Journal of Computer and System Sciences

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1 Journal of Computer and System Scences 78 (0) 4 Contents lsts avalable at ScenceDrect Journal of Computer and System Scences Computng optmal outcomes under an expressve representaton of settngs wth externaltes Vncent Contzer a,,tuomassandholm b a Department of Computer Scence, Duke Unversty, Durham, NC 7708, USA b Computer Scence Department, Carnege Mellon Unversty, Pttsburgh, PA 53, USA artcle nfo abstract Artcle hstory: Receved 3 August 009 Receved n revsed form 9 August 00 Accepted 0 February 0 Avalable onlne 5 March 0 Keywords: Expressve markets Externaltes Representaton When a decson must be made based on the preferences of multple agents, and the space of possble outcomes s combnatoral n nature, t becomes necessary to thnk about how preferences should be represented, and how ths affects the complexty of fndng an optmal (or at least a good) outcome. We study settngs wth externaltes, whereeach agent controls one or more varables, and how these varables are set affects not only the agent herself, but also potentally the other agents. For example, one agent may decde to reduce her polluton, whch wll come at ost to herself, but wll result n a beneft for all other agents. We formalze how to represent such domans and show that n a number of key specal cases, t s NP-complete to determne whether there exsts a nontrval feasble soluton (and therefore the maxmum socal welfare s completely napproxmable). However, for one mportant specal case, we gve an algorthm that converges to the soluton wth the maxmal concesson by each agent (n a lnear number of rounds for utlty functons that addtvely decompose nto pecewse constant functons). Maxmzng socal welfare, however, remans NP-hard even n ths settng. We also demonstrate a specal case that can be solved n polynomal tme usng lnear programmng. 0 Elsever Inc. All rghts reserved.. Introducton In many domans, a decson needs to be made based on the preferences of multple agents. Often, the space of possble outcomes s combnatoral n nature, so that t becomes necessary to consder how preferences should be represented, as well as to desgn algorthms for fndng an optmal (or at least a good) outcome. (For a recent overvew of such work, see [7].) Combnatoral auctons (for an overvew, see []) are ommon example. In such an aucton, there are multple tems to be allocated among the agents, so an outcome s defned by a specfcaton of whch bundle of tems each agent gets (plus, perhaps, payments to be made by or to the agents). Varants such as combnatoral reverse auctons (where the auctoneer seeks to procure a set of tems) and combnatoral exchanges (where the agents trade tems among themselves) have also receved attenton [30,38,9,3,36]. A pervasve assumpton n ths work (wth a very recent excepton [5]) has been that there are no allocatve externaltes: that s, no agent cares what happens to an tem unless that agent herself receves the tem. Ths s nsuffcent to model A short, early conference verson appeared n the proceedngs of the 005 AAAI conference. Ths work has been supported by the Natonal Scence Foundaton under grants IIS-0678, IIS , IIS-083, IIS , IIS , and CAREER IIS ; t has also been supported by two Sloan Fellowshps. * Correspondng author. E-mal addresses: contzer@cs.duke.edu (V. Contzer), sandholm@cs.cmu.edu (T. Sandholm) /$ see front matter 0 Elsever Inc. All rghts reserved. do:0.06/j.jcss

2 V. Contzer, T. Sandholm / Journal of Computer and System Scences 78 (0) 4 3 stuatons where a bdder who does not wn an tem stll cares whch other bdder wns t for example, ths may be the case f the tem s a nuclear weapon []. Recently, some work n sponsored-search auctons, where multple advertsement slots on a page are for sale, has started to consder the role of externaltes [3,,5,7,4,6,8,33,]. Here, the dea s that the attenton that the user pays to one ad can depend on whch other ads are shown smultaneously. More generally, and more closely related to ths paper, there are many mportant domans where actons taken by one agent affect many other agents. For example, f one agent takes on a task, such as buldng a brdge, many other agents may beneft from ths (and the extent of ther beneft n general depends on how the brdge s bult, for example, on how heavy a load t can support). Smlarly, f ompany reduces ts polluton, many ndvduals may beneft, even f they have nothng to do wth the goods that the company produces. An acton s effect on an otherwse unnvolved agent s utlty s commonly known as an externalty (for a dscusson, see [6]). When makng decsons based on the preferences of multple agents, externaltes must be taken nto account, so that (potentally complex) arrangements can be made that are truly to every agent s beneft. One doman n whch externaltes play a fundamental role, and that fts n the framework descrbed n ths paper, s the desgn of expressve mechansms for donatng to (say) chartable causes [9,4]. The basc dea here s as follows. If one agent donates to harty, then another agent who also cares about ths charty benefts. For that reason, t may happen that, even f each ndvdual agent does not care enough about the charty to gve money to t by herself, t s nevertheless possble that all of the agents prefer a jont arrangement n whch each agent gves ertan amount to the charty. Ths s because, thanks to the arrangement, each ndvdual agent s donaton s effectvely multpled by the number of agents. Ths opens up the possblty of mechansms that take everyone s preferences over the chartes as nput, and then determne an arrangement for how much each agent should pay. Externaltes play a fundamental role here: an agent gvng to harty mposes an externalty on the other agents who care about ths charty, and ths s why the agents can beneft from a jont arrangement. In ths paper, we study whether optmal (or at least good) outcomes can be effcently computed, under a qute general representaton of settngs wth externaltes. To our knowledge, ths s the frst such study of a general representaton of settngs wth externaltes. A common objectve s to maxmze socal welfare, whch s the sum of the agents utltes. However, n most settngs, there are constrants that must be satsfed. Typcally, there are voluntary partcpaton constrants, meanng that no agent s made worse off by partcpatng n the mechansm. Addtonally, f only the agents themselves know ther preferences, and the agents are self-nterested (the settng of mechansm desgn), then there may be ncentve compatblty constrants, meanng that no agent should be able to make herself better off by msreportng her preferences. After ntroducng our basc representaton scheme for settngs wth externaltes, we study the computatonal complexty of the followng problem: gven the agents preferences, fnd a good (f possble, an optmal) outcome that satsfes the voluntary partcpaton constrants. Ths problem s analogous to the wnner determnaton problem n combnatoral auctons and exchanges, whch conssts of fndng an optmal allocaton of the tems, gven the bds. The wnner determnaton problem n combnatoral auctons and exchanges has receved a tremendous amount of prevous attenton (for example [34, 3,35,38,5,8,39,0,0,9,4]). In ths paper, we wll mostly focus on restrcted settngs that cannot model, e.g., fully general combnatoral auctons and exchanges, so that we do not nhert all of the complextes from those settngs (whch would trvalze our results). Also, n ths frst research on the topc, we do not consder any ncentve compatblty constrants that s, we take the agents reported preferences at face value. Ths s reasonable when the agents preferences are common knowledge; when there are other reasons to beleve that the agents report ther preferences truthfully (for example, for ethcal reasons, or because the party reportng the preferences s concerned wth the global welfare rather than the agent s ndvdual utlty) ; or when we are smply nterested n fndng outcomes that are good relatve to the reported preferences (for example, because we are an optmzaton company that gets rewarded based on how good the outcomes that we produce are relatve to the reported preferences). Nevertheless, we beleve that ncentve compatblty s an mportant topc for future research, and we wll dscuss t at the end of the paper n Secton 9. As we noted, we do mpose voluntary partcpaton constrants. The rest of ths paper s organzed as follows. In Secton, we defne our representaton and the basc problems that we study under ths representaton. We show that the problem of fndng a nontrval feasble soluton s hard n a number of specal cases, ncludng when each agent controls only one varable (Secton 3); when there are only negatve externaltes and each agent controls at most two varables (Secton 4); and when there are only negatve externaltes and there are only two agents, but there s no constrant on how many varables they control (Secton 7). In Secton 5, we gve an algorthm for the case where there are only negatve externaltes and each agent controls only one varable. Under mnmal assumptons, ths algorthm fnds or converges to the feasble outcome wth the maxmal concessons by each agent; moreover, gven some addtonal assumptons (under whch the hardness results proven n other sectons stll hold), the algorthm requres only a lnear number of rounds. Nevertheless, n Secton 6, we show that fndng the socal welfare maxmzng outcome remans hard even n ths settng. Fnally, n Secton 8, we show that the socal welfare maxmzng outcome can be found n polynomal tme usng lnear programmng f all the utlty functons are pecewse lnear and concave. Very recent research has studed the relatonshp between expressveness and socal welfare []. For example, n a large organzaton, when a representatve of a department wthn the organzaton s asked what the department s needs are, t s possble that ths representatve acts n the organzaton s best nterest, rather than the department s.

3 4 V. Contzer, T. Sandholm / Journal of Computer and System Scences 78 (0) 4. Defntons We formalze the problem settng as follows. Defnton. In a settng wth externaltes: there are n agents,,...,n; each agent controls m varables x, x,...,xm R 0 ; and each agent has a utlty functon u : R M R (where M = n j= m j). Here, u (x,...,xm,...,x n,...,xm n n ) represents agent s utlty for any gven settng of all the varables. In general, one can also mpose constrants on whch values for (x,...,xm ) agent can choose, but we wll refran from dong so here. (We can effectvely exclude certan values by makng the utltes for them very negatve.) We say that the default outcome s the one where all the x j are set to 0, 3 and we requre wthout loss of generalty that all agents utltes are 0 at the default outcome. Thus, the voluntary partcpaton constrant states that every agent s utlty should be nonnegatve. Defnton. An outcome (x,...,xm,...,x n,...,xm n n ) s feasble (aka. satsfes voluntary partcpaton) fforevery, wehave u (x,...,xm,...,x n,...,xm n n ) 0. Wthout any restrctons placed on t, ths setup s very general. For nstance, we can model a (mult-tem, mult-unt) combnatoral exchange wth t. We recall that n ombnatoral exchange, each agent has an ntal endowment of a number of unts of each tem, as well as preferences over endowments (possbly ncludng tems not currently n the agent s possesson). The goal s to fnd some reallocaton of the tems (possbly together wth a specfcaton of payments to be made and receved) so that no agent s left worse off, and some objectve s maxmzed under ths constrant. We can model ths n our framework as follows: for each agent, for each tem n that agent s possesson, for each other agent, let there be a varable representng how many unts of that tem the former agent transfers to the latter agent. If payments are allowed, then we addtonally need varables representng the payment from each agent to each other agent. We note that ths framework allows for allocatve externaltes, that s, for the expresson of preferences over whch of the other agents receves a partcular tem. Of course, f the agents can have nonlnear preferences over bundles of tems (there are complementartes or substtutabltes among the tems), then, barrng some specal concse representaton, specfyng the utlty functons requres an exponental number of values. 4 We need to make some assumpton about the structure of the utlty functons f we do not want to specfy an exponental number of values. For most of ths paper, we make the followng assumpton, whch states that the effect of one varable on an agent s utlty s ndependent of the effect of another varable on that agent s utlty. We note that ths assumpton dsallows the model of ombnatoral exchange that we just gave, unless there are no complementartes or substtutabltes among the tems. Ths s not a problem nsofar as our prmary nterest here s not so much n combnatoral exchanges as t s n more natural, smpler externalty problems, such as aggregatng preferences over polluton levels. We note that ths restrcton makes the hardness results that we present later much more nterestng (wthout the restrcton, the results would have been unsurprsng gven known hardness results for combnatoral exchanges). However, for some of our postve results, we wll not actually need ths assumpton for example, for convergence results for an algorthm that we wll present. Defnton 3. u addtvely decomposes (across varables) fu (x,...,xm,...,x n,...,xm n n ) = n mk k= j= uk, j (x j k ). That s, the agent has a separate component utlty functon for each varable, and the agent s overall utlty s the sum of these components. When utlty functons addtvely decompose, we wll sometmes be nterested n the specal cases where the u k, j are step functons (denoted δ x a, whch evaluates to 0 f x < a and to otherwse), or pecewse constant functons (lnear combnatons of step functons). 5 In addton, we wll focus strctly on settngs where the hgher an agent sets her own varables, the worse t s for herself. We wll call such settngs concessons settngs. So, f the agents were to act ndependently, then each agent would selfshly set all her varables to 0 (the default outcome). 3 Ths s wthout loss of generalty because the varables x j outcome. If these changes can be both postve and negatve for some real-world varable, we can model ths wth two varables x j can be used to represent the changes n the real-world varables relatve to the default, x j, the dfference between whch represents the change n the real-world varable. 4 Thus, the fact that determnng the exstence of a nontrval feasble soluton for ombnatoral exchange s NP-complete [38] does not mply that determnng the exstence of a nontrval feasble soluton n our framework s NP-complete, because there s an exponental blowup n representaton sze. 5 For these specal cases, t may be conceptually desrable to make the domans of the varables x j paper for the sake of consstency. dscrete, but we wll refran from dong so n ths

4 V. Contzer, T. Sandholm / Journal of Computer and System Scences 78 (0) 4 5 Defnton 4. A concessons settng s a settng wth externaltes where for any (x,...,xm,...,x n,...,xm n n ) R M,forany, j m, and for any ˆx j > x j,wehaveu (x,...,xm,...,ˆx j,...,x n,...,xm n n ) u (x,...,xm,...,x j,...,x n,...,xm n n ). Thus, n oncessons settng, an agent s utlty s monotoncally weakly decreasng n that agent s own varables. In parts of ths paper, we wll be nterested n the followng addtonal assumpton, whch states that the hgher an agent sets her varables, the better t s for the others. (For nstance, the more ompany reduces ts polluton, the better t s for all others nvolved.) Defnton 5. A concessons settng has only negatve externaltes f for any (x,...,xm,...,x n,...,xm n n ) R M, for any, j m,foranyˆx j > x j, and for any k, u k(x,...,xm,...,ˆx j,...,x n,...,xm n n ) u k (x,...,xm,...,x j,...,x n,...,xm n n ). Thus, when there are only negatve externaltes, an agent s utlty s monotoncally weakly ncreasng n the other agents varables. We defne trval settngs of varables as settngs that are ndstngushable from settng them to 0. Defnton 6. The value r s trval for varable x j to 0. That s, for any x,...,xm j,...,x, x j+ xn,...,xm n n ) = u k (x,...,xm j,...,x, 0, x j+ to a trval value. f t does not matter to anyone s utlty functon whether x j s set to r or,...,xn,...,xm n n, and for any k, wehaveu k (x,...,xm j,...,x, r, x j+,...,,...,xn,...,xm n n ). A settng of all the varables s trval f each varable s set We are now ready to defne the followng two computatonal problems that we wll study. Defnton 7 (FEASIBLE-CONCESSIONS). We are gven oncessons settng. We are asked whether there exsts a nontrval feasble soluton. Defnton 8 (SW-MAXIMIZING-CONCESSIONS). We are gven oncessons settng. We are asked to fnd a feasble soluton that maxmzes socal welfare (among feasble solutons). The followng smple proposton shows that f the frst problem s hard, then the second problem s hard to approxmate to any rato. Proposton. Suppose that FEASIBLE-CONCESSIONS s NP-hard even under some constrants on the nstance (but no constrant that prohbts addng another dummy agent that derves postve utlty from any nontrval settng of the varables of the other agents). Then, t s NP-hard to approxmate SW-MAXIMIZING-CONCESSIONS to any postve rato, even under the same constrants. Proof. We reduce an arbtrary FEASIBLE-CONCESSIONS nstance to an SW-MAXIMIZING-CONCESSIONS nstance that s dentcal, except that a sngle addtonal agent has been added that derves postve utlty from any nontrval settng of the varable(s) of the other agents, and to whose varables all agents are completely ndfferent (they cannot derve any utlty from the new agent s varable(s)). If the orgnal nstance has no nontrval feasble soluton, then nether does the new nstance, and the maxmum socal welfare that can be obtaned s 0. On the other hand, f the orgnal nstance has a nontrval feasble soluton, then the new nstance has a feasble soluton wth postve socal welfare: the exact same soluton s stll feasble, and the new agent wll get postve utlty (and the others, nonnegatve utlty). It follows that any algorthm that approxmates SW-MAXIMIZING-CONCESSIONS to some postve rato wll return a socal welfare of 0 f there s no soluton to the FEASIBLE-CONCESSIONS problem nstance, and postve socal welfare f there s a soluton and thus the algorthm could be used to solve an NP-hard problem. It would appear that these problems (or, n the case of SW-MAXIMIZING-CONCESSIONS, ts decson varant) should naturally at least fall n the class NP, because an outcome should be able to serve as ertfcate. Nevertheless, we cannot say ths wthout makng some assumpton about how the utlty functons are represented. We now gve a weak suffcent condton. Defnton 9. A famly of concessons nstances has the outcomes-are-certfcates (OAC) property f: we can wthout loss of generalty restrct our attenton to a set of outcomes such that these outcomes can be represented n polynomal space, gven an outcome, we can compute each agent s utlty n polynomal tme, and gven an outcome, we can compute n polynomal tme whether ths outcome corresponds to a trval settng of the varables.

5 6 V. Contzer, T. Sandholm / Journal of Computer and System Scences 78 (0) 4 For example, when the agents utlty functons addtvely decompose, and ther utlty functons for ndvdual varables are pecewse constant functons (where the ponts of dscontnuty and the correspondng values are explctly gven, say, as ratonal numbers), the OAC property holds: we can wthout loss of generalty restrct our attenton to outcomes where each varable s set to a value at whch a dscontnuty occurs for some agent (or to 0) because any other value wll be equvalent to some such value and because these values are gven explctly n the nput, these outcomes can be represented n polynomal space. Moreover, gven an outcome, a smple lookup suffces to compute each agent s utlty for each varable. Fnally, to determne whether an outcome s trval, t suffces to check whether there exsts an agent that receves nonzero utlty from at least one of the varables values. We note that we can approxmate any contnuous functon wth a pecewse constant functon, wth the caveat that we may need an nfnte number of peces to approxmate the tal end of the functon (n practce, we can smply gnore values of the varables that are too large to occur n practcal solutons). Proposton. For any famly of concessons nstances that satsfes the OAC property, FEASIBLE-CONCESSIONS and the decson varant of SW-MAXIMIZING-CONCESSIONS (does there exst a feasble soluton wth socal welfare K?) are n NP. Proof. In each case, the outcomes to whch we can restrct our attenton wll serve as the certfcates; by assumpton, these certfcates have polynomal length. Also, by assumpton, we can determne n polynomal tme whether t s a trval outcome. To determne whether an outcome s feasble, we compute each agent s utlty (whch, by assumpton, we can do n polynomal tme), and check whether t s at least 0. Ths shows that FEASIBLE-CONCESSIONS s n NP. Moreover, because we can compute the agents utltes effcently, we can also compute the socal welfare effcently. Ths shows that the decson varant of SW-MAXIMIZING-CONCESSIONS s n NP. 3. Hardness wth postve and negatve externaltes We frst show that f we do not make the assumpton that there are only negatve externaltes, then determnng whether a nontrval feasble soluton exsts s NP-complete even when each agent controls only one varable. In ths paper, when membershp n NP s straghtforward, we just gve the hardness proof. Also, when each agent controls only one varable, no superscrpt j on the varables or the component utlty functons s necessary. Theorem. FEASIBLE-CONCESSIONS s NP-complete (assumng OAC for NP membershp), even when all utlty functons decompose addtvely (and all the components u k are step functons), and each agent controls only one varable. Proof. We reduce an arbtrary satsfablty nstance (gven by varables V and clauses C) to the followng FEASIBLE- CONCESSIONS nstance. Let the set of agents be as follows. For each varable v V,lettherebeanagenta v, controllng a sngle varable x av. Also, for every clause c C, let there be an agent, controllng a sngle varable x ac. Fnally, let there be a sngle addtonal agent a 0 controllng x a0. Let all the utlty functons decompose addtvely, as follows. (We recall that the notaton δ x a evaluates to 0 f x < a, and to otherwse.) For any v V, u a v a v (x av ) = δ xav. Foranyv V, u a 0 a v (x a0 ) = δ xa0. Foranyc C, u (x ac ) = (n(c) V )δ xac where n(c) s the number of varables that occur n c n negated form. For any c C, u a 0 (x a0 ) = ( V )δ xa0. Foranyc C and v V where +v occurs n c, u a v (x av ) = δ xav. For any c C and v V where v occurs n c, u a v (x av ) = δ xav. u a 0 a 0 (x a0 ) = C δ xa0. Foranyc C, u a 0 (x ac ) = δ xac. All the other functons are 0 everywhere. We proceed to show that the nstances are equvalent. Frst suppose there exsts a soluton to the satsfablty nstance. Then, let x av = fv s set to true n the soluton, and x av = 0fv s set to false n the soluton. Let x ac = forallc C, and let x a0 =. Then, the utlty of every a v s at least + = 0. Also, the utlty of a 0 s C + C =0. And, the utlty of every s n(c) V + V + pt(c) nt(c) = n(c) + pt(c) nt(c), where pt(c) s the number of varables that occur postvely n c and are set to true, and nt(c) s the number of varables that occur negatvely n c and are set to true. Of course, pt(c) 0 and nt(c) n(c); and f at least one of the varables that occur postvely n c s set to true, or at least one of the varables that occur negatvely n c s set to false, thenpt(c) nt(c) n(c) +, so that the utlty of s at least n(c) n(c) + = 0. But ths s always the case, because the assgnment satsfes the clause. So there exsts a soluton to the FEASIBLE-CONCESSIONS nstance. Now suppose there exsts a soluton to the FEASIBLE-CONCESSIONS nstance. If t were the case that x a0 <, then for all the a v we would have x av < (ora v would have a negatve utlty), and for all the we would have x ac < (because otherwse the hghest utlty possble for s n(c) V < 0). So the soluton would be trval. It follows that x a0. Thus, n order for a 0 to have nonnegatve utlty, t follows that for all c C, x ac. Now, let v be set to true f x av, and to false f x av <. So, the utlty of every s n(c) V + V + pt(c) nt(c) = n(c) + pt(c) nt(c). Inorderfor ths to be nonnegatve, we must have (for any c) that ether nt(c)<n(c) (at least one varable that occurs negatvely n c s set to false) orpt(c)>0 (at least one varable that occurs postvely n c s set to true). Therefore, we have a satsfyng assgnment.

6 V. Contzer, T. Sandholm / Journal of Computer and System Scences 78 (0) Hardness wth only negatve externaltes Next, we show that even f we do make the assumpton of only negatve externaltes, then determnng whether a nontrval feasble soluton exsts s stll NP-complete, even when each agent controls at most two varables. Theorem. FEASIBLE-CONCESSIONS s NP-complete (assumng OAC for NP membershp), even when there are only negatve externaltes, all utlty functons decompose addtvely (and all the components are step functons), and each agent controls at most two varables. Proof. We reduce an arbtrary satsfablty nstance to the followng FEASIBLE-CONCESSIONS nstance. Let the set of agents be as follows. For each varable v V,lettherebeanagenta v, controllng varables x + a v and x a v. Also, for every clause c C, let there be an agent, controllng a sngle varable x ac. Let all the utlty functons decompose addtvely, as follows: For any v V, u a v,+ a v (x + a v ) = C δ x + av, and ua v, a v (x a v ) = C δ x av.foranyv V and c C, u a v (x ac ) = δ xac. Foranyc C, u (x ac ) = δ xac. Foranyc C and v V where +v occurs n c, u a v,+ (x + a v ) = δ x + av ; and for any c C and v V where v occurs n c, u a v, (x a v ) = δ x av. All the other functons are 0 everywhere. We proceed to show that the nstances are equvalent. Frst suppose there exsts a soluton to the satsfablty nstance. Then, let x + a v = fv s set to true n the soluton, and x + a v = 0 otherwse; and, let x a v = fv s set to false n the soluton, and x a v = 0otherwse.Letx ac = forallc C. Then, the utlty of every a v s C + C =0. Also, the utlty of every s at least + (because all clauses are satsfed n the soluton, there s at least one +v c wth x + a v =, or at least one v c wth x a v = ). So there exsts a soluton to the FEASIBLE-CONCESSIONS nstance. Now suppose there exsts a soluton to the FEASIBLE-CONCESSIONS nstance. At least one of the x + a v or at least one of the x a v must be set nontrvally ( ), because otherwse no x ac can be set nontrvally. But ths mples that for any clause c C, x ac (for otherwse the a v wth a nontrval settng of her varables would have negatve utlty). So that none of the have nonnegatve utlty, t must be the case that for any c C, ether there s at least one +v c wth x + a v, or at least one v c wth x a v. Also, for no varable v V can t be the case that both x + a v and x a v, as ths would leave a v wth negatve utlty. But then, lettng v be set to true f x + a v, and to false otherwse, must satsfy every clause. So there exsts a soluton to the satsfablty nstance. 5. An algorthm for the case of only negatve externaltes and one varable per agent We have shown that wth both postve and negatve externaltes, fndng a nontrval feasble soluton s hard even when each agent controls only one varable; and wth only negatve externaltes, fndng a nontrval feasble soluton s hard even when each agent controls at most two varables. In ths secton we show that these results are, n a sense, tght, by gvng an algorthm for the case where there are only negatve externaltes and each agent controls only one varable. 6 Under some mnmal assumptons, ths algorthm wll return (or converge to) the maxmal feasble soluton, that s, the soluton n whch the varables are set to values that are as large as possble. Moreover, n the case of pecewse constant functons, t wll return ths soluton n a lnear number of teratons and hence n polynomal tme. (Snce our hardness results so far were for pecewse constant functons, ths mples that those results are tght.) Although the settng for ths algorthm may appear very restrcted, t stll allows for the soluton of many nterestng problems. For example, consder governments negotatng over how much to reduce ther countres carbon doxde emssons, for the purpose of reducng global warmng. As another example, consder agents negotatng over how much to reduce ther use of ommon resource such as ommuncaton network (where heavy use slows down the network). We wll not requre the assumpton of decomposng utlty functons n ths secton (except where stated). The followng clam shows the sense n whch the maxmal soluton s well defned n the settng under dscusson: there cannot be multple maxmal solutons, and under ontnuty assumpton, a maxmal soluton exsts. Lemma. In oncessons settng wth only negatve externaltes and n whch each agent controls only one varable, let x, x,...,x n and x, x,...,x n be two feasble solutons. Then max{x, x }, max{x, x },...,max{x n, x n } s also a feasble soluton. Moreover, f all the utlty functons are contnuous, then, lettng X be the set of values for x that occur n some feasble soluton, sup(x ), sup(x ),...,sup(x n ) s also a feasble soluton. Proof. For the frst clam, we need to show that every agent receves nonnegatve utlty n the proposed soluton. Suppose wthout loss of generalty that x x. Then, we have u (max{x, x }, max{x, x },...,max{x, x },...,max{x n, x n }) = u (max{x, x }, max{x, x },...,x,...,max{x n, x n }) u (x, x,...,x,...,x n ), where the nequalty stems from the fact that there are only negatve externaltes. But the last expresson s nonnegatve because the frst soluton s feasble. 6 After the conference verson of ths paper, Ghosh and Mahdan, who were at that pont not aware of ths work, ndependently dscovered effectvely the same algorthm n ther more specfc framework for mechansms for donatons to chartes [4].

7 8 V. Contzer, T. Sandholm / Journal of Computer and System Scences 78 (0) 4 For the second clam, we wll fnd a sequence of feasble solutons that converges to the proposed soluton. By contnuty, any agent s utlty at the lmt pont must be the lmt of that agent s utlty n the sequence of feasble solutons; and because these solutons are all feasble, ths lmt must be nonnegatve. For each agent, let{(x, j, x, j,...,x, j n )} j N be a sequence of feasble solutons wth lm j x, j = sup(x ). By repeated applcaton of the frst clam, we have that (for any j) max {x, j }, max {x, j },...,max {x, j n } s a feasble soluton, gvng us a new sequence of feasble solutons. Moreover, because ths new sequence domnates every one of the orgnal sequences, and for each agent there s at least one orgnal sequence where the -th element converges to sup(x ), the sequence converges to the soluton sup(x ), sup(x ),...,sup(x n ). We are now ready to present the algorthm. Frst, we gve an nformal descrpton. The algorthm proceeds n stages; n each stage, for each agent, t elmnates all the values for that agent s varable that would result n a negatve utlty for that agent regardless of how the other agents set ther varables (gven that they use values that have not yet been elmnated). Algorthm.. for := to n {. X 0 := R 0 (alternatvely, X 0 := [0, M] where M s some upper bound)} 3. t := 0 4. repeat untl (( ) X t = Xt ) { 5. t := t + 6. for := to n { 7. X t := {x X t : x X t, x X t,...,x X t, x + X t +,...,x n Xn t : u (x, x,...,x,...,x n ) 0}}} The set updates n Step 7 of the algorthm are smple to perform, because all the X t always take the form [0, r], [0, r), or R 0 (because we are n oncessons settng), and n Step 7 t never hurts to choose values for x, x,...,x, x +,...,x n that are as large as possble (because we have only negatve externaltes). Roughly, the goal of the algorthm s for sup(x t ), sup(xt ),...,sup(xt n ) to converge to the maxmal feasble soluton (that s, the feasble soluton such that all of the varables are set to values at least as large as n any other feasble soluton). We now show that the algorthm s sound, n the sense that t does not elmnate values of the x that occur n feasble solutons. Lemma. Suppose we are runnng Algorthm n oncessons settng wth only negatve externaltes where each agent controls only one varable. If for some t, r / X t, then there s no feasble soluton wth x set to r. Proof. We wll prove ths by nducton on t. Fort = 0 ths s vacuously true. Now suppose we have proved t true for t = k; we wll prove t true for t = k +. By the nducton assumpton, all feasble solutons le wthn X k Xk n.but f r X k+, ths means exactly that there s no feasble soluton n X k Xk n wth x = r. It follows there s no feasble soluton wth x = r at all. However, the algorthm s not complete, n the sense that (for some unnatural functons) t does not elmnate all the values of the x that do not occur n feasble solutons. Proposton 3. Suppose we are runnng Algorthm n oncessons settng wth only negatve externaltes where each agent controls only one varable. For some (dscontnuous) utlty functons (even ones that decompose addtvely), the algorthm wll termnate wth nontrval X t even though the only feasble soluton s the zero soluton. Proof. Consder the followng symmetrc example: u (x ) = x for x <, u (x ) = otherwse; u (x ) = (x ) for x <, u (x ) = otherwse; u (x ) = (x ) for x <, u (x ) = otherwse; u (x ) = x for x <, u (x ) = otherwse. There s no feasble soluton wth x orx, because the correspondng agent s utlty would defntely be negatve. In order for agent to have nonnegatve utlty we must have (x ) x. Unless they are both zero, ths mples x > x. Smlarly, n order for agent to have nonnegatve utlty we must have (x ) x, and unless they are both zero, ths mples x > x. It follows that the only feasble soluton s the zero soluton. Unfortunately, n the algorthm, we frst get X = X =[0, ); thenalso,wegetx = X =[0, ) (for any x <, we can set x = x < and agent wll get utlty 0, and smlarly for agent ). So the algorthm termnates wth nontrval X t.

8 V. Contzer, T. Sandholm / Journal of Computer and System Scences 78 (0) 4 9 However, f we make some reasonable assumptons on the utlty functons (specfcally, that they are ether contnuous or pecewse constant), then the algorthm s complete, n the sense that t wll (eventually) remove any values of the x that are too large to occur n any feasble soluton. Thus, the algorthm converges to the maxmal feasble soluton. (Ths does not mean that t necessarly termnates, and as a result we cannot gve a runtme for the algorthm n the contnuous case.) We wll present the case of contnuous utlty functons frst. Theorem 3. Suppose we are runnng Algorthm n oncessons settng wth only negatve externaltes where each agent controls only one varable. Suppose that all the utlty functons are contnuous. Also, suppose that all the X 0 are ntalzed to [0, M]. Then, all the X t are closed sets. Moreover, f the algorthm termnates after the t-th teraton of the repeat loop, then sup(xt ), sup(xt ),...,sup(xt n ) s the maxmal feasble soluton. If the algorthm does not termnate, then lm t sup(x t ), lm t sup(x t ),...,lm t sup(x t n ) s the maxmal feasble soluton. Proof. Frst we show that all the X t are closed sets, by nducton on t. Fort = 0, the clam s true, because [0, M] s a closed set. Now suppose they are all closed for t = k; we wll show them to be closed for t = k +. In the step n the algorthm n whch we set X k+, n the choce of x,...,x, x +,...,x n, we may as well always set each of these x j to sup(x k j ) (whch s nsde Xk because X k s closed by the nducton assumpton), because ths wll maxmze agent s utlty. j j It follows that X k+ ={x : u (sup(x k ),...,sup(xk ), x, sup(x k + ),...,sup(xk n )) 0}. But because u s contnuous, ths set must be closed by elementary results from analyss. Now we proceed to show the second clam. Because each X t s closed, t follows that sup(x t ) Xt. Ths mples that, for every agent, there exst x X t, x X t,...,x X t, x + X t +,...,x n Xn t such that u (x, x,...,sup(x t ),...,x n) 0. Because for every agent, X t = X t (the algorthm termnated), ths s equvalent to sayng that there exst x X t, x X t,...,x X t, x + X t +,...,x n Xn t such that u (x, x,...,sup(x t ),...,x n) 0. Of course, for each of these x, we have x sup(x t ). Because there are only negatve externaltes, t follows that u (sup(x t ), sup(xt ),...,sup(xt ),...,sup(xt n )) u (x, x,...,sup(x t ),...,x n) 0. Thus, sup(x t ), sup(xt ),...,sup(xt n ) s feasble. It s also maxmal by Lemma. Fnally, we prove the thrd clam. For any agent, for any t, we have u (sup(x t ), sup(x t ),...,lm t sup(x t ),...,sup(xn t )) u (sup(x t ), sup(x t ),...,sup(x t ),...,sup(xt n )) (because the X t are decreasng n t, and we are n oncessons settng). The last expresson evaluates to a nonnegatve quantty, usng the same reasonng as n the proof of the second clam wth the fact that sup(x t ) Xt. But then, by contnuty, 0 lm t (u (sup(x t ), sup(x t ),..., lm t sup(x t ),...,sup(xt n ))) = u (lm t sup(x t ), lm t sup(x t ),...,lm t sup(x t ),...,lm t sup(xn t )) = u (lm t sup(x t ), lm t sup(x t ),..., lm t sup(x t ),..., lm t sup(xn t )). It follows that lm t sup(x t ), lm t sup(x t ),...,lm t sup(xn t ) s feasble. It s also maxmal by Lemma. We observe that pecewse constant functons are not contnuous, and thus Theorem 3 does not apply to the case where the utlty functons decompose addtvely and the component utlty functons are pecewse constant. Nevertheless, the algorthm works on such utlty functons, and we can even prove that the number of teratons s lnear n the number of peces. There s one caveat: the way we have defned pecewse constant functons (as lnear combnatons of step functons δ x a ), the maxmal feasble soluton s not well defned (the set of feasble ponts s never closed on the rght, that s, t does not nclude ts least upper bound). To remedy ths, call a feasble soluton quas-maxmal f there s no feasble soluton that s larger (that s, all the x are set to values that are at least as large) and that gves some agent a larger utlty. Hence, a quas-maxmal feasble soluton s maxmal for all ntents and purposes. Theorem 4. Suppose we are runnng Algorthm n oncessons settng wth only negatve externaltes where each agent controls only one varable. If all the utlty functons decompose addtvely and all the components u k are pecewse constant wth fntely many steps (the range of the u k s fnte), then the algorthm wll termnate after at most T teratons of the repeat loop, where T s the total number of steps n all the self-components u (that s, the sum of the szes of the ranges of these functons). Moreover, f the algorthm termnates after the t-th teraton of the repeat loop, then any soluton (x, x,...,x n ) wth for all, x arg max x X t j u j (x ),s a quas-maxmal feasble soluton. Proof. If for some and t, X t X t, t must be the case that for some value r ntherangeofu,thepremageofths value s n X t X t (t has just been elmnated from consderaton). Informally, one of the steps of the functon u has been elmnated from consderaton. Because ths must occur for at least one agent n every teraton of the repeat loop before termnaton, t follows that there can be at most T teratons before termnaton. Now, f the algorthm termnates after the t-th teraton of the repeat loop, and a soluton (x, x,...,x n ) wth for all, x arg max x X t j u j (x ) s chosen, t follows that each agent derves as much utlty from the other agents varables as s possble wth the sets X t (because of the assumpton of only negatve externaltes, any settng of a varable that maxmzes the total utlty for the other agents also maxmzes the utlty for each ndvdual other agent). We know that

9 0 V. Contzer, T. Sandholm / Journal of Computer and System Scences 78 (0) 4 for each agent, there s at least some settng of the other agents varables wthn the X t that wll gve agent enough j utlty to compensate for the settng of ts own varable (by the defnton of X t and usng the fact that X t j = Xt j, as the algorthm has termnated); and thus t follows that the utlty maxmzng settng s also enough to make s utlty nonnegatve. So the soluton s feasble. It s also quas-maxmal by Lemma. Algorthm can be extended to cases where some agents control multple varables, by nterpretng x n the algorthm as the vector of agent s varables (and ntalzng the X 0 as Cartesan products of sets). However, the next proposton shows that ths extenson of Algorthm fals. (Ths s perhaps not surprsng n lght of our earler hardness result, Theorem, buttsstllnstructvetoseeexactlyhow t fals.) Proposton 4. Suppose we are runnng the extenson of Algorthm just descrbed n oncessons settng wth only negatve externaltes. When some agents control more than one varable, the algorthm may termnate wth nontrval X t even though the only feasble solutons are trval solutons, even when all of the utlty functons decompose addtvely and all of the components u k, j are step functons (or contnuous functons). Proof. Let each of three agents control two varables, wth utlty functons as follows. (We recall once agan that the notaton δ x a evaluates to 0 f x < a, and to otherwse.) u, (x ) = 3δ x ; u, (x ) = 3δ x ; u, (x ) = 3δ x ; u, (x ) = 3δ x ; u 3, 3 (x 3 ) = 3δ x 3 ; u 3, 3 (x 3 ) = 3δ x 3 ; u, (x ) = δ x ; u 3, (x 3 ) = δ x 3 ; u, (x ) = δ x ; u 3, (x 3 ) = δ x 3 ; u, 3 (x ) = δ x ; u, 3 (x ) = δ x. Increasng any one of the varables to a value of at least wll decrease the correspondng agent s utlty by 3, and wll rase only one other agent s utlty, by. It follows that there s no nontrval feasble soluton, because any nontrval soluton wll have negatve socal welfare (total utlty), and hence at least one agent must have negatve utlty. In the algorthm, after the frst teraton, t becomes clear that no agent can set both her varables to values of at least (because each agent can derve at most 4 < 6 utlty from the other agents varables). Nevertheless, for any agent, t stll appears possble at ths stage to set ether (but not both) of her varables to a value of at least. Unfortunately, n the next teraton, ths stll appears possble (because each of the other agents could set the varable that s benefcal to ths agent to a value of at least, leadng to a utlty of 4 > 3 for ths agent). It follows that the algorthm gets stuck wth nontrval X t. These component utlty functons are easly made contnuous, whle changng nether the algorthm s behavor on them nor the set of feasble solutons for nstance, by makng each functon lnear on the nterval [0, ]. In the next secton, we dscuss maxmzng socal welfare, under the same condtons under whch we showed Algorthm to be successful n fndng the maxmal feasble soluton (whch does not necessarly maxmze welfare). 6. Maxmzng socal welfare remans hard In oncessons settng wth only negatve externaltes where each agent controls only one varable, the algorthm we provded n the prevous secton returns the maxmal feasble soluton, n a lnear number of rounds for utlty functons that decompose addtvely nto pecewse constant functons. However, ths may not be the most desrable soluton. For nstance, we may be nterested n the feasble soluton wth the hghest socal welfare (that s, the hghest sum of the agents utltes). In ths secton we show that fndng ths soluton remans hard, even n the settng n whch Algorthm fnds the maxmal soluton fast.

10 V. Contzer, T. Sandholm / Journal of Computer and System Scences 78 (0) 4 Theorem 5. The decson varant of SW-MAXIMIZING-CONCESSIONS (does there exst a feasble soluton wth socal welfare K?) s NP-complete (assumng OAC for NP membershp), even when there are only negatve externaltes, all utlty functons decompose addtvely (and all the components u k are step functons), and each agent controls only one varable. Proof. We reduce an arbtrary EXACT-COVER-BY-3-SETS nstance (gven by a set S and subsets S, S,...,S q ( S =3) wth whch to cover S, wthout any overlap) to the followng SW-MAXIMIZING-CONCESSIONS nstance. Let the set of agents be as follows. For every S there s an agent a S. Also, for every element s S there s an agent a s.everyagentontrols a sngle varable x a. Let all the utlty functons decompose addtvely, as follows: For any S, u a S a S (x as ) = 7δ xas. ForanyS and for any s, u a s a S (x as ) = 7δ xas. Foranys, u a s a s (x as ) = δ xas. Foranys and for any S wth s S, u a S a s (x as ) = c(s) δ x as, where c(s) s the number of sets S wth s S. All the other functons are 0 everywhere. (We may assume wthout loss of generalty that c(s) : f c(s) = 0 then the orgnal problem nstance s clearly nfeasble, and f c(s) = then t s clear that the one set S that contans s must be used n the cover, and so we can reduce t to a smpler problem nstance.) Let the target socal welfare be K = 7q( S ) + 7 S. We proceed to show that the two nstances are equvalent. 3 Frst, suppose there exsts a soluton to the EXACT-COVER-BY-3-SETS nstance. Then, let x as = 0fS s n the cover, and x as = otherwse. For all s, letx as =. Then a S receves a utlty of 7 S f S s n the cover, and 7( S ) otherwse. Furthermore, for all s S, a s receves a utlty of (c(s) ) c(s) = 0 (because for exactly c(s) ofthec(s) subsets S wth s n t, the correspondng agent has her varable set to : the only excepton s the subset S that contans s and s n the cover). It follows that all the agents receve nonnegatve utlty, and the total utlty (socal welfare) s 7q( S ) + 7 S 3. So there exsts a soluton to the SW-MAXIMIZING-CONCESSIONS nstance. Now, suppose that there exsts a soluton to the SW-MAXIMIZING-CONCESSIONS nstance. We frst observe that f for some s S, x as <, the total utlty (socal welfare) can be at most 7q( S ) + S < 7q( S ) + 7 S 3 (because each a S can receve at most 7( S ), and each a s can receve at most c(s), and because c(s) ths can be at most ). So c(s) t must be the case that x as foralls S. It follows that, n order for none of these a s to have nonnegatve utlty, for every s S, there are at least c(s) subsetss wth x as and s S. In other words, for every s S, there s at most one subset S wth s S wth x as <. In other words agan, the subsets S wth x as < are dsjont (and so there are at most S S of them). However, f there were only k 3 3 subsetss wth x as <, then the total utlty (socal welfare) can be at most 7q( S ) + 7k + S 3k (each a S receves at least 7( S ), and they receve no more unless they are among the k, n whch case they receve an addtonal 7; and every a s receves 0 unless s s n none of the k dsjont subsets S, n whch case the agent wll receve at most ; but of course there can be at most S 3k such agents). But 7q( S )+7k + S 3k 7q( S )+ S +4( S S ) = 7q( S )+7 4, whch s less than the target. It follows there 3 3 are exactly S 3 dsjont subsets S wth x as < an exact cover. So there exsts a soluton to the EXACT-COVER-BY-3-SETS nstance. 7. Hardness wth only two agents So far, we have not assumed any bound on the number of agents. A natural queston to ask s whether such a bound makes the problem easer to solve. In ths secton, we show that the problem of determnng the exstence of a nontrval feasble soluton n oncessons settng wth only negatve externaltes remans NP-complete even wth only two agents (when there s no restrcton on how many varables each agent controls). Theorem 6. FEASIBLE-CONCESSIONS s NP-complete (assumng OAC for NP membershp), even when there are only two agents, there are only negatve externaltes, and all utlty functons decompose addtvely (and all the components u k, j are step functons). Proof. We reduce an arbtrary KNAPSACK nstance (gven by r pars (c, v ), where all the c and v are postve; ost constrant C; and a value objectve V ) to the followng FEASIBLE-CONCESSIONS nstance wth two agents. Agent controls only one varable, x. Agent controls r varables, x, x,...,xr. Agent s utlty functon s u (x, x, x,...,xr ) = V δ x + r j= v jδ j x. Agent s utlty functon s u (x, x, x,...,xr ) = Cδ x r j= c jδ j x. We proceed to show that the nstances are equvalent. Suppose there s a soluton to the KNAPSACK nstance, that s, a subset S such that j S c C and j S v V. Then, let x =, and for any j r, letx j = δ j S (that s, x j = f j S, and x j = 0 otherwse). Then u (x, x, x,...,xr ) = V + j S v j 0. Also, u (x, x, x,...,xr ) = C j S c j 0. So there s a soluton to the FEASIBLE-CONCESSIONS nstance. Now suppose there s a soluton to the FEASIBLE-CONCESSIONS nstance, that s, a nontrval settng of the varables (x, x, x,...,xr ) such that u (x, x, x,...,xr ) 0 and u (x, x, x,...,xr ) 0. If t were the case that x <, then ether all of agent s varables are set to values smaller than (n whch case we have a trval soluton), or at least one of agent s varables s set to a nontrval value (n whch case agent gets negatve utlty because the settng of x s worthless to hm). It follows that x. Thus, n order for agent to get nonnegatve utlty, we must have r j= v jδ j x V.Let

11 V. Contzer, T. Sandholm / Journal of Computer and System Scences 78 (0) 4 S ={j: x j }. Then t follows that j S v j V. Also, n order for agent to get nonnegatve utlty, we must have j S c j = r j= c jδ j x C. So there s a soluton to the KNAPSACK nstance. 8. A specal case that can be solved to optmalty usng lnear programmng Fnally, n ths secton, we demonstrate a specal case n whch we can fnd the feasble outcome that maxmzes socal welfare (or other lnear objectves) n polynomal tme, usng lnear programmng. For the sake of makng thngs defnte and smple, we assume that all the components of the utlty functons are represented as pecewse lnear functons (consstng of a fnte number of segments). Crucally, we assume that these functons are concave. For ths result we wll need no addtonal assumptons (no bounds on the number of agents or varables per agent, etc.). Theorem 7. If all of the utlty functons decompose addtvely, and all of the components u k, j are pecewse lnear (consstng of a fnte number of segments) and concave, then SW-MAXIMIZING-CONCESSIONS can be solved n polynomal tme usng lnear programmng. Proof. Let the varables of the lnear program be the x j j and the uk, k, each of whch s a sngle-dmensonal real-valued varable. (We wrte them n bold font to ndcate that they are varables n the lnear program; n partcular, t s mportant to dstngush the varable u k, j from the functon u k, j ( ): the latter s part of the nput.) Of course, to obtan a sensble soluton, thevaluesoftheu k, j should be determned by the values of the x j j, namely, uk, = u k, j (x j ) should hold. Ths s not a lnear k k constrant, but we can capture t wth ollecton of lnear constrants, usng a standard lnear programmng trck. Before we do so, we note that the objectve (socal welfare) can be wrtten as n = n k= mk j= uk, j Also, the feasblty (voluntary partcpaton) constrants can be wrtten as: for any, werequre n k= these constrants are lnear n the varables. All that remans to do s to make sure that the u k, j the x j j varables. Due to the objectve of maxmzng socal welfare, a solver wll always set the uk, k, whch s lnear n the varables. mk j= uk, j 0. Agan, varables take the correct values wth respect to the values of varables to values that are as hgh as possble, so we only need to make sure that they are not set too hgh. That s, we only need to add constrants to ensure that u k, j u k, j (x j j ). Here, we wll use the fact that the functons uk, ( ) are pecewse lnear and k concave. Consder a lnear segment of the functon u k, j ( ); letl( ) be the lnear functon that concdes wth ths segment. Because u k, j ( ) s concave, the nonlnear constrant u k, j u k, j (x j j ) mples the lnear constrant uk, l(x j k k ).Conversely,f we add all such lnear constrants (one for each of the fntely many segments), collectvely they wll mply the nonlnear constrant u k, j u k, j (x j ). Hence, we can replace each of the nonlnear constrants wth an equvalent collecton of lnear k constrants. These constrants complete the lnear program. 9. Conclusons and future research In combnatoral auctons and smlar settngs, a no-externaltes assumpton s commonly made. However, more recently, externaltes have been recevng more attenton. For example, n sponsored-search auctons, the attenton that one wnnng advertser gets from the user n general depends on whch other advertsers have won a slot. In other settngs, externaltes play an even greater role. Novel mechansms for determnng how much each agent should gve to certan chartable causes [9,4] rely on the fact that an agent may be wllng to gve more f ths nduces others to gve more as well. Thus, these mechansms fundamentally rely on an externalty namely, that one agent derves utlty from seeng another agent gve money to harty. More generally, t s clear that falng to model externaltes comes at a sgnfcant cost n welfare n many settngs. For nstance, when an agent s decdng whether to buld a publc good such as a brdge, many other agents may be affected by ths decson, as they could make use of the brdge. As another example, ompany settng ts polluton level may affect the health and safety of many. Ths paper, to our knowledge, s the frst that consders a general representaton of settngs wth externaltes and studes the problem of computng good or even optmal outcomes wthn ths framework. We showed that when both postve and negatve externaltes occur, determnng whether a nontrval feasble soluton exsts s NP-complete even when each agent controls only one varable and all the utlty functons decompose addtvely nto step functons. We then showed that wth only negatve externaltes, determnng whether a nontrval feasble soluton exsts s NP-complete even when each agent controls at most two varables and all the utlty functons decompose addtvely nto step functons. We then gave an algorthm for the case where there are only negatve externaltes and each agent controls only one varable, ntended to fnd the feasble soluton wth the varables set to values that are as large as possble. We showed that, although the algorthm may fal wth certan dscontnuous utlty functons, t ether termnates at or converges to the maxmal soluton wth contnuous utlty functons; and for the case where the utlty functons decompose addtvely nto pecewse constant functons, t always termnates correctly, n a lnear number of rounds. We also showed why the natural generalzaton of the algorthm to cases where some agents control more than one varable may fal even when all the utlty functons decompose addtvely nto step functons or contnuous functons. We then showed that