such that f(x) g(x) As a quck proof of ths fact, observe that f for all x, f(x) > g(x) then t would be the case that E[f(X)] > E[g(X)] nstead of the other way around A key step n obtanng the lower bound s n defnng a dstrbuton over bd vectors on whch any truthful aucton obtans the same expected revenue Defnton 5 Let the random vector of bds B (n) be n d bds generated from the dstrbuton wth each bd B satsfyng Pr[B > z] = 1/z for all z 1 Lemma 2 For B (n) defned above, any truthful aucton, A, has expected revenue satsfyng, [ ] E A(B (n) ) n Proof Consder a truthful aucton A Let T be the prce offered to bdder n the bd-ndependent mplementaton of A T s a random varable dependng on A and B and therefore T and B are ndependent random varables Let P be the prce pad by bdder, e, 0 f B < T and T otherwse For t 0, E[P T = t] = t Pr[B > t T = t] = t Pr[B > t] 1, snce B s ndependent of T Therefore E[P ] 1 and E [ A(B (n) ) ] = E[P ] n For the nput B (n) an aucton attemptng to maxmze the proft of the seller has no reason to ever offer prces less than one The proof of the above lemma shows that any aucton that always offers prces of at least one has expected revenue exactly n 31 The n = 2 Case To gve an outlne for how our man proof wll proceed, we frst present a proof that the compettve rato for a two bdder aucton s at least 2 Of course, the fact that the 1-tem Vckrey aucton acheves ths compettve rato means that ths result s tght The proof we gve below smplfes the proof of the same result gven n [2] Lemma 3 E [ F (2) (B (2) ) ] = 4 Proof From the defnton of F (2), F (2) (B (2) ) = 2 mnb (2) Therefore, for z 2, Pr [ F (2) (B (2) ) > z ] = Pr[B 1 > z/2 B 2 > z/2] = 4/z 2 Usng the defnton of expectaton for non-negatve contnuous random varables of E[X] = 0 Pr[X > x] dx we have [ ] E F (2) (B (2) ) = (4/z 2 )dz = 4 Lemma 4 The optmal compettve rato for a two bdder aucton s 2 The proof of ths lemma follows drectly from Lemmas 2 and 3, and Fact 1

7 32 The General Case For the general case, as n the two bdder case, we must compute the expectaton of F (2) (B (n) ) Lemma 5 For n bds from the above dstrbuton, the expected value of F (2) s [ ] ( ) 1 ( ) 1 E F (2) (B (n) n 1 ) = n n n 1 1 Proof In ths proof we wll get a closed form expresson for Pr [ F (2) (B (n) ) > z ] and then ntegrate to obtan the expected value Note that all bds are at least one and therefore, we wll assume that z n Clearly for z < n, Pr [ F (2) (B (n) ) > z ] = 1 Let V be a random varable for the value of the th largest bd, eg, V 1 = max B To get a formula for Pr [ F (2) (B (n) ) ], we defne a recurrence based on the random varable F n,k defned as F n,k = max(k + )V Intutvely, F n,k represents the optmal sngle prce revenue from B (n) and an addtonal k consumers each of whch has a value equal to the hghest bd, V 1 To defne the recurrence, fx n, k, and z and defne the events H for 1 n Intutvely, the event H represents the fact that bdders n B (n) and the k addtonal consumers have bd hgh enough to equally share z, whle no larger set of j > bdders of B (n) can do the same H = V z/(k + ) n j=+1 V j < z/(k + j) ( ) ( ) n k + Pr[H ] = Pr[F n,k+ < z] z Note that events H are dsjont and that F n,k s at least z f and only f one of the H occurs Thus, [ n ] Pr[F n,k > z] = Pr H = Pr[H ] = =1 =1 =1 ( ) ( ) n k + Pr[F n,k+ < z] (1) z Equaton (1) defnes a two dmensonal recurrence The base case of ths recurrence s gven by F 0,k = 0 We are nterested n F (2) (B (n) ) whch s the same as F n,0 except that we gnore the H 1 case Ths gves [ ] Pr F (2) (B (n) ) > z = Pr[F n,0 > z] Pr[H 1 ] = Pr[F n,0 > z] n z Pr[F n 1,1 < z] (2)

8 To obtan Pr [ F (2) (B (n) ) ] we can solve the recurrence for F n,k gven by Equaton (1) We wll show that the soluton s: ( ) n ( ) z k z k n Pr[F n,k > z] = 1 (3) z z k Note that (3) s correct for n = 0 We show that t s true n general nductvely Substtutng n our proposed soluton (3) nto (1) we obtan: Pr[F n,k > z] = =1 ( n = z k n z n ) ( ) ( k + z k z =1 ) n ( ) z k n z k z ( ) n (k + ) (z k ) n 1 (4) We now apply the followng verson of Abel s Identty [11]: (x + y) n = x j=0 ( ) n (x + j) j 1 (y j) n j j Makng the change of varables, j = n, x = z k n, and y = k + n we get: z n n z k n = ( ) n (k + ) (z k ) n 1 =0 We subtract out the = 0 term and plug ths dentty nto (4) to get Pr[F n,k > z] = z k n ( z n ) z n (z k)n 1 z k n ( ) n ( ) z k z k n = 1 z z k Thus, our closed form expresson for the recurrence s correct Recall our goal s to compute Pr [ F (2) (B (n) ) > z ] Equaton (3) shows that Pr[F n,0 > z] = n/z Ths combned wth Equaton (2) and Equaton (3) gves the followng for z n: [ ] Pr F (2) (B (n) ) > z = n z n z Pr[F n 1,1 < z] = n z Pr[F n 1,1 > z] ( = n ( ) n 1 ( ) ) z 1 z n 1 z z z 1 Recall that for z n, Pr [ F (2) (B (n) ) > z ] = 1 To complete ths proof, we use the formula E [ F (2) (B (n) ) ] = Pr [ F (2) (B (n) ) > z ] dz = n+ 0 n Pr[ F (2) (B (n) ) > z ] dz

9 In the form above, ths s not easly ntegrable; however, we can transform t back nto a bnomal sum whch we can ntegrate: [ ] Pr F (2) (B (n) ) > z = n [ ] E F (2) (B (n) ) = n + n ( ) ( ) 1 n 1 z 1 ( ) ( 1 n 1 n z 1 ( ) 1 ( 1 n 1 = n n n 1 1 ) dz ) Theorem 2 The compettve rato of any aucton on n bdders s 1 ( ) 1 ( ) 1 n 1 n 1 1 Ths theorem comes from combnng Lemma 2, Lemma 5, and Fact 1 Of course, for the specal case of n = 2 ths gves the lower bound of 2 that we already gave For n = 3 ths gves a lower bound of 13/6 A lower bound for the compettve rato of the best aucton for general n s obtaned by takng the lmt In the proof of the man theorem to follow, we use the followng fact Fact 2 For 1 k K, 0 < a k < 1, then K k=1 (1 a k) 1 K k=1 a k Theorem 3 The compettve rato of any aucton s at least 242 Proof We prove ths theorem by showng that, lm n ( 1 ( ) 1 ( ) ) 1 n 1 = 1 + n 1 1 ( 1) ( 1)( 1)! (5) After whch, routne calculaton shows that the rght hand sde of the above equaton s at least 242 whch gves the theorem To prove that (5) holds, t s suffcent to show that ( 1 + ) ( ( 1) 1 ( 1)( 1)! ( ) 1 1 n 1 ( ) ) n 1 1 = O ( ) 1 n

10 We proceed as follows: ( ) ( 1 + ( 1) ( ) ( 1)( 1)! n 1 ( ) 1 ( ) 1 ( 1)( 1)! n 1 n 1 1 ( ) = n(n 1) (n + 2) 1 ( 1)( 1)! n 1 ( ( = ) ( 1 2 ( 1)( 1)! n n 2 j 1 1 ( 1)( 1)! n j=1 ( ) 2 = 1 3 ( 1)( 1)! n n ( 1)( 1)! 1 n ( ) ) n 1 1 ) ( 1 2 )) n 3 ( 1)( 1)! Snce ( 1)! grows exponentally, 3 ( 1)( 1)! s bounded by a constant and we have the desred result 4 Lower Bounds versus Upper Bounds As mentoned earler, the lower bound of 242 for large n does not match the compettve rato of the best known aucton (currently 339 [3]) In ths secton, we brefly consder the ssue of matchng upper bounds for small values of n For n = 2 the 1-tem Vckrey aucton obtans the optmal compettve rato of 2 (see Secton 23) It s nterestng to note that for the n = 2 case the optmal aucton always uses sale prces chosen from the set of nput bds (n partcular, the second hghest bd) Ths motvates the followng defnton Defnton 6 We say an aucton, A, s restrcted f on any nput the sale prces are drawn from the set of nput bd values, unrestrcted otherwse Whle the Vckery aucton s a restrcted aucton, the Vckrey aucton wth reserve prce r, whch offers the hghest bdder the greater of r and the second hghest bd value, s not restrcted as r may not necessarly be a bd value Desgnng restrcted bd-ndependent auctons s easer than desgnng general ones as the set of sale prces s determned by the nput bds However, as we show next, even for the n = 3 case the optmal restrcted aucton s compettve rato s worse than that of the optmal unrestrcted aucton Lemma 6 For n = 3, no restrcted truthful aucton, BI f, can acheve a compettve rato better than 5/2

11 Proof Because BI f s restrcted, f(a, b) {a, b} For h > 1 and a hb, let p = suppr[f(a, b) = b] a,b For ǫ close to zero, let a and b be such that a > hb and Pr[f(a, b) = b] p ǫ The expected revenue for the aucton on {a, b + ǫ, b} s at most b + ǫ + pb Here, the b + ǫ an upper bound on the payment from the a bd and the pb s an upper bound on the expected from the b + ǫ bd (as p s an upper bound on the probablty that ths bd s offered prce b) Note that F (2) = 3b so the compettve rato obtaned by takng the lmt as ǫ 0 s at least 3/(1 + p) An upper bound for the expected revenue for the aucton on {a + ǫ, a, b} s 2pb+(1 p+ǫ)a The pb+(1 p+ǫ)a s from the a+ǫ and the pb s from the a bd For large h, F (2) = 2a so the compettve rato s at least 2h/(2pb+h(1 p+ǫ)) The lmt as ǫ 0 and h gves a bound on the compettve rato of 2/(1 p) Settng these two ratos equal we obtan an optmal value of p = 1/5 whch obtans a compettve rato of 5/2 Ths lower bound s tght as the followng lemma shows Lemma 7 For a b, the bd-ndependent aucton, BI f wth { b wth probablty 1/5 f(a, b) = a otherwse acheves a compettve rato of 5/2 for three bdders We omt the proof as t follows va an elementary case analyss It s nterestng to note that the above aucton s essentally performng a 1-tem Vckrey aucton wth probablty 4/5 and a 2-tem Vckrey aucton wth probablty 1/5 Lemma 8 An unrestrcted three bdder aucton can acheve a better compettve rato than 5/2 Proof For a b, the bd ndependent aucton BI f wth { b wth probablty 15/23 b a 3b/2 3b/2 wth probablty 8/23 f(a, b) = { b wth probablty 3/23 a > 3b/2 a wth probablty 20/23 has compettve rato 23 We omt the elementary case analyss Recall that the lower bound on the compettve rato for three bdders s 13/6 217 Obtanng the optmal aucton for three bdders remans an nterestng open problem

12 5 Conclusons We have proven a lower bound of 242 on the compettve rato of any truthful aucton The algorthmc technque used, that of lookng at dstrbutons of bdders on whch all auctons perform the same and boundng the expected value of the metrc (eg, F (2) ), s natural and useful for other aucton related problems There s a strange artfact of the compettve framework that we employ here (and that whch s used n pror work [2,3]) As we showed, the optmal worst case aucton for sellng two tems s the 1-tem Vckrey aucton Ths aucton only sells one tem, yet we had two tems Our optmal restrcted aucton for three tems never sells more that two tems Yet, under our compettve framework t s not optmal to run ths optmal restrcted aucton for three tems when there are only two tems for sale As t turns out, ths s not a problem when usng a dfferent but related metrc, V opt, defned as the k-tem Vckrey aucton that obtans the hghest proft, e, V opt (b) = max ( 1)b (for b b +1 ) Acknowledgements We would lke to thank Amos Fat for many helpful dscussons References 1 Goldberg, A, Hartlne, J, Wrght, A: Compettve Auctons and Dgtal Goods In: Proc 12th Symp on Dscrete Algorthms, ACM/SIAM (2001) Fat, A, Goldberg, A, Hartlne, J, Karln, A: Compettve Generalzed Auctons In: Proc 34th ACM Symposum on the Theory of Computng, ACM Press, New York (2002) 3 Goldberg, A, Hartlne, J: Compettveness va Concensus In: Proc 14th Symp on Dscrete Algorthms, ACM/SIAM (2003) 4 Vckrey, W: Counterspeculaton, Auctons, and Compettve Sealed Tenders Journal of Fnance 16 (1961) Nsan, N, Ronen, A: Algorthmc Mechansm Desgn In: Proc of 31st Symposum on Theory of Computng, ACM Press, New York (1999) Archer, A, Tardos, E: Truthful mechansms for one-parameter agents In: Proc of the 42nd IEEE Symposum on Foundatons of Computer Scence (2001) 7 Segal, I: Optmal Prcng Mechansm wth Unknown Demand Amercan Economc Revew 93 (2003) Lehmann, D, O Callaghan, L, Shoham, Y: Truth Revelaton n Approxmately Effcent Combnatoral Auctons In: Proc of 1st ACM Conf on E-Commerce, ACM Press, New York (1999) Mrrlees, J: An Exploraton nto the Theory of Optmal Income Taxaton Revew of Economcs Studes 38 (1971) Goldberg, A, Hartlne, J, Karln, A, Saks, M, Wrght, A: Compettve auctons and dgtal goods Games and Economc Behavor (2002) Submtted for publcaton An earler verson avalable as InterTrust Techncal Report STAR- TR Abel, N: Bewes enes Ausdrucks von welchem de Bnomal-Formel en enzelner Fall st Crelles Journal für de Rene und Angewandte Mathematk 1 (1826)