Fair and Efficient Division among Families
FFair and Efficient Division Among Families
Sophie Bade and Erel Segal-HaleviOctober 21, 2020
Abstract
Is efficiency consistent with fairness? Our approach to this ques-tion concerns the case where multiple individuals with diverse prefer-ences are bound to consume the same bundle. Families are our leadexample: the father, mother and children get to consume the samegarden, kitchen, and vacations. We adapt each of the three most pop-ular principles of fairness: envy-freeness, egalitarian-equivalence andthe fair-share guarantee, in three different ways to the world of fami-lies. For any given criterion of fairness, an allocation is unanimous-fair if it is fair according to every individual member of each family, it is aggregate-fair if it is fair according to a particular aggregation of allfamily members’ preferences, and it is collective-fair if it is fair ac-cording to the family’s — typically incomplete — preferences, thatrank a bundle above another bundle if and only if each member of thefamily ranks the first bundle above the second.While efficiency is generally incompatible with unanimous egali-tarian equivalence, and incompatible with unanimous envy-freenessin economies with three or more families, unanimously envy-free ef-ficient allocations always exist in economies with just two families.The unanimous fair share guarantee is easy to achieve: Under genericconditions the set of efficient allocations with the fair share guaranteecontains some collectively envy-free and some collectively egalitarianequivalent allocations. We use modified versions of the traditionalmarket equilibrium approach and lexicographic optimization to es-tablish our results. a r X i v : . [ ec on . T H ] O c t Introduction
Several siblings inherit their parents’ goods. The problem whether thesegoods can be shared fairly and efficiently has been extensively studied in eco-nomics. A classic positive result by Foley (1967) shows that Pareto-efficientdivisions which are also fair in the sense that no sibling envies another one,exist under mild conditions. But now assume that each sibling is marriedand has children, so that the goods have to be divided among families, whosemembers consume the same bundle. All family members live in the samehouse, use the same furniture, and enjoy the same trees and flowers in thebackyard. Standard theory then does not apply since the family membersmay have different preferences. While the husband in one family may preferhis family’s bundle to bundles obtained by his siblings’ families (and thusfeel no envy), his wife may envy her sister in law’s family.As another example, consider the division of disputed resources, such asland, water, or oil, among neighboring countries. Different citizens of thesame country may have different preferences over these resources. Giventhat different citizens need not agree on the fairness of any given dispute-settlement agreement, such agreements may be hard to achieve. The sharingof office space, administrative staff and computing time among a set of teamsin a company provides yet another example, where different members of anygiven team may disagree on the fairness of an allocation.These examples raise two questions: (a) What divisions should be con-sidered “fair” in an economy of families? (b) When do fair and efficientallocations exist?We consider three classic criteria of fairness. An allocation has the fair-share guarantee (FS) if no agent likes the average bundle better thantheir own bundle; if each agent likes their bundle at least as much as anyother agent’s bundle then the allocation is envy-free (EF) ; the allocation is egalitarian-equivalent (EE) if there exists a reference bundle that each agent Also known as proportionality . For any of the three fairness criteria (FS, EF or EE), we would ideallylike all members in all families to agree that the allocation is fair accordingto the given criterion. An allocation that is fair according to every individualfamily member is unanimous-fair . Theorem 3.4 shows that such unanimous-FS Pareto optima exist under mild compactness assumptions. To prove thisresult we normalize all agents utilities of the average bundle to 1 and showthat an allocation that lexicographically maximizes the smallest utilities isunanimous-FS as well as Pareto optimal. Propositions 4.2 and 5.1 show thatefficiency may clash with respectively unanimous-EF or unanimous-EE. Weuse plain vanilla examples with strictly monotonic and convex preferencesto prove both these propositions. While the non-existence of unanimous-EE Pareto optima (Proposition 5.1) applies as long as there are at least twofamilies, Theorem 4.1 shows that Proposition 4.2 applies only when there areat least three families; any economy with just two families has a unanimous-EF Pareto optimum.Instead of requiring that allocations are fair from the vantage point ofeach individual member of a family we could require that allocations are fairaccording to some particular aggregations of all family members’ preferences.We could, for example, assume that families use the Nash bargaining or theutilitarian criterion to mitigate conflict between its members. An allocationis then aggregate-fair if it satisfies the selected fairness notion according tothe aggregate preferences of all families. Since unanimous fairness impliesaggregate fairness for any aggregation, aggregate-FS Pareto optima alwaysexist. Since the family allocation problem with aggregate preferences reducesto the well-studied individual allocation problem, it is therefore easy to findPareto optima that are either aggregate-EF or aggregate-EE. Such allocationsmay, however, violate the — arguably most basic criterion of fairness — The rationale behind EE is that, from an egalitarian perspective, an ideal division isone in which each agent gets an equal bundle. However, an allocation in which each agentgets exactly 1 /n of all resources is usually not efficient. To reconcile egalitarianism withefficiency, an allocation is defined as egalitarian equivalent if there exists an (infeasible)allocation where all agents consume the same bundle, such that each agent is indifferentbetween that bundle and their consumption in the allocation (Pazner and Schmeidler,1978). collective preference which ranksone bundle above another bundle if the first bundle is preferred to the secondby all its members. Any allocation that is fair according to these collectivepreferences is collective-fair . Theorem 4.4 then shows that collective-EF andunanimous-FS Pareto optima exist for any number of families whose membershave locally-non-satiated strictly convex preferences. In analogy, Theorem5.2 shows that collective-EE and unanimous-FS Pareto optima exist for anynumber of families whose members have strictly monotonic strictly convexpreferences. Our strategy of proof uses the fact that any aggregate-fairallocation is also collective-fair (as shown in Lemma 3.3). The specific aggre-gators we use to establish Theorems 4.4 and 5.2 assume that families eitheruse the Rawlsian notion of fairness of Nash-bargaining to aggregate theirmembers’ preferences into some complete and transitive joint preference ofthe family.The strong nexus between fairness and market equilibrium in standardeconomies, where market equilibria from equal endowments have the fairshare guarantee and are envy-free (Foley, 1967), loses its force in the economyof families. Proposition 4.3 shows that, even if unanimous-EF Pareto optimaexist, it may be impossible to find them as market equilibria from equalendowments, as families with divergent preferences may need larger budget-sets than families with homogeneous preferences to attain a unanimous-EFallocation. Say three families care about only two goods: donations to charityand the acquisition of modern art. For a family which cares about both notto envy two other families who respectively only care about charity or art,the family with members of either sort needs to spend at least as muchon art and charity as do the families who only spend on only one of thetwo. So markets with equal endowments do not guarantee fair outcomes ineconomies with families. Safeguards for families with discordant preferencesor for individuals within families may be required to obtain fair allocations. Without strict convexity, it is possible to guarantee any two of the three conditions(collective-EE, unanimous-FS and Pareto-optimality) but not all three (Theorem 5.3). Related work
The modern study of fairness in economics was initiated by Steinhaus (1948),who proved the existence of fair share allocations of a heterogeneous good(“cake”). Since then, fairness has been extensively studied in economics(Young, 1995; Moulin, 2004; Thomson, 2011) as well as in other disciplinessuch as mathematics and computer science.The existence of envy-free
Pareto-optima was initially proved as a conse-quence of the existence of market equilibrium (Foley, 1967). If all preferencesare convex and strictly monotonic, then there exists a market equilibriumfrom equal incomes, and it is both Pareto-optimal and envy-free. Varian(1974) showed that no-envy Pareto-optima may exist even when a compet-itive equilibrium does not. The convexity of preferences can be replaced bya different condition: for each weakly-Pareto-optimal vector of utilities, theset of allocations with this vector of utilities is a singleton. Svensson (1983)and Diamantaras (1992) then showed that it is sufficient that the latter setof allocations be convex or respectively contractible. Diamantaras (1992)’sresult even applies to economies with public goods. Svensson (1994) andBogomolnaia et al. (2017) showed that envy free Pareto optima exist underyet further relaxed conditions.The egalitarian equivalence criterion was introduced by Pazner and Schmei-dler (1978). They proved that a Pareto-optimal egalitarian-equivalent allo-cation exists even in economies with production, in contrast to envy-freePareto-optima (Vohra, 1992).
Formally the families in this paper are agents with incomplete preferences(representable by vector valued utilities). Viewed through this lens, our papercontributes to the literature on behavioral welfare economics pioneered byFon and Otani (1979); Bernheim and Rangel (2007, 2009); Mandler (2014,2017); Fleurbaey and Schokkaert (2013). Consider an economy where allagents view all options through a set of different frames, as suggested by5alant and Rubinstein (2008). The notions of unanimous and collective fair-ness then respectively require that an allocation is fair according to eachframe of an agent or according to the incomplete preferences of the agent.If the agent uses different frames in different points of time or to evalu-ate uncertainty we can apply our results to economies where agents have β − δ -preferences following Laibson (1997); ODonoghue and Rabin (1999) orexhibit Knightian uncertainty following Bewley (2002). The existence of fair allocations when some of the goods are public has beenstudied e.g. by Diamantaras (1992); Diamantaras and Wilkie (1994, 1996);Guth and Kliemt (2002). There, each good is either private (consumed by asingle agent) or public (consumed by all agents).We conversely consider club goods , that is goods that are public inside afamily — but private outside (i.e, all family members enjoy the same bundle,but members of other families cannot enjoy it). Club goods were popularizedby Buchanan (1965) and studied in various contexts. The literature on clubgoods studies questions such as the optimal number of members in a club,optimal number of clubs, optimal quantity of club-good provision, pricingpolicies and exclusion mechanisms (Sandler and Tschirhart, 1980; Hillman,1993; Sandler and Tschirhart, 1997; Loertscher and Marx, 2017; Macken-zie and Trudeau, 2019). Some papers have studied fairness in the contextof discrimination prevention, i.e., when competition between different clubsprevents clubs from discriminating against some of its members by chargingthem higher fees (Sandler and Tschirhart, 1980). As far as we know, fairallocation of goods among different clubs has not been considered yet. A modern example of a club good is information: information is partlyexcludable (via intellectual property law), but once it is given to a group,it is not rival. Therefore our work may have implications on fair divisionof information, for example, dividing training samples for machine learning In contrast to the literature on club goods, in this paper we do not consider congestioneffects: the family sizes are fixed, and all members of a family consume the same bundleregardless of the family size. Several recent works (Manurangsi and Suksompong, 2017; Suksompong, 2018;Segal-Halevi and Suksompong, 2019; Segal-Halevi and Nitzan, 2019; Ky-ropoulou et al., 2019) study the algorithmic problem of fairly dividing asingle heterogeneous resource (“cake”) or several indivisible goods amongfamilies. They show that, in most settings, unanimous-fairness (even itsweak variant — unanimous-FS) may not be attainable. To circumvent thisnon-existence problem, they require allocations to be democratically insteadof unanimously fair; an allocation, in turn, is democratically fair if a cer-tain fraction of all members in each family perceives it as fair. The setup ofthese problems are quite different than ours. Cake-cutting usually requiresthat each family receives a connected piece, or at least a small number ofconnected components. In indivisible goods allocation, fairness usually can-not be guaranteed even without any additional constraints, so the focus ison finding approximately-fair allocations. In contrast, our model of fully di-visible goods always allows for fair allocations, and the challenge is to findallocations that are both fair and efficient.Recently, Ghodsi et al. (2018) studied fair division of rooms and rentamong families of tenants, using three notions of fairness termed strong,aggregate and weak. Their strong-fairness is our unanimous-EF; their weak-fairness means that at least one agent does not envy, which is similar (butnot identical) to our collective-EF; their aggregate-fairness is a special caseof our notion of aggregate-EF in which the aggregate function of each familyis the sum of the utilities of the members.It is important to distinguish our family fairness notions from two verydifferent notions of group fairness .(a) One notion of group-fairness involves the standard resource-allocationsetting in which each individual receives an individual bundle (Berliant et al.,1992; H¨usseinov, 2011; Dall’Aglio et al., 2009; Dall’Aglio and Di Luca, 2014;Todo et al., 2011; Mouri et al., 2012). A group-envy-free division is defined We are grateful to Hal Varian for this insight.
7s a division in which no coalition of individuals can take the pieces allocatedto another coalition with the same number of individuals and re-divide thepieces among its members such that all members are weakly better-off. Inour setting the families are fixed and agents do not form coalitions on-the-fly;the challenge arises from the fact that all individuals in each family consumethe same bundle.(b) A second notion of group-fairness comes from an entirely differentfield — artificial intelligence. Consider an AI system that automaticallydetects potential criminals based on their personal traits. If such a systemreports significantly more suspects with a certain skin-color, this might beconsidered a violation of group-fairness — the members of the group withthat particular skin-color are treated unfairly. There is a growing literatureon various definitions of group-fairness in this context; see, for example,Dwork et al. (2012), H´ebert-Johnson et al. (2017) and the references therein.
There is a finite set I of individuals; the individuals are partitioned intofamilies F . Individual i ’s family is φi , any family f ∈ F may have multiplemembers or just one. There are G different homogeneous divisible goods and R G + is the set of consumption bundles; the total endowment is e ∈ R G + with e (cid:29) The average bundle e := | F | e defines each family’s fair share ofthe total endowment. An allocation is a vector x = { x f } f ∈ F of consumptionbundles x f ∈ R G + whose sum does not exceed the aggregate endowment: (cid:80) f ∈ F x f ≤ e . X is the set of all allocations.Each member i of some family f consumes the same bundle x f , andeach individual i ’s preference over any two allocations x and x (cid:48) only dependshis family’s consumption bundles x φi and x (cid:48) φi in the given allocations. Acontinuous utility function u i : R G + → R represents individual i ’s preferenceover bundles in R G + . An economy is formally defined by ( I, F, R G + , { u i } i ∈ I , e ). For any two vectors x, x (cid:48) ∈ R m for some integer m , say x ≥ x (cid:48) if x i ≥ x (cid:48) i for all i , x > x (cid:48) if x ≥ x (cid:48) but not x = x (cid:48) and x (cid:29) x (cid:48) if x i > x (cid:48) i for all i . i ’s preference is • strictly monotonic if x > x (cid:48) implies u i ( x ) > u i ( x (cid:48) ) for all x, x (cid:48) ∈ R G ; • locally non-satiated if for every x ∈ R G + and (cid:15) >
0, there exists some x (cid:48) ∈ R G + such that u i ( x (cid:48) ) > u i ( x ) and such that the Euclidean distancebetween x and x (cid:48) does not exceed (cid:15) . • strictly convex if u i ( x ) ≥ u i ( x (cid:48) ) together with x (cid:54) = x (cid:48) implies u i ((1 − α ) x + αx (cid:48) ) > u i ( x (cid:48) ) for all x, x (cid:48) ∈ R G + and α ∈ (0 , • convex if u i ( x ) > u i ( x (cid:48) ) implies u i ((1 − α ) x + αx (cid:48) ) > u i ( x (cid:48) ) for all x, x (cid:48) ∈ R G + and α ∈ (0 , We consider three different notions of family preferences. Unanimous prefer-ence is the strictest of them. We say that a preference statement unanimously holds for family f if the statement holds for each member of f . The samepreference statement collectively holds for family f if it holds according tofamily f ’s (typically incomplete) collective preference (cid:37) colf . The latter inturn ranks a bundle x above a different bundle x (cid:48) if and only if each mem-ber of the family agrees with this ranking. Without such an agreement (cid:37) colf does not rank the two bundles. Formally, we have x (cid:37) colf x (cid:48) , if and only if u i ( x ) ≥ u i ( x (cid:48) ) holds for all i ∈ f . If u i ( x ) > u i ( x (cid:48) ) and u j ( x (cid:48) ) > u j ( x ) holdsfor two different members i and j of family f then (cid:37) colf does not rank x and x (cid:48) and we write, x (cid:46)(cid:47) colf x (cid:48) . The family is collectively indifferent ( x ∼ colf x (cid:48) ) ifand only if each of its members is indifferent ( u i ( x ) = u i ( x (cid:48) ) for all i ∈ f ).Comparing the two notions we see that x is unanimously preferred to x (cid:48) ifand only if x (cid:37) colf x (cid:48) . A difference between the two notions arises in negations.Family f , for example, finds x unanimously no worse than x (cid:48) if and only if u i ( x ) (cid:54) < u i ( x (cid:48) ) for all i ∈ f , which holds if and only if x (cid:37) colf x (cid:48) . Conversely,family f finds x collectively no worse than x (cid:48) if and only if x (cid:54)≺ colf x (cid:48) , whichholds if and only if either x (cid:37) colf x (cid:48) or x (cid:46)(cid:47) colf x (cid:48) .9 complete and transitive preference relation (cid:37) aggf is called an aggregator of family f if it ranks some bundle x above x (cid:48) whenever all the members of f hold this preference, and ranks x strictly above x (cid:48) whenever additionallyat least one member of f holds this strict preference. Formally, (cid:37) aggf isan aggregator of f if for any x, x (cid:48) ∈ R G + : u i ( x ) ≥ u i ( x (cid:48) ) for all i ∈ f implies x (cid:37) aggf x (cid:48) ; where x (cid:31) aggf x (cid:48) must hold if the preceding statement on individualpreferences holds strictly for at least one member of f . We also call a vectorof aggregators (cid:37) agg : = ( (cid:37) aggf ) f ∈ F an aggregator. If a function g : R | F | → R is strictly increasing in all its components, then g (( u i ( · ) i ∈ f )) : X → R represents an aggregator of family f . The following example illustrates therelations between the different notions of family preferences:
Example 3.1 (Comparing bundles)
Consider a family f consisting of ahusband and a wife. Say two different bundles x and x (cid:48) respectively yieldutility vectors (1 ,
2) and (4 ,
1) to the husband and the wife. According tothe aggregation represented by U f ( x ) := (cid:112) u h ( x ) + u w ( x ) for all x ∈ R G + , thefamily is indifferent between the two bundles. Since different family membersprefer different bundles, we have x (cid:46)(cid:47) colf x (cid:48) , so the family’s collective prefer-ence over the two bundles is incomplete. The family cannot unanimouslyrank the two bundles. Conversely if the two bundles x and x (cid:48) respectivelyyield utility vectors (9 ,
1) and (3 ,
1) then the family prefers bundle x accord-ing to all notions of preference defined above.The relations between the different notions of preferences generalize andwe have the following Lemma: Lemma 3.2
Fix two bundles x, x (cid:48) ∈ R G + and a family f . Then f unani-mously prefers x to x (cid:48) if and only if f aggregate-prefers x to x (cid:48) according toevery aggregator (cid:37) aggf . Proof f unanimously prefers x to x (cid:48) means that u i ( x ) ≥ u i ( x (cid:48) ) holds foreach i ∈ f . Then the definition of aggregate preferences directly implies x (cid:37) aggf x (cid:48) for every aggregator (cid:37) aggf . Conversely, say | f | ≥ u j ( x (cid:48) ) >u j ( x ) for some individual j ∈ f . The latter implies that for some sufficiently No such function g represents the lexicographic aggregator defined below. α > αu j ( x (cid:48) ) + (cid:80) i ∈ f \{ j } u i ( x (cid:48) ) > αu j ( x ) + (cid:80) i ∈ f \{ j } u i ( x ). Thepreceding function represents an aggregate preference of family f , for which x (cid:37) aggf x (cid:48) does not hold. (cid:3) The proofs of our main results rely on aggregators which correspond todifferent notions of justice a family may subscribe to. To define these threeaggregators normalize each individual’s utility u i so that u i ( e ) = 0.A Rawlsian family focusses on its least-well-off members. Only if mul-tiple bundles in the choice set of a Rawlsian family maximize the minimalutility experience by any of its members, does a Rawlsian family consider thesecond lowest utility experienced by any of its members. Family memberswith higher utilities are only considered in the comparison of two bundles, ifthe two bundles yield the same utilities to the members who derive the leastutility from the given bundles. So a Rawlsian family lexicographically max-imizes the smallest utilities experienced by any of its members. To formallydefine the Rawlsian (or lexicographic) aggregator (cid:37) lexf , fix any two bundles x and x (cid:48) . Order the members of f as i , . . . i | f | and i (cid:48) , . . . , i (cid:48)| f | , such that foreach k < | f | agent i k experiences (weakly) lower utility than agent i k +1 at x and agent i (cid:48) k experiences (weakly) lower utility than agent i (cid:48) k +1 at x (cid:48) . If u i k ( x ) = u i (cid:48) k ( x (cid:48) ) holds for all k , then x ∼ lexf x (cid:48) . If not, say k is the minimalindex for which u i k ( x ) (cid:54) = u i (cid:48) k ( x (cid:48) ). We then have x (cid:31) lexf x (cid:48) if u i k ( x ) (cid:54) = u i (cid:48) k ( x (cid:48) )and x (cid:48) (cid:31) lexf x otherwise (Dubins and Spanier, 1961; Moulin, 2004). A family that mediates internal conflict using Nash bargaining, with thefair share e as the outside option, strives to maximize the product of allits members’ utilities. The aggregator (cid:37)
Nashf represents such a family. Todefine (cid:37)
Nashf say that U Nashf ( x ) : = Π i ∈ f u i ( x ) for all i ∈ f . Noting that thisfunction only represents an aggregator for bundles x with u i ( x ) > define (cid:37) Nashf by using (cid:37) lexf for all bundles x with u j ( x ) ≤ j ∈ f . The aggregator (cid:37) lexf is not representable by a utility function. The normalization u i ( e ) is essential here. With this definition an individual’s utility u i ( x ) of a bundle x coincides with the difference between u i ( x ) and this individual’s utilityof the outside option: u i ( e ). We for example have U Nashf ( x ) = U Nashf ( x (cid:48) ) for any two bundles x and x (cid:48) with u i ( x ) = 0 = u j ( x (cid:48) ) for some i, j ∈ f . (cid:37) Nashf x (cid:48) ⇔ (cid:40) U Nashf ( x ) ≥ U Nashf ( x (cid:48) ) , if u i ( x ) , u i ( x (cid:48) ) > i ∈ fx (cid:37) lexf x (cid:48) , otherwise . A utilitarian family strives to maximize the sum of all its members’ util-ities. The utilitarian aggregator (cid:37)
Sumf for family f is represented by theutility function U Sumf ( · ) = (cid:80) i ∈ f u i ( · ). If each family member has strictlymonotonic (locally non-satiated) preferences, then the utilitarian preferenceof the family is strictly monotonic (locally non-satiated). A triplet ( p , x , x ) with x , x ∈ X and p ∈ R G is a market equilibrium from x , if for each family and each bundle x (cid:48) ∈ R G + (a) p x f ≤ p x f , and: (b) x (cid:48) (cid:31) f x f implies p x (cid:48) > p x f . So ( p , x , x ) is a market equilibrium from x if each bundle x f is affordable for that family and any bundle x (cid:48) a familystrictly prefers to x f is unaffordable for that family. If x = ( e, . . . , e ), then( p , x , x ) is called a market equilibrium from equal endowments .An allocation x is Pareto dominated by a different allocation x (cid:48) if for everyindividual i : u i ( x (cid:48) φi ) ≥ u i ( x φi ), and for at least one individual j : u j ( x (cid:48) φj ) >u j ( x φj ). Equivalently, x is Pareto-dominated by x (cid:48) if for every family f : x (cid:48) f (cid:37) colf x f , and for at least one family: x (cid:48) f (cid:31) colf x f . An allocation x is Pareto-optimal in a set S if it is not Pareto-dominated by any allocation in the set S . An allocation that is Pareto optimal in the set of all allocations X issimply called Pareto-optimal (efficient). In the context of individual consumption an allocation is envy free if no indi-vidual envies any other individual. An allocation has the fair share guarantee if each individual prefers his bundle to the fair share. Finally an allocationis egalitarian equivalent if there exists some reference bundle r such that noindividual has a strict preference over this bundle and his allocated bundle.We apply the three notions of family preferences to the three criteria offairness to derive a total of nine notions of family fairness. An allocation is12 nanimous -fair if it is fair according to the preferences of every member inevery family. The same allocation is collective -fair if it is fair according tothe collective preferences (cid:37) colf of every family f . Finally, the allocation is (cid:37) agg - aggregate -fair if it is fair according to each family’s aggregator (cid:37) aggf . • The allocation x is unanimous-envy-free if there is no pair of individuals i, i (cid:48) such that u i ( x φi (cid:48) ) > u i ( x φi (cid:48) ); it is (cid:37) agg - aggregate-envy-free if thereis no pair of families f, f (cid:48) such that x f (cid:48) (cid:31) aggf x f ; it is collective-envyfree if there is no pair families f, f (cid:48) such that x f (cid:48) (cid:31) colf x f . • For x to be unanimous - aggregate -, or collective -egalitarian equiva-lent, each family f must consider its bundle x f to be not better orworse than some some reference bundle r . For x to be unanimous-egalitarian equivalent u i ( r ) = u i ( x φi ) must hold for all i . For x to be (cid:37) agg -aggregate egalitarian equivalent r ∼ aggf x f must hold for all f . Fi-nally x is collective-egalitarian equivalent if either r ∼ colf x f of r (cid:46)(cid:47) colf x f holds for all f . • For x to satisfy the unanimous -, (cid:37) agg - aggregate - or collective fair shareguarantee the following conditions have to hold for all families f , u i ( x φi ) ≥ u i ( e ) for all i ∈ f , x (cid:37) aggf e , and x f (cid:37) colf e .The relations between the different notions of family preferences implyrelations between the different notions of fairness. While the equivalence ofcollective-FS and unanimous-FS immediately follows from the above defini-tion, we summarize some further relations between the different notions offairness in the following Lemma 3.3. Lemma 3.3 a) An allocation is unanimous-EE (unanimous-EF, unan-imous-FS) if-and-only-if it is (cid:37) agg -aggregate-EE ( (cid:37) agg -aggregate-EF, (cid:37) agg -aggregate-FS) for every aggregator (cid:37) agg .b) An allocation is collective-EE (collective-EF) if-and-only-if it is (cid:37) agg -aggregate-EE ( (cid:37) agg -aggregate-EF) for at least one aggregator (cid:37) agg . Proof
Fix an allocation x ∈ X and a family f . Whether or not an allocation is (cid:37) agg -fair depends on the particular aggregator used.
13) The three statements directly follow from from Lemma 3.2 which statesthat family f unanimously prefers one bundle to a different bundle ifand only if the family (cid:37) aggf -aggregate prefers the former bundle to thelatter for every aggregator (cid:37) aggf .b) If x is (cid:37) agg -aggregate-EE with a reference bundle r , then each family f is (cid:37) agg -indifferent between r and their allocated bundle under x . So wein particular have x f ∼ aggf r . If u i ( x f ) = u i ( r ) holds for each i ∈ f , thenwe have x f ∼ colf r . If not then then u j ( x f ) > u j ( r ) and u j (cid:48) ( x f ) < u j (cid:48) ( r )must hold for two members j and j (cid:48) of f , as the aggregator (cid:37) aggf wouldotherwise strictly rank x f and r . But the existence of j, j (cid:48) ∈ f impliesthat x f (cid:46)(cid:47) colf r , so x is collective-EE.Conversely, suppose x is collective-EE with a reference bundle r . Theneach family f , either u i ( r ) = u i ( x f ) for each i ∈ f or u j ( x f ) > u j ( r )as well as u j (cid:48) ( x f ) < u j (cid:48) ( r ) for two members j and j (cid:48) of f . In the for-mer case, x f ∼ aggf r holds for any aggregator. In the latter case, wecan choose positive numbers α i for each i such that (cid:80) i ∈ f α i u i ( x f ) = (cid:80) i ∈ f α i u i ( r ). Since each α i is positive, the function (cid:80) i ∈ f α i u i ( · ) repre-sents an aggregator (cid:37) aggf for family f . By construction, this aggregatorsatisfies x f ∼ aggf r . Hence, x is (cid:37) agg -aggregate-EE.Now suppose that x is (cid:37) agg -aggregate-EF. Fix any family f (cid:48) (cid:54) = f . If u i ( x f ) = u i ( x f (cid:48) ) holds for each i ∈ f , then we have x f ∼ colf x f (cid:48) . If not,then u i ( x f ) > u i ( x f (cid:48) ) must hold for at least one agent i ∈ f , as theaggregator (cid:37) aggf would otherwise rank x f (cid:48) strictly above x f . Agent i ’spreference u i ( x f ) > u i ( x f (cid:48) ) then implies that x f (cid:48) (cid:31) colf x f does not hold.So family f does not collectively prefer x f (cid:48) . Since the families f and f (cid:48) were chosen arbitrarily, x is collective-EF.Conversely, suppose x is collective-EF. For each family f , normalize theagents’ utilities such that u i ( x f ) = 0 for all i ∈ f . Choose some α > α min i ∈ f u i ( x f (cid:48) ) + U Sumf ( x f (cid:48) ) ≤ (cid:48) (cid:54) = f . Firstly note that the function α min i ∈ f u i ( x f (cid:48) ) + U Sumf ( x f (cid:48) )represents an aggregator (cid:37) aggf for each family. Secondly the allocation x is (cid:37) aggf -aggregate-EF, since α min i ∈ f u i ( x f ) + U Sum ( x f ) = 0 holds foreach family by the normalization of all individual utilities. This completes the proof. (cid:3)
The theory of aggregate fairness is no different from the theory of indi-vidual fairness: we just have to interpret the individuals in standard fairnessresults as families with the preferences (cid:37) aggf . It is therefore our main goalto establish conditions under which there are Pareto optima which satisfythe much stronger unanimous-fairness conditions. We start by showing thatunanimous-FS Pareto optima always exist. The following theorem sets thestage for our two main positive results which show that unanimous-FS Paretooptima which are in addition either collective-EF or collective-EE (Theorems4.4 and 5.2) exist under broad conditions.
Theorem 3.4
Any economy has a unanimous-FS Pareto optimum.
Proof
Define a lexicographic ranking (cid:37)
LEX over allocations along the linesof the definition of (cid:37) lexf . Normalize all agents’ utilities such that u i ( e ) = 0 forall i ∈ I . Then lexicographically rank any two allocations x , x (cid:48) according tothe utilities they yield to all agents in the economy — not just the agents inone family. Since X is compact and since all u i are continuous, there existsa (cid:37) LEX -optimal allocation x ∗ in X , which is also Pareto optimal. Since( e, . . . , e ) is feasible, we have x ∗ (cid:37) LEX ( e, . . . , e ). So u i ( x ∗ φi ) ≥ u i ( e ) holdsfor all i , so x ∗ is unanimous-FS. (cid:3) Such an α exists since x f (cid:37) colf x f (cid:48) implies that either u i ( x f ) = u i ( x f (cid:48) ) for all i ∈ f , sothat α min i ∈ f u i ( x f (cid:48) ) + U Sumf ( x f (cid:48) ) = 0 for any α , or min i ∈ f u i ( x f (cid:48) ) < i ∈ f u i ( x f ),so that for large enough α all comparisons between x f and x f (cid:48) are decided based on theminimal utilities experienced by family members. Note that the aggregator defined here (cid:37) agg depends on the allocation x . It may wellbe that in some economy with multiple collective-EE allocations there exists no aggregatorsuch that all allocations are aggregate-EF according to this one aggregator. .5 Examples with two goods If there are only two goods in an economy, we call these two goods y and z .We assume that any such economy is endowed with | F | units of each good,so that e = (1 , Marginal rate of substitution.
If a utility function u i : R → R isdifferentiable, we define the marginal rate of substitution between y and z , M RS i ( y, z ), as: M RS i ( y, z ) := d ( u i ( y, z )) dy (cid:30) d ( u i ( y, z )) dz . Single-crossing property
Two different strictly monotonic preferencesrepresented by u i and u j satisfy the single crossing property if any twoindifference-curves defined by u i ( y, z ) = α and u j ( y, z ) = β for some α, β ∈ R have at most one point in common. So u i and u j have the single-crossingproperty if for all bundles ( y, z ) and ( y (cid:48) , z (cid:48) ) and either u i = u , u j = u (cid:48) or u i = u (cid:48) , u j = u :If y (cid:48) > y and z (cid:48) < z then u ( y (cid:48) , z (cid:48) ) ≥ u ( y, z ) ⇒ u (cid:48) ( y (cid:48) , z (cid:48) ) > u (cid:48) ( y, z )If y (cid:48) < y and z (cid:48) > z then u (cid:48) ( y (cid:48) , z (cid:48) ) ≥ u (cid:48) ( y, z ) ⇒ u ( y (cid:48) , z (cid:48) ) > u ( y, z ) . If one individuals’ marginal rate of substitution is higher than the other’sat each possible bundle ( y, z ), then u i and u j satisfy the single crossingproperty. An agent’s preference has a Cobb-Douglas utility representation if u i ( y, z ) = y α z − α holds for some α ∈ (0 , Theorem 3.5
Consider a two-good economy with differentiable utilities. Let x be an allocation in the interior of X . Then x is Pareto optimal if and only f: (cid:92) f ∈ F [min i ∈ f M RS i ( y f , z f ) , max i ∈ f M RS i ( y f , z f )] (cid:54) = ∅ . Theorem 3.5 has been shown to hold in (much) more general environ-ments by Fon and Otani (1979); Mandler (2014), so we omit its proof here.The statement can also be proven with some simple modifications of thearguments in Mas-Colell (1974).
In this section, we prove that unanimous-EF Pareto optima exist under mildconditions when there are only two families, but might not exist with morethan two families. Even if unanimous-EF Pareto optima exist, it mightbe impossible to find them as market equilibria from equal endowments.The market approach, however, turns out to be useful to find collective-EF, unanimous-FS Pareto optima. The different examples used to provethe impossibility results do not rely on extreme conditions: each of theseexamples has only two goods and three families, two of which are singles.All individuals have plain vanilla preferences — strictly convex and strictlymonotonic.
Theorem 4.1
Any economy with exactly two families and convex prefer-ences has a unanimous-EF and unanimous-FS Pareto optimum.
Proof
By Theorem 3.4, there exists an unanimous-FS Pareto optimum x . To see that x is unanimous-EF, suppose by contradiction that someindividual i in family f envies family f (cid:48) , so that u i ( x f ) < u i ( x f (cid:48) ) = u i ( e − x f ).By convexity, u i ( x f ) must also be smaller than u i ( x f + ( e − x f )). But thelatter bundle is exactly the fair share e — a contradiction to the assumptionthat x is unanimous-FS. (cid:3) Proposition 4.2
Some economies with three families have no unanimous-EF Pareto optimum.
Proof
Consider a two-good-economy with four agents I : = { h, w, s, s (cid:48) } belonging to three families F : = (cid:8) { h, w } , { s } , { s (cid:48) } (cid:9) . Each individual i ∈ I has a strictly monotonic preference represented by a differentiable utility u i : R → R , and all preferences satisfy a single-crossing property so thatfor any two bundles ( y, z ), ( y (cid:48) , z (cid:48) ): If y (cid:48) > y and z (cid:48) < z then: u h ( y (cid:48) , z (cid:48) ) ≥ u h ( y, z ) ⇒ u s (cid:48) ( y (cid:48) , z (cid:48) ) > u s (cid:48) ( y, z ) ,u s (cid:48) ( y (cid:48) , z (cid:48) ) ≥ u s (cid:48) ( y, z ) ⇒ u s ( y (cid:48) , z (cid:48) ) > u s ( y, z ) ,u s ( y (cid:48) , z (cid:48) ) ≥ u s ( y, z ) ⇒ u w ( y (cid:48) , z (cid:48) ) > u w ( y, z ) . If y (cid:48) < y and z (cid:48) > z then: u w ( y (cid:48) , z (cid:48) ) ≥ u w ( y, z ) ⇒ u s ( y (cid:48) , z (cid:48) ) > u s ( y, z ) ,u s ( y (cid:48) , z (cid:48) ) ≥ u s ( y, z ) ⇒ u s (cid:48) ( y (cid:48) , z (cid:48) ) > u s (cid:48) ( y, z ) ,u s (cid:48) ( y (cid:48) , z (cid:48) ) ≥ u s (cid:48) ( y, z ) ⇒ u h ( y (cid:48) , z (cid:48) ) > u h ( y, z ) . For example, we can take agents with Cobb-Douglas preferences with different coeffi-cients, such that the coefficients of the two singles s, s (cid:48) are between the coefficient of thehusband h and the coefficient of the wife w . w ife, h usband and two singles s and s (cid:48) . Say ( y f , z f ) is the bundle consumedby the f amily consisting of the husband and wife. The single-crossing prop-erty implies that the indifference curves of the husband, wife and the twosingles through ( y f , z f ) only cross at that point. For s to not envy the fam-ily, his consumption ( y s , z s ) must lie on or above his green indifference curve.So suppose that ( y s , z s ) (cid:54) = ( y f , z f ) holds in addition to the latter condition.Then the husband envies s if z s > z f while the wife envies s if z s < z f .Hence s must consume ( y f , z f ). The same is true for s (cid:48) . But then the allo-cation cannot be Pareto-optimal, since the two singles have different MRS-sat ( y f , z f ). 19ssume furthermore that any unanimous-EF allocation must be inte-rior. Consider an unanimous-EF allocation x = (( y f , z f ) , ( y s , z s ) , ( y s (cid:48) , z s (cid:48) )).For s not to envy f , we must have: u s ( y s , z s ) ≥ u s ( y f , z f )Since all preferences are strictly monotonic, and since h and w may not envy s , there are only three options regarding the relation of ( y f , z f ) and ( y s , z s ): • If y s > y f and z s < z f , then u s ( y s , z s ) ≥ u s ( y f , z f ) and single-crossingimply that u w ( y s , z s ) > u w ( y f , z f ) so the wife envies s . • If y s < y f and z s > z f , then u s ( y s , z s ) ≥ u s ( y f , z f ) and single-crossingimply that u h ( y s , z s ) > u h ( y f , z f ) so the husband envies s .So we must have ( y s , z s ) = ( y f , z f ).Applying the same arguments to s (cid:48) we see that also ( y s (cid:48) , z s (cid:48) ) = ( y f , z f )holds, so ( y s (cid:48) , z s (cid:48) ) = ( y s , z s ). But then the allocation x cannot be Pareto-optimal, as the the two singles satisfy the single crossing property and musttherefore consume different bundles in any interior Pareto optimum. (cid:3) The classic proof that envy-free Pareto optima with the fair share guaranteeexist (Foley, 1967), argues that market equilibria from equal endowmentsyield such allocations. Since the fair share is each individual’s endowment,each individual must weakly prefer his or her equilibrium consumption to thefair share. Since each individual faces the same budget set, each individualcould choose any other individual’s bundle. So no individual envies any other.This technique does not fare as well in economies with families. Proposition4.2 already shows that unanimous-EF Pareto optima need not exist. Buteven if such allocations exist, they may not arise as a market equilibria fromequal endowments. This for example holds when all preferences have Cobb-Douglas utility representations. roposition 4.3 Some economies have unanimous-EF and unanimous-FSPareto optima, even though no such allocation arises as a market equilibriumfrom equal endowments.
Proof
Consider a two-good-economy with four agents I : = { h, w, s, s (cid:48) } belonging to three families f : = { h, w } , { s } and { s (cid:48) } . The husband h andthe single person s have the same Cobb-Douglas preference with coefficient α h = , the wife w and the single person s (cid:48) have the same Cobb-Douglaspreference with coefficient α w = .We first show that this economy has unanimous-FS Pareto-optima. For (cid:0) ( y s , z s ) , ( y s (cid:48) , z s (cid:48) ) , ( y f , z f ) (cid:1) to be an allocation y s + y s (cid:48) + y f = 3 and z s + z s (cid:48) + z f = 3 have to hold. For the singles and the family members not to envy eachother y s z s = y f z f and y s (cid:48) z s (cid:48) = y f z f must hold. If (cid:0) ( y s , z s ) , ( y s (cid:48) , z s (cid:48) ) , ( y f , z f ) (cid:1) is on the boundary of X then all members of at least one family have zeroutility. Since the members of this family must not envy the other families,each agent must have utility zero in the allocation (cid:0) ( y s , z s ) , ( y s (cid:48) , z s (cid:48) ) , ( y f , z f ) (cid:1) which can therefore not be Pareto optimal. So (cid:0) ( y s , z s ) , ( y s (cid:48) , z s (cid:48) ) , ( y f , z f ) (cid:1) must be in the interior of X and z s y s = M RS s ( y s , z s ) = M RS s (cid:48) ( z s (cid:48) , y s (cid:48) ) =2 y s (cid:48) z s (cid:48) must hold. The resulting system of 5 equations in 6 unknowns has acontinuum of solutions. To obtain a concrete unanimous-FS and unanimous-EF Pareto optimum, we additionally impose the symmetry condition y f = z f .The resulting system has a unique solution x ∗ , illustrated in Figure 2.Now, suppose that some unanimous-EF Pareto optimum (cid:0) ( y s , z s ), ( y s (cid:48) , z s (cid:48) ),( y f , z f ) (cid:1) could be obtained as a market equilibrium from equal endowments.By Pareto optimality and the single crossing property we have ( y s , z s ) (cid:54) =( y s (cid:48) , z s (cid:48) ), so that either ( y f , z f ) (cid:54) = ( y s , z s ) or ( y f , z f ) (cid:54) = ( y s (cid:48) , z s (cid:48) ) (or both)must hold. W.l.o.g. suppose that the former is true. Since all individuals’preferences are strictly convex, single s strictly prefers ( y s , z s ) to all otherbundles in the budget set. We in particular have u s ( y s , z s ) > u s ( y f , z f ).Since u h = u s , the husband then envies s — a contradiction. To obtain anunanimous-EF Pareto optimum in a market equilibrium, the family musthave a higher income than the singles. (cid:3) Family size is not the relevant factor in the above proof, intra-family het-erogeneity is: the same non-existence result holds if we replace each single21igure 2:
An illustration of Proposition 4.3. The three blue discs denote the symmetricunanimous-EF Pareto-optimum, and we have y ∗ s ≈ . z ∗ s ≈ . y ∗ s (cid:48) ≈ . z ∗ s (cid:48) ≈ . y ∗ f ≈ . z ∗ f ≈ . s are indifferent between the bundles of the family and single s , the wife and single s (cid:48) are indifferent between the bundles of the family and single s (cid:48) .The husband and single s share the bold indifference curve; the wife and single s (cid:48) share thedashed indifference curve. Since each individual (weakly) prefers their own consumption tothe bundles of the others, the allocation is unanimous-EF. The allocation is efficient, sincethe MRS of s and s (cid:48) at their bundles, represented by the thick dotted line lies between thehusband’s and the wife’s MRSs at their family’s bundle (represented by the thin dottedlines). To support this allocation as a market equilibrium, both singles would have to facethe thick dotted line as their budget constraint. But the family’s bundle is not affordablewith this budget. We now show that collective-EF, unanimous-FS, and efficiency are compati-ble if all agents’ preferences are strictly convex. We use the market equilib-rium approach to establish the existence of a (cid:37) lex -aggregate-EF unanimous-FS Pareto optimum. By Lemma 3.3 this (cid:37) lex -aggregate-EF allocation iscollective-EF. Theorem 4.4
If all individuals’ preferences are strictly convex and locallynon-satiated, then there exist unanimous-FS, (cid:37) lexf -aggregate-EF, and collec-tive-EF Pareto optima.
The proof uses the market equilibrium approach. To facilitate we assumethat each family f strives to maximize the utility U min f which evaluates con-sumption bundles according to the minimal utility experienced by any familymember, so U min f ( x f ) = min i ∈ f u i ( x f ) for all x f ∈ R G + . Note that U min f doesnot represent an aggregator. However, Lemma 4.5 below shows that the ag-gregator (cid:37) lexf can — under circumstances assumed in Theorem 4.4 — usedinterchangeably with preference represented by the utility U min f . Lemma 4.5
Suppose all individuals’ preferences are strictly convex and X f is a non-empty, convex and compact subset of R G + . Then the bundle x ∗ f is To see that market equilibria using arbitrary aggregators need not produce unanimous-FS allocations consider a two-good-economy with four agents I : = { h, w, h (cid:48) , w (cid:48) } belongingto two families f : = { h, w } and f (cid:48) = { h (cid:48) , w (cid:48) } . Each individual i ∈ I has a Cobb-Douglas utility with coefficients α h = . α w = . α w (cid:48) = .
4, and α h (cid:48) = .
1. Say eachfamily prefers bundles that incur higher utility to the husband. Only if the husbandis indifferent between two bundles, the family chooses the bundle preferred by the wife.Then (cid:0) (1 , , (( . , . , ( . , . , ( e, e ) (cid:1) is a market equilibrium from equal endowments. Atthe price p = 1, each husband gets his best affordable bundle. However both wives prefer e = ( . , .
5) to their respective bundles: . . . . < . . . . = . lexf -maximal in X f if and only if it is U min f -maximal in X f . There is aunique (cid:37) lexf -maximal bundle in X f . Proof
Fix ( u i ) i ∈ f and X f as set out in the Lemma. Since X f is non-emptyand compact and since all agents’ utilities u i are continuous, there exist U min f -maxima in X f . Since each u i represents a strictly convex preference, thefunction U min f too represents a strictly convex preference. Since X f is convex,there exists a unique U min f -maximal bundle x ∗ f in X f . By the definition of (cid:37) lexf the set of (cid:37) lexf -maxima in X f is a subset of the set of U min f -maxima in X f , so that x ∗ f is also the unique (cid:37) lexf -maximum in X f . (cid:3) We are now ready to prove Theorem 4.4.
Proof of Theorem 4.4.
Since all individuals’ preferences are strictly con-vex, locally non-satiated, and representable by continuous utilities, each util-ity U min f is continuous and represents strictly convex and locally non-satiatedpreference. So the economy where each family has preferences representedby U min f has a market equilibrium from equal endowments ( p , x ∗ , ( e, . . . , e )). Claim 1: x ∗ is unanimous-FS. Since e is in the budget set of eachfamily, min i ∈ f u i ( x ∗ f ) = U min f ( x ∗ f ) ≥ U min f ( e ) = min i ∈ f u i ( e ) = 0 holds foreach family f . So we have u i ( x ∗ f ) ≥ i and x ∗ satisfies unanimous-FS. Claim 2: x ∗ is (cid:37) lexf -aggregate-EF. Since x ∗ is an equilibrium alloca-tion, x ∗ f is for each family f the U min f -maximal bundle in their budget set.Since any budget set is convex, compact and nonempty and since all agentshave strictly convex preferences, Lemma 4.5 implies that x ∗ f is for each family f the (cid:37) lexf -maximal bundle in their budget set. Since each family has thesame budget set, the allocation x ∗ is (cid:37) lex -aggregate-EF. Claim 3: x ∗ is collective-EF. Since (cid:37) lexf is an aggregator of f for everyfamily f , Claim 2 together with Lemma 3.3 implies that x ∗ is collective-EF. Claim 4: x ∗ is Pareto optimal. Suppose that some allocation x (cid:48) didPareto improve on x ∗ , so that u i ( x (cid:48) φi ) ≥ u i ( x ∗ φi ) for all i and u j ( x (cid:48) φj ) > u j ( x ∗ φj )for some j . By the definition of (cid:37) lexf , we then get x (cid:48) f (cid:37) lexf x ∗ f for all f ∈ F and x (cid:48) f (cid:48) (cid:31) lexf (cid:48) x ∗ f (cid:48) for at least one family f (cid:48) , in particular for the family φj . ByLemma 4.5, x ∗ f is for each family f the unique (cid:37) lexf -maximal choice in their24udget set. So p ∗ · x (cid:48) f (cid:48) > p ∗ · x ∗ f (cid:48) holds for any family with x (cid:48) f (cid:48) (cid:31) lexf x ∗ f (cid:48) while x (cid:48) f = x ∗ f holds for all other families. Since all families preferences are locallynon-satiated we have p ∗ · x ∗ f = p ∗ · e for each family f . Summing over allfamilies, we obtain (cid:80) nf =1 p ∗ · x (cid:48) f > p ∗ · e , a contradiction to the feasibility of x (cid:48) which requires (cid:80) nf =1 x (cid:48) f ≤ e . (cid:3) Example 4.6 below shows that the statement in Theorem 4.4 does nothold if the Rawlsian aggregator (cid:37) lex is replaced by an arbitrary aggregator (cid:37) agg . The two-good economy in the example consists of one husband-and-wife-family as well as two singles. The preferences of the two singles differ somuch that at least one of the singles consumes only one good in any Paretooptimum. According to wife’s preferences the goods are close complements.Conversely the husband’s as well as the family’s aggregate preferences valueone good, say y , much more than the other. Overall the economy is con-structed such that the husband-wife family must (for some small (cid:15) ) consumeat least 1 − (cid:15) of each good for the wife to prefer her family’s bundle to thefair share. This minimal consumption by the family then implies that thesingle, who values y more highly than z , must in any Pareto optimum con-sume significantly more than half of the total endowment of y for him not toenvy the couple. For the couple not to envy this single, the couple must alsoconsume more than half the total endowment of y — a contradiction. Example 4.6
Consider a two-good economy with four agents I : = { h, w, s, s (cid:48) } belonging to three families F : = (cid:8) { h, w } , { s } , { s (cid:48) } (cid:9) . All individuals havestrictly monotonic, strictly convex preferences. We in particular assume: • w considers y and z to be close complements: for w to prefer herfamily’s bundle to the fair share y f > .
99 and z f > .
99 must hold. • h ’s utility increases only marginally in z consumption. • The family aggregator emphasizes h ’s utility, so that the family aggregate-envies single s at any allocation with y f < y s − . For example, u w ( y f , z f ) = 4 − y − f − z − f . For example, u h ( y f , z f ) = y . f + z . f / For example, U f ( y f , z f ) = 500 u h ( y f , z f ) + u w ( y f , z f ) / The utilities of the two singles are explicitly given by u s ( y s , z s ) = √ y s + 2 + √ z s + 5 and u s (cid:48) ( y s (cid:48) , z s (cid:48) ) = √ y s (cid:48) + 5 + √ z s (cid:48) + 2.The singles’ marginal rates of substitution at any allocation ( x, y ) are M RS s ( y s , z s ) = (cid:114) z s + 5 y s + 2 M RS s (cid:48) ( y s (cid:48) , z s (cid:48) ) = (cid:114) z s (cid:48) + 2 y s (cid:48) + 5 . Since each single can at most consume 3 units of either good, we have forany interior allocation ( y, z ) that
M RS s ( y s , z s ) ∈ (1 , , M RS s (cid:48) ( y s (cid:48) , z s (cid:48) ) ∈ ( 12 , . Theorem 3.5 then implies that the economy has no interior Pareto optima;either z s = 0 or y s (cid:48) = 0 or both must hold. The fair-share guarantee incombination with z s = 0 requires √ y s + 2 + √ ≥ √ √ y s ≥ .
78. Similarly, in the case that y s (cid:48) = 0 we must havethat z s (cid:48) ≥ . z s = 0 and y s ≥ .
78. For family f not to envy single s , y f ≥ . y s (cid:48) = 0 and z s (cid:48) ≥ .
78. For the wife w to prefer herfamily’s bundle to e , as required by unanimous-FS, we must have z f ≥ . s consumes at most 3 − . − .
78 = 0 .
23 units of z . For single s to prefer his bundle to e we must have (cid:112) y s + 2 + √ .
23 + 5 ≥ √ √ ⇒ y s ≥ . s we must have y f ≥ .
57. In either casewe obtain y s + y f > y .We do not know whether Theorem 4.4 can be extended to any otheraggregator except (cid:37) lex . In the preceding section we showed that it does not take much for unanimous-EF and efficiency to clash. Unanimous-EE is even more restrictive: even26conomies with two families need not have any unanimous-EE Pareto optima.To understand this difference between egalitarian equivalence in the standardmodel and unanimous-EE, fix an arbitrary bundle r and a family consistingof a husband and wife with differing preferences. While there generally aremany bundles x such that a single agent is indifferent between x and r , itis much harder to find bundles x such that both the husband and the wifeare indifferent between x and r . Indeed, if the husband’s and the wife’spreferences have the single crossing property then x = r is the only bundlesuch that the husband and the wife are indifferent between x and r . In theupcoming non-existence proof we specify an economy with two such coupleswhich must consume the same bundle in any unanimous-EE allocation. Weobtain a contradiction by differentiating the preferences of the two couplesenough, so that they must consume different bundles in any interior Paretooptimum. Proposition 5.1
Some economies with two families have no unanimous-EEPareto optimum.
Proof
Consider a two-good economy with four individuals I : = { h, w, h (cid:48) , w (cid:48) } belonging to two families f = { h, w } and f (cid:48) = { h (cid:48) , w (cid:48) } . All four individualshave strictly monotone and strictly convex preferences, satisfying the single-crossing property such that for each bundle ( y, z ): M RS w ( y, z ) < M RS h ( y, z ) < M RS h (cid:48) ( y, z ) < M RS w (cid:48) ( y, z )Assume that x is an unanimous-EE Pareto optimum and let r be its ref-erence bundle. By the single crossing property there is exactly one bundle( y f , z f ) such that h and w are indifferent between that bundle and r , namely( y f , z f ) = r . By the same token, f (cid:48) must consume r too, so that x = ( r, r ).Since max i ∈ f M RS i ( r ) < min i ∈ f (cid:48) M RS i ( r ) the allocation is by Theorem 3.5not Pareto optimal. The proof is illustrated in Figure 3. (cid:3) An illustration of Proposition 5.1. Family 1 has two memberswith solid indifference curves; Family 2 has two members with dotted indif-ference curves. To be indifferent between their bundle to some r , they mustconsume exactly r . But this allocation cannot be Pareto optimal — a Paretoimprovement can be attained by giving the solid family r + ( (cid:15), − (cid:15) ) and givingthe dotted family r + ( − (cid:15), (cid:15) ) , for some small (cid:15) . .2 Aggregate and collective egalitarian-equivalent Paretooptima satisfying unanimous fair share While unanimous-EE Pareto optima exist only rarely, collective-EE Paretooptima that are also unanimous-FS exist under quite general conditions:
Theorem 5.2
If all individuals’ preferences are strictly monotonic and strictlyconvex, then there exist unanimous-FS, (cid:37)
Nashf -aggregate-EE, and collective-EE Pareto optima.
Just as in the proof of Theorem 4.4, it is convenient to replace preference (cid:37)
Nashf with a representable preference, that leads to the same choices withinthe context of Theorem 5.2. To do so recall that (cid:37)
Nashf is represented by U Nashf ( x f ) = Π i ∈ f u i ( x f ) on the set of bundles that each member of f strictlyprefers to e . To extend U Nashf to all possible consumption bundles, define U Nashf ( x f ) := (cid:40) Π i ∈ f u i ( x f ) if u i ( x f ) ≥ i ∈ f − . The utility U Nashf does not represent an aggregation of family f ’s preferences:we for example have U Nashf ( x f ) = 0 for any bundle with u i ( x f ) ≥ i ∈ f and u j ( x f ) = 0 for some j ∈ f . We can nevertheless replace (cid:37) Nashf with the preference represented by U Nashf in the upcoming proof as boththese preferences make the same choices from any set containing e . Proof of Theorem 5.2
For any t ≥ V ( t ) : = max x ∈ X min f ∈ F (cid:0) U Nashf ( x f ) − U Nashf ( t · e ) (cid:1) . To see that V ( t ) is well-defined, say E : = { x ∈ X | u i ( x φi ) ≥ } is the set ofall unanimous-FS allocations. Since individual preferences are continuousand convex and since X is convex and compact, its subset E is convex andcompact too. Since min f ∈ F (cid:0) U Nashf ( x f ) − U Nashf ( t · e ) (cid:1) is continuous on E ,max x ∈ E min f ∈ F (cid:0) U Nashf ( x f ) − U Nashf ( t · e ) (cid:1) is well defined. Since E ⊆ X and Recall the normalization u i ( e ) = 0 for all i . f ∈ F (cid:0) U Nashf ( x f ) − U Nashf ( t · e ) (cid:1) < min f ∈ F (cid:0) U Nashf ( x (cid:48) f ) − U Nashf ( t · e ) (cid:1) holds for any pair of allocations x ∈ E and x (cid:48) ∈ X \ E , V ( t ) equalsmax x ∈ E min f ∈ F (cid:0) U Nashf ( x f ) − U Nashf ( t · e ) (cid:1) and is therefore well-defined. Claim 1: If V (1) = 0 then ( e, . . . , e ) is a unanimous-FS, (cid:37) Nashf -aggregate-EE as well as collective-EE Pareto optimum.
Clearly( e, . . . , e ) is unanimous-FS and unanimous-EE (and therefore collective aswell as (cid:37)
Nash -aggregate-EE). To see that ( e, . . . , e ) is also Pareto optimalsuppose some allocation x (cid:48) ∈ X did Pareto dominate ( e, . . . , e ). Since x (cid:48) Pareto dominates ( e, . . . , e ), u i ( x (cid:48) φi ) ≥ i . Since E is convex, x (cid:48)(cid:48) : = x (cid:48) + ( e, . . . , e ) ∈ E . By the strict convexity of individual preferences u i ( x (cid:48)(cid:48) φi ) > min( u i ( x (cid:48) φi ) , u i ( e )) = 0. By the definition of U Nashf , U Nashf ( x (cid:48)(cid:48) f ) > f ∈ F , and therefore V (1) = max x ∈ E min f ∈ F (cid:0) U Nashf ( x f ) − U Nashf ( e ) (cid:1) ≥ min f ∈ F U Nashf ( x (cid:48)(cid:48) f ) > . For the remainder assume that V (1) >
0. Note that V ( | F | ) < x ∈ X such that each family f is indifferent between x f and the full endowment of the economy e = | F | e . The continuityof V together with V (1) > V ( | F | ) < t ∗ > V ( t ∗ ) = 0. Define the allocation x ∗ such thatmin f ∈ F (cid:0) U Nashf ( x ∗ f ) − U Nashf ( t ∗ · e ) (cid:1) = 0. Claim 2: u i ( x ∗ φi ) > for each i so x ∗ is unanimous-FS. Since t ∗ > U Nashf ( x ∗ f ) > U Nashf the latter canonly hold if u i ( x ∗ φi ) > i . Claim 3: x ∗ is Pareto optimal. Suppose not, suppose x ∗ were Paretodominated by some allocation x (cid:48) ∈ X . So we have u i ( x (cid:48) φi ) ≥ u i ( x ∗ φi ) forall i and u j ( x (cid:48) φj ) > u j ( x ∗ φj ) for at least one j . Since all preferences arestrictly monotonic, since u i ( x ∗ φi ) > i , and since U Nashφj ( x (cid:48) φj ) >U Nashφj ( x ∗ φj ) ≥ U Nashφj ( t ∗ e ), there exists a small amount of resources (cid:15) suchthat U Nashφj ( x (cid:48) φj − (cid:15) ) > U Nashφj ( t ∗ e ) as well as u i ( x (cid:48) φj ) > i ∈ φj .Now define a new allocation x (cid:48)(cid:48) by redistributing the resources (cid:15) from φj x (cid:48) , formally x (cid:48)(cid:48) φj : = x (cid:48) φj − (cid:15) and x (cid:48)(cid:48) f : = x (cid:48) f + (cid:15) | F |− for all f (cid:54) = φj . By strict monotonicity U Nashf ( x (cid:48)(cid:48) f ) > U Nashf ( x (cid:48) f ) ≥ U Nashf ( t ∗ e ) holds for all f (cid:54) = φj while U Nashφj ( x (cid:48)(cid:48) φj ) > U Nashφj ( t ∗ e ) holds byconstruction. We in sum obtain a contradiction to the definition of t ∗ as V ( t ∗ ) ≥ min f ∈ F (cid:0) U Nashf ( x (cid:48)(cid:48) f ) − U Nashf ( t ∗ · e ) (cid:1) > Claim 4: U Nashf ( x ∗ f ) = U Nashf ( t ∗ · e ) for all f ∈ F . Suppose not,suppose U Nashf (cid:48) ( x ∗ f (cid:48) ) > U Nashf (cid:48) ( t ∗ e ) for some f (cid:48) ∈ F . The proof of the presentclaim follows the same steps as the proof of the preceding claim. To obtaina contradiction to the definition of t ∗ , we redistribute a small amount ofresources from family f (cid:48) to all other families to obtain a new allocation x (cid:48)(cid:48) with U Nashf ( x (cid:48)(cid:48) f ) > U Nashf ( t ∗ e ) for all f . Claim 5: x ∗ is (cid:37) Nash -aggregate- EE.
Since U Nashf represents (cid:37)
Nashf for all bundles x f with u i ( x f ) > i ∈ f , Claims 2 and 4 togetherimply that x ∗ f ∼ Nashf t ∗ e holds for each f . So x ∗ is (cid:37) Nash -aggregate-EE.
Claim 6: x ∗ is collective- EE. Since (cid:37)
Nashf is an aggregation of (cid:37) f thepreceding claim together with Lemma 3.3 implies that x ∗ is collective-EE. (cid:3) If we weaken the assumption of strictly convex preferences to an assump-tion of merely convex preferences in Theorem 5.2, we can only guarantee anytwo properties out of { Pareto optimality, unanimous-FS, collective-EE } . Inthe following proposition we first show that economies which allow for lin-ear preferences need not have any unanimous-FS, collective-EE Pareto op-tima. The second part shows that it is indeed possible to guarantee any twoproperties out of { Pareto optimality, unanimous-FS, collective-EE } when theassumption of strictly convex preferences is relaxed to that of convex prefer-ences. Proposition 5.3 (a) Some three-family economies with strictly monotonicand (weakly) convex preferences have no unanimous-FS and collective-EEPareto optimum. (b) There always exist collective-EE Pareto optima, unan-imous-FS Pareto optima, and collective-EE unanimous-FS allocations.
Proof of Proposition 5.3: Non-existence of Pareto-optimal FScollective-EE allocations. roof (a) Consider a two good economy with five agents I : = { h, w, h (cid:48) , w (cid:48) , s } belonging to three families f : = { h, w } , f (cid:48) : = { h (cid:48) , w (cid:48) } and { s } , where: • The single s and the husbands h and h (cid:48) have identical linear preferences: u h ( y, z ) = u h (cid:48) ( y, z ) = u s ( y, z ) = y + z . • The wives w and w (cid:48) have different Cobb-Douglas preferences: u w ( y, z ) = y / z / and u w (cid:48) ( y, z ) = y / z / .For the single and both husbands to enjoy the fair-share guarantee, y f + z f = y f (cid:48) + z f (cid:48) = y s + z s = 1 must hold. The only Pareto-optimal allocationsatisfying these equations gives family f (1 / , /
2) and family f (cid:48) (3 / , / r := ( y r , z r ) be the corresponding reference bundle. Since thesingle must be indifferent between his bundle and r , y r + z r must equal 2. Thismeans that both husbands are indifferent between their family’s bundle and r . But, regardless of which r we pick, at least one wife prefers her family’sbundle to r . Her family then collectively strictly prefers its bundle to r .(b) Theorem 3.4 proved the existence of unanimous-FS Pareto optima.The equal allocation is collective-EE and unanimous-FS. To see that therealways exists a collective-EE Pareto optimum, revisit the proof of Theo-rem 5.2 replacing Nash-bargaining-aggregation with utilitarian aggregation,so that each family f ’s preference is represented by U sumf : R G + → R with U sumf ( x f ) : = (cid:80) i ∈ f u i ( x f ) for each x f ∈ R G + keeping the normalization that u i ( e ) = 0 holds for each individual i . Amending the definition of V ( t ) to thedifferent aggregator, translate the arguments in the above proof to see thatalso in the present case there exists some t ◦ such that V ( t ◦ ) = 0. As abovedefine an allocation x ◦ such that 0 = min f ∈ F (cid:0) U Sumf ( x ◦ f ) − U Sumf ( t ◦ · e ) (cid:1) .Without concern for x ◦ being unanimous-FS, the proof that x ◦ is efficient,aggregate-EE and collective-EE transfers verbatim from the second caseabove. (cid:3) In the example used to prove part (a), the present technique identifies an allocation x ∗ which the two husbands consider inferior to the fair share e . Alternatives
We have studied three adaptations of three different fairness concepts for fam-ily economies. We gave a range of conditions under which even the strongestadaptation — unanimous fairness — can be achieved: In economies with justtwo families some Pareto optima are unanimous-EF as well as unanimous-FS;unanimous-FS and efficiency are generically compatible with either collective-EF or with collective-EE. So when do Pareto optima exist that are collective-EF as well as collective-EE? It turns out that such allocations exist only veryrarely: Thomson (1990) showed that egalitarian equivalent Pareto optima aregenerically not envy-free when there are at least three agents with completepreferences. Since complete preferences are a special case of collective pref-erences, Thomson (1990)’s non-existence result directly transfers to familyeconomies.Some of our negative examples depend on quite divergent preferenceswithin families. For instance, the example of an economy without unanimous-EF Pareto optima (Proposition 4.2) relies on the fact that the two most dis-similar people in the economy are married. Since this particular economy hasunanimous-EF Pareto optima for a different family structure where similarpeople form couples, one may wonder whether economies with assortativelymatched families always have unanimous-EF Pareto optima.To see that this is not the case consider a two-good-economy with three(heterosexual) couples. Say that all individuals have Cobb-Douglas prefer-ences, where m < m < w < m < w < w holds for m i and w i thecoefficients of man and woman i . Consider the assortative matching whereman m i and woman w i get married, so f i = { m i , w i } for all i = 1 , ,
3. Since w can’t envy f and since w and m cannot envy f , families f and f must consume the same bundle. Mutatis mutandis we see that also f and f must consume the same bundle. So all families consume the same bun-dle. A contradiction arises since f and f must, by Theorem 3.5, consumedifferent bundles.If we allow any arbitrary grouping into pairs an unanimous-EF Pareto op-timum exists: simply match the two agents with the highest two coefficients,with the middle coefficients and the lowest coefficients. Then find a market34quilibrium from equal endowments given that each pair maximizes its aver-age utility. The question whether (and how) this observation generalizes isopen. Our Theorems 4.4 and 5.2 respectively show that aggregate-EF accord-ing to the Rawlsian aggregator and aggregate-EE according to the Nash-bargaining-aggregator are consistent with unanimous-FS as well as with ef-ficiency. We show in Example 4.6 that there are economies and aggregatorsthat do not have any unanimous-FS Pareto optimum that is aggregate-EFaccording to the given aggregator. The question for which other aggregatorssuch aggregate-EE or aggregate-EF unanimous-FS Pareto optima exist re-mains open. We conjecture that such Pareto optima exist for any “sufficientlyconvex” aggregator.Our notions of preference-aggregation considers (cid:37) aggf an aggregate prefer-ence of family f , if (cid:37) aggf is complete and transitive and if x (cid:37) i x (cid:48) for all i ∈ f implies x (cid:37) aggf x (cid:48) where the latter preference must hold strictly if x (cid:31) j x (cid:48) inaddition holds for at least one j ∈ f . But many natural aggregation rulesfor preferences, such as majority rule or the Borda count, do not yield tran-sitive aggregate preferences. So one could adapt the notions of fairness usingone of these aggregation rules. In that vein, we could consider an allocationmajority-EF if it is envy free for at least half of all members of each family.Conversely an allocation would violate majority-EF if there exist two families f and f (cid:48) such that the majority of family members in f envies family f (cid:48) . Fora practical application think of democratic countries which may approve anallocation by referendum. We leave this question for further study. We are grateful for the support of the Minerva foundation through theARCHES prize. Erel is grateful to the Israel Science Foundation for grant Recently Ghodsi et al. (2018) and Segal-Halevi and Suksompong (2020) showed thatsuch ad-hoc grouping guarantees the existence of envy-free allocations in the respectivecontexts of rent division and cake-cutting (they did not consider Pareto-efficiency). Majority fairness was studied in the context of cake-cutting (Segal-Halevi and Nitzan,2019) and indivisible item allocation (Segal-Halevi and Suksompong, 2019). We are grateful to Shane Auerbach, Amit Goyal and Kitsune Cavalryfor participating in the discussion.
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