1、Efficient Egalitarian EquivalentAllocations over a Single GoodMarco LiCalzi Antonio NicoloDept. of Applied Mathematics Dept. of EconomicsUniversity of Venice University of Padua(October 2005)Abstract. This paper studies efficient and egalitarian allocations over asingle heterogeneous and infinitely
2、divisible good. We prove the existence ofsuch allocations using only measure-theoretic arguments. Under the addi-tional assumption of complete information, we identify a sufficient conditionon agents preferences that makes it possible to apply the Pazner-Schmeidlerrule for uniquely selecting an effi
3、cient egalitarian equivalent allocation. Fi-nally, we exhibit a simple procedure that implements the Pazner-Schmeidlerselection in a subgame-perfect equilibrium.Keywords: egalitarism, equity, fair division, implementation, no-envy.JEL Classification Numbers: D630, C700, D300.MathSci Subject Classifi
4、cation: 91B32, 91B54, 28B05.Correspondence to:Antonio Nicolo Department of Economics, University of PaduaVia del Santo 3337123 Padua, ItalyPhone: +39 049-827-4285Fax: +39 049-827-4211E-mail: antonio.nicolounipd.it We thank the seminar audience at Boston University for their comments. The re-search o
5、n this paper greatly benefited from the warm hospitality of Boston University.Financial support from MIUR is acknowledged.1 IntroductionA fair division problem arises when two or more agents are called to divide agood over which they claim equal rights. The oldest known examples includeAbraham and L
6、ot arguing over land division (Genesis 13), and Prometheusand Zeus disputing a pile of meat (Hesiods Theogony). A recent dramaticexample is the carving of Bosnia and Herzegovina as an independent entitywithin the Dayton Accords that put an end to a 3-year civil war over thespoils of the former Repub
7、lic of Yugoslavia.There are several situations where the solution of a fair division problemcannot call on instruments like prices, monetary compensations, or auctions.This may be due to liquidity constraints; or to the psychological difficultyto bring a dispute down to monetary evaluations; or to p
8、olitical constraints,as in the case of Bosnia and Herzegovina; or to the presence of judiciallyenforceable rights such as under U.S. law “to seek partition in kind,or physical division, of jointly owned land”; see Miceli and Sirmans (2000).This paper studies the problem of fair division when the dis
9、pute mustbe resolved using division in kind. We are interested in devising a proce-dure that can help the parties to reach an outcome that is both fair andefficient. We assume that the disputed object is a single infinitely divisi-ble good over which agents have heterogeneous preferences and that th
10、ereare no consumption externalities. The canonical example is the divisionof a cake, when agents have different (additive) preferences over differentslices; see Steinhaus (1948). A less obvious example is the case of a finite(or countable) number of homogeneous infinitely divisible goods, where thea
11、ggregate endowment is viewed as the single heterogeneous good at stake;see Chambers (2005, Section 5).There are two main ordinal concepts in the fair division literature. Thefirst is the envy-free principle which states that each party should (weakly)prefers its share to anyone elses. This was propo
12、sed by Gamow and Stern(1958, pp. 117119), but became widely known after Foley (1967). Anyefficient envy-free allocation is ex post stable because no one desires to ex-change what he received with anyone elses share. However, this solutionconcept suffers from a multiplicity problem that makes it less
13、 satisfactoryfrom an ex-ante, or procedural, point of view. There are in general manyefficient envy-free allocations, and each of them provides different payoffsto the agents. Therefore, they are likely to disagree on how to select oneamong these allocations. The divide-and-choose mechanism under co
14、mpleteinformation, for instance, selects among all the efficient envy-free allocationsthe division that maximizes the payoff to the divider so conflict is likelyto shift over how the divider is chosen.An alternative normative concept is the egalitarian equivalent criterionwhich states that each part
15、y should be indifferent between getting his shareand some reference bundle, identical for all agents. This was introduced by1Pazner and Schmeidler (1978) to overcome the problem that efficient no-envyallocation may not exist at all for economies with non-convex preferencesor with production. As diff
16、erent reference bundles lead to different shares,the multiplicity problem over efficient and egalitarian equivalent allocationsresurfaces in the choice of the reference bundle. Pazner and Schmeidler(1978) suggests to circumvent the difficulty by focusing only on those ref-erence bundles that are pro
17、portional to the total endowment. (Assumingefficiency, this leads to a unique selection.) Sprumont and Zhou (1999) ax-iomatizes this “Pazner-Schmeidler” rule for exchange economies with con-vex preferences where the endowment is a finite number of homogeneousinfinitely divisible goods.It is not imme
18、diately obvious how to extend the “PaznerSchmeidler”rule when the endowment is a single heterogeneous good. Consider thedivision of a contested cake among a group of people who have equal claimson it. The agents may evaluate the value of a piece of the cake along differentattributes: its crust, its
19、filling, its weight, the number of strawberries on it,and so on. The challenge is how to make sure that all relevant attributes areproportionally represented in the reference bundle. Moreover, if the partiesthemselves agree that a criterion should be represented whenever an agentcares about it, is t
20、here a way to elicit this strategic information from eachparty?We answer these questions under the assumption that each agent canpartition the disputed cake into a finite (or countable) number of parcelsthat he (but not necessarily the other parties) view as homogeneous. Theintuition is the followin
21、g. Each agent divides the cakes in as many parcelsas he likes. Equally sized morsels from the same parcel carry the same util-ity to the agent, so that each parcel is a homogeneous good for the agent.Note that equally sized morsels from two different parcels may carry dif-ferent utility to him; and,
22、 similarly, equally sized morsels from an agentsparcel may give different utilities to another agent. Consider now the com-mon refinement of all the agents partitions. Each parcel in this new andfiner partition is a homogeneous good for each party. This brings us back tothe standard setting for the
23、Pazner-Schmeidler rule. Hence, we choose thereference bundle among those that are proportional to this common refine-ment. Under efficiency, the selection of the reference bundle to define theegalitarian equivalent allocation is again unique.Clearly, in the search for a procedure to implement the ef
24、ficient egali-tarian equivalent allocation with respect to this special reference bundle, wealso need to overcome the difficulty to devise a game in which each agentmust announce his own (truthful) partition of the cake. Lying over onespartition may lead to a different reference bundle and hence to
25、a bettershare for the liar. We provide a simple procedure which implements thedesired outcome as a subgame-perfect equilibrium, under the assumptionthat agents have complete information about their preferences. The proce-2dure is simple in the sense of Thomson (2005). It generalizes a mechanismsugge
26、sted in Crawford (1979) as ameliorated in Demange (1984). Theirmechanism derives an efficient egalitarian equivalent allocation for a finitecollection of homogenous goods. Our procedure simultaneously “discovers”the right way to partition the heterogeneous good.The paper is organized as follows. Sec
27、tion 2 describes our model, whichis a standard version of the classical setup for cake division problems. Sec-tion 3 proves the existence of efficient egalitarian equivalent allocations fora single heterogeneous good using only measure-theoretic assumptions; theonly other existence result we are awa
28、re of is more general in scope butrequires additional topological assumptions; see Berliant et alii (1992). Sec-tion 4 describes the assumptions that define the economic environment overwhich our procedure can be applied. Section 5 states the implementationresult. Long proofs are relegated in the ap
29、pendix.2 The modelOur model is an abstraction of the classical problem where a cake (or a pieceof land) must be allocated among several agents. There is a measurablespace (,F), where is the object to be divided among the n agents andF is a -algebra over . We say that an element of F is a parcel and
30、thatan F-measurable subset of a parcel is a morsel, which is a nicer term than“subparcel”. Any subset of mentioned in the following is an element ofF, and hence a parcel.For n 2, let N = 1,2,.,n be the (finite) set of agents. Agentshave preferences over parcels of . Each agent i is endowed with a ut
31、ilityfunction ui : F R+ that is a nonatomic probability measure on F. (Sincepreferences are invariant up to a positive rescaling of the utility function,ui() = 1 is only a normalization.) A measure ui is nonatomic if, for eachparcel A and each x in (0,u(A), there exists another parcel B A suchthat u
32、i(B) = x. Hence, the range of each ui is the (convex) interval 0,1.A utility function u over parcels is absolutely continuous with respect toanother measure over F if (A) = 0 implies u(A) = 0 for any parcel A.Clearly, any utility function ui is absolutely continuous with respect to themeasure = summ
33、ationtextni=1 ui. We make the assumption that the utility functionsare mutually absolutely continuous; that is, if ui(A) = 0 for some parcelA, then uj(A) = 0 for any agent j. Since agents agree on the null parcels,we say that a parcel has zero (or positive) measure without specifying ameasure.An (or
34、dered) partition P = (p1,.,pk) of constituted only by parcelsis called a parceled k-partition. An allocation X = (x1,.,xn) is a parceledn-partition, where xi is the parcel assigned to agent i in N. An allocation Xis efficient (or weakly efficient, respectively)ifthereexistsnootherallocation3Y = (y1,
35、.,yn) such that ui(yi) ui(xi) for all i, with the strict inequalityholding for some i (or ui(yi) ui(xi) for all i). Any efficient allocation isalso weakly efficient. The converse is true under our assumption that agentshave preferences that are mutually absolutely continuous; see Akin (1995,Lemma 9)
36、.There are several criteria to evaluate the fairness of an allocation. Forinstance, an allocation X is proportional if ui(xi) (1/n) for all i; andit is equitable if ui(xi) = uj(xj) for all i and j. These two notions offairness hinge on the demanding assumption that interpersonal preferencesare compa
37、rable. The main fairness criteria based on ordinal preferences aretwo. An allocation X is envy-free if ui(xi) ui(xj) for all i and j, and itis egalitarian equivalent (for short, EE) if there exists a reference parcel Asuch that ui(xi) = ui(A) for all i. Any envy-free allocation is proportional,but t
38、he converse is true only if n = 2.Under our setup, the following existence results are known. Dubins andSpanier (1961) proves the existence of efficient and proportional allocationsfor preferences which may not be mutually absolutely continuous. It notesthat adding this assumption ensures that all e
39、fficient allocations are equi-table. Maccheroni and Marinacci (2003) gives sufficient conditions to extendthe existence result for proportional allocations when the utility functionsare subadditive. Weller (1985) proves the existence of weakly efficient andenvy-free allocations; efficiency follows i
40、mmediately under mutual absolutecontinuity.More existence results are known under related setups, which addition-ally assume that is a subset of Rk. For instance, Stromquist (1980) provesthe existence of envy-free allocations for a planar cake using a larger classof preferences, but restricting the
41、set of admissible partitions. Berliant etalii (1992) has several results. It gives a stronger version of Wellers (1985)result assuming that the utility functions are absolutely continuous with re-spect to the Lebesgue measure. And it proves the existence of efficient andegalitarian equivalent alloca
42、tions for a general class of preferences that musthowever be continuous in a complicated topology described in Berliant andDunz (2004).3 Existence of efficient EE partitionsThis section proves the existence of efficient and egalitarian equivalent al-locations in our setup. Contrary to Berliant et al
43、ii (1992), we make notopological assumptions so that the proof does not rely on the structure ofRk.We need a few definitions. Let u = (u1,.,un) be the vector of the nagents utility functions on the measurable space (,F). The set R(u) =u(A) : AF in Rn is the range of u. The range of u spans the vecto
44、r4of utilities that the agents can achieve if they are all given the same par-cel. By assumption, each ui is a nonatomic probability measure. Then, byLyapunovs convexity theorem, R(u) is a compact and convex subset of Rn.Let be the set of all parceled n-partitions of . The set RP(u) =(u1(x1),., un(x
45、n) : X in Rn is the partition range of u. The parti-tion range is sometimes called the Individual Pieces Set; see Barbanel (2005).The partition range of u spans the vector of utilities that the agents canachieve by dividing up the cake according to some allocation. Dvoretzky etalii (1951) derives fr
46、om Lyapunovs convexity theorem a more general result,which implies that the partition range is also a compact and convex subsetof Rn.We also call on the following three lemmata, which assume that u isa (finite) vector of nonatomic probability measures. The first two resultscorrespond to Corollary 1.
47、1 and Lemma 5.3 in Dubins and Spanier (1961),respectively. They do not require preferences that are mutually absolutelycontinuous.Lemma 1 Given an integer k and positive weights 1,.,k with summationtextj j =1, there exists a parceled k-partition X = (x1,.,xk) such that ui(xj) = jfor all i = 1,.,n an
48、d j = 1,.,k.For k = n and j = 1/n for all j, this implies that the partition range of ualways contains the point (1/n,.,1/n). The next lemma, instead, concernsthe range of u and implies that it always contains the whole diagonal.Lemma 2 For any t in 0,1 there exists a parcel At such that ui(At) = tf
49、or each i.Our last lemma characterize efficient allocations when preference are mu-tually absolutely continuous; see Barbanel and Zwicker (1997, Theorem 1).Section 7C in Barbanel (2005) discusses the case where mutual absolutecontinuity does not hold.Lemma 3 If preferences are mutually absolutely continuous, an allocationis efficient if and only if it maximizes a convex combination of the utilityfunctions.The following is the main result in this section.Theorem 1 There exists at least one allocation which is eff
