Expressve Negotaton over Donatons to Chartes Vncent Contzer Carnege Meon Unversty 5000 Forbes Avenue Pttsburgh, PA 523, USA contzer@cs.cmu.edu Tuomas Sandhom Carnege Meon Unversty 5000 Forbes Avenue Pttsburgh, PA 523, USA sandhom@cs.cmu.edu ABSTRACT When donatng money to a (say, chartabe) cause, t s possbe to use the contempated donaton as negotatng matera to nduce other partes nterested n the charty to donate more. Such negotaton s usuay done n terms of matchng offers, where one party promses to pay a certan amount f others pay a certan amount. However, n ther current form, matchng offers aow for ony mted negotaton. For one, t s not mmedatey cear how mutpe partes can mae matchng offers at the same tme wthout creatng crcuar dependences. Aso, t s not mmedatey cear how to mae a donaton condtona on other donatons to mutpe chartes, when the donator has dfferent eves of apprecaton for the dfferent chartes. In both these cases, the mted expressveness of matchng offers causes economc oss: t may happen that an arrangement that woud have made a partes (donators as we as chartes) better off cannot be expressed n terms of matchng offers and w therefore not occur. In ths paper, we ntroduce a bddng anguage for expressng very genera types of matchng offers over mutpe chartes. We formuate the correspondng cearng probem (decdng how much each bdder pays, and how much each charty receves), and show that t s NP-compete to approxmate to any rato even n very restrcted settngs. We gve a mxed-nteger program formuaton of the cearng probem, and show that for concave bds, the program reduces to a near program. We then show that the cearng probem for a subcass of concave bds s at east as hard as the decson varant of near programmng. Subsequenty, we show that the cearng probem s much easer when bds are quasnear for surpus, the probem decomposes across chartes, and for payment maxmzaton, a greedy approach s optma f the bds are concave (athough ths atter probem s weay NP-compete when the bds are not concave). For the quasnear settng, we study the mechansm desgn queston. We show that an ex-post effcent mechansm s Supported by NSF under CAREER Award IRI-970322, Grant IIS-9800994, ITR IIS-008246, and ITR IIS-02678. Permsson to mae dgta or hard copes of a or part of ths wor for persona or cassroom use s granted wthout fee provded that copes are not made or dstrbuted for proft or commerca advantage and that copes bear ths notce and the fu ctaton on the frst page. To copy otherwse, to repubsh, to post on servers or to redstrbute to sts, requres pror specfc permsson and/or a fee. EC 04, May 7 20, 2004, New Yor, New Yor, USA. Copyrght 2004 ACM -583-7-0/04/0005...$5.00. mpossbe even wth ony one charty and a very restrcted cass of bds. We aso show that there may be benefts to nng the chartes from a mechansm desgn standpont. Categores and Subject Descrptors F.2 [Theory of Computaton]: Anayss of Agorthms and Probem Compexty; J.4 [Computer Appcatons]: Soca and Behavora Scences Economcs Genera Terms Agorthms, Economcs, Theory Keywords Expressve Negotaton, Donatons to Chartes, Maret Cearng, Mechansm Desgn. INTRODUCTION When money s donated to a chartabe (or other) cause (hereafter referred to as charty), often the donatng party gves uncondtonay: a fxed amount s transferred from the donator to the charty, and none of ths transfer s contngent on other events n partcuar, t s not contngent on the amount gven by other partes. Indeed, ths s currenty often the ony way to mae a donaton, especay for sma donatng partes such as prvate ndvduas. However, when mutpe partes support the same charty, each of them woud prefer to see the others gve more rather than ess to ths charty. In such scenaros, t s sensbe for a party to use ts contempated donaton as negotatng matera to nduce the others to gve more. Ths s done by mang the donaton condtona on the others donatons. The foowng exampe w ustrate ths, and show that the donatng partes as we as the chartabe cause may smutaneousy beneft from the potenta for such negotaton. Suppose we have two partes, and 2, who are both supporters of charty A. To ether of them, t woud be worth $0.75 f A receved $. It foows nether of them w be wng to gve uncondtonay, because $0.75 < $. However, f the two partes draw up a contract that says that they w each gve $0.5, both the partes have an ncentve to accept ths contract (rather than have no contract at a): wth the contract, the charty w receve $ (rather than $0 wthout a contract), whch s worth $0.75 to each party, whch s greater than the $0.5 that that party w have to gve. Effectvey, each party has made ts donaton condtona on the other party s donaton, eadng to arger donatons and greater happness to a partes nvoved.
One method that s often used to effect ths s to mae a matchng offer. Exampes of matchng offers are: I w gve x doars for every doar donated., or I w gve x doars f the tota coected from other partes exceeds y. In our exampe above, one of the partes can mae the offer I w donate $0.5 f the other party aso donates at east that much, and the other party w have an ncentve to ndeed donate $0.5, so that the tota amount gven to the charty ncreases by $. Thus ths matchng offer mpements the contract suggested above. As a rea-word exampe, the Unted States government has authorzed a donaton of up to $ bon to the Goba Fund to fght AIDS, TB and Maara, under the condton that the Amercan contrbuton does not exceed one thrd of the tota to encourage other countres to gve more [23]. However, there are severa severe mtatons to the smpe approach of matchng offers as just descrbed.. It s not cear how two partes can mae matchng offers where each party s offer s stated n terms of the amount that the other pays. (For exampe, t s not cear what the outcome shoud be when both partes offer to match the other s donaton.) Thus, matchng offers can ony be based on payments made by partes that are gvng uncondtonay (not n terms of a matchng offer) or at east there can be no crcuar dependences. 2. Gven the current nfrastructure for mang matchng offers, t s mpractca to mae a matchng offer depend on the amounts gven to mutpe chartes. For nstance, a party may wsh to specfy that t w pay $00 gven that charty A receves a tota of $000, but that t w aso count donatons made to charty B, at haf the rate. (Thus, a tota payment of $500 to charty A combned wth a tota payment of $000 to charty B woud be just enough for the party s offer to tae effect.) In contrast, n ths paper we propose a new approach where each party can express ts reatve preferences for dfferent chartes, and mae ts offer condtona on ts own apprecaton for the vector of donatons made to the dfferent chartes. Moreover, the amount the party offers to donate at dfferent eves of apprecaton s aowed to vary arbtrary (t does need to be a doar-for-doar (or n-doarfor-doar) matchng arrangement, or an arrangement where the party offers a fxed amount provded a gven (stre) tota has been exceeded). Fnay, there s a cear nterpretaton of what t means when mutpe partes are mang condtona offers that are stated n terms of each other. Gven each combnaton of (condtona) offers, there s a (usuay) unque souton whch determnes how much each party pays, and how much each charty s pad. However, as we w show, fndng ths souton (the cearng probem) requres sovng a potentay dffcut optmzaton probem. A arge part of ths paper s devoted to studyng how dffcut ths probem s under dfferent assumptons on the structure of the offers, and provdng agorthms for sovng t. Typcay, arger organzatons match offers of prvate ndvduas. For exampe, the Amercan Red Cross Lberty Dsaster Fund mantans a st of busnesses that match ther customers donatons [8]. Towards the end of the paper, we aso study the mechansm desgn probem of motvatng the bdders to bd truthfuy. In short, expressve negotaton over donatons to chartes s a new way n whch eectronc commerce can hep the word. A web-based mpementaton of the deas descrbed n ths paper can factate vountary reaocaton of weath on a goba scae. Adtonay, optmay sovng the cearng probem (and thereby generatng the maxmum economc wefare) requres the appcaton of sophstcated agorthms. 2. COMPARISON TO COMBINATORIAL AUCTIONS AND EXCHANGES Ths secton dscusses the reatonshp between expressve charty donaton and combnatora auctons and exchanges. It can be spped, but may be of nterest to the reader wth a bacground n combnatora auctons and exchanges. In a combnatora aucton, there are m tems for sae, and bdders can pace bds on bundes of one or more tems. The auctoneer subsequenty abes each bd as wnnng or osng, under the constrant that no tem can be n more than one wnnng bd, to maxmze the sum of the vaues of the wnnng bds. (Ths s nown as the cearng probem.) Varants ncude combnatora reverse auctons, where the auctoneer s seeng to procure a set of tems; and combnatora exchanges, where bdders can both buy and and se tems (even wthn the same bd). Other extensons ncude aowng for sde constrants, as we as the specfcaton of attrbutes of the tems n bds. Combnatora auctons and exchanges have recenty become a popuar research topc [20, 2, 7, 22, 9, 8, 3, 3, 2, 26, 9, 25, 2]. The probems of cearng expressve charty donaton marets and cearng combnatora auctons or exchanges are very dfferent n formuaton. Nevertheess, there are nterestng paraes. One of the man reasons for the nterest n combnatora auctons and exchanges s that t aows for expressve bddng. A bdder can express exacty how much each dfferent aocaton s worth to her, and thus the gobay optma aocaton may be chosen by the auctoneer. Compare ths to a bdder havng to bd on two dfferent tems n two dfferent (one-tem) auctons, wthout any way of expressng that (for nstance) one tem s worthess f the other tem s not won. In ths scenaro, the bdder may wn the frst tem but not the second (because there was another hgh bd on the second tem that she dd not antcpate), eadng to economc neffcency. Expressve bddng s aso one of the man benefts of the expressve charty donaton maret. Here, bdders can express exacty how much they are wng to donate for every vector of amounts donated to chartes. Ths may aow bdders to negotate a compex arrangement of who gves how much to whch charty, whch s benefca to a partes nvoved; whereas no such arrangement may have been possbe f the bdders had been restrcted to usng smpe matchng offers on ndvdua chartes. Agan, expressve bddng s necessary to acheve economc effcency. Another parae s the computatona compexty of the cearng probem. In order to acheve the fu economc effcency aowed by the maret s expressveness (or even come cose to t), hard computatona probems must be soved n combnatora auctons and exchanges, as we as n the charty donaton maret (as we w see).
3. DEFINITIONS Throughout ths paper, we w refer to the offers that the donatng partes mae as bds, and to the donatng partes as bdders. In our bddng framewor, a bd w specfy, for each vector of tota payments made to the chartes, how much that bdder s wng to contrbute. (The contrbuton of ths bdder s aso counted n the vector of payments so, the vector of tota payments to the chartes represents the amount gven by a donatng partes, not just the ones other than ths bdder.) The bddng anguage s expressve enough that no bdder shoud have to mae more than one bd. The foowng defnton maes the genera form of a bd n our framewor precse. Defnton. In a settng wth m chartes c,c 2,...,c m, a bd by bdder b j s a functon v j : R m R. The nterpretaton s that f charty c receves a tota amount of π c, then bdder j s wng to donate (up to) v j(π c,π c2,...,π cm ). We now defne possbe outcomes n our mode, and whch outcomes are vad gven the bds that were made. Defnton 2. An outcome s a vector of payments made by the bdders (π b,π b2,...,π bn ), and a vector of payments receved by the chartes (π c,π c2,...,π cm ). A vad outcome s an outcome where n. m (at east as much money s coected π bj j= π c = as s gven away); 2. For a j n, π bj v j(π c,π c2,...,π cm ) (no bdder gves more than she s wng to). Of course, n the end, ony one of the vad outcomes can be chosen. We choose the vad outcome that maxmzes the objectve that we have for the donaton process. Defnton 3. An objectve s a functon from the set of a outcomes to R. 2 After a bds have been coected, a vad outcome w be chosen that maxmzes ths objectve. One exampe of an objectve s surpus, gven by n π bj j= m π c. The surpus coud be the profts of a company = managng the expressve donaton maretpace; but, aternatvey, the surpus coud be returned to the bdders, or gven to the chartes. Another objectve s tota amount donated, gven by m π c. (Here, dfferent weghts coud aso = be paced on the dfferent chartes.) Fndng the vad outcome that maxmzes the objectve s a (nontrva) computatona probem. We w refer to t as the cearng probem. The forma defnton foows. Defnton 4 (DONATION-CLEARING). We are gven a set of n bds over chartes c,c 2,...,c m. Addtonay, we are gven an objectve functon. We are ased to fnd an objectve-maxmzng vad outcome. How dffcut the DONATION-CLEARING probem s depends on the types of bds used and the anguage n whch they are expressed. Ths s the topc of the next secton. 2 In genera, the objectve functon may aso depend on the bds, but the objectve functons under consderaton n ths paper do not depend on the bds. The technques presented n ths paper w typcay generaze to objectves that tae the bds nto account drecty. 4. A SIMPLIFIED BIDDING LANGUAGE Specfyng a genera bd n our framewor (as defned above) requres beng abe to specfy an arbtrary rea-vaued functon over R m. Even f we restrcted the possbe tota payment made to each charty to the set {0,, 2,...,s}, ths woud st requre a bdder to specfy (s+) m vaues. Thus, we need a bddng anguage that w aow the bdders to at east specfy some bds more concsey. We w specfy a bddng anguage that ony represents a subset of a possbe bds, whch can be descrbed concsey. 3 To ntroduce our bddng anguage, we w frst descrbe the bddng functon as a composton of two functons; then we w outne our assumptons on each of these functons. Frst, there s a utty functon u j : R m R, specfyng how much bdder j apprecates a gven vector of tota donatons to the chartes. (Note that the way we defne a bdder s utty functon, t does not tae the payments the bdder maes nto account.) Then, there s a donaton wngness functon w j : R R, whch specfes how much bdder j s wng to pay gven her utty for the vector of donatons to the chartes. We emphasze that ths functon does not need to be near, so that uttes shoud not be thought of as expressbe n doar amounts. (Indeed, when an ndvdua s donatng to a arge charty, the reason that the ndvdua donates ony a bounded amount s typcay not decreasng margna vaue of the money gven to the charty, but rather that the margna vaue of a doar to the bdder hersef becomes arger as her budget becomes smaer.) So, we have w j(u j(π c,π c2,...,π cm )) = v j(π c,π c2,...,π cm ), and we et the bdder descrbe her functons u j and w j separatey. (She w submt these functons as her bd.) Our frst restrcton s that the utty that a bdder derves from money donated to one charty s ndependent of the amount donated to another charty. Thus, u j(π c,π c2,...,π cm )= m u j(π c ). (We observe that ths = does not mpy that the bd functon v j decomposes smary, because of the nonnearty of w j.) Furthermore, each u j must be pecewse near. An nterestng speca case whch we w study s when each u j s a ne: u j(π c )= a jπ c. Ths speca case s justfed n settngs where the scae of the donatons by the bdders s sma reatve to the amounts the chartes receve from other sources, so that the margna use of a doar to the charty s not affected by the amount gven by the bdders. The ony restrcton that we pace on the payment wngness functons w j s that they are pecewse near. One nterestng speca case s a threshod bd, where w j s a step functon: the bdder w provde t doars f her utty exceeds s, and otherwse 0. Another nterestng case s when such a bd s partay acceptabe: the bdder w provde t doars f her utty exceeds s; but f her utty s u<s, she s st wng to provde ut doars. s One mght wonder why, f we are gven the bdders utty functons, we do not smpy maxmze the sum of the uttes rather than surpus or tota donated. There are severa reasons. Frst, because affne transformatons do not affect utty functons n a fundamenta way, t woud be poss- 3 Of course, our bddng anguage can be trvay extended to aow for fuy expressve bds, by aso aowng bds from a fuy expressve bddng anguage, n addton to the bds n our bddng anguage.
be for a bdder to nfate her utty by changng ts unts, thereby mang her bd more mportant for utty maxmzaton purposes. Second, a bdder coud smpy gve a payment wngness functon that s 0 everywhere, and have her utty be taen nto account n decdng on the outcome, n spte of her not contrbutng anythng. 5. AVOIDING INDIRECT PAYMENTS In an nta mpementaton, the approach of havng donatons made out to a center, and havng a center forward these payments to chartes, may not be desrabe. Rather, t may be preferabe to have a partay decentrazed souton, where the donatng partes wrte out checs to the chartes drecty accordng to a souton prescrbed by the center. In ths scenaro, the center merey has to verfy that partes are gvng the prescrbed amounts. Advantages of ths ncude that the center can eep ts ega status mnma, as we as that we do not requre the donatng partes to trust the center to transfer ther donatons to the chartes (or requre some compcated verfcaton protoco). It s aso a step towards a fuy decentrazed souton, f ths s desrabe. To brng ths about, we can st use the approach descrbed earer. After we cear the maret n the manner descrbed before, we now the amount that each donator s supposed to gve, and the amount that each charty s supposed to receve. Then, t s straghtforward to gve some specfcaton of who shoud gve how much to whch charty, that s consstent wth that cearng. Any greedy agorthm that ncreases the cash fow from any bdder who has not yet pad enough, to any charty that has not yet receved enough, unt ether the bdder has pad enough or the charty has receved enough, w provde such a specfcaton. (A of ths s assumng that π bj = π c. In the case b j c where there s nonzero surpus, that s, π bj > π c,we b j c can dstrbute ths surpus across the bdders by not requrng them to pay the fu amount, or across the chartes by gvng them more than the souton specfes.) Nevertheess, wth ths approach, a bdder may have to wrte out a chec to a charty that she does not care for at a. (For exampe, an envronmenta actvst who was usng the system to ncrease donatons to a wdfe preservaton fund may be requred to wrte a chec to a group supportng a rght-wng potca party.) Ths s ey to ead to compants and noncompance wth the cearng. We can address ths ssue by ettng each bdder specfy expcty (before the cearng) whch chartes she woud be wng to mae a chec out to. These addtona constrants, of course, may change the optma souton. In genera, checng whether a gven centrazed souton (wth zero surpus) can be accompshed through decentrazed payments when there are such constrants can be modeed as a MAX-FLOW probem. In the MAX-FLOW nstance, there s an edge from the source node s to each bdder b j, wth a capacty of π bj (as specfed n the centrazed souton); an edge from each bdder b j to each charty c that the bdder s wng to donate money to, wth a capacty of ; and an edge from each charty c to the target node t wth capacty π c (as specfed n the centrazed souton). In the remander of ths paper, a our hardness resuts appy even to the settng where there s no constrant on whch bdders can pay to whch charty (that s, even the probem as t was specfed before ths secton s hard). We aso generaze our cearng agorthms to the partay decentrazed case wth constrants. 6. HARDNESS OF CLEARING THE MARKET In ths secton, we w show that the cearng probem s competey napproxmabe, even when every bdder s utty functon s near (wth sope 0 or n each charty s payments), each bdder cares ether about at most two chartes or about a chartes equay, and each bdder s payment wngness functon s a step functon. We w reduce from MAX2SAT (gven a formua n conjunctve norma form (where each cause has two teras) and a target number of satsfed causes T, does there exst an assgnment of truth vaues to the varabes that maes at east T causes true?), whch s NP-compete [7]. Theorem. There exsts a reducton from MAX2SAT nstances to DONATION-CLEARING nstances such that. If the MAX2SAT nstance has no souton, then the ony vad outcome s the zero outcome (no bdder pays anythng and no charty receves anythng); 2. Otherwse, there exsts a souton wth postve surpus. Addtonay, the DONATION-CLEARING nstances that we reduce to have the foowng propertes:. Every u j s a ne; that s, the utty that each bdder derves from any charty s near; 2. A the u j have sope ether 0 or ; 3. Every bdder ether has at most 2 chartes that affect her utty (wth sope ), or a chartes affect her utty (wth sope ); 4. Every bd s a threshod bd; that s, every bdder s payment wngness functon w j s a step functon. Proof. The probem s n NP because we can nondetermnstcay choose the payments to be made and receved, and chec the vadty and objectve vaue of ths outcome. In the foowng, we w represent bds as foows: ({(c,a )},s,t) ndcates that u j (π c )=a π c (ths functon s 0 for c not mentoned n the bd), and w j(u j)=t for u j s, w j(u j) = 0 otherwse. To show NP-hardness, we reduce an arbtrary MAX2SAT nstance, gven by a set of causes K = {} = {(, )} 2 over a varabe set V together wth a target number of satsfed causes T, to the foowng DONATION-CLEARING nstance. Let the set of chartes be as foows. For every tera L, there s a charty c. Then, et the set of bds be as foows. For every varabe v, there s a bd b v = ({(c +v, ), (c v, )}, 2, ). For every tera, there s 4 V a bd b =({(c,)}, 2, ). For every cause = {, } K, 2 there s a bd b =({(c,), (c 2, )}, 2, ). Fnay, 8 V K there s a snge bd that vaues a chartes equay: b 0 = ({(c, ), (c 2, ),...,(c m,)}, 2 V + T, + ). We 8 V K 4 6 V K show the two nstances are equvaent. Frst, suppose there exsts a souton to the MAX2SAT nstance. If n ths souton, s true, then et π c =2+ T ; otherwse 8 V 2 K πc = 0. Aso, the ony bds that are not accepted (meanng the threshod s not met) are the b where s fase, and the b such that both of, 2 are fase. Frst we show that no bdder whose bd s accepted pays more than she s wng to. For each b v, ether c +v or c v receves at east 2, so ths bdder s threshod has been met.
For each b, ether s fase and the bd s not accepted, or s true, c receves at east 2, and the threshod has been met. For each b, ether both of, 2 are fase and the bd s not accepted, or at east one of them (say )strue (that s, s satsfed) and c receves at east 2, and the threshod has been met. Fnay, because the tota amount receved by the T 8 V K chartes s 2 V +, b0 s threshod has aso been met. The tota amount that can be extracted from the accepted bds s at east V ( )+ V +T + + )= 4 V 8 V K 4 6 V K 2 V + T + T > 2 V +, so there s postve 8 V K 6 V K 8 V K surpus. So there exsts a souton wth postve surpus to the DONATION-CLEARING nstance. Now suppose there exsts a nonzero outcome n the DONATION-CLEARING nstance. Frst we show that t s not possbe (for any v V ) that both b +v and b v are accepted. For, ths woud requre that π c+v + π c v 4. The bds b v,b +v,b v cannot contrbute more than 3, so we need another at east. It s easy seen that for any other v, acceptng any subset of {b v,b +v,b v } woud requre that at east as much s gven to c +v and c v as can be extracted from these bds, so ths cannot hep. Fnay, a the other bds combned can contrbute at most K + + <. It foows that we can nterpret the outcome n the DONATION-CLEARING nstance 8 V K 4 6 V K as a parta assgnment of truth vaues to varabes: v s set to true f b +v s accepted, and to fase f b v s accepted. A that s eft to show s that ths parta assgnment satsfes at east T causes. Frst we show that f a cause bd b s accepted, then ether b or b 2 s accepted (and thus ether or 2 s set to true, hence s satsfed). If b s accepted, at east one of c and c 2 must be recevng at east ; wthout oss of generaty, say t s c, and say corresponds to varabe v (that s, t s +v or v). If c does not receve at east 2, b s not accepted, and t s easy to chec that the bds b v,b +v,b v contrbute (at east) ess than s pad to c +v and c +v. But ths s the same stuaton that we anayzed before, and we now t s mpossbe. A that remans to show s that at east T cause bds are accepted. We now show that b 0 s accepted. Suppose t s not; then one of the b v must be accepted. (The souton s nonzero by assumpton; f ony some b are accepted, the tota payment from these bds s at most K <, whch s not 8 V K enough for any bd to be accepted; and f one of the b s accepted, then the threshod for the correspondng b v s aso reached.) For ths v, b v,b +v,b v contrbute (at east) ess than the tota payments to c 4 V +v and c v. Agan, the other b v and b cannot (by themseves) hep to cose ths gap; and the b can contrbute at most K <. 8 V K 4 V It foows that b 0 s accepted. Now, n order for b 0 to be accepted, a tota of 2 V + T 8 V K must be donated. Because s not possbe (for any v V ) that both b +v and b v are accepted, t foows that the tota payment by the b v and the b can be at most 2 V. 4 Addng b 0 s payment of + to ths, we st need 4 6 V K T 2 from the b 8 V K. But each one of them contrbutes at most, so at east T of them must be accepted. 8 V K Coroary. Uness P=NP, there s no poynoma-tme agorthm for approxmatng DONATION-CLEARING (wth ether the surpus or the tota amount donated as the objectve) wthn any rato f(n), where f s a nonzero functon of the sze of the nstance. Ths hods even f the DONATION- CLEARING structures satsfy a the propertes gven n Theorem. Proof. Suppose we had such a poynoma tme agorthm, and apped t to the DONATION-CLEARING nstances that were reduced from MAX2SAT nstances n Theorem. It woud return a nonzero souton when the MAX2SAT nstance has a souton, and a zero souton otherwse. So we can decde whether arbtrary MAX2SAT nstances are satsfabe ths way, and t woud foow that P=NP. (Sovng the probem to optmaty s NP-compete n many other (noncomparabe or even more restrcted) settngs as we we omt such resuts because of space constrant.) Ths shoud not be nterpreted to mean that our approach s nfeasbe. Frst, as we w show, there are very expressve fames of bds for whch the probem s sovabe n poynoma tme. Second, NP-competeness s often overcome n practce (especay when the staes are hgh). For nstance, even though the probem of cearng combnatora auctons s NP-compete [20] (even to approxmate [2]), they are typcay soved to optmaty n practce. 7. MIXED INTEGER PROGRAMMING FORMULATION In ths secton, we gve a mxed nteger programmng (MIP) formuaton for the genera probem. We aso dscuss n whch speca cases ths formuaton reduces to a near programmng (LP) formuaton. In such cases, the probem s sovabe n poynoma tme, because near programs can be soved n poynoma tme []. The varabes of the MIP defnng the fna outcome are the payments made to the chartes, denoted by π c, and the payments extracted from the bdders, π bj. In the case where we try to avod drect payments and et the bdders pay the chartes drecty, we add varabes π c,b j ndcatng how much b j pays to c, wth the constrants that for each c, π c π c,b j ; and for each b j, π bj π c,b j. Add- b j c tonay, there s a constrant π c,b j = 0 whenever bdder b j s unwng to pay charty c. The rest of the MIP can be phrased n terms of the π c and π bj. The objectves we have dscussed earer are both near: surpus s gven by n π bj m π c, and tota amount donated j= = s gven by m π c (coeffcents can be added to rep- = resent dfferent weghts on the dfferent chartes n the objectve). The constrant that the outcome shoud be vad (no defct) s gven smpy by: n m π c. π bj j= = For every bdder, for every charty, we defne an addtona utty varabe u j ndcatng the utty that ths bdder derves from the payment to ths charty. The bdder s tota
utty s gven by another varabe u j, wth the constrant that u j = m u j. = Each u j s gven as a functon of π c by the (pecewse near) functon provded by the bdder. In order to represent ths functon n the MIP formuaton, we w merey pace upper boundng constrants on u j, so that t cannot exceed the gven functons. The MIP sover can then push the u j varabes a the way up to the constrant, n order to extract as much payment from ths bdder as possbe. In the case where the u j are concave, ths s easy: f (s,t ) and (s +,t + ) are endponts of a fnte near segment n the functon, we add the constrant that u j t + πc s s + s (t + t ). If the fna (nfnte) segment starts at (s,t ) and has sope d, we add the constrant that u j t + d(π c s ). Usng the fact that the functon s concave, for each vaue of π c, the tghtest upper bound on u j s the one correspondng to the segment above that vaue of π c, and therefore these constrants are suffcent to force the correct vaue of u j. When the functon s not concave, we requre (for the frst tme) some bnary varabes. Frst, we defne another pont on the functon: (s +,t + )=(s +M,t + dm), where d s the sope of the nfnte segment and M s any upper bound on the π cj. Ths has the effect that we w never be on the nfnte segment agan. Now, et x,j be an ndcator varabe that shoud be f π c s beow the th segment of the functon, and 0 otherwse. To effect ths, frst add a constrant x,j =. Now, we am to represent π c as a =0 weghted average of ts two neghborng s,j. For 0 +, et λ,j be the weght on s,j. We add the constrant + =0 λ,j λ,j =. Aso, for 0 +, we add the constrant x +x (where x and x + are defned to be zero), so that ndeed ony the two neghborng s,j have nonzero weght. Now we add the constrant π c = + =0 s,j λ,j, and now the λ,j must be set correcty. Then, we can set u j = + t,j λ,j. (Ths s a standard MIP technque [6].) =0 Fnay, each π bj s bounded by a functon of u j by the (pecewse near) functon provded by the bdder (w j). Representng ths functon s entrey anaogous to how we represented u j as a functon of π c. (Agan we w need bnary varabes ony f the functon s not concave.) Because we ony use bnary varabes when ether a utty functon u j or a payment wngness functon w j s not concave, t foows that f a of these are concave, our MIP formuaton s smpy a near program whch can be soved n poynoma tme. Thus: Theorem 2. If a functons u j and w j are concave (and pecewse near), the DONATION-CLEARING probem can be soved n poynoma tme usng near programmng. Even f some of these functons are not concave, we can smpy repace each such functon by the smaest upper boundng concave functon, and use the near programmng formuaton to obtan an upper bound on the objectve whch may be usefu n a search formuaton of the genera probem. 8. WHY ONE CANNOT DO MUCH BETTER THAN LINEAR PROGRAMMING One may wonder f, for the speca cases of the DONATION- CLEARING probem that can be soved n poynoma tme wth near programmng, there exst speca purpose agorthms that are much faster than near programmng agorthms. In ths secton, we show that ths s not the case. We gve a reducton from (the decson varant of) the genera near programmng probem to (the decson varant of) a speca case of the DONATION-CLEARING probem (whch can be soved n poynoma tme usng near programmng). (The decson varant of an optmzaton probem ass the bnary queston: Can the objectve vaue exceed o? ) Thus, any speca-purpose agorthm for sovng the decson varant of ths speca case of the DONATION- CLEARING probem coud be used to sove a decson queston about an arbtrary near program just as fast. (And thus, f we are wng to ca the agorthm a ogarthmc number of tmes, we can sove the optmzaton verson of the near program.) We frst observe that for near programmng, a decson queston about the objectve can smpy be phrased as another constrant n the LP (forcng the objectve to exceed the gven vaue); then, the orgna decson queston concdes wth asng whether the resutng near program has a feasbe souton. Theorem 3. The queston of whether an LP (gven by a set of near constrants 4 ) has a feasbe souton can be modeed as a DONATION-CLEARING nstance wth payment maxmzaton as the objectve, wth 2v chartes and v + c bds (where v s the number of varabes n the LP, and c s the number of constrants). In ths mode, each bd b j has ony near u j functons, and s a partay acceptabe threshod bd (w j(u) =t j for u s j, otherwse w j(u) = ut j s j ). The v bds correspondng to the varabes menton ony two chartes each; the c bds correspondng to the constrants menton ony two tmes the number of varabes n the correspondng constrant. Proof. For every varabe x n the LP, et there be two chartes, c +x and c x. Let H be some number such that f there s a feasbe souton to the LP, there s one n whch every varabe has absoute vaue at most H. In the foowng, we w represent bds as foows: ({(c,a )},s,t) ndcates that u j (π c )=a π c (ths functon s 0 for c not mentoned n the bd), and w j(u j)=t for u j s, w j(u j)= u jt otherwse. s For every varabe x n the LP, et there be a bd b x = ({(c +x, ), (c x, )}, 2H, 2H c ). For every constrant v x sj n the near program, et there be a bd bj = r j ({(c x,r j )} :r j >0 {(c+x, rj )} :r j <0,( r j )H sj,). Let the target tota amount donated be 2vH. Suppose there s a feasbe souton (x,x 2,...,x v)tothe LP. Wthout oss of generaty, we can suppose that x H for a. Then, n the DONATION-CLEARING nstance, 4 These constrants must ncude bounds on the varabes (ncudng nonnegatvty bounds), f any.
for every, et π c+x = H + x, and et π c x = H x (for a tota payment of 2H to these two chartes). Ths aows us to extract the maxmum payment from the bds b x a tota payment of 2vH c. Addtonay, the utty of bdder b j s now r j (H x )+ r j (H+x )= :r j >0 :r j <0 ( r j )H r j x ( r j )H sj (where the ast nequaty stems from the fact that constrant j must be sat- sfed n the LP souton), so t foows we can extract the maxmum payment from a the bdders b j, for a tota payment of c. It foows that we can extract the requred 2vH payment from the bdders, and there exsts a souton to the DONATION-CLEARING nstance wth a tota amount donated of at east 2vH. Now suppose there s a souton to the DONATION- CLEARING nstance wth a tota amount donated of at east vh. Then the maxmum payment must be extracted from each bdder. From the fact that the maxmum payment must be extracted from each bdder b x, t foows that for each, π c+x + π c x 2H. Because the maxmum extractabe tota payment s 2vH, t foows that for each, π c+x + π c x =2H. Let x = π c+x H = H π c x. Then, from the fact that the maxmum payment must be extracted from each bdder b j, t foows that ( r j )H s j r j πc :r j x + r j πc >0 :r j +x = r j (H x )+ <0 :r j >0 r j (H +x ) = ( r j )H r j x. Equvaenty, :r j <0 r j x s j. It foows that the x consttute a feasbe souton to the LP. 9. QUASILINEAR BIDS Another cass of bds of nterest s the cass of quasnear bds. In a quasnear bd, the bdder s payment wngness functon s near n utty: that s, w j = u j. (Because the unts of utty are arbtrary, we may as we et them correspond exacty to unts of money so we do not need a constant mutper.) In most cases, quasnearty s an unreasonabe assumpton: for exampe, usuay bdders have a mted budget for donatons, so that the payment wngness w stop ncreasng n utty after some pont (or at east ncrease sower n the case of a softer budget constrant). Nevertheess, quasnearty may be a reasonabe assumpton n the case where the bdders are arge organzatons wth arge budgets, and the chartes are a few sma projects requrng reatvey tte money. In ths settng, once a certan sma amount has been donated to a charty, a bdder w derve no more utty from more money beng donated from that charty. Thus, the bdders w never reach a hgh enough utty for ther budget constrant (even when t s soft) to tae effect, and thus a near approxmaton of ther payment wngness functon s reasonabe. Another reason for studyng the quasnear settng s that t s the easest settng for mechansm desgn, whch we w dscuss shorty. In ths secton, we w see that the cearng probem s much easer n the case of quasnear bds. Frst, we address the case where we are tryng to maxmze surpus (whch s the most natura settng for mechansm desgn). The ey observaton here s that when bds are quasnear, the cearng probem decomposes across chartes. Lemma. Suppose a bds are quasnear, and surpus s the objectve. Then we can cear the maret optmay by cearng the maret for each charty ndvduay. That s, for each bdder b j, et π bj = π b. Then, for each charty c j c, maxmze ( b j π b j ) π c, under the constrant that for every bdder b j, π b j u j(π c ). Proof. The resutng souton s certany vad: frst of a, at east as much money s coected as s gven away, because π bj π c = π b π c = (( π b j c b j c j b ) c c j b j π c ) and the terms of ths summaton are the objectves of the ndvdua optmzaton probems, each of whch can be set at east to 0 (by settng a the varabes are set to 0), so t foows that the expresson s nonnegatve. Second, no bdder b j pays more than she s wng to, because u j π bj = u j(π c ) π b = (u c c j j(π c ) π b ) and the terms of ths c j summaton are nonnegatve by the constrants we mposed on the ndvdua optmzaton probems. A that remans to show s that the souton s optma. Because n an optma souton, we w extract as much payment from the bdders as possbe gven the π c, a we need to show s that the π c are set optmay by ths approach. Let πc be the amount pad to charty π c n some optma souton. If we change ths amount to π c and eave everythng ese unchanged, ths w ony affect the payment that we can extract from the bdders because of ths partcuar charty, and the dfference n surpus w be u j(π c ) u j(πc ) π c + πc. Ths expresson s, of b j course, 0 f π c = πc. But now notce that ths expresson s maxmzed as a functon of π c by the decomposed souton for ths charty (the terms wthout π c n them do not matter, and of course n the decomposed souton we aways set π b = u j j(π c )). It foows that f we change π c to the decomposed souton, the change n surpus w be at east 0 (and the souton w st be vad). Thus, we can change the π c one by one to the decomposed souton wthout ever osng any surpus. Theorem 4. When a bds are quasnear and surpus s the objectve, DONATION-CLEARING can be done n near tme. Proof. By Lemma, we can sove the probem separatey for each charty. For charty c, ths amounts to maxmzng ( u j(π c )) π c as a functon of π c. Because a b j ts terms are pecewse near functons, ths whoe functon s pecewse near, and must be maxmzed at one of the ponts where t s nondfferentabe. It foows that we need ony chec a the ponts at whch one of the terms s nondfferentabe. Unfortunatey, the decomposng emma does not hod for payment maxmzaton. Proposton. When the objectve s payment maxmzaton, even when bds are quasnear, the souton obtaned by decomposng the probem across chartes s n genera not optma (even wth concave bds).
Proof. Consder a snge bdder b pacng the foowng quasnear bd over two chartes c and c 2: u (π c )=2π c for 0 π c, u (π c )=2+ πc otherwse; u 2 4 (π c2 )= π c. The decomposed souton s 2 πc = 7, 3 πc 2 = 0, for a tota donaton of 7. But the souton 3 πc =,π c2 =2s aso vad, for a tota donaton of 3 > 7. 3 In fact, when payment maxmzaton s the objectve, DONATION-CLEARING remans (weay) NP-compete n genera. (In the remander of the paper, proofs are omtted because of space constrant.) Theorem 5. DONATION-CLEARING s (weay) NPcompete when payment maxmzaton s the objectve, even when every bd s concerns ony one charty (and has a stepfuncton utty functon for ths charty), and s quasnear. However, when the bds are aso concave, a smpe greedy cearng agorthm s optma. Theorem 6. Gven a DONATION-CLEARING nstance wth payment maxmzaton as the objectve where a bds are quasnear and concave, consder the foowng agorthm. Start wth π c =0for a chartes. Then, ettng d b j u j (πc ) γ c = dπ c (at nondfferentabe ponts, these dervatves shoud be taen from the rght), ncrease π c (where c arg max c γ c ), unt ether γ c s no onger the hghest (n whch case, recompute c and start ncreasng the correspondng payment), or u j = π c and γ c <. Fnay, b j c et π bj = u j. (A smar greedy agorthm wors when the objectve s surpus and the bds are quasnear and concave, wth as ony dfference that we stop ncreasng the payments as soon as γ c <.) 0. INCENTIVE COMPATIBILITY Up to ths pont, we have not dscussed the bdders ncentves for bddng any partcuar way. Specfcay, the bds may not truthfuy refect the bdders preferences over chartes because a bdder may bd strategcay, msrepresentng her preferences n order to obtan a resut that s better to hersef. Ths means the mechansm s not strategy-proof. (We w show some concrete exampes of ths shorty.) Ths s not too surprsng, because the mechansm descrbed so far s, n a sense, a frst-prce mechansm, where the mechansm w extract as much payment from a bdder as her bd aows. Such mechansms (for exampe, frst-prce auctons, where wnners pay the vaue of ther bds) are typcay not strategy-proof: f a bdder reports her true vauaton for an outcome, then f ths outcome occurs, the payment the bdder w have to mae w offset her gans from the outcome competey. Of course, we coud try to change the rues of the game whch outcome (payment vector to chartes) do we seect for whch bd vector, and whch bdder pays how much n order to mae bddng truthfuy benefca, and to mae the outcome better wth regard to the bdders true preferences. Ths s the fed of mechansm desgn. In ths secton, we w brefy dscuss the optons that mechansm desgn provdes for the expressve charty donaton probem. 0. Strategc bds under the frst-prce mechansm We frst pont out some reasons for bdders to msreport ther preferences under the frst-prce mechansm descrbed n the paper up to ths pont. Frst of a, even when there s ony one charty, t may mae sense to underbd one s true vauaton for the charty. For exampe, suppose a bdder woud e a charty to receve a certan amount x, but does not care f the charty receves more than that. Addtonay, suppose that the other bds guarantee that the charty w receve at east x no matter what bd the bdder submts (and the bdder nows ths). Then the bdder s best off not bddng at a (or submttng a utty for the charty of 0), to avod havng to mae any payment. (Ths s nown n economcs as the free rder probem [4]. Wth mutpe chartes, another nd of manpuaton may occur, where the bdder attempts to steer others payments towards her preferred charty. Suppose that there are two chartes, and three bdders. The frst bdder bds u (π c )= fπ c, u (π c ) = 0 otherwse; u 2 (π c2 )=fπ c2, u 2 (π c2 ) = 0 otherwse; and w (u )=u f u, w (u )= + 00 (u ) otherwse. The second bdder bds u 2(π c )= fπ c, u (π c ) = 0 otherwse; u 2 2(π c2 ) = 0 (aways); w 2(u 2)= 4 u2 f u2, w2(u2) = + (u2 ) otherwse. 4 00 Now, the thrd bdder s true preferences are accuratey represented 5 by the bd u 3(π c )=fπ c, u 3(π c )=0 otherwse; u 2 3(π c2 )=3fπ c2, u 2 3(π c ) = 0 otherwse; and w 3(u 3)= 3 u3 f u3, w3(u3) = + (u3 ) otherwse. Now, t s straghtforward to chec that, f the thrd 3 00 bdder bds truthfuy, regardess of whether the objectve s surpus maxmzaton or tota donated, charty w receve at east, and charty 2 w receve ess than. The same s true f bdder 3 does not pace a bd at a (as n the prevous type of manpuaton); hence bdder 2 s utty w be n ths case. But now, f bdder 3 reports u 3(π c ) = 0 everywhere; u 2 3(π c2 )=3fπ c2, u 2 3(π c2 ) = 0 otherwse (ths part of the bd s truthfu); and w 3(u 3)= u3 f u3, 3 w 3(u 3)= otherwse; then charty 2 w receve at east 3, and bdder 3 w have to pay at most. Because up to 3 ths amount of payment, one unt of money corresponds to three unts of utty to bdder 3, t foows hs utty s now at east 3 =2>. We observe that n ths case, the strategc bdder s not ony affectng how much the bdders pay, but aso how much the chartes receve. 0.2 Mechansm desgn n the quasnear settng There are four reasons why the mechansm desgn approach s ey to be most successfu n the settng of quasnear preferences. Frst, hstorcay, mechansm desgn has been been most successfu when the quasnear assumpton coud be made. Second, because of ths success, some very genera mechansms have been dscovered for the quasnear settng (for nstance, the VCG mechansms [24, 4, 0], or the dagva mechansm [6, ]) whch we coud appy drecty to the expressve charty donaton probem. Thrd, as we saw n Secton 9, the cearng probem s much easer n 5 Formay, ths means that f the bdder s forced to pay the fu amount that hs bd aows for a partcuar vector of payments to chartes, the bdder s ndfferent between ths and not partcpatng n the mechansm at a. (Compare ths to bddng truthfuy n a frst-prce aucton.)
ths settng, and thus we are ess ey to run nto computatona troube for the mechansm desgn probem. Fourth, as we w show shorty, the quasnearty assumpton n some cases aows for decomposng the mechansm desgn probem over the chartes (as t dd for the smpe cearng probem). Moreover, n the quasnear settng (une n the genera settng), t maes sense to pursue soca wefare (the sum of the uttes) as the objectve, because now ) unts of utty correspond drecty to unts of money, so that we do not have the probem of the bdders arbtrary scang ther uttes; and 2) t s no onger possbe to gve a payment wngness functon of 0 whe st affectng the donatons through a utty functon. Before presentng the decomposton resut, we ntroduce some terms from game theory. A type s a preference profe that a bdder can have and can report (thus, a type report s a bd). Incentve compatbty (IC) means that bdders are best off reportng ther preferences truthfuy; ether regardess of the others types (n domnant strateges), or n expectaton over them (n Bayes-Nash equbrum). Indvdua ratonaty (IR) means agents are at east as we off partcpatng n the mechansm as not partcpatng; ether regardess of the others types (ex-post), or n expectaton over them (ex-nterm). A mechansm s budget baanced f there s no fow of money nto or out of the system n genera (ex-post), or n expectaton over the type reports (ex-ante). A mechansm s effcent f t (aways) produces the effcent aocaton of weath to chartes. Theorem 7. Suppose a agents preferences are quasnear. Furthermore, suppose that there exsts a snge-charty mechansm M that, for a certan subcass P of (quasnear) preferences, under a gven souton concept S (mpementaton n domnant strateges or Bayes-Nash equbrum) and a gven noton of ndvdua ratonaty R (ex post, ex nterm, or none), satsfes a certan noton of budget baance (ex post, ex ante, or none), and s ex-post effcent. Then there exsts such a mechansm for any number of chartes. Two mechansms that satsfy effcency (and can n fact be apped drecty to the mutpe-charty probem wthout use of the prevous theorem) are the VCG (whch s ncentve compatbe n domnant strateges) and dagva (whch s ncentve compatbe ony n Bayes-Nash equbrum) mechansms. Each of them, however, has a drawbac that woud probaby mae t mpractca n the settng of donatons to chartes. The VCG mechansm s not budget baanced. The dagva mechansm does not satsfy ex-post ndvdua ratonaty. In the next subsecton, we w nvestgate f we can do better n the settng of donatons to chartes. 0.3 Impossbty of effcency In ths subsecton, we show that even n a very restrcted settng, and wth mnma requrements on IC and IR constrants, t s mpossbe to create a mechansm that s effcent. Theorem 8. There s no mechansm whch s ex-post budget baanced, ex-post effcent, and ex-nterm ndvduay ratona wth Bayes-Nash equbrum as the souton concept (even wth ony one charty, ony two quasnear bdders, wth dentca type dstrbutons (unform over two types, wth ether both utty functons beng step functons or both utty functons beng concave pecewse near functons)). The case of step-functons n ths theorem corresponds exacty to the case of a snge, fxed-sze, nonexcudabe pubc good (the pubc good beng that the charty receves the desred amount) for whch such an mpossbty resut s aready nown [4]. Many smar resuts are nown, probaby the most famous of whch s the Myerson-Satterthwate mpossbty resut, whch proves the mpossbty of effcent batera trade under the same requrements [5]. Theorem 7 ndcates that there s no reason to decde on donatons to mutpe chartes under a snge mechansm (rather than a separate one for each charty), when an effcent mechansm wth the desred propertes exsts for the snge-charty case. However, because under the requrements of Theorem 8, no such mechansm exsts, there may be a beneft to brngng the chartes under the same umbrea. The next proposton shows that ths s ndeed the case. Proposton 2. There exst settngs wth two chartes where there exsts no ex-post budget baanced, ex-post effcent, and ex-nterm ndvduay ratona mechansm wth Bayes-Nash equbrum as the souton concept for ether charty aone; but there exsts an ex-post budget baanced, ex-post effcent, and ex-post ndvduay ratona mechansm wth domnant strateges as the souton concept for both chartes together. (Even when the condtons are the same as n Theorem 8, apart from the fact that there are now two chartes.). CONCLUSION We ntroduced a bddng anguage for expressng very genera types of matchng offers over mutpe chartes. We formuated the correspondng cearng probem (decdng how much each bdder pays, and how much each charty receves), and showed that t s NP-compete to approxmate to any rato even n very restrcted settngs. We gave a mxed-nteger program formuaton of the cearng probem, and showed that for concave bds (where utty functons and payment wngness functon are concave), the program reduces to a near program and can hence be soved n poynoma tme. We then showed that the cearng probem for a subcass of concave bds s at east as hard as the decson varant of near programmng, suggestng that we cannot do much better than a near programmng mpementaton for such bds. Subsequenty, we showed that the cearng probem s much easer when bds are quasnear (where payment wngness functons are near) for surpus, the probem decomposes across chartes, and for payment maxmzaton, a greedy approach s optma f the bds are concave (athough ths atter probem s weay NP-compete when the bds are not concave). For the quasnear settng, we studed the mechansm desgn queston of mang the bdders report ther preferences truthfuy rather than strategcay. We showed that an ex-post effcent mechansm s mpossbe even wth ony one charty and a very restrcted cass of bds. We aso showed that even though the cearng probem decomposes over chartes n the quasnear settng, there may be benefts to nng the chartes from a mechansm desgn standpont. There are many drectons for future research. One s to bud a web-based mpementaton of the (frst-prce) mechansm proposed n ths paper. Another s to study the computatona scaabty of our MIP/LP approach. It s aso
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