Axiomatizing Political Philosophy of Distributive Justice: Equivalence of No-envy and Egalitarian-equivalence with Welfare-egalitarianism

Transcription

1 Axomatzng Poltcal Phlosophy of Dstrbutve Justce: Equvalence of No-envy and Egaltaran-equvalence wth Welfare-egaltaransm Duygu Yengn May 22, 2012 Abstract We analyze the poltcal phlosophy queston of what knd of welfare dfferentals are allowed f we respect prvate ownershp rghts over self and publc ownershp of the external world. We consder cases where agents have equal rghts over the publcly owned external world resources but are ndvdually responsble for ther preferences. We dscuss that these ownershp rghts are naturally reflected by the two fundamental axoms of farness: no-envy and egaltaranequvalence. We show that these axoms characterze the welfare-egaltaran mechansms (that are decson-eff cent and ncentve compatble) n a model of allocatng dscrete resources when money transfers are possble. We also relate no-envy and egaltaran-equvalence to the equalty of what debate and buld a lnk between resource and opportunty egaltaransm, and welfareegaltaransm JEL Classfcatons: C79, D61, D63. Key words: egaltaransm, egaltaran-equvalence, no-envy, dstrbutve justce, equalty of opportunty, resource egaltaransm, prvate ownershp of the self and publc ownershp of external world, NIMBY problems, allocaton of ndvsble goods and money, dscrete publc goods, strategy-proofness, populaton monotoncty, far dvson, the Groves mechansms. 1 Introducton We nvestgate how much nequalty n fnal outcomes (welfares) s allowed f one follows the poltcal phlosophy that respects the prvate ownershp rghts over self (preferences, sklls, costs, etc.) whle stll assumng socety has collectve rghts over the external world (collectve resources, populaton, technology, condtons of the economy etc.). Ths poltcal phlosophy theory was ntroduced by Cohen (1986) and an axomatc nvestgaton of ts mplcatons was frst carred out by Mouln and Roemer (1989) n a model wth a publcly owned producton technology and prvately owned sklls. They formulated axoms to reflect these property rghts and the mechansm characterzed turned out to be welfare-egaltaran (all agents should experence the same welfare). Hence, n the economc settng they study, surprsngly, even f one adapts the ntermedate poltcal poston by grantng prvate ownershp rghts over the self, one stll arrves at the radcal egaltaran result of equalzng welfares, overwrtng the effects of self ownershp rghts. I thank Prof. John Roemer for the dscusson on the topc. I am also grateful to Prof. Wllam Thomson for hs gudance and advce. Ths paper s based on the frst part of a prevous workng paper ( that has been dvded nto two papers for the clarty of exposton and coherence of content. A paper based on the second part s forthcomng n the B.E. Journal of Theoretcal Economcs, Yengn (2011a). School of Economcs, The Unversty of Adelade, SA 5005, Australa; e-mal: duygu.yengn@adelade.edu.au. 1

2 Smlar notons appled n dfferent settngs, may not characterze the same solutons due to the dfferent economc nformaton conveyed n dfferent models (for examples, see Secton 5.2 and Footnote 12 n Yengn, 2011a). It s a legtmate queston to ask whether Mouln and Roemer s concluson on the effects of publc and prvate property rghts holds only under partcular model specfcatons and wth partcular axoms, or t s a more general fact that would hold true n dfferent economc envronments and under dfferent axoms. Our results here are sgnfcant to show that the latter s the case. The model we nvestgate dffers from that of Mouln and Roemer (1989) n several aspects: there s no producton, nstead, a set of collectvely owned heterogenous dscrete resources and monetary transfers are to be dstrbuted among a fnte set of agents who has quaslnear preferences. Another dfference s that we assume that preferences are prvate nformaton whch s not the case n Mouln and Roemer (1989), even though they express n footnote 2 that such an assumpton seems sutable snce t s natural to take preferences as the prvate property of self. Our model analyzes the allocaton of ndvsble goods and money, but t s general enough to ncorporate varous nterestng real lfe examples. Examples nclude auctons held to allocate water enttlements to farmers; the allocatons of fshng or polluton permts, allocaton of communty housng or chartable goods and money among the needy, managng the use of commonly owned ndvsble goods n cooperatve enterprses such as cooperatve supported agrculture, allocaton of nhertance among hers; mposton of tasks as n government requstons and emnent doman proceedngs 1 ; and allocaton of dscrete publc goods 2 (e.g., stng problem of noxous facltes also known as the not-n-my-backyard, NIMBY problem). Wthout loss of generalty, we focus on the case of allocatng tasks whch agents are collectvely responsble to perform, as n emnent doman or NIMBY problem. Allocaton of ndvsble goods and money s an extensvely studed problem n the far dvson lterature, yet t has not been analyzed from the poltcal phlosophy vewpont of ownershp rghts over self and external world. Hence, our paper s novel n specfyng the model specfc axoms whch reflect the prvate ownershp of the self and publc ownershp of the external world and brdge the dfferent vews on equalty of what n the problem of allocatng unproduced collectvely owned goods. As Roemer (1986) dscusses, dfferent economc settngs requre dfferent axoms to reflect the model-specfc nformaton. As the model specfcatons change, the axoms that express the same underlyng notons would have to dffer. For nstance, the model consdered by Mouln and Roemer (1989) ncludes a collectvely owned technology; hence, the axom they use to approprately reflect the noton of publc ownershp of the external world s technology monotoncty: when publcly owned technology mproves, no one s worse off. Ths axom s not sutable for our model snce here, there s no queston of producton, nstead, we study the dstrbuton of a gven set of collectvely owned dscrete resources and money and we also allow for populaton to vary. Thus, our axoms on publc ownershp of the external world need to reflect both () the publc ownershp rghts over the allocated resources and the fact that () populaton s a publcly owned external world factor. Naturally, axoms dfferent than the ones consdered by Mouln and Roemer (1989) are needed 1 Government requston s government s demand to use goods and servces of the cvlans usually n tmes of natonal emergency such as natural dsasters and wars. Emnent doman s government s rght to seze prvate property, wthout the owner s consent, for publc use such as to buld a road or a publc utlty. In both cases, owners are legally guaranteed to receve just monetary compensaton. 2.e., Determnng whch agents wll supply what dscrete publc goods when all agents derve the same beneft but only the supplers bear the costs. Taxes and subsdes can facltate the sharng of total cost. Examples are locatng state captals, parks, nternatonal arports, or publc bads such as a nuclear facltes. See Yengn (2011b) for a detaled dscusson. Note that how many or whch publc goods to be produced s already gven, the only queston s whch agents wll supply the publc goods (e.g., whch localty wll host the noxous faclty). 2

3 to reflect these ownershp rghts. To address fact (), we mpose populaton monotoncty: when populaton, a publcly owned external world factor, changes, snce no agent s responsble for ths change, welfare of all agents should be affected n the same drecton. In Mouln and Roemer (1989), sklls are prvately owned and the axom they use to reflect ths noton s that the more hghly sklled agent should be at least as well off as the less sklled one. We take preferences of ndvduals as prvately owned. Ths assumpton would lead to two facts: (I) each agent s held responsble for her own preference and (II) agents preferences are ther prvate nformaton. These two facts need to be expressed n our axoms supportng the prvate ownershp of self. We ncorporate fact (II) n the desgn of our model as an assumpton on preferences. Hence, we need an axom of ncentve compatblty. Although, a strong requrement, strategy-proofness (reportng the true preferences s a weakly domnant strategy for all agents) s the most reasonable requrement when preferences are strctly under prvate ownershp; that s, nether any agent nor the center knows other agents preferences or the lkelhood of others preferences. Next, we nvestgate whch axoms are the best canddates to reflect that () resources to be allocated are collectvely owned that s agents have equal rghts or responsbltes over the allocated resources (publc ownershp of the external world) and (I) each agent s held responsble for her own preference (prvate ownershp of self). We propose that the two natural axoms that reflect these rghts are egaltaran-equvalence and no-envy and we wll dscuss how these two axoms reflect the two theores of dstrbutve justce, respectvely: equalty of resources and equalty of opportuntes. The argument on property rghts over the external world and self can be traced back to the equalty of what (Sen, 1980) debate n poltcal phlosophy. On the one end of the debate s welfare-egaltaransm. Ths theory supports publc ownershp of not only the external world but also self on the grounds that preferences, sklls etc. are nfluenced by external factors outsde agents control such as socal condtonng or bologcal determnaton (Cohen, 1989; Roemer, 1998). Thus, no agent should be punshed or rewarded for her self (e.g., her preference) and welfare dfferentals that arse from prvate ownershp of the self are not justfed. On the other end of the debate le the two well-known theores of justce: equalty of resources (Dworkn, 2000) and equalty of opportuntes (Arneson, 1989; Cohen, 1989; Roemer, 1998; and Kolm, 1996). Suppose that the natural mplcaton of a collectvely owned external world s to grant people equal rghts and responsbltes over t. Hence, ether the external world resources should be equally dstrbuted (equalty of resources) or people should be gven equal opportuntes to access them (equalty of opportuntes). The welfare each person enjoys from her equal share depends only on her own preferences and prvate ownershp of self s also respected. The welfare dfferences among agents would be solely due to the dfferences n ther preferences, f every agent were assgned the same bundle. Ths equal-resource allocaton ensures that each agent has an equal share of the jontly owned resources whle each agent bears the consequences of her own preferences. However, such an allocaton composed of dentcal bundles s not always feasble due to the heterogenety of tasks. Stll, we can choose a feasble allocaton whch s Pareto-ndfferent to a hypothetcal reference allocaton composed of dentcal bundles (egaltaran-equvalence, Pazner and Schmedler, 1978). Alternatvely, one may argue that collectve ownershp of the external world means grantng agents equal access to t. Pck an allocaton and magne an opportunty set consstng of all the bundles comprsng ths allocaton. Let each agent choose her most preferred bundle from ths common set. An allocaton s envy-free (Foley, 1967), f each agent chooses the bundle that s ntended for her n the allocaton. Each agent enjoys equalty of opportunty over the collectvely owned resources; however, the welfare of an agent only depends on her choce, that s she s held responsble for her own preference and prvate ownershp of self s respected. Our frst Theorem shows that, when there are at least three agents and costs over tasks are add- 3

4 tve or subaddtve 3 or when there s a sngle task, then under an eff cent assgnment of the objects and strategy-proofness, egaltaran-equvalence and no-envy s equvalent to welfare-egaltaransm. Our result s carred over to economes wth two agents (Theorem 2), f we addtonally mpose populaton monotoncty. Thus, n an entrely dfferent model than that of Mouln and Roemer (1989) and usng dfferent axoms that reflect the ownershp rghts over self and external world, we also show that the mechansms characterzed are welfare-egaltaran. Thus, our paper together wth Mouln and Roemer (1989) ndcates a sgnfcant ssue: t may be a general phenomenon that the jont mplcaton of the axoms whch reflect publc ownershp of the external world and prvate ownershp of self s the radcal poltcal poston that places self under publc ownershp and releves ndvduals from beng responsble for ther own preferences. Our results here are also related to two papers whch also study the same model as n ths paper. Under an eff cent assgnment of the tasks and strategy-proofness, Pápa (2003) characterzes, on the subaddtve doman, the class of envy-free mechansms, and Yengn (2010) characterzes, on any doman, the class of egaltaran-equvalent mechansms. The characterzed classes of mechansms n these two papers, are farly large and are not equvalent to welfare-egaltaran class. Hence, by Pápa (2003) and Yengn (2010), nether opportunty egaltaransm nor resource egaltaransm ndvdually mples welfare-egaltaransm. 4. However, as an alternatve nterpretaton of our Theorem 1, under the condtons of Theorem 1, when mposed jontly, the two axoms that support equalty of resources and equalty of opportuntes together result n equalty of welfares. There s an extensve lterature studyng egaltaran solutons (see Gnés and Marhuenda, 2000, and references theren). Smlar to Mouln and Roemer (1989), most of the characterzatons of egaltaransm n the lterature rely on monotoncty and soldarty axoms (see, for nstance, our companon paper Yengn; 2012a, forthcomng). Unlke the prevous lterature, here, the axoms we consder to characterze the welfare-egaltaran solutons are not soldarty axoms, but the two most fundamental equty concepts n the far dvson lterature, namely, no-envy and egaltaranequvalence. To sum up, our contrbuton s threefold: to provde an alternatve foundaton for welfareegaltaransm based on no-envy and egaltaran-equvalence; to show the compatblty of these two central farness notons n an mportant model wth prvate nformaton where ther jont mplcaton has not been nvestgated, and to relate these axoms to poltcal phlosophy on dstrbutve justce, property rghts, and the equalty of what debate. Hence, we follow an axomatc method to answer the essental questons posed n poltcal phlosophy n a general problem of allocatng collectvely owned resources. In secton 2, we present the model and defne the mechansms. In Secton 3, we present our characterzatons. Secton 4 dscusses the budget propertes of the mechansms and presents concludng remarks. All proofs are n the Appendx. 2 Prelmnares A fnte set of ndvsble tasks s to be allocated among a fnte set of agents by a center. All tasks must be allocated. An agent can be assgned ether no task, a sngle task, or more than one task. Each task s assgned to only one agent. Let A be the fnte set of tasks, wth A 1, and α, β be typcal elements of A. 3 Valuatons over dscrete goods are superaddtve. 4 Roemer (1986) obtans the equvalence between resource-egaltaransm and welfare-egaltaransm. However, n Roemer (1986), the approach s dfferent and equvalence result s obtaned by assumng that the nternal trats such as sklls and preferences of agents are part of the resources to equalze. In our model, we do not use such an assumpton. 4

5 There s an nfnte set of potental agents ndexed by the postve natural numbers N {1, 2,...}. In any gven problem, only a fnte number of them are present. Let N be the set of subsets of potental agents wth at least two agents. Let n 2 and N wth N = n be a typcal element of N. The number of agents may be smaller than, equal to, or greater than the number of tasks. Let 2 A be the set of subsets of A. Each agent has a cost functon c : 2 A R + wth c ( ) = 0. 5 We refer to such a cost functon as unrestrcted. Let C un be the set of all such functons. Our results can be easly adapted to a settng n whch for each A 2 A wth A, c (A) > 0. If for each A (2 A \ ), c (A) = c ({α}), then c s addtve. If for each par {A, A } 2 A α A wth A A =, c (A A ) c (A) + c (A ), then c s subaddtve, and f for each {A, A } 2 A wth A A =, c (A A ) c (A) + c (A ), then c s superaddtve. Let C ad, C sub, and C sup be the classes of addtve, subaddtve, and superaddtve cost functons, respectvely. Let C be a generc element of {C un, C ad, C sub, C sup } and C N be the n fold Cartesan product of C. For each N N, a cost profle for N s a lst c (c 1,..., c n ). Let C N be the doman of cost profles where for each N, c C. A cost profle defnes an economy. Let c, c, ĉ be typcal economes wth assocated agent sets N, N, N. For each N N and each N, let c be the cost profle of the agents n N\{}. For each par {N, N } N such that N N and each c C N, let c N be the restrcton of c to N : c N (c ) N. There s a perfectly dvsble good we call money. Let t denote agent s consumpton of the good. We call t agent s transfer: f t > 0, t s a transfer from the center to ; f t < 0, t s a transfer from to the center. The center assgns the tasks and determnes each agent s transfer. Agent s utlty when she s assgned the set of tasks A 2 A (note that A may be empty) and consumes t R s u(a, t ; c ) = c (A ) + t. An assgnment for N s a lst (A ) N such that A = A and for each par {, j} N, A A j =. For each N N, let A(N) be the set of all possble assgnments for N. A transfer profle for N s a lst (t ) N R N. An allocaton for N s a lst (A, t ) N where (A ) N s an assgnment and (t ) N s a transfer profle for N. A mechansm s a functon ϕ (A, t) defned over the unon C N that assocates wth each economy an allocaton: for each N N, each c C N, and each N, ϕ (c) (A (c), t (c)) 2 A R. For each N N and each c C N, let W (c) be the mnmal total cost among all possble assgnments for N. That s, { } W (c) = mn c (A ) : (A ) N A(N). N Snce there s no restrcton on the sze of ndvdual or total transfer (restrctons on the sze of total transfer are dscussed n Secton 4), every allocaton s Pareto-domnated by another allocaton wth hgher transfers. On the other hand, snce utltes are quas-lnear, gven a cost profle c, an allocaton that mnmzes the total cost s Pareto-eff cent for c among all allocatons wth the same, or smaller, total transfer. Our frst axom requres mechansms to choose only such allocatons. 5 As usual, R + denotes the set of non-negatve real numbers. N N N N N 5

6 Assgnment-Eff cency: For each N N and each c C N, c (A (c)) = W (c). Snce costs are prvate nformaton, an assgnment-eff cent mechansm assgns the tasks so that the actual total cost s mnmal only f agents report ther true costs. Smlarly, truthful revelaton of costs s essental n order to determne far allocatons. Then, a desrable property for a mechansm s that no agent should ever beneft by msrepresentng her costs (Gbbard, 1973; Satterthwate, 1975). Strategy-Proofness: For each N N, each N, each c C N, and each c C, u(ϕ (c); c ) u(ϕ (c, c ); c ). The Egaltaran mechansms choose, for each economy, an eff cent assgnment of the tasks. We work wth sngle-valued mechansms and assume that each Egaltaran mechansm s assocated wth a te-breakng rule τ that determnes whch of the eff cent assgnments (f there are more than one) s chosen. Let T be the set of all possble te-breakng rules. Let γ : N R be an arbtrary functon that assocates each populaton wth a real number and Γ be the set of all such functons. The Egaltaran mechansm assocated wth γ Γ and τ T, E γ,τ : Let E γ,τ (A τ, t γ,τ ) be such that for each N N and each c C N, (A τ (c)) N s an eff centassgnment for c and for each N, t γ,τ (c) = j N\{} N c j (A τ j (c)) + γ(n). The transfers of an Egaltaran mechansm have a smple structure: each agent pays the sum of the costs ncurred by the other agents at the eff cent assgnment chosen by the mechansm and receves a sum of money γ(n) R that may depend on the populaton, but s ndependent of the cost profle n the economy. That s, n all economes wth the same agent set N, the amount that agents receve s γ(n) R. Dfferent choces for γ correspond to dfferent selectons from the class of Egaltaran mechansms. Let E γ {E γ,τ τ T }. Note that for each γ Γ, the mechansms n E γ are Pareto-ndfferent. That s, the partcular te-breakng rule used s rrelevant n the determnaton of the utltes. Let E {E γ γ Γ} be the class of the Egaltaran mechansms. The mechansms n ths class equalze welfare of all agents: Welfare-Egaltaransm: For each N N, each par {, j} N, and each c C N, u(ϕ (c); c ) = u(ϕ j (c); c j ). Lemma 1. A mechansm s assgnment-eff cent, strategy-proof, and welfare-egaltaran on f and only f t belongs to E. 3 The Results C N N N Egaltaran-equvalence requres that only those allocatons such that each agent s ndfferent between her assgned bundle and a common reference bundle (consstng of a reference set of tasks and a reference transfer) should be chosen. Egaltaran-Equvalence: For each N N and each c C N, there are a reference set of tasks (whch may be empty) R(c) 2 A and a reference transfer r(c) R such that for each N, u(ϕ (c); c ) = u((r(c), r(c)); c ). 6

7 Another central farness noton s no-envy, whch requres that each agent should fnd her bundle at least as desrable as any other agent s bundle. Hence, gven the opportunty of choosng among all the bundles compromsng an allocaton, an agent should choose her assgned bundle. No-Envy: For each N N, each par {, j} N, and each c C N, u(ϕ (c); c ) u(ϕ j (c); c ). When costs are subaddtve or addtve, not only the Egaltaran mechansms, but also the followng mechansms satsfy these two axoms. 7

8 The Extended-egaltaran mechansm assocated wth γ Γ and τ T, Ê γ,τ : Let Ê γ,τ (A τ, t γ,τ ) be such that for each N N and each c C N, (A τ (c)) N s an eff centassgnment for c; and for each N N and each N, f N > 2, f N = 2, t γ,τ (c) = t γ,τ (c) for each c C N, ether () t γ,τ (c) = t γ,τ (c) for each c C N, or () t γ,τ (c) = c j (A) c j (A τ j (c)) + γ(n), where j N\{}, for each c CN. That s, when there are more than two agents, the transfers of Ê γ,τ are same as the transfers of an Egaltaran mechansm E γ,τ. However, when there are only two agents, then the mechansm has two optons for transfers, opton () or (). For some pars of agents, the transfers can be chosen equal to the transfers of E γ,τ (opton ()); and for other pars, the transfers can be chosen to be as n (). 6 Note that for a gven two-agent populaton N, for all economes c that pertan to populaton N, the mechansm should stck to one type of transfer: ether () or (). That s, for a gven N N wth N = 2, ether () apples for each c C N or, () apples for each c C N. Let Ê {Êγ,τ γ Γ, τ T } be the class of such mechansms. Note that E Ê. Now, we present our frst characterzaton. Theorem 1. On the subaddtve doman, an assgnment-eff cent and strategy-proof mechansm s egaltaran-equvalent and envy-free f and only f t belongs to Ê. By Theorem 1, on the subaddtve doman and when there are at least three agents, under assgnment-eff cency and strategy-proofness, the two axoms that support equalty of resources and equalty of opportuntes, namely, egaltaran-equvalence and no-envy, together mply equalty of welfares. 7 Hence, the axomatc method we followed produced a surprsng and notable answer to the long gong poltcal phlosophy debate on equalty of what. Suppose new agents jon some ntal populaton. The cost of an eff cent assgnment n the larger populaton s at most as large as the one n the smaller populaton. Hence, a populaton ncrease s good news for the socety. Snce none of the agents n the ntal populaton s responsble for the populaton ncrease (a change n the publcly owned external world), all of them should be at least as well off n the larger populaton as n the smaller one (Thomson, 1983). Populaton Monotoncty: For each par {N, N } N such that N N, each N, and each c C N, u(ϕ (c); c ) u(ϕ (c N ); c ). In Lemma 3 n the Appendx, we characterze the mechansms that are assgnment-eff cent, strategy-proof, and populaton monotonc. Our next Theorem states that under assgnment eff cency and strategy-proofness, on the subaddtve doman, egaltaran-equvalence, no-envy, and populaton monotoncty together mply welfare-egaltaransm n economes wth any number of agents. That s, f we add populaton monotoncty n Theorem 1, then class (Ê \E) s ruled out completely and we are left wth a subclass of E. 6 Note that when there are only two agents, the transfers specfed by Êγ,τ n opton () dffer from the transfers of a Pvotal (Vckrey) mechansm by the amount γ(n). 7 Note that, separately, nether egaltaran-equvalence nor no-envy mples equalty of welfares. 8

9 Theorem 2. On the subaddtve doman, an assgnment-eff cent and strategy-proof mechansm s egaltaran-equvalent, envy-free, and populaton monotonc f and only f t s an egaltaran mechansm n E γ where γ : N R s such that for each par {N, N} N wth N N, γ(n) γ(n ). (1) In our model, external world s comprsed of jont resources and populaton. Due to the quaslnearty of preferences, prvate ownershp of self requres bearng the effects of one s own cost functon. One can also nterpret cost functon of an agent as an ndcator of her skll: the lower the costs a person generates to perform tasks, the more sklled she s n performng those tasks. A more sklled agent would enjoy a hgher welfare from the same bundle than a less sklled agent. Each of the axoms used n Theorems 1 and 2 reflect the ownershp rghts of publc over the external world and of ndvdual over her self. Arguments for egaltaran-equvalence, no-envy, and populaton monotoncty are already presented n the Introducton. Strategy-proofness reflects the fact that preferences are n the doman of prvate ownershp of self; hence, even n cases where the center knows the preferences, t should treat them as prvately held nformaton. 8 Assgnmenteff cency may also be related to the jont ownershp of the external world resources: each agent should have the same rght to propose a new allocaton of the dscrete resources as long as the proposal doesn t harm anyone else and total transfer does not ncrease. The characterzatons n Theorems 1 and 2 ndcate that, on the subaddtve doman, a mechansm respectng the aforementoned ownershp rghts should actually equalze welfares and gnore the prvate ownershp of self. Thus, usng entrely dfferent axoms and a dfferent model, we arrve at the same concluson as Mouln and Roemer (1989). The results n Theorems 1 and 2 only hold when the costs are addtve or subaddtve, or when there s a sngle object to assgn as s generally the case n a NIMBY problem. Note that the mechansms n Ê are egaltaran-equvalent on every doman. However, by Pápa (2003), on the unrestrcted doman, no assgnment-eff cent and strategy-proof mechansm s envy-free. Characterzng assgnment-eff cent, strategy-proof, and envy-free mechansms on the superaddtve doman s an open queston. Characterzatons on ths doman are both techncally and notatonally complex; hence, generally not studed. 9 4 Concludng Remarks 4.1 Budget Propertes and Welfare lower bounds The class E we characterzed here s a sub-class of the well-known class of Groves mechansms (Vckrey, 1961; Clarke, 1971; and Groves, 1973) due to Lemma 1 and the followng result (the proof of Lemma 2 follows from Holmström, 1979, snce for each N N, C N s convex): Lemma 2. A mechansm s assgnment-eff cent and strategy-proof on C N f and only f t s a Groves mechansm. The obvous lmtaton of the Groves mechansms s that they do not balance the budget. That s, total transfer does not add up to a requred fxed amount n all economes. Hence, f we requre 8 See also footnote 2 n Mouln and Roemer (1989). 9 Papers analyzng Groves mechansms generally restrct attenton to ether the sngle-object case and to the addtve doman (Porter et al. 2004; Atlamaz and Yengn, 2008), or to the allocaton of homogenous objects where each agent can receve at most one object (Ashlag and Serzawa, forthcomng, Ohseto, 2006), or to the subaddtve doman (Pápa, 2003). N N 9

10 truthful revelaton of the preferences/costs, we need to pay the prce of budget mbalances. Ths drawback s mtgated by the fact that mechansms n E generate defcts bounded above (surpluses bounded below). Yengn (2011a) consders a class of upper bounds on defct that are lnear functons of the total cost of an eff cent assgnment. M = {m k,t : N N C N R c N N C N, m k,t (c) = kw (c) + T wth k (n 1) and T R}. One ntutve example of ths class s the defct upper bound that requres total compensaton to agents not to exceed the total cost they ncur (k = 1, T = 0) : for each c C N, t (c) W (c). Such a defct upper bound s appealng when the center wants to partally remburse the agents for ther costs such as n emnent doman proceedngs. Other ntutve examples of M nclude the one that requres total defct not to exceed a constant sum T R (k = 0), and the one that requres there s no budget-defct (k = T = 0). Theorem 2 n Yengn (2011a) shows that the class of welfare-egaltaran Groves mechansms can be parttoned nto subclasses {E γ } γ Γ ; and for each γ Γ, E γ s characterzed by a partcular upper bound on defct (m k,t M wth T = N γ(n)) and a correspondng welfare-lower-bound (no agent s worse off than the case where she s assgned all the tasks and compensated by an amount of money γ(n)) 10. Hence, under assgnment-eff cency and strategy-proofness, when the center knows the maxmal defct t can tolerate (or mnmum surplus t wants to generate) as well as the mnmal welfare t wants to guarantee to agents, then a partcular subclass E γ emerges. For nstance, takng T = 0, a Groves mechansm generates no defct (or a defct up to kw (c) wth k (n 1)) and guarantees each agent a utlty at least as much as her stand-alone utlty when she performs all tasks and receves no compensaton f and only f t s an Egaltaran mechansm E γ such that for each N N, γ(n) = 0. Also, by Yengn (2010), an egaltaran-equvalent Groves mechansm ensures that n any populaton N N, no matter what the costs of the agents are, the defct never exceeds T R, f and only f t s an Egaltaran mechansm E γ,τ such that for each N N, γ(n) T N.11 Hence, assumng that egaltaran-equvalence reflects resource-egaltaransm, for Groves mechansms wth defcts never exceedng a fxed amount, resource egaltaransm mples welfare-egaltaransm. If tasks are mposed on agents as n government requstons and emnent doman, then agents do not have the opton of refusng ther task assgnments, even f they may experence a negatve utlty. Also, f agents are collectvely responsble for the completon of tasks (as n NIMBY problems), then they are responsble for bearng the costs of the tasks and t may be natural that they should end up wth negatve utltes. In such cases, mechansms n E are very appealng snce they respect the welfare lower bounds we consdered n Yengn (2011a) that are analogous to weak socal partcpaton constrants. But f one nssts on that no agent should experence a negatve utlty level (ndvdual ratonalty), the mechansms n E do not satsfy ths property. However, f there s an upper bound on the cost that any agent may ncur, then there are ndvdually ratonal Egaltaran mechansms: Suppose there exsts K R + such that for each N and each A 2 A, c (A) K. Then, on the doman of cost profles comprsed of such cost functons, an Egaltaran mechansm s ndvdually ratonal f and only f t belongs to E γ where γ : N R s such that for each N N, γ(n) K. To see ths, by ndvdual ratonalty, for each N N and each 10 γ compensaton-welfare-lower-bound 11 Among all egaltaran-equvalent Groves mechansms that satsfy ths bounded defct condton, the ones for whch ths nequalty holds as an equalty Pareto-domnate the others. Also, note that an Egaltaran mechansm E γ,τ generates no-defct when for each N N, γ(n) 0. N N N 10

11 c C N, u(e γ,τ (c); c ) 0, that s, γ(n) W (c). Snce ths s true for each c C N, and γ(n) s ndependent of c, we have γ(n) max (c)} = K. c CN{W 4.2 Compatblty of Equty Notons Most characterzatons of welfare-egaltaran solutons n the lterature rely on soldarty axoms (see, for references, Gnés and Marhuenda, 2000; Yengn, 2011a). Here, we presented an alternatve foundaton for welfare-egaltaransm manly based on no-envy and egaltaran-equvalence. 12 Although, these axoms are the two most essental notons of justce n far dvson problems, ther compatblty s not guaranteed n several models: In tme dvson problems (dvson of a one-dmensonal, non-homogeneous, and atomless contnuum, when each agent s to receve an nterval), no egaltaran-equvalent mechansm s envy-free (Thomson, 1996). As showed by Postlewate (quoted by Danel, 1978), there are well-behaved exchange economes where all egaltaranequvalent and Pareto-eff cent allocatons volate no-envy. Under budget balance, no-envy s ncompatble wth egaltaran-equvalence n queueng problems (Chun, 2006); and n a more general model of allocatng ndvsble goods and money where each agent can be assgned at most one object (Thomson, 1990). Theorem 2 also brngs good news n showng the compatblty of populaton monotoncty and no-envy, whch doesn t exst n most other models, for nstance, n the problem of allocatng an nfntely dvsble good over whch agents have sngle-peaked preferences (Thomson, 1995b); n exchange economes when Pareto-eff cency s requred (Km, 2004); n the problem of allocatng ndvsble goods and money where each agent can be assgned at most one object and budgetbalance s requred (Alkan, 1994, Tadenuma and Thomson, 1993). In our companon paper, Yengn (2011a), we characterze the Egaltaran mechansms by soldarty (when costs of some agents ncrease, then no one s better off) and one of the followng: order preservaton (agents wth lower costs are weakly better off), egaltaran-equvalence, or on the subaddtve doman, no-envy. To sum up, n our settng several equty axoms that are ncompatble n many other models can be satsfed jontly. However, the prce of the compatblty s to overwrte any welfare dfferentals arsng from prvate ownershp of self as evdenced by Theorems 1 and 2 here, as well as our results n Yengn (2011a). In other words, for Groves mechansms, jont mplcaton of most farness axoms s welfare-egaltaransm as dsplayed n Table 1. On the subaddtve doman: f N 3, egaltaran-equvalence+no-envy E (Theorem 1). egaltaran-equvalence +no-envy+populaton monotoncty E (Theorem 2). no-envy+soldarty E (Yengn, 2011a). On any doman: order-preservaton+soldarty E (Yengn, 2011a). egaltaran-equvalence+soldarty E (Yengn, 2011a). m k,t bounded-defct+γ compensaton-welfare-lower-bound E (Yengn, 2011a). egaltaran-equvalence+constant-uper-bound-on-defct(no-defct) E (Yengn, 2010). Table 1: Results under assgnment-eff cency and strategy-proofness. One can desgn Egaltaran mechansms that satsfy all of the axoms lsted n Table 1. Hence, n our model, the Egaltaran mechansms appear to be the best canddates to satsfy several dfferent 12 To our knowledge, no smlar characterzaton of welfare-egaltaransm based on these two axoms exsts n other models. 11

12 equty and soldarty requrements as well as generatng bounded defcts and respectng certan welfare bounds (see Yengn, 2011a). These results renforce the mportance of the class E n the economc settng we study. Note that the Pvotal/Vckrey mechansms, whch have been the focus of most of the lterature on the Groves mechansms, volate egaltaran-equvalence, populaton monotoncty, soldarty, bounded-defct, and no-defct. 5 Appendx For each N, let h be a real-valued functon defned over the unon N N C N such that for each N N wth N and each c C N, h depends only on c. Let h = (h ) N and H be the set of all such h. Let G h,τ (A τ, t h,τ ) be a mechansm such that for each N N and each c C N, (A τ (c)) N s an eff cent-assgnment for c and for each N, t h,τ (c) = c j (A τ j (c)) + h (c ) = W (c) + c (A τ (c)) + h (c ). (2) j N\{} The mechansm G h,τ s called a Groves mechansm. The transfer of each agent determned by a Groves mechansm has two parts. Frst, each agent pays the total cost ncurred by all other agents at the assgnment chosen by the mechansm. Second, each agent receves a sum of money h (c ) R that does not depend on her own cost c. We have the followng equaton whch wll be of much use. 13 For each N N, each N, and each c C N, u(g h,τ (c); c ) = W (c) + h (c ). (3) Note that E γ,τ = G h,τ, where for each N N, each N, and each c C N, h (c ) = γ(n). Smlarly, Êγ,τ = G h,τ such that for each N N and each N, f N > 2, then h (c ) = γ(n) for each c C N, (4) f N = 2, then Proof of Theorem 1: () ether h (c ) = γ(n) for each c C N, (5) () or h (c ) = γ(n) + c j (A), where j N\{}, for each c C N. (6) If Part: Pck an assgnment-eff cent and strategy-proof mechansm. By Lemma 2, t s a Groves mechansm G h,τ for some h H and τ T. Let G h,τ Ê. Then, there s γ : N R such that G h,τ = Êγ,τ as descrbed above. We wll show that G h,τ s (a) envy-free on the subaddtve doman, and (b) egaltaran-equvalent on every doman. (a) Assume, by contradcton, that G h,τ s not envy-free on the subaddtve doman. Then, there are N N, c Csub N, and {, j} N such that u(gh,τ (c); c ) < u(g h,τ j (c); c ). Ths nequalty, (2), and (3) together mply W (c) + h (c ) < c (A τ j (c)) + t h,τ j (c), = c (A τ j (c)) W (c) + c j (A τ j (c)) + h j (c j ). (7) 13 By (3), for each h H, the mechansms n {G h,τ } τ T are Pareto-ndfferent. 12

13 Frst, consder cases (4) and (5). By (7), c (A τ j (c)) < c j(a τ j (c)). Ths nequalty mples that (c) and snce c s subaddtve, we have A τ j c (A τ (c) A τ j (c)) c (A τ (c)) + c (A τ j (c)) < c (A τ (c)) + c j (A τ j (c)). Then, t would be less costly than W (c), f was assgned (A τ (c) Aτ j (c)) and j was assgned no task, whch contradcts that A τ (c) s an eff cent assgnment. Now, consder case (6). By (7), c j (A) < c (A τ j (c)) + c j (A τ j (c)) + c (A). (8) By (8), A τ j (c) A. Snce c s subaddtve and A = Aτ (c) Aτ j (c), then c (A) c (A τ (c))+c (A τ j (c)). Ths nequalty and (8) together mply c j (A) < c (A τ (c)) + c j(a τ j (c)). Then, t would be less costly than W (c), f j was assgned all the tasks, whch contradcts that A τ (c) s an eff cent assgnment. (b) Now, we show that G h,τ s egaltaran-equvalent. Let N N and c C N. Frst, consder cases (4) and (5). By (3), for each N, u(g h,τ (c); c ) = W (c) + γ(n). Let R(c) = and r(c) = W (c)+γ(n). Then, for each N, u(g h,τ (c); c ) = c (R(c))+r(c). Hence, G h,τ s egaltaran-equvalent. Next, consder case (6). By (3), for each N, u(g h,τ for j N\{}. Let R(c) = A and r(c) = W (c) + (c); c ) = W (c) + γ(n) + c j (A) c (A) + γ(n). Then, for each N, u(g h,τ (c); c ) = c (R(c)) + r(c). Hence, G h,τ s egaltaran-equvalent. Only-f Part: Pck an assgnment-eff cent and strategy-proof mechansm. By Lemma 2, t s a Groves mechansm G h,τ for some h H and τ T. Let G h,τ be egaltaran-equvalent and envy-free on the subaddtve doman. By Theorem 1 n Yengn (2010) (adapted to our varable populaton settng), f a Groves mechansm s egaltaran-equvalent, then for economes wth dfferent populatons, dfferent reference set of tasks can be chosen; but for economes wth the same populaton N, the same reference set of tasks R(N) must work. Moreover, by Yengn (2010), a Groves mechansm s egaltaran-equvalent f and only f for each N N, there are a real number γ(n) R and a reference set R(N) 2 A such that for each N, h (c ) = γ(n) + c j (R(N)) for each c C N. (9) j N\{} By Theorem 1 n Pápa (2003) (adapted to our varable-populaton and undesrable-objects settng), f G h,τ s envy-free on the subaddtve doman, then there s a lst of functons ndexed by populatons, {σ N } N N wth σ N : R + R such that for each N N, each N, and each c C N sub, N By (9) and (10), for each N N and each par {, j} N, h (c ) = σ N (W (c )), (10) σ N (W (c )) σ N (W (c j )) = c j (R(N)) c (R(N)) for each c C N sub. (11) We wll prove that G h,τ = Êγ,τ by showng, on the subaddtve doman, the equvalence of equaton (9) to (4), (5), and (6). To acheve ths, we need to consder the followng two cases: Case 1: For each N N wth N > 2, there s γ(n) R such that (9) holds for R(N) = for each c Csub N. That s, equalty (9) s equvalent to (4). 13

14 Proof of Case 1: Let N N wth N > 2. By egaltaran-equvalence, there are γ(n) R and R(N) 2 A such that (9) holds for each c C N sub. Clam: For each c C N sub and each par {, j} N, c (R(N)) = c j (R(N)). Note that the Clam holds for each c Csub N ; and by (9), R(N) s same for each c CN sub. Ths s possble f and only f R(N) =. Ths would prove that Case 1 holds. Now, we wll show that the Clam s true: Assume, by contradcton to the clam, that there s c Csub N such that for some par {, j} N, c (R(N)) c j (R(N)). (12) Wthout loss of generalty, let W (c j ) W (c ). Let c j (R(N)) c (R(N)). Let ĉ Cad N follows: () for each A 2 A, ĉ (A) = A W (c j) A, () there s ε > max{, 0} such that for each A 2 A, ĉ j (A) = A ( W (c ) A ) + + ε, be as () for each k N\{, j} and each A 2 A, ĉ k (A) = A W (c ) A. Note that + ε > max{, 0}. Hence, for each A (2 A \ ), ĉ j (A) > ĉ k (A). By () and (), for each A (2 A \ ), ĉ j (A) ĉ (A) = ( ) W (c ) W (c j ) A + + ε, A >. (13) By () and (), (I) W (ĉ ) = ĉ k (A) = W (c ) for some k N\{, j}. Snce W (c j ) W (c ), by () and (), (II) W (ĉ j ) = ĉ (A) = W (c j ). By (I), (II), and (11), c j (R(N)) c (R(N)) = = ĉ j (R(N)) ĉ (R(N)). Ths equalty and (13) together mply R(N) =. Ths mples c j (R(N)) = c (R(N)), whch contradcts (12). Hence, Clam must be true. Case 2: For each N N wth N = 2, there s γ(n) R such that (9) holds ether for R(N) = for each c Csub N ; or for R(N) = A for each c CN sub. That s, equalty (9) s equvalent to (5) or (6). Proof of Case 2: Let N N wth N = 2. Wthout loss of generalty, let N = {, j}. By egaltaran-equvalence, there are γ(n) R and R(N) 2 A such that (9) holds for each c Csub N. We wll show that R(N) {, A}. Let c Csub N and ĉ = (c, ĉ j ) Csub N be such that (I) ĉ j(a) = c j (A) and (II) for each A (2 A \{, A}), ĉ j (A) c j (A). By (I), W (ĉ ) = c j (A) = W (c ). Ths equalty and (10) together mply h (ĉ ) = h (c ). Ths equalty and (9) together mply ĉ j (R(N)) = c j (R(N)). Ths equalty and (II) together mply that R(N) {, A}. Ths proves Case 2. 14

15 Lemma 3. A Groves mechansm G h,τ s populaton monotonc f and only f for each par {N, N } N such that N N, each N, and each c C N, h (c ) h (c N \{}). (14) Proof of Lemma 3: Let h H be as n (14). Let {N, N } N be such that N N, N, and c C N. Snce W (c N ) W (c), by (3) and (14), u(g h,τ (c); c ) u(g h,τ (c N ); c ). Hence, G h,τ s populaton monotonc. Conversely, let G h,τ be a populaton monotonc Groves mechansm. Assume, by contradcton, that there are {N, N } N such that N N, N, and c C N for whch h (c ) < h (c N \{}). (15) For each A (2 A \ ), let P (A) = {A 1, A 2,..., A k } be a partton of A nto k A non-empty subsets. That s, for each par {A, A } P (A), A A = and A P (A) A = A. Let P(A) {P (A) : P (A) {A}} be the set of all parttons of A except for the partton {A}. Let ĉ be such that for each α A, ĉ ({α}) = ĉ (A) = mn{ mn {c j(a)}, mn { j N\{} P (A) P(A) mn {c j({α})}, and each A 2 A, j N\{} A P (A) ĉ (A )}}. (16) As an llustraton of calculaton of (16), suppose A = {α, β, θ}, N = {, k, l}, and c C N. The followng table presents c and the correspondng ĉ. {α} {β} {θ} {α, β} {α, θ} {β, θ} A c k c l ĉ To see how ĉ s calculated, consder A = {a, β}. There are two ways to partton A : P (A) = {{a}, {β}} and P (A) = {A}. Here, P(A) = {P (A)} and ĉ (A ) = ĉ ({a}) + ĉ ({β}) = 15. Snce mn {c j(a)} = 13 < 15, by (16), ĉ (A) = 13. j N\{} A P (A) Let ĉ = (ĉ, c ) C N. Snce ĉ = c and ĉ N \{} = c N \{}, by (15) h (ĉ ) < h (ĉ N \{}). (17) By (16), for each A 2 A, ĉ (A) mn {c j(a)}. Ths nequalty and the fact that ĉ s addtve j N\{} or subaddtve together mply W (ĉ) = W (ĉ N ) = ĉ (A). These equaltes, populaton monotoncty, and (3) together mply h (ĉ ) h (ĉ N \{}), whch contradcts (17). Proof of Theorem 2: Let E γ,τ be an Egaltaran mechansm such that γ s as n (1). By Lemmas 1 and 2, and Theorem 1, E γ,τ s an egaltaran-equvalent Groves mechansm that s envy-free on the subaddtve doman. By Lemma 3, E γ,τ s populaton monotonc. Conversely, let G h,τ be a Groves mechansm that s egaltaran-equvalent, populaton monotonc, and envy-free on the subaddtve doman. By Lemma 3, h H s as n (14). By Theorem 1, G h,τ belongs to Ê. 15

16 Assume that G h,τ (Ê\E). Then, by (6), there s N = {, j} N such that h (c j ) = c j (A) + γ(n ) for each (c, c j ) C N sub. Let N = {, j, k} N. Then, by (4), h (c j, c k ) = γ(n) for each c = (c, c j, c k ) C N sub. Let ε > max{0, γ(n ) γ(n)}. Let ĉ j C ad be such that for each A (2 A \{ }), ĉ j (A) = A ( γ(n) γ(n ) + ε ). (18) A Snce (c, ĉ j ) C N sub, by (6), (I) h (ĉ j ) = ĉ j (A) + γ(n ). Snce (c, ĉ j, c k ) Csub N, by (4), (II) h (ĉ j, c k ) = γ(n). By (18), ĉ j (A) = γ(n) γ(n ) + ε. Ths equalty, (I), and (II) together mply h (ĉ j ) = γ(n) + ε > h (ĉ j, c k ). Ths nequalty contradcts (14). Hence, G h,τ / (Ê\E). Let G h,τ E. Then, there s γ : N R such that G h,τ = E γ,τ. Note that for each par {N, N } N such that N N, each N, and each c C N, h (c ) = γ(n) and h (c N \{}) = γ(n ). By (14), γ(n) γ(n ). Hence, E γ,τ E γ where γ : N R s as n (1). 6 References Alkan A (1994): Monotoncty and envyfree assgnments. Economc Theory 4: Arneson R (1989): Equalty of opportunty for welfare. Phlosophcal Studes 56: Ashlag I, Serzawa S (2011): Characterzng Vckrey allocaton rule by anonymty. Socal Choce and Welfare, forthcomng. Atlamaz M, Yengn D (2008): Far Groves mechansms. Socal Choce and Welfare 31: Clarke EH (1971): Mult-part prcng of publc goods. Publc choce 11: Chun Y (2006): No-envy n queueng problems. Economc Theory 29: Cohen GA (1986): Self-ownershp, world ownershp, and equalty. Socal Phlosophy and Polcy. 3: Cohen GA (1989): On the currency of egaltaran justce. Ethcs 99: Danel T (1978): Ptfalls n the theory of farness Comment. Journal of Economc Theory 19: Dworkn R (2000): Soveregn vrtue: the theory and practce of equalty. Harvard Unversty Press. Cambrdge, Mass. Foley D (1967): Resource allocaton and publc sector. Yale Economc Essays 7: Gbbard A (1973): Manpulaton of votng schemes: A general result. Econometrca 41: Gnés M, Marhuenda F (2000): Welfarsm n economc domans. Journal of Economc Theory 93: Groves T (1973): Incentves n teams. Econometrca 41: Holmström B (1979): Groves scheme on restrcted domans. Econometrca 47: Ohseto S. (2006): Characterzatons of strategy-proof and far mechansms for allocatng ndvsble goods. Economc Theory 29: Km H (2004): Populaton monotonc rules for far allocaton problems. Socal Choce and Welfare 23: Kolm SC (1996): Modern theores of justce. MIT Press. Cambrdge, Mass. Mouln H, Roemer J (1989). Publc ownershp of the external world and prvate ownershp of self. Journal of Poltcal Economy. 97: Pápa S (2003): Groves sealed bd auctons of heterogeneous objects wth far prces. Socal Choce and Welfare 20: Pazner A, Schmedler D (1978): Egaltaran equvalent allocatons: A new concept of economc equty. The Quarterly Journal of Economcs 92:

17 Porter R, Shoham Y, Tennenholtz M (2004): Far mposton. Journal of Economc Theory 118: Rawls J (1971): A theory of justce. Belknap Press. Cambrdge, MA. Roemer J (1986): Equalty of resources mples equalty of welfare. Quarterly Journal of Economcs 101: Roemer J (1995): Equalty and responsblty. Boston Revew 20:3-16. Roemer J (1996): Theores of dstrbutve justce. Cambrdge (Mass.), Harvard Unversty Press. Roemer J (1998): Equalty of opportunty. Cambrdge (Mass.), Harvard Unversty Press. Satterthwate M (1975): Strategy-proofness and Arrow s condtons: Exstence and correspondence theorems for votng procedures and socal welfare functons. Journal of Economc Theory 10: Sen A (1980): Equalty of what? In Tanner Lectures on Human Values, Vol. I. Cambrdge Unversty Press. Tadenuma K, Thomson W (1993): The far allocaton of an ndvsble good when monetary compensatons are possble. Mathematcal Socal Scences 25: Thomson W (1983): The far dvson of a fxed supply among a growng populaton. Mathematcs of Operatons Research 8: Thomson W (1990): On the non exstence of envy-free and egaltaran-equvalent allocatons n economes wth ndvsbltes. Economcs Letters 34: Thomson W (1995): Populaton-monotonc solutons to the problem of far dvson when preferences are sngle-peaked. Journal of Economc Theory 5: Thomson W (1996): On the problem of tme dvson. Unpublshed manuscrpt, Unversty of Rochester. Vckrey W (1961): Counterspeculaton, auctons, and compettvely sealed tenders. Journal of Fnance 16:8-37. Yengn D (2010): Egaltaran-equvalent Groves mechansms n the allocaton of heterogeneous objects. Socal Choce and Welfare,onlne frst. Yengn D (2011a): Characterzng welfare-egaltaran mechansms wth soldarty when valuatons are prvate nformaton. The B.E. Journal of Theoretcal Economcs, forthcomng. Yengn D (2011b): Identcal preferences lower bound for allocaton of heterogenous tasks and NIMBY problems. Journal of Publc Economc Theory, forthcomng. 17