How To Find Out If An Equal Cost Sharing Mechanism Is A Maximal Welfare Loss Minimizer

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1 On Equal Cost Sharing in the Provision of an Excludable Public Good JORDI MASSÓ ANTONIO NICOLÒ ARUNAVA SEN TRIDIB SHARMA LEVENT ÜLKÜ June 0 Abstract: We study the e ciency and fairness roerties of the equal cost sharing mechanism in the rovision of a binary and excludable ublic good. According to the maximal welfare loss criterion, equal cost sharing is otimal within the class of strategyroof, individually rational and no-budgetde cit mechanisms only when there are agents. In general the equal cost Acknowledgements: We thank María Angeles de Frutos, Rajat Deb, Emilio Gutiérrez, Rich McLean, Hervé Moulin, and William Thomson for their comments. We are esecially grateful to Shinji Ohseto for his comments and helful suggestions. We are grateful to an Associate Editor and three referees of this journal for many invaluable comments and suggestions. Jordi Massó acknowledges nancial suort from the Sanish Ministry of Economy and Cometitiveness, through the Severo Ochoa Programme for Centers of Excellence in R&D (SEV ) and FEDER grant ECO (Gruo Consilidado-C), and from the Generalitat de Catalunya, through the rize "ICREA Academia" for excellence in research and grant SGR He also acknowledges the suort of MOVE, where he is an a liated researcher. Antonio Nicolò s work is artially suorted by the roject "Intelligent Preference reasoning for multi-agent decision making" (Univ. of Padua). Jordi Massó is at the Universitat Autònoma de Barcelona and Barcelona GSE. Deartament d Economia i Història Economica, Edi ci B, UAB. 089 Bellaterra (Barcelona), Sain. jordi.masso@uab.es. Antonio Nicolò is at the Diartimento di Scienze Economiche. Università degli Studi di Padova. Via del Santo. 5, Padova, Italy. antonio.nicolo@unid.it. Arunava Sen is at the Indian Statistical Institute. 7, S.J.S. Sansanwal Marg, New Delhi 006, India. asen@isid.ac.in Tridib Sharma is at the Centro de Investigación Económica, ITAM, Av. Santa Teresa No. 90 Col. Héroes de Padierna, C.P Mexico D.F. sharma@itam.mx Levent Ülkü is at the Centro de Investigación Económica, ITAM, Av. Santa Teresa No. 90 Col. Héroes de Padierna, C.P Mexico D.F. levent.ulku@itam.mx.

2 sharing mechanism is no longer otimal in this class: we rovide a class of mechanisms obtained by symmetric erturbations of equal cost sharing with strictly lower maximal welfare loss. We show that if one of two ossible fairness conditions is additionally imosed, equal cost sharing mechanism regains otimality. JEL Classi cation: D7 Keywords: Excludable ublic good, Equal cost sharing, Maximal welfare loss

3 Introduction An excludable and non-rivalrous binary ublic good is a good that can be used by several agents in a grou, with the ossibility of excluding some agents from its consumtion. However once an agent gains access to the ublic good his valuation is indeendent of who else has access to it. A grou of agents, each rivately informed about his valuation of the good, has to jointly decide on its rovisioning. The good costs a xed amount to roduce indeendent of the number of users. Two decisions have to be made: (i) the set of users if the ublic good is rovided, and (ii) the list of agents contributions (or individual rices). An examle of the kind of roblem we have in mind is the rovision of online classes. Imagine a world-renowned scientist who is o ered to teach several advanced online lectures in his area of exertise. The lectures would be recorded and students would be able to follow them at any time and any lace. Of course, one would rst need to enroll by aying a rice to be able to virtually attend these lectures, which takes the form of, say, visiting a website and entering an individualized assword. In this sense exclusion is feasible and, erhas more imortantly, imlementable. As imortantly, an enrolled student s value does not deend on who else has access to the lectures, hence there are no allocation externalities. In other words, following the lectures does not entail rivalry in consumtion. Of course teaching takes time and its cost on the lecturer would tyically deend on the size of his class in a lecture hall. But in the online context the time value of the lecturer would be indeendent of who he is virtually addressing, as all he would have to do would be to lecture in front of a camera. Hence his comensation could reasonably be designed indeendent of the users of the service he rovides. Now (i) who among the grand set of otential students should have access to these online lectures, and (ii) how should they share the comensation of the lecturer? These decisions will tyically deend on agents valuations; however, since these are rivate information they have to be elicited from the agents. In other words, the mechanism that mas ro les of valuations into allocations or decisions must be incentive comatible. We imose the requirement of strategyroofness that guarantees that no agent can gain by misreresenting For instance, Deb and Razzolini (999a, 999b), Dobzinski et al. (008), Moulin and Shenker (00), Mutuswami (005, 008), Osheto (000, 005), Olszewski (004), and Yu (007).

4 his true valuation irresective of his beliefs about the valuations of other agents. A second basic requirement is that the mechanism be individually rational, i.e. an agent who is a user cannot be charged more than his valuation while a non-user cannot make a ositive ayment. This requirement ensures that all agents articiate voluntarily in the decision-making rocess. Finally, we shall require that there be no-budget-de cit, i.e. agents contributions should cover the cost of rovisioning the ublic good. A simle and attractive mechanism that can be used to rovide an excludable binary ublic good is the equal cost sharing mechanism. It can be imlemented by auction-like indirect mechanisms. 4 At every ro le of valuations, the equal cost sharing mechanism selects the allocation that maximizes (in the set-inclusion sense) the grou of users subject to the following requirements: (i) each user s contribution is the cost of rovision divided by the number of users, (ii) this contribution is no greater than his valuation, and (iii) all non-users ay zero. It is easy to verify that this mechanism is well-de ned at every ro le and satis es strategy-roofness, individual rationality and no-budget-de cit. Our goal, in this aer, is to throw additional light on this mechanism. In articular, we investigate notions of e ciency and fairness associated with this mechanism. It is well-known that strategyroofness, individual rationality and nobudget-de cit are incomatible with e ciency. Since there is no rivalry in consumtion, excluding an agent with a strictly ositive valuation is not ef- cient when aggregate valuation is above cost. Since individual rationality requires that the contribution aid by each agent is not larger than his valuation of the ublic good, agents will have incentives to mis-reort their own valuation in order to lower their contribution. As a way out one could reduce rices, ossibly all the way to zero. But then one would violate the requirement of no-de cit. In view of this incomatibility, a second-best aroach is increasingly being adoted in the literature on mechanism design. This aroach identi es a mechanism that minimizes the maximal welfare loss in the class of all strategyroof and individually rational mechanisms. 5 4 See Deb and Razzolini (999a, 999b). 5 The worst-case welfare objective function is a well-established and widely-used criterion. For alications in related areas, see Moulin and Shenker (00), Moulin (008), and Juarez (008a, 008b) in the context of ublic good rovision. See also Koutsouias and Paadimitriou (999), Roughgarden (00), and Roughgarden and Tardos (00) in the comuter science literature on the rice of anarchy, introduced to measure the e ects of sel sh routing in a congested network. 4

5 The welfare loss of a mechanism at a ro le of valuations is the di erence between the aggregate welfare of the rst-best and the aggregate welfare of the mechanism evaluated at the ro le. The maximal welfare loss of a mechanism is the suremum, taken over all ro les of valuations, of its welfare loss. Then, each mechanism is evaluated according to its maximal welfare loss and the goal is to select a mechanism that minimizes it. Since we are interested in mechanisms satisfying individual rationality, it turns out that ine ciencies arise from the exclusion (as users) of some (or all) agents who have strictly ositive valuations. The maximal welfare loss of a mechanism is then the sum of the valuations of all non-users of the good at the reference ro le which maximizes this sum. We show that when there are two agents, the equal cost sharing mechanism minimizes maximal welfare loss in the class of all strategyroof, individually rational and no-budget-de cit mechanisms. However this result does not hold in general: we construct a class of feasible mechanisms which outerform equal cost sharing in terms of maximal welfare loss. However, the equal cost sharing mechanism satis es certain additional fairness roerties, which fail in the class we construct. We identify two such roerties which we call weak demand monotonicity and weak free entry. When there are more than two agents, these two roerties come into con ict with our second-best notion of e ciency. In the class of all strategy-roof, individually rational and no-de cit mechanisms which satisfy weak demand monotonicity or weak free entry, we show that the equal cost sharing mechanism minimizes maximal welfare loss. Thus, if one wants to imrove uon the level of "e ciency" generated by the equal cost sharing mechanism, one has to give u on certain notions of fairness. We say that a mechanism violates weak free entry if there is a tye ro le where an agent with a lower evaluation of the good is a user at the cost of the exclusion of a higher tye agent. Hence this condition is a weak notion of fairness in the sense that it is imlied by the standard conditions of free entry 6 or envyfreeness 7. Weak demand monotonicity requires that when the valuation of all agents (weakly) increases for the ublic good, the set of users should not lose any member. This notion of fairness is imlied by demand monotonicity 8. These imlications are discussed in Section 5. 6 See, for examle, Deb and Razzolini (999a, 999b). 7 See, for examle, Srumont (0). 8 See, for examle, Ohseto (000) and Deb and Razzolini (999). 5

6 Our aer comlements the closely related aer of Dobzinski et al. (008). They show that the equal cost sharing mechanism is a maximal welfare loss minimizer in the class of mechanisms that are strategyroof, budget balanced and that satisfy an axiom called equal treatment. The authors write: "An interesting research roblem is to characterize the class of mechanisms obtained by droing the (admittedly strong) equal treatment condition." We do not o er such a characterization because we do not claim that the equal cost sharing mechanism is the unique maximal welfare loss minimizer. However as mentioned earlier, in the two agent case, we show that the equal cost sharing mechanism is a maximal welfare loss minimizer in the larger class of mechanisms when the equal treatment axiom is droed and budget-balancedness is relaxed to the no-de cit condition. For more than two agents, the examle that we rovide shows that the result in Dobzinski et al. (008) does not hold when budget balancedness is relaxed to the no-budget-de cit condition (but equal treatment is maintained). The Dobzinski et al. result is imortant because budget balancedness imlies a certain notion of e ciency. Surluses generate wastage when valuable resources are not made use of. If however, surluses can be committed to other uses or layers (not under consideration), then budget balancedness may seem to be a strong restriction. On the other hand, under our fairness criteria, the budget balanced equal cost sharing mechanism is indeed a maximal welfare loss minimizer. Another related aer is Moulin and Shenker (00). They consider the rovision of a binary, excludable ublic good when the cost function is a submodular function of the set of users. They show that the mechanism associated with the Shaley value cost sharing formula (which corresonds to the equal cost sharing mechanism for the case of a binary ublic good with xed cost of rovision) is the unique mechanism that minimizes maximum welfare loss in the class of mechanisms that are de ned from a cross monotonic cost sharing method and are grou strategyroof, individually rational, non-subsidizing (the cost shares are non negative), budget balanced, and that satisfy consumer sovereignty. Cross monotonicity requires the rice aid by a user to weakly decrease when the set of users exands. 9 Grou strategyroofness is an incentive comatibility requirement when coalitions of agents are allowed to coordinate messages for mutual bene t. Consumer 9 Cross monotonicity is related to the axiom of oulation monotonicity introduced and analyzed in Thomson (98a, 98b). 6

7 sovereignty ensures that every agent has a valuation that guarantees his articiation irresective of the valuations of other agents. Thus our result alies to a more secialized setting than that in Moulin and Shenker (00) but establishes the otimality of the equal-cost sharing social choice function amongst a much broader class of mechanisms. We also note that strategyroofness is a more comelling axiom than grou strategyroofness from a decision-theoretic ersective. If agents are ignorant of the valuations of other agents, assumtions about the ability of coalitions to coordinate their messages for mutual bene t require stronger justi cation. Grou strategyroofness can also be a demanding requirement in this setting - see Juarez (008b). Model An excludable binary ublic good is to be rovided to a club of agents. The grand set of agents is N = f; :::; ng and a club is a ossibly emty subset of N. If the good is rovided to no agent, i.e., if the club is emty, no cost is incurred. Providing the good to any nonemty club generates a cost of, regardless of the size or the membershi structure of the club. Each agent i N has a tye t i which gives his value for being in a club. We assume that t i can take on any nonnegative value, in other words, the tye sace is T i = < +. A tye ro le is a vector t < n +, and is exibly denoted t = (t i ; t i ) for any i, where t i is a list of the tyes of all agents excet for i. Agents ayo s are quasilinear in money and there are no informational or allocative externalities. Hence i s ayo is determined by his own tye and ayment only. In articular it does not deend on who else might be included in the club. To be recise, suose that club S is formed, i s tye is t i and his ayment is i. Then his ayo is I i (S)t i i where I i (S) is the indicator function, i.e., I i (S) = if i S and I i (S) = 0 otherwise. A mechanism is a function m = (S m ; m ) : < n +! N < n, which, for every tye ro le t < n +, determines a (ossibly emty) club S m (t), and a ayment m i (t) for every agent i N. While refering to a articular mechanism m = (S m ; m ), we will usually dro the suerscrit m, and write m = (S; ). Furthermore we will feel free to use functions S() and f i ()g i=;:::;n in reference to a given mechanism m whenever no confusion should arise. Note that agents inside or outside the club could be making or receiving ayments. 7

8 We are interested in a domain of mechanisms which satisfy three basic feasibility requirements: strategyroofness, individual rationality and nobudget-de cit. De nition A mechanism m = (S; ) is feasible if it satis es the following three conditions.. strategyroofness (SP): for every i; t = (t i ; t i ) and t 0 i, I i (S(t))t i i (t) I i (S(t 0 i; t i ))t i i (t 0 i; t i ).. individual rationality (IR): for every i and t, I i (S(t))t i i (t) 0.. no-budget-de cit (NBD): for every t, P in i(t) whenever S(t) 6=?, and P in i(t) 0 otherwise. Let be the class of feasible mechanisms. Notice that our feasibility conditions are all ex ost, eliminating the need to include a rior on the joint tye sace in the model. Strategyroofness says that it is a weakly dominant strategy for the agent to reort his true tye in the revelation game induced by the mechanism. Individual rationality imlies that at every tye ro le each agent receives at least his tye-indeendent ayo from an outside otion, which we normalize to zero. Finally no-budget-de cit imlies that at every tye ro le the cost of the club is covered by agents ayments. We next give without roof a classical result on the characterization of strategyroof and individually rational mechanisms. (See Myerson (98) for examle.) Proosition A mechanism m = (S m ; m ) is strategyroof and individually rational if and only if for every i N, there exist functions m i : < + n! < + [ fg and h m i : < n +! < + such that for every t = (t i ; t i ). if t i > m i (t i ), then i S m (t) and m i (t) = m i (t i ) h m i (t i ),. if t i < m i (t i ), then i 6 S f (t) and m i (t) = h m i (t i ),. if t i = m i (t i ), then either [i S m (t) and m i (t) = m i (t i ) h m i (t i )] or [i 6 S m (t) and m i (t) = h m i (t i )]. 8

9 Thus if m is strategyroof, then whether i will belong to the club at a tye ro le or not deends on how his tye comares to a cuto m i (t i ) determined by other agents tyes. If his tye is above the cuto, then i belongs to the club. Furthermore his ayment is the di erence between his cuto and an amount h m i (t i ) which, again, is only deendent on others tyes. If his tye is below the cuto, then he does not belong to the club and his ayment is the negative of h m i (t i ). If, on the other hand, a mechanism m is generated by functions f m i ; h m i g i=;:::;n with these features, then it is strategyroof. Furthermore a strategyroof mechanism m is individually rational if and only if the functions h m i are all nonnegative valued. In this aer, we will always break the tie in case of Proosition in favor of including the agent in the club. None of our results deend on this restriction. Furthermore, whenever convenient, we will dro the suerscrit m in the functions m i and h m i associated with a strategyroof and individually rational mechanism m. The no-budget-de cit requirement in our notion of feasibility imoses a nontrivial condition on the cuto s and lum sum amounts associated with a strategyroof and individually rational mechanism. A mechanism m is feasible according to De nition if and only if, at every t, S(t) 6=? imlies X X i (t i ) h i (t i ), and is(t) in S(t) =? imlies h i (t i ) = 0 for all i N where the functions f i ; h i g generate m as in Proosition. Note that a feasible mechanism necessarily balances the budget whenever it forms the emty club: i (t) = h i (t i ) = 0 for all i. To the best of our knowledge, there is no tractable characterization of the class of feasible mechanisms. We view this as the main di culty in front of mechanism design in our environment. Maximal Welfare Loss E ciency The classical notion of e ciency dictates in our model that a mechanism should either include all in the club, or else, exclude all from the club. More recisely, m is e cient if S(t) = N whenever P in t i, and S(t) =? otherwise. Hence under e ciency the ublic good is essentially non-excludable. This stems from the fact that once the club has some member, the marginal net bene t of including another is nonnegative. 9

10 Unfortunately this notion of e ciency is incomatible with our feasibility constraints. In other words, there is no m which is e cient. Even though this is a famous result, let us review the argument as it alies in our environment for the secial case of two agents. We will concentrate on three tye vectors, (; 0), (0; ) and (; ). Suose that m is feasible and e cient. Then S(; 0) = S(0; ) = S(; ) = f; g by e ciency. Furthermore (; 0) = (0; ) = 0 by IR. Now SP gives (; ) = (; ) = 0 as well, leading to a budget de cit at (; ) and hence to a contradition with feasibility. 0 Given this imossibility, the literature has turned to relaxing feasibility and/or e ciency requirements on mechanisms. Of our interest is a relaxation of e ciency called maximal welfare loss e ciency. In order to de ne this notion, we need some rearations. First-best welfare at a tye ro le t, which we will denote W (t), is the total welfare in the society achievable under comlete information. Hence nx o W (t) = max t i ; 0 : in First-best welfare serves as the benchmark to evaluate the erformance of a mechanism under incomlete information. For any m and any tye ro le t, let W m (t) be the welfare generated by m at t, i.e., W m (t) = X t X is m i (t) Now let W L m (t) = W (t) in m i (t): W m (t): Thus W L m (t) is the welfare loss of m at t in comarison to the rst-best. Finally let MW L m denote the maximal welfare loss (MWL) of m taken over all tye ro les, i.e., MW L m = su W L m (t): t< n + For any m; m 0, we will say that m is MWL suerior to m 0 if MW L m < MW L m0. 0 According to our de nition of e ciency the grand club must be formed at tye vectors (; 0) and (0; ); although the emty club would have induced the same total welfare of zero. However the argument would still work if we relaced (; 0) and (0; ) with (; ") and ("; ) where " is a small ositive number. 0

11 Our mechanism designer will evaluate each mechanism m on the basis of its maximal welfare loss MW L m and will be interested in emloying a mechanism whose maximal welfare loss is minimal. De nition For any domain 0, we will say that m is maximal welfare loss e cient in 0 if m arg min m 0 MW L m. In what follows, we will de ne and investigate the MWL e ciency roerties of a articular mechanism, the equal cost sharing mechanism, which we will denote f. Equal Cost Sharing We begin by de ning the equal cost sharing mechanism, which will be the main focus of our aer. For any set S N, let #S denote the cardinality of S. De nition The equal cost sharing mechanism f = (S; ) is de ned as follows: for every t S(t) = [ fs N : for all i S, t i #S g i (t) = #S(t) if i S(t); 0 otherwise. Clearly f. The mechanism f forms the largest club whose members can individually rationally and equally share cost. Indeed, for every t, t i =#S(t) for all i S(t), in other words, the set fs N : for all i S, t i =#Sg of clubs is closed under the union oeration. To see this, take two clubs T and T 0 in fs N : for all i S, t i =#Sg. Note that for all i T [ T 0, either t i =#T =#(T [ T 0 ) or t i =#T 0 =#(T [ T 0 ). Note that f is in articular budget-balanced, i.e., the sum of agents contributions exactly cover the cost of the club. Furthermore f has the following Strategyroofness of f follows because the following monotonicity roerty is satis ed: if i S f (t i ; t i ) and t i < t 0 i, then i Sf (t 0 i ; t i).

12 anonymity roerty: the associated cuto functions and their arguments are indeendent of the names of the agents. If there are three agents, for examle, (x; y) = (x; y) = (x; y) = (y; x): To roceed, we will calculate MW L f. Lemma The maximal welfare loss of the equal cost sharing mechanism f is P n k= =k. Proof. We will rst show that W L f (t) < P n k= =k. If P in t i <, then P W (t) = W f (t) = 0 giving W L f (t) = 0 as well. So take any t such that in t i so that W (t) = P in t i. There are two cases. If S f (t) = N, then W f (t) = P in t i = W (t) and W L f (t) = 0. If #S f (t) = n 0 f0; ; :::; n g, then let (t () ; t () ; :::; t (n n0 )) be a listing of the tyes of agents in NnS f (t) from highest to lowest, with ties broken arbitrarily. We claim that t (k) < for every k = ; :::; n n n 0 +k 0. If not, let k be the largest k f; :::; n n 0 g such that t (k). It follows that t n 0 +k (k) n 0 for +k all k k and S f (t) should have had n 0 + k instead of n 0 members, a contradiction. Hence W L f (t) = X i6s f (t) t i = nx n 0 k= t (k) < nx n 0 k= =(n 0 + k) nx =k giving W L f (t) < P n k= =k as we wanted to show. Hence for all t, W L f (t) < P n k= =k giving MW Lf P n k= =k. To nish, we establish the reverse inequality by taking a sequence of tye vectors at which f generates welfare losses converging to P n k= =k. For all su ciently small " > 0, let t " = ( "; "; :::; "). Note S f (t " ) =? n and if P in t" i, which haens if " is small enough, then W L f (t " ) = PiN t" i = P n k= =k n". Now as " # 0, W Lf (t " )! P n k= =k, indicating that MW L f = su t W L f (t) P n k= =k. We will now show that when there are only two agents, there is no feasible mechanism in whose MWL is less than that of f. Proosition The equal cost sharing mechanism f is maximal welfare loss e cient in if N = f; g. k=

13 Proof. By Lemma, MW L f = =. Suose MW L g < = for some g. We rst claim that for all su ciently small " > 0, we must have S g ( "; ") 6=?. Otherwise W g ( "; ") = 0 and W L g ( "; ") = " giving MW L g lim "#0 W L g ( "; ") = =, a contradiction. Similarly we must also have S g ( "; ") 6=? for all su ciently small " > 0. Fix such ". It follows that S g ( "; ") = S g ( "; ") = f; g as neither individual alone could individually rationally cover the cost of a nonemty club at these tye ro les. Now SP and IR imly S g ( "; ") = f; g g ( "; ") = g ( "; ") " and g ( "; ") = g ( "; ") "; leading to a budget de cit at the tye vector ( "; "). Hence g could not have been feasible. It turns out, however, that when N contains three agents or more, f is no longer maximal welfare loss e cient in. To establish this, we will later resent an examle of a feasible mechanism which is MWL suerior to f. We will rst show, however, that if certain notions of fairness are imosed in the de nition of feasibility, then f regains MWL otimality. In other words, we will show that there exist subsets generated by fairness considerations, where f is MWL e cient even with n > agents. To this end, we introduce two distinct and arguably rather weak notions of fairness. De nition 4 A mechanism m = (S; ) satis es weak demand monotonicity (wdm) if S(t) S(t 0 ) whenever t i t 0 i for all i N. De nition 5 A mechanism m = (S; ) satis es weak free entry (wfe) if for every i; j and t, i S(t) whenever j S(t) and t i > t j. Note that neither condition has an imosition on how ayments are determined. In a discussion section to follow, we will argue that these two conditions are indeendent and that they are weaker than the demand monotonicity, free entry and envy-freeness conditions that aear in the literature. Let wdm and wf E denote the classes of feasible mechanisms satisfying wdm and wfe resectively. It is clear that f wdm \ wf E. Imosing either wdm or wfe is su cient to obtain the MWL e ciency of f.

14 Proosition The equal cost sharing mechanism f is maximal welfare loss e cient in wdm [ wf E. Proof. We will rove this result here for the secial case when N = f; ; g. The argument extends to n > agents at the cost of some investment in notation as we show in Aendix. By Lemma, MW L f = 5. Suose that MW 6 Lg < 5 for some g 6 wdm [ wf E. Our roof will rely on how g necessarily behaves at tye ro les ( "; "; "), ( "; "; "), and ( "; "; ") and their ermutations for small enough but strictly ositive ". We recall from Lemma that MW L f obtains at all 6 ermutations of the ro le ( "; "; ") as " falls to 0. We rst claim that for " > 0 small enough, g must form a nonemty club at the tye ro les ( "; "; ") and ( "; "; "). If not, the welfare loss of g at these ro les would be 5 ", and taking the limit as " 6 falls to zero would give MW L g = 5, a contradiction. Now x a su ciently 6 small " > 0 such that S g ( "; "; ") and S g ( "; "; ") are both nonemty. We now claim that S g ( "; "; "): There are two cases to consider. First suose that g wdm. If S g ( "; "; ") or S g ( "; "; "), then S g ( "; "; ") by wdm. If not, then S g ( "; "; ") = f; g = S g ( "; "; ") by feasibility. Now SP and IR give f; g S g ( "; "; "); g ( "; "; ") = g ( "; "; ") <, and g ( "; "; ") = g ( "; "; ") < : 4

15 Hence S g ( "; "; ") so that no-budget-de cit obtains. Suose now that g wf E. Then S g ( "; "; ") since otherwise S g ( "; "; ") = f; g and wfe fails. Similarly S g ( "; "; "). Consequently f; g S g ( "; "; "), g ( "; "; ") = g ( "; "; ") <, and g ( "; "; ") < g ( "; "; ") <. Thus we must have, once again, S g ( "; "; ") to clear the budget. The exact same reasoning alies in showing that S g ( "; "; ") and S g ( "; "; "). Thus S g ( "; "; ") = f; ; g by SP. However the ayments at this tye ro le do not cover the cost of the club. For examle g ( "; "; ") = g ( "; "; ") <. Similarly g ( "; "; ") and g ( "; "; ") are both strictly less than. Hence g could not have been feasible. It is instructive to juxtaose the roofs of Proositions and in order to understand the comlications that arise with three or more agents. Both roofs rely on exloiting the behavior of mechanisms at hand at certain critical tye ro les. For small enough " > 0, take the ro le ( "; ") in the two-agent case, and the ro le ( "; "; ") in the three-agent case. To beat f in MWL, an alternate mechanism must form nonemty clubs at these ro les. Now in the former scenario, this nonemty club is necessarily the grand coalition f; g. In the latter, however, the nonemty club is either the grand coalition f; ; g, or one of the doubletons f; g and f; g. It is recisely this multilicity of ossible clubs at critical ro les that creates the comlications with three or more agents. We need, in order to attain a budget de cit at the ro le ( "; "; "); that at every ermutation of the ro le ( "; "; ") the agent with the lowest tye " to belong to the club formed. Unfortunately feasibility of a mechanism alone may fail to lead to this conclusion as our Examle below shows. As a result we need wfe or wdm for the argument to work. 5

16 In order to show that Proosition is tight, in other words, to show that a failure of wdm and wfe leads to the subotimality of f, we will next resent a class of feasible mechanisms obtained from small erturbations of f. In doing so we will make use of the anonymity roerty, which is also satis ed by the erturbations by construction. Examle Suose N = f; ; g and take " (0; ). Consider the following erturbation of the equal cost sharing mechanism f, which we call 4 f " and resent together with f for comarison uroses. Both f anf f " are anonymous and the diagrams give the cuto s of some k N as a function of (t i ; t j ) where i, j and k are distinct agents in N. t j t j φ k = / / + ε φ k = / φ = k ε / / φ k = φ k = / φ k = / / / + ε / ε φ k = φ = / +ε k φ = k + ε φ = / + ε k φ = / ε k φ k = / t i ε + ε + ε t i FIGURE : f when n = FIGURE : when n = f ε In both diagrams the arguments of k are omitted to economize on sace. For both mechanisms, at any vector (t i ; t j ) which lies at the border of two or more distinct regions, the e ective cuto is the smallest one. At (t i ; t j ) = ( ; ) for examle, f imoses a cuto of on agent k. Similarly at (t i; t j ) = ( + "; ), f " imoses the cuto " on k. Both mechanisms charge each member of any nonemty club exactly his cuto. Hence their h i functions are zero-valued. We would like to oint out various features of the mechanism f " in Examle. 6

17 . Generated by symmetric cuto functions as given in the lot, f " satis- es SP. Furthermore since the associated h i functions are zero-valued, IR obtains and members of any club ay their cuto s. In Aendix, we exhibit how f " behaves in detail by artitioning < + into 6 distinct sets of tye vectors and analyzing them individually.. As " vanishes, the mechanism f " converges (ointwise) to f. Hence we interret f " to be a erturbation of f.. In order to calculate MW L f ", take the tye vectors ( "; ; ) and ( + "; + "; ) or any one of their ermutations. At tye vectors su ciently close to these vectors from below, we will now show that f " forms the emty club. Let us verify this leisurely. First take ( "; ; ) and a su ciently small > 0. Omitting subscrits note that the cuto s imosed by f " are ( " ; ) =, ( " ; ) =, and ( ; ) = " Hence if t is su ciently close to ( "; ; ) from below, then S(t) =?, causing a welfare loss of 5 ". 6 Simiarly If we take ( + "; + "; ) and a small > 0, we get ( + " ; + " ) =, ( + " ; ) =. Note that as " <, +" <. Hence at tye vectors su ciently close 4 to ( +"; +"; ) from below, no agent meets his cuto and the emty club is formed. The associated welfare loss this time is + 4" < 5 as 6 " <. In either case MW 4 Lf " < MW L f. Aendix demonstrates in detail the calculation of MW L f ". 4. An interesting question is to identify that " for which MW L f " is the lowest among ff " : 0 < " < g. This obtains by equating the two 4 7

18 candidates for MW L f " 5 above: " = 6 other words inf MW L f " = MW L f =0 = 4 0<"< 5 : 4 + 4", which gives " = 0. In 5. Our erturbation f " fails both wdm and wfe. For examle, f " forms the club f; g at the tye ro le ( + "; + "; ): However when all tyes are larger at (; ; + ") f " excludes and forms the club f; g. This is a failure of wdm. Furthermore, the fact that agent does not belong to the club f; g at ( + "; + "; ) even though t > t is a failure of wfe. This was of course to be exected because of Proosition above. Any mechanism which satis es wdm or wfe could not ossibly be suerior to f in MWL. 6. We would like to re-emhasize that f " is by construction anonymous: for every i; t and ermutation on f; ; g, f " i (t ; t ; t ) = f " (i)(t () ; t () ; t () ): Let A be the class of anonymous and feasible mechanisms. Thus the equal cost sharing mechanism f is not MWL e cient in A as f " A. This notion of anonymity is, of course, stronger than anonymity in utility terms, which aears in Srumont (0). It aears in Dobzinski et al. (008) where it is called equal treatment. 7. As oosed to the equal sharing mechanism f, its erturbation f " is not budget balanced. However the budget surlus induced by f " is at most 6" and this is not large enough to o set the e ciency gain in the worst-case scenario. Details are in Aendix. 4 Discussion To the best of our knowledge, wdm and wfe are novel modi cations of closely related (and stronger) conditions that aear in the literature. Note that there is no logical imlication between our conditions. To see this suose N = f; g. Suose that S(t) = fg if t and S(t) =? otherwise, with agent being charged the full cost of the club whenever he is in the club. This mechanism is in, it satis es wdm and fails wfe. On the other hand the mechanism given by S(t) = fig if t j t i and S(t) =? otherwise, 8

19 with the agent in the club covering the cost of the club fully belongs to, satis es wfe but fails wdm. Thus the class wdm [ wf E on which f e is MWL e cient is a strict suerset of both wdm and wf E. We note that Proosition is not a corollary to Proosition, as there exist feasible mechanisms outside wdm [ wf E in -agent environments. As an examle, consider the mechanism de ned by S(t) =? if t <, S(t) = f; g, (t) = and (t) = 0 if t < and S(t) = fg and (t) = if t : Our wdm condition is a straightforward weakening of the demand monotonicity condition that aears in Ohseto (000), which requires, on to of our wdm, that S(t) = S(t 0 ) whenever t i t 0 i for every i S(t) and t 0 i t i for every i 6 S(t). Furthermore for any feasible mechanism, our wfe condition is imlied by the free entry condition which aears in Deb and Razzolini (999). A mechanism m satis es free entry if for every i; j and t, i S(t) whenever j S(t) and t i > j (t). Suose that m is a feasible mechanism which satis es free entry. If for some t, i and j, j S m (t) and t i > t j, then t i > m j (t) as well by individual rationality. Consequently i S m (t) by free entry. This establishes that m satis es wfe as well. A di erent condition that imlies wfe is the classical notion of envyfreeness (see, for examle, Srumont 0). A mechanism m is envy-free if for every t, i and j, I i (S(t))t i i (t) I j (S(t))t i j (t). Suose that m is a feasible mechanism. If at some tye ro le t, j S(t) and i 6 S(t) even though t i > t j, then i envies j since I i (S(t))t i i (t) = 0 < t i t j I j (S(t))t i j (t) where the weak inequality follows by individual rationality. Hence if m is an envy-free and feasible mechanism, it also satis es wfe. 5 Conclusion Equal cost sharing mechanism is a simle and aealing rocedure with desirable incentive roerties. We show in this aer that, in general, it is not maximal welfare loss e cient. Hence a mechanism designer with the worstcase-scenario in mind, may ot to emloy a di erent mechanism, erhas a 9

20 small erturbation of equal cost sharing which may lead to budget surlus as we exhibit in Examle. If the designer is restricted to use a mechanism with certain fairness roerties (as embodied in our weak demand monotonicity or weak free entry conditions), however, then equal cost sharing can not be imroved uon. We leave for future work the investigation of the consequences of relacing the no-budget-de cit condition in our feasibility de nition with strict budget balance. As we remarked above, our erturbation of the equal cost sharing mechanism which fares better in terms of maximal welfare loss leads to budget surluses. Hence it may very well be that equal cost sharing mechanism is maximal welfare loss e cient within the class of feasible and budget balanced mechanisms. We also leave for future work the ambitious question of how to solve the mechanism design roblem min m MW Lm. We erceive the main di culty here to be the absence of a tractable characterization of the set of strategyroof, individually rational and no-budgetde cit mechanisms. References [] Deb, R. and Razzolini, L. "Auction-like mechanisms for ricing excludable ublic goods," Journal Economic Theory 88, (999a). [] Deb, R. and Razzolini, L. "Voluntary cost sharing for an excludable ublic roject," Mathematical Social Sciences 7, -8 (999b). [] Dobzinski, S., Mehta, A., Roughgarden, T., and Sundararajan, M. "Is Shaley cost sharing otimal?" In SAGT 08 Proceedings of the st International Symosium on Algorithmic Game Theory (008). [4] Juarez, R. "The worst absolute loss in the roblem of the commons: random riority versus average cost," Economic Theory 4, (008a). 0

21 [5] Juarez, R. "Otimal grou strategyroof cost sharing: budget-balance vs. e ciency," mimeo. University of Hawaii (008b). [6] Juarez, R. "Grou strategy roof of cost sharing: mechanisms without indi erences," mimeo (008c). [7] Koutsouias, E. and Paadimitriou, C. "Worst-case equilibria." In the 6th Annual Symosium on Theoretical Asects of Comuter Science (999). [8] Mehta, A., Roughgarden, T., and Sundararajan, M. "Beyond Moulin mechanisms." In Proceedings of the 8th ACM Conference on Electronic Commerce (EC), -0 (007). [9] Moulin H. "The rice of anarchy of serial, average and incremental cost sharing," Economic Theory 6, (008). [0] Moulin, H. and Shenker, S. "Strategyroof sharing of submodular costs: budget balance versus ec/ ciency," Economic Theory 8, 5-5 (00). [] Mutuswami, S. "Strategyroofness, non-bosiness, and grou strategyroofness in a cost model," Economics Letters 89, 8-88 (005). [] Mutuswami, S. "Strategyroof cost sharing of a binary good and the egalitarian solution," Mathematical Social Sciences 48, 7-80 (008). [] Myerson, R. "Otimal auction design," Mathematics of Oerations Research 6, 58-7 (98). [4] Nisan, N. "Introduction to mechanism design (for comuter scientists)." In Algorithmic Game Theory, edited by N. Nisan, T. Roughgarden, E. Tardos, and V.V. Vazirani. Cambridge University Press, New York (008). [5] Ohseto, S. "Characterizations of strategy-roof mechanisms for excludable versus nonexcludable ublic rojects," Games and Economic Behavior, 5-66 (000). [6] Ohseto, S. "Augmented serial rules for an excludable ublic good," Economic Theory 6, (005).

22 [7] Olszewski, W. "Coalition strategy-roof mechanisms for rovision of excludable ublic goods," Games and Economic Behavior 46, 88-4 (004). [8] Roughgarden, T. "The rice of anarchy is indeendent of the network toology," Symosium on Theory of Comuting (00). [9] Roughgarden, T. and Sundararajan, M. "New trade-o s in cost-sharing mechanisms." In Proceedings of the 8th Annual ACM Symosium on the Theory of Comuting (STOC) (006). [0] Roughgarden, T. and Tardos, E. "How bad is sel sh routing?," Journal of the Association for Comuting Machinery 49, 6-59 (00). [] Srumont, Y. "Poulation monotonic allocation schemes for cooerative games with transferable utility," Games and Economic Behavior, (990). [] Srumont, Y. "Constrained-otimal Strategy-roof Assignment: Beyond the Groves Mechanisms," Journal of Economic Theory (0), htt://dx.doi.org/0.06/j.jet [] Thomson, W. "The fair division of a xed suly among a growing oulation," Mathematics of Oerations Research 8, 9-6 (98a). [4] Thomson, W. "Problems of fair division and the egalitarian rincile," Journal of Economic Theory, -6 (98b). [5] Yu, Y. "Serial cost sharing of an excludable ublic good available in multile units," Social Choice and Welfare 9, (007).

23 Aendix : Proof of Proosition Let N = f; :::; ng where n >. Here we will generalize the argument resented in the main text for n =. To this end we will need to develo some notation. For any tye vector t < n +, let P (t) be the set of all ermutations of t. In other words t 0 P (t) i there is a bijection : N! N such that for every i, t i = t 0 (i). Now for any " (0; ) and any k = ; :::; n, de ne the n tye vector " if i k Hence t k i (") = " if k < i i t (") = ( "; "; "; :::; n "); t (") = ( "; "; t n (") = ( "; :::; "): "; :::; n "); Now let T k (") = P (t k (")) for every ; :::; k: Note that k also indicates the number of agents whose tyes are " in every element of T k ("). Furthermore if k >, then for every t T k ("), there exists some t 0 T k (") such that t 0 t i = i if t 0 i 6= ", and k " if t 0 i = ". k Hence for every k > and every t T k ("), there exist k vectors t ; :::; t k T k (") such that t is obtained by increasing the tye of the agent with k " in any one of t ; :::; t k to ". If " is small enough, any mechanism which has a lower maximal welfare loss than equal cost sharing must form a nonemty club at all members of T k (") for every k. We record this rather straightforward observation next. Lemma If MW L g < MW L f, then there exists " > 0 such that for every " (0; ") and t S n k= T k ("), S g (t) 6=?.

24 Proof. P Take a mechanism g with MW L g < MW L f. Recall that MW L f = n i= =i. By construction of the sets ft k (")g k=;:::;n, if t T k (") and t 0 T k ("), then P n i= (t0 i t i ) = > 0 for k >. Consequently for k any list ft k g k=;:::;n where t k T k (") for every k, the number P n i= tk i is increasing in k. Thus if S g (t k ) =? for all k, then W L g (t n ) > > W L g (t ). So it su ces to show that S g (t) 6=? for every t T ("). Suose, towards a contradiction, that for every " > 0, there exists some " (0; ") and t(") T (") such that S g (t(")) =?. Then there exists a sequence f" q g & 0 and a corresonding sequence of tye vectors t q T (" q ) such that S g (t q ) =? for all q: Note that such t q is a ermutation of the tye vector ( "; "; :::; "). Then W L g (t q )! P n n i= =i, a contradiction to the hyothesis MW L g < P n i= =i. Thus our suosition is false: there exists " > 0 such that if " (0; ") and t T ("), S g (t) 6=?. From now on we will x such small " (0; ") and to economize on notation we will write T k instead of T k ("). We will also denote by t the tye vector ( "; :::; "), the unique element of T n. Our goal is to show that if () g is strategyroof and individually rational, () MW L g < P n k= and () g k satis es wdm or wfe, then g runs a budget de cit at some t S n k= T k. To begin, take a strategyroof and individually rational mechanism g such that MW L g < P n k=. k Case : Weak Demand Monotonicity satis es wdm. Write Suose that g additionally T n = ft ;n ; t ;n ; :::; t n;n g; where for every k, t k;n k = " and t k;n n i = " for all i 6= k. Note that the only di erence between the tye ro les t k;n and t is agent k s tye: t i = if i 6= k, and " > t k;n k if i = k. t k;n i This observation has an imortant consequence, which we will use recursively below. Note that S g (t k;n ) 6=? by the Lemma above. Now if k S g (t k;n ), then, by SP and IR, k S g (t ) and k (t ) = k (t k;n ) n ": 4

25 There are two ossibilities: () k S g (t k;n ) for every k and g runs a budget de cit at t, in which case the roof for Case is comlete, or () k 0 6 S g (t k0 ;n ) for some k 0. Suose the latter case, and in articular, without loss of generality, k 0 =. Now consider the following subset of T n : T n () = ft T n : t = n "g. Hence t T n () i n agents have tye ", agent has tye " n and a di erent agent has tye ". Now to in down this di erent agent, n we write T n () = ft ;n ; t ;n ; :::; t n;n g: where t k;n k = ". Note that for every k >, t ;n n = t k;n = " n and t ;n i = if i 6= k, and " > t k;n k if i = k. t k;n i Note that since t ;n i t k;n i for all i and since we suosed 6 S g (t ;n ) above, wdm imlies 6 S g (t k;n ) for any k = ; :::; n. However S g (t k;n ) 6=? by the Lemma. Now using the exact same logic of the revious aragrah, if some k S g (t k;n ), then, by SP and IR, k S g (t ;n ) and k (t ;n ) = k (t k;n ) n ": Once again, there are two ossibilities: () k S g (t ;n ) for all k >, leading to a budget de cit and the termination of the roof of Case, or () k 0 6 S g (t k0 ;n ) for some k 0 >. Suose the latter case and, without loss of generality, k 0 =. Continuing in this fashion, suose k 6 S g (t k;n k ) for all k = ; :::; n and write T (; ; :::; n ) = ft n ; ; t n; g. Now wdm imlies that agents ; :::; n should belong to neither S g (t n ; ) nor S g (t n; ). However, since a club must be formed at both tye vectors, we must have S g (t n ; ) = S g (t n; ) = fn ; ng. 5

26 This follows because t n ; n = t n; n = " and t n ; n = t n; n = ". Thus at either tye vector no agent can individually rationally nance the club alone. Hence, by SP and IR, n ; n S g (t n ; ), g n (t n ; ) = g n (t n ; ) ", and g n(t n; ) = g n(t n ; ) ": But now to avoid a budget de cit at t n ; agent n must be included in S g (t n ; ) which is the contradiction we need to nish the roof of Case. Case : Weak Free Entry Now suose that g additionally satis es wfe. De ne for every k = ; ::::; n and every t T k Nk k (t) = fi N : t i = "g and Nk+(t) k = fi N : t i "g. k + Note that Nk k(t) and N k+ k (t) di er by a unique element, and this is the agent whose tye is ". Call this agent i (t), the agent with the highest tye k+ at t which is not ". It follows that N k k+(t) = N k k (t) [ fi (t)g: We will claim here that for every k = ; :::; n and every t T k, N k k+ (t) Sg (t). This is a consequence of wfe and we will ostone the roof for a little while. Given this claim, the roof of Case and therefore of Proosition can easily be nished. We have for all t T n N = Nn n (t) S g (t) where the equality is by construction of the sets Nk+ k (t) and the set inclusion is by the claim. Since S g (t) N as well by de nition, S g (t) = N for all t T n. By SP and IR, therefore S g (t ) = N and g i (t ) n " for all i, 6

27 leading to a budget de cit at t. All that remains now is to rove the claim. We will use induction. Let k = and take any t T. Since S g (t) contains some agent, it contains the agent with tye ". In other words, N (t) S g (t). However we can not have N (t) = S g (t) as this would either violate IR, or lead to a budget de cit. Hence S g (t) contains other agents and one of these must be i (t) by wfe. Thus N (t) [ fi (t)g = N (t) S g (t): Now suose, as induction hyothesis, that for some k f; :::; ng, N k k (t) S g (t) for all t T k. Take any t T k. There exist, by construction of the sets T ; :::; T n, tye vectors t ; :::; t k T k such that for every l = ; :::; k t l t i = i if i 6= i (t l ), and " if i = i (t l ). It follows that Furthermore by SP and IR i (t l ) N k k (t l ) S g (t l ): i (t l ) S g (t) and i (t l ) k " for every l. Note that fi (t l ) : l = ; :::; kg = Nk k(t), thus N k k(t) Sg (t). But Nk k (t) can not cover the cost of the club at t. Hence either there is budget de cit at t, or there is at least one more member in S g (t). By wfe this member is i (t) and Nk k (t) [ fi (t)g = Nk+(t) k S g (t) roving the claim. This nishes the roof of Proosition. 7

28 Aendix : Perturbation of Equal Cost Sharing In this aendix, we will elaborate on the family of mechanisms f " which we introduced in Examle. Suose that N = f; ; g and take " (0; ). 4 We would like to re-emhasize that f "! f ointwise as "! 0 and that f ", just like f, is an anonymous mechanism. We will describe f " in a series of diagrams, each corresonding to a articular interval of t values while t and t freely vary. To this end we will artition the joint tye sace < + into 6 arts, T ; :::; T 6. On T = f(t ; t ; t ) < + : t [0; T = f(t ; t ; t ) < + : t [ ")g, and "; + ")g f " is given by: t t {} = {} = {,} = / + ε = / ε / {,} = / = / / + ε / / + ε {,} = / = / = / + ε = / ε {,} Ø {} = Ø {} = t + ε + ε f ε in T in T f ε t Note that f " behaves identical to f in T. However this behavior ends at t = ". In articular su tt W L f " (t) = 5 6 " and this suremum is given by tye vectors (; ; ") and ( ; ; "). Note that T does not allow W L f " (t) to reach MW L f = 5 6 at any t T. Next consider the sace T of tye vectors. This is a rather thin slice as t takes values in an interval of measure ". On T, su tt W L f " (t) = + 4" 8

29 which falls short of MW L f = 5 as we took " <. The suremum on T 6 4 obtains close (from below) to the critical tye vectors ( + "; ; + ") and (; + "; + "). Imortantly, in T clubs f; g and f; g are formed in order to cover the area where f obtains MWL. However at any t where S(t) = f; g, t + " > t and at any t where S(t) = f; g, t + " > t. Hence f " excludes higher tye agents from clubs at the bene t of lower tye agents, and therefore violate wfe. Furthermore, f " also violates wdm: when all tyes increase the club formed could change from f; g to f; g or from f; g to f; g. This is to be exected in light of Proosition. Any mechanism g such that MW L g < MW L f needs to violate both wfe and wdm when n >. We move on to higher values for t. On f " is given by: T = f(t ; t ; t ) < + : t [ + "; )g and T 4 = f(t ; t ; t ) < + : t [ ; + ")g t t {} = {,} {,,} {,,} {,,} / + ε / / + ε / ε = / ε = / + ε = / ε = / + ε = / ε Ø = / + ε = / + ε = / + ε = / ε = / + ε = / + ε = / + ε = / ε = / ε {,,} = / ε = / ε = / + ε = / + ε = / ε {,,} = / + ε = / + ε {,} = / ε {,,} / + ε / / + ε {} = = / + ε = / + ε = / ε = / {,} = / Ø {,,} = / + ε = / + ε = / + ε {,} = / = / {,,} {,,} = / + ε = / + ε = / ε ε + ε + ε t + ε + ε t f ε in T f ε in T 4 9