Please cite as: Velez, R. A., Sharing an increase of the rent fairly, Soc Choice Welf (special issue in honor of William Thomson) 48(1), 59-80 (2017) The final publication is available at link.springer.com

Sharing an increase of the rent fairly Rodrigo A. Velez∗ b

Department of Economics, Texas A&M University, College Station, TX 77843 USA

July 20, 2016

Abstract We characterize the family of non-contestable budget-monotone rules for the allocation of objects and money as those obtained by maximizing a maxmin social welfare function among all non-contestable allocations. We provide three additional seemingly independent approaches to construct these rules. We present three applications of this characterization. First, we show that one can “rectify” any non-contestable rule without losing non-contestability. Second, we characterize the preferences that admit, for each budget, a non-contestable allocation satisfying a minimal or maximal individual consumption of money constraint. Third, we study continuity properties of the non-contestable correspondence. JEL classification: D61; D63; D70. Keywords: solidarity; resource monotonicity; allocation of objects and money; non-contestable allocations.

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Introduction

A group of roommates pay the rent of their apartment in such a way that no roommate prefers the room of another roommate if she has to pay the other roommate’s share of the rent. The landlord increases the rent. This paper characterizes the ways in which the roommates can redistribute the house and rent payments, so they all, in welfare terms, contribute to the higher rent without compromising the initial equitability property of the allocation. Formally, we study an assignment problem with money. There are n agents who are endowed with n objects. Agents consume exactly one object and an ∗ Thanks to William Thomson, two anonymous referees, an associate editor, and the audience at SCW16 for useful comments and discussions. All errors are my own. [email protected];

https://sites.google. om/site/rodrigoavelezswebpage/home

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amount of an infinitely divisible good that we refer to as money. We assume that, ceteris paribus, more money is better for each agent. We refer to the aggregate consumption of money at an allocation as its budget. This model accommodates the allocation of rooms and rent described above, the division of a partnership, and the allocation of tasks and salary among workers. We study systematic ways of making recommendations for each possible budget. We refer to each such a function as a “rule.” We axiomatically study rules. We require two axioms. The first is non-contestabilty, i.e., no agent should prefer the allotment of another agent to her own (Foley, 1967).1 This axiom has been central in the theory of equitable allocation (see Thomson, 2010, for a survey). It captures the idea of equal opportunity (Kolm, 1971) and precludes a form of clear-cut bias (Varian, 1974). The second axiom is budget-monotonicity, i.e., the welfare of each agent should be an increasing function of the budget (Moulin and Thomson, 1988). This axioms captures the idea of solidarity when budget changes. The roommates case illustrates it. If the rent to be collected increases, agents should all be responsible for the change and thus be affected in a comparable way. Since money is desirable to each agent, this leads to the requirement that each agent’s welfare decreases as the rent to be collected increases. Alkan et al. (1991) introduce four approaches to construct non-contestable budget-monotone rules. The first is to fix a continuous representation of preferences and for each budget maximize the minimum utility across agents among all non-contestable allocations. The second is again to fix a continuous representation of preferences and for each budget minimize the maximum utility across agents among all non-contestable allocations. The third is to fix a list of continuous and increasing functions, one for each object, and then for each budget maximize the minimal transformed consumption of money among all non-contestable allocations. The fourth is again to fix a list of continuous and increasing functions, one for each object, and then for each budget minimize the maximal transformed consumption of money among all non-contestable allocations. Our main result, Theorem 1, states that each of these four approaches spans the whole family of non-contestable budget-monotone rules. Thus, these seemingly independent constructions are actually equivalent: by choosing the parameters of one of these rules, one is implicitly choosing the parameters for the other constructions. Our contribution goes beyond showing that each of the four constructions above are equivalent. Alkan et al. (1991)’s proof that these rules are budget1 This axioms is commonly referred to as “no-envy.” See Velez (2016) for a discussion why “non-contestability” reflects better the normative content of the axiom.

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monotone is based on linear programming and linear approximation techniques. We present a new constructive proof of this result, which is of independent interest. Indeed, we show that one can obtain budget-monotonicity properties of the non-contestable set directly from the existence of non-contestable allocations results available in the literature (e.g. Svensson, 1983, 1987; Velez, 2016) without using approximation techniques. We present three applications of Theorem 1. First, we show how one can “rectify” a non-contestable rule that fails to be budget-monotone. A rule is budget monotone at a certain budget, if each agent is worse off at each lower budget and better off at each higher budget. We show that one can “iron” any non-contestable rule preserving both non-contestability and the rule’s recommendations for budgets at which it is budget-monotone. That is, given an arbitrary non-contestable rule, say f , one can find a non-conestable rule that is “weakly budget-monotone” and coincides with f for the budgets at which f is budget-monotone (Theorem 3). Then, we show that given an arbitrary noncontestable rule, say f , and a positive δ, one can find a non-contestable rule that is budget-monotone and coincides with f on the budgets at which f is budget-monotone up to a set of Lebesgue measure δ (Theorem 4). Our second application is to characterize the set of preferences that admit for each budget a non-contestable allocation satisfying a minimum or maximum individual consumption of money constraint (Proposition 1). Our third application is to show that the non-contestable correspondence not only is continuous in welfare space, but also admits a continuous selection, a feature that is usually difficult to guarantee for a correspondence that is not convex (Michael, 1956). This allows us to show the existence of non-contestable budget-monotone efficient rules in the extension of our model where there are more objects than agents (Sec. 4.3). Our approach to study rules and their properties, as opposed to studying isolated allocation problems, can be traced back to Thomson (1983). Previous literature has explored the performance of allocation rules when one considers other dimensions of assignment problems with money as variable. Most notably, Alkan et al. (1991) and Alkan (1994) study solidarity notions under the arrival of objects; Alkan (1994) and Tadenuma and Thomson (1995b) study solidarity notions under the arrival of new agents; and Tadenuma and Thomson (1991) study the consistency axiom, which can be interpreted as a form of solidarity when allocations have to be reassessed for a subset of agents with their corresponding initial allotments (Thomson, 2012).

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2

Model

2.1

Environment

Let N be a set of n agents and A a set of n objects. Generic agents are i , j , and generic objects are α, β . Each agent consumes one object and an amount of a perfectly divisible good we refer to as money. Agent i ’s generic consumption bundle is (xi , α) where xi ∈ R and α ∈ A. Each agent has complete and transitive preferences on consumption bundles. Agent i ’s preference is R i and fixed throughtout. The symmetric and asymmetric parts of Ri are denoted by I i and Pi respectively. Preferences are money monotone: ceteris paribus, each agent prefers a higher consumption of money. Preferences also satisfy the compensation assumption: for each bundle (xi , α) and each object β , there is xi′ such that (xi , α) Ii (xi′ , β ).2 Let R be the domain of these preferences. The profile of preferences is R ≡ (R i )i ∈N ∈ R n . With exception of Sec.5, where we discuss incentives issues, we consider the preference profile R as fixed and do not quantify it in our results. An allocation is a pair z ≡ (x , µ ) where x ≡ (xi )i ∈N ∈ Rn and µ : N → A is a bijection. Agent i ’s bundle at z is z i ≡ (xi , µi ), where xi is her consumption of money and µi her assigned object. The set of allocations is Z . We denote the set of allocations whose aggregate consumption of money is m ∈ R, i.e., (x , µ) P such that i ∈N xi = m, by Z m . Each agent’s preferences induce a preference on allocations: for each pair {z , z ′ } ⊆ Z , z Ri z ′ if z i Ri z i′ . For an allocation z ≡ (x , µ) we denote the consumption of money of the agent who receives object α ∈ A at z by x α . For each i ∈ N , each pair {α, β } ∈ A, and each xα ∈ R, let q i (x α , α, β ) ∈ R be the amount such that (xα , α) Ii (qi (xα , α, β ), β ). For each m ∈ R and each z ≡ (x , µ) ∈ Z m , let d i j (z ) ≡ qi (xi , µi , µ j ) − x j . For each x , y ∈ R2 , seg(x , y ) ≡ {z ∈ R2 : z = λx + (1 − λ)y , λ ∈ [0, 1]} is the segment from x to y .

2.2

Socially desirable allocation rules

We search for socially desirable systematic ways to provide recommendations for each possible configuration of resources in our environment. We refer to such an object as a rule, i.e., a function that assigns to each possible aggregate consumption of money, say m, an allocation in Z m . We denote the generic rule by f : R → Z . We denote the consumption bundle assigned to agent i ∈ N at f (m) by fi (m). 2

Money-monotonicity and the compensation assumption imply continuity.

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By studying rules we can evaluate our recommendations with two different perspectives. First, we can formulate axioms, i.e., properties of allocations that can be justified in normative terms. The following two axioms capture the idea that, in welfare terms, an allocation should not be unambiguously dominated by another feasible allocation and should not be unambiguously biased towards any agent: z ∈ Z m is efficient if there is no z ′ ∈ Z m such that for each i ∈ N , z ′ Ri , z and for at least one j ∈ N , z ′ Pj z ; z ∈ Z is non-contestable if for each pair {i , j } ⊆ N , z i Ri z j (Foley, 1967; Varian, 1974). In our environment, non-contestable allocations exist (Svensson, 1983; Alkan et al., 1991; Velez, 2016) and are efficient (Svensson, 1983). We denote the set of non-contestable allocations by F and the set of non-contestable allocations with aggregate consumption of money m by Fm . We say that a rule f is noncontestable if for each m, f (m) is non-contestable. If f is non-contestable we write f ∈ F . The following simple but fundamental result will be used repeatedly throughout the paper; see Alkan et al. (1991) for its proof. Lemma 1 (Decomposition Lemma; Alkan et al. (1991)). Let z ≡ (x , µ) ∈ F and b) ∈ F . Then, both µ and µ b are bijections between: zb ≡ ( xb, µ (i) {i ∈ N : z i Pi zbi } and {α ∈ A : xα > xbα }. (ii) {i ∈ N : z i Ii zbi } and {α ∈ A : xα = xbα }. (iii) {i ∈ N : zbi Pi z i } and {α ∈ A : xbα > xα }.

Second, by studying rules instead of individual allocations we can articulate the idea of “solidarity,” i.e., given a change in the conditions that describe the resource allocation problem, agents should share the welfare gains or losses in comparable ways. Since interpersonal comparisons of utility are difficult, an uncontroversial requirement is that each agent’s welfare change in the same direction (Thomson, 1983). In our model, in which the aggregate consumption of money is the only parameter that can change and money is desirable to all agents, this solidarity principle takes a simple form:

A rule f is budget-monotone at m ∈ R if (1) for each l < m , each agent prefers f (m) to f (l ); and (2) for each h > m , each agent prefers f (h ) to f (m); f is budget-monotone if it is budget-monotone at each m ∈ R. We now introduce six families of rules based on intuitive criteria of distributive justice. The first four families allow us to make a simple description of the non-contestable and budget-monotone rules. The other two families of rules are non-contestable but may violate budget-monotonicity; in Sec. 4.1 we show 5

how we can reconcile these intuitive approaches to fair allocation with budgetmonotonicity. An non-contestable rule f is: • Value-Maxmin (Alkan et al., 1991):3 if there is a continuous representation of preferences u ≡ (u i )i ∈N such that for each m, n o f (m) ∈ argmax min u i (z i ) . i ∈N

z ∈Fm

• Money-Maxmin (Alkan et al., 1991; Velez, 2011): if there is a profile of continuous strictly increasing functions g ≡ (g α )α∈A where for each α, g α : R → R and for each m, n o f (m) ∈ argmax min g α (xα ) . z ≡(x ,µ)∈Fm

α∈A

• Value-Minmax (Alkan et al., 1991): if there is a continuous representation of preferences u ≡ (u i )i ∈N such that for each m, n o f (m) ∈ argmin max u i (z i ) . i ∈N

z ∈Fm

• Money-Minmax (Alkan et al., 1991; Velez, 2011): if there is a profile of continuous strictly increasing functions g ≡ (g α )α∈A where for each α, g α : R → R and for each m, n o f (m) ∈ argmin max g α (xα ) . z ≡(x ,µ)∈Fm

α∈A

• Utilitarian: if there is a continuous representation of preferences u ≡ (u i )i ∈N such that for each m, « ¨ X u i (z i ) . f (m) ∈ argmax z ∈Fm

i ∈N

• Equal-compensation rule (Tadenuma and Thomson, 1995b): if for each m, § ª ′ ′ f (m) ∈ z ∈ Fm : for each z ∈ Fm , min d i j (z ) ≥ min d i j (z ) . i , j ∈N , i 6= j

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i , j ∈N , i 6= j

This family of rules can be seen as the implementation of the Rawlsian principle of distributive justice (Rawls, 1972) constrained by efficiency and no-envy.

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3

Budget-monotone rules

Each of the families of value-maxmin, money-maxmin, value-minmax, and money-minmax rules defined in Sec. 2.2 coincides with the family of non-contestable budget-monotone rules. Theorem 1. Let f be non-contestable. The following are equilivalent: 1. f is budget-monotone 2. f is Value-Maxmin. 3. f is Money-Maxmin. 4. f is Value-Minmax. 5. f is Money-Minmax. We first prove that statement 1 implies statements 2-5 in Theorem 1. The intuition why statement 1 implies statements 2 and 4 is the following (the proof that statement 1 implies statements 3 and 5 is based on a similar argument). Suppose that f is non-contestable and budget-monotone. Let wi be a continuous representation of agent i ’s preferences and m 7→ vi (m) ≡ wi ( fi (m)) the image in agent i ’s utility space of f ’s recommendations. Since f is budgetmonotone, this function is strictly increasing. The main steps of our argument consist on proving that vi is continuous and that the function u i ≡ vi−1 ◦ wi which assigns to an arbitrary bundle, say z i , the budget m for which agent i would be indifferent between her assignment fi (m) and z i is well-defined and continuous. Since f is budget-monotone (higher budget makes everybody better off ), u i represents agent i ’s preferences. Moreover, u i has the property that for each m, u i ( fi (m)) = m. Thus, since non-contestable allocations are efficient, it follows that f is a value-Maxmin and value-Minmax rule with respect to representation (u i )i ∈N . Proof. Let f be non-contestable and budget monotone. We first prove that f is value-maxmin and value-minmax. We construct a profile of continuous utility functions u ≡ (u i )i ∈N representing R such that for each m, and each pair {i , j } ⊆ N , u i ( fi (m)) = u j ( f j (m)). We claim that f (m) is a value-maxmin rule for u. Suppose by contradiction that there is m ∈ R and z ∈ Fm such that mini ∈N u i (z i ) > mini ∈N u i ( fi (m)). Then, for each i ∈ N , u i (z i ) > u i ( fi (m)). Thus, f (m) is not efficient. Thus, f (m) 6∈ F (Svensson, 1983). This is a contradiction. A symmetric argument shows that f is a value-minmax rule for u. For each i ∈ N , let wi be a continuous representation of Ri and vi ≡ wi ◦ fi . We 7

claim that each vi is a continuous function. Let m ∈ R and {mk }k ∈N be a sequence converging to m as k → ∞. We prove that as k → ∞, vi (mk ) → vi (m). Since {mk }k ∈N converges, it is bounded. Thus, since f is budget-monotone, {vi (mk )}k ∈N is bounded. Thus, it suffices to show that if a subsequence of {vi (mk )}k ∈N , which we denote by {vi (mk )}k ∈N for convenience, converges, its limit is vi (m).4 Now, since {mk }k ∈N is bounded and f is non-contestable, { f (mk )}k ∈N is bounded in RN ×A N (endowed with the box topology). Select a subsequence of { f (mk )}k ∈N that converges as k → ∞ and let z ≡ (x , µ) be its limit. Since each wi is continuous, as k → ∞, vi (mk ) → wi (z i ) . Since preferences are continuous, then z ∈ F . Since each subsequence of {mk }k ∈N converges to m, then P i ∈N x i = m and z ∈ Fm . There are two cases. Case 1: there is a subsequence of {mk }k ∈N , {f mk }k ∈N , such that for each k ∈ fk ≤ m. Then, for each k ∈ N, vi (f N, m mk ) ≤ vi (m). Thus, for each i ∈ N , wi (z i ) ≤ vi (m). Thus, for each i ∈ N , wi (z i ) = vi (m) = wi ( fi (m)), for otherwise z is neither efficient nor non-contestable. Case 2: there is a subsequence of {mk }k ∈N , {f mk }k ∈N , such that for each k ∈ fk ≥ m. Then, for each k ∈ N, vi (f N, m mk ) ≥ vi (m). Thus, for each i ∈ N , wi (z i ) ≥ vi (m). Thus, for each i ∈ N , wi (z i ) = vi (m) = wi ( fi (m)), for otherwise f (m) is neither efficient nor non-contestable. Let i ∈ N and (xi , α) ∈ R×A. Let u i (xi , α) ≡ m such that wi (xi , α) = wi ( f (m)). We claim that u i is well-defined. Since preferences are money-monotone and satisfy the compensation assumption, there is l ∈ R such that for each z ∈ Fl , wi (z i ) ≤ wi (xi , α); symmetrically, there is h ∈ R such that for each z ∈ Fh , wi (xi , α) ≤ wi (z i ). Thus, vi (l ) ≤ wi (xi , α) ≤ vi (h ). Since vi is continuous, there is m ∈ R such that vi (m) = wi ( fi (m)). Since f is budget-monotone, vi is strictly increasing and such m is unique. We claim that u i represents Ri . Let (xi , α) and (xi′ , α′ ) be such that (xi′ , α′ ) Ri (xi , α). We claim that u i (xi′ , α′ ) ≥ u i (xi , α). Suppose by means of contradiction that u i (xi , α) > u i (xi′ , α′ ). Since f is budgetmonotone, wi ( fi (u i (xi , α))) > wi ( f (u i (xi′ , α′ ))). Thus, wi (xi , α) > wi (xi′ , α′ ). This contradicts (xi′ , α′ ) Ri (xi , α). Now, let (xi , α) and (xi′ , α′ ) be such that u i (xi′ , α′ ) ≥ u i (xi , α). Since f is budget-monotone, wi ( fi (u i (xi′ , α′ ))) ≥ wi ( f (u i (xi , α))). Thus, (xi′ , α′ ) Ri (xi , α). By construction, for each m ∈ R and each pair {i , j } ⊆ N , u i ( fi (m)) = u j ( f j (m)) = m. Finally, we claim that u i is continuous. Let {(xik , αk )}k ∈N be a convergent sequence in R × A and let (xi , α) be its limit as k → ∞. Without loss of generality we can assume that the sequence {αk }k ∈N is constant. We claim that u i (xik , α) converges to u i (xi , α). For each k ∈ N, let mk ≡ u i (xik , α). Since {xik }k ∈N is convergent, it is bounded. Since preferences are money-mono4

Let {xk }k ∈N be a bounded sequence of real numbers and x ∈ R. If all convergent subsequences of {xk }k ∈N converge to x , then the sequence {xk }k ∈N is convergent and its limit is x .

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tone, {mk }k ∈N is bounded. Thus, it suffices to show that if a subsequence of {mk }k ∈N , which we denote by {mk }k ∈N for convenience, converges, its limit is u i (xi , α). Let m be the limit as k → ∞ of {mk }k ∈N . For each k ∈ N, let z k ≡ f (mk ). Since {mk }k ∈N is bounded, so is {z k }k ∈N . Thus, one can select a convergent subsequence of {z k }k ∈N . Let z be the limit of such a convergent subsequence. Since wi is continuous, {wi (z ik )}k ∈N converges as k → ∞ to wi (z i ). Since for each k ∈ N, wi (z ik ) = wi (xik , α), then {wi (xik , α)}k ∈N converges as k → ∞ to wi (xi , α). Thus, wi (z i ) = wi (x , α). From our argument above, wi (z i ) = wi ( fi (m)). Thus, wi (xi , α) = wi ( fi (m)) and u i (xi , α) = m. We now prove that f is money-maxmin and money-minmax. We construct a profile of continuous increasing functions g ≡ (g α )α∈A where for each α, g α : R → R, with the following property. For each m and each pair {α, β } ⊆ A, if f (m) ≡ (x , µ), then g α (xα ) = g β (xβ ). We claim that f is a money-maxmin rule for such a profile of functions. Suppose by contradiction that there is m ∈ R and z ≡ (y , µ) ∈ Fm such that minα∈A g α (yα ) > min Pα∈A g α (xα ). Since each g α is increasing, then for each α ∈ A, yα > xα . Thus, α∈A yα > m. This is a contradiction. A symmetric argument shows that f is a money-minmax rule for g . For each α ∈ A and each m ∈ R, let yα (m) ≡ xα where f (m) = (x , µ). Since f is budget-monotone, by the Decomposition Lemma (Lemma 1), each yα is a strictly increasing function. Since f ∈ F , yα is unbounded. We claim that each yα is continuous. Let {mk }k ∈N be a convergent sequence and m its limit as k → ∞. Let z ≡ (x , µ) ≡ f (m). We prove that {yα (mk )}k ∈N converges as k → ∞ to yα (m) = xα . Since {mk }k ∈N converges, it is bounded. Thus, since yα is strictly increasing, {yα (mk )}k ∈N is bounded. Thus, it suffices to show that if a subsequence of {yα (mk )}k ∈N , which we denote by {yα (mk )}k ∈N for convenience, converges, its limit as k → ∞ is yα (m). Since {mk }k ∈N is bounded and f is none k )}k ∈N be a concontestable, { f (mk )}k ∈N is bounded in RN ×A N . Let {e z k ≡ ( xek , µ e) be its limit as k → ∞. Since vergent subsequence of { f (mk )}k ∈N and ze ≡ ( xe , µ fk → m, ze ∈ Fm . There preferences are continuous, ze ∈ F . Since as k → ∞, m are two cases. If there is a convergent subsequence of { f (mk )}k ∈N , { f (f mk )}k ∈N , fk ≤ m. Let ze ≡ ( xe, µ e) be the limit as k → ∞ of such that for each k ∈ N, m { f (f mk )}k ∈N . Since f is budget-monotone, Lemma, for Pby the Decomposition P each k ∈ N, xe k ≤ x . Thus, xe ≤ x . Since α∈A xα = α∈A xeα = m, then xe = x . Since {yα (mk )}k ∈N is convergent and { xeαk }k ∈N is one of its subsequences, as k → ∞, yα (mk ) → xeα = xα . The argument is symmetric if there is a convergent fk ≥ m. Let subsequence of { f (mk )}k ∈N , { f (f mk )}k ∈N , such that for each k ∈ N, m α ∈ A. For each xα ∈ R, let g α (xα ) ≡ m such that the consumption of money of the agent who receives object α at f (m) is xα . As above, by the Intermediate Value Theorem, g α is well-defined. By construction, for each m and each pair {α, β } ⊆ A, if f (m) ≡ (x , µ), g α (xα ) = g β (xβ ) = m. Since f is budget-monotone, 9

by the Decomposition lemma each g α is strictly increasing. Finally, we prove that each g α is continuous. Let {xαk }k ∈N be a convergent sequence and let xα be its limit as k → ∞. We prove that as k → ∞, g α (xαk ) → g α (xα ). Since {xαk }k ∈N converges, it is bounded. Since {xαk }k ∈N is bounded and g α is increasing, {g α (xαk )}k ∈N is bounded. Thus, it suffices to show that if a subsequence of {mk ≡ g α (xαk )}k ∈N , which we denote by {mk ≡ g α (xαk )}k ∈N for convenience, converges, its limit as k → ∞ is g α (xα ). Let m be the limit as k → ∞ of {mk }k ∈N . We proved above that yα is continuous. Thus, as k → ∞, xαk = yα (mk ) → yα (m). Thus, yα (m) = xα . Thus, g α (xα ) = m. We now discuss the proof that each of numerals 2-5 imply numeral 1 in Theorem 1. This was first stated and proved by Alkan et al. (1991). A preference is “piece-wise linear” if it is represented by a function whose marginal utility of money is constant over a finite set of intervals.5 Alkan et al. (1991) base their work on linear programming applied to linear preference profiles, the local linearization of piece-wise linear preferences, and the approximation of a continuous preference by piece-wise linear preferences. We present an alternative direct proof based only on the existence of non-contestable allocations for a continuous economy. The key to our proof is the following result.6 Theorem 2 (strict monotonicity). Let m ∈ R, z ∈ Fm , and ǫ > 0. Then, (i) there is an allocation in Fm +ǫ that each agent prefers to z , and (ii) There is an allocation in Fm −ǫ that is strictly worse than z for each agent. Proof. We prove statement (i). The proof of statement (ii) is symmetric. Let m, z ≡ (x , µ) and ǫ as in the statement of the theorem. We prove (i) by induction over n, the number of agents. The statement is trivial when n = 1. Suppose then that the statement is true when there are n − 1 agents and consider an economy with n agents. Let z ′ ≡ (x ′ , µ′ ) ∈ Fm +ǫ/2. Existence of this allocation is guaranteed by Velez (2016, Theorem 6(i)); one can also prove this from Svensson (1983) in a similar way that one one proves Velez (2016, Theorem 6(i)) from Velez (2016, Theorem 1).7 Let A 1 ≡ {α ∈ A : xα′ > xα }. Since ǫ > 0, A 1 6= ;. Let 5 The domain of piece-wise linear preferences can be defined without reference to utility representations by interpolating a finite set of ordered indifference sets. 6 (i) in the Strict Monotonicity Theorem was first stated and proved by Alkan et al. (1991). Our contribution here is to provide a direct proof of it that requires no linear approximation. 7 The existence of z ′ ≡ (x ′ , µ′ ) ∈ Fm +ǫ/2 is the essential step in our proof. In their construction, Alkan et al. (1991) prove both the strict monotonicity theorem and existence of non-contestable allocations from linear approximations. An earlier version of this paper presented a proof of Theorem 2 based on the following corollary of Svensson (1983): for each positive budget there are non-contestable allocations at which each agent consumes a non-negative amount of money whenever each agent is indifferent among all bundles that have zero money.

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N1 ≡ {i ∈ N : µ′i ∈ A 1 }, A 2 ≡ A \ A 1 , and N2 ≡ N \ N1 . Suppose that N2 6= ;, for otherwise the theorem would follow. By the Decomposition Lemma, both µ and µ′ are bijections between N1 and A 1 and between N2 and A 2 , and for each ′ agent i ∈ N1 , z i′ Pi z i . Let 0 < δ1 < ǫ/2 be such that P δ1 < min{xα − xα : α ∈ A 1 }. By definition of A 1 , there is such a δ1 . Let m1 ≡ i ∈N1 xi and m2 = m − m1 . Applying our induction hypothesis to the economy with set of agents N1 endowed with A 1 and budget m1 + δ1 we can find an allocation z 1 ≡ (x 1 , µ1 ) that is noncontestable for this economy, and such that for each i ∈ N1 , z i1 Pi z i . Thus, for each i ∈ N1 and each j ∈ N2 , z i1 Pi z j . Thus, there is 0 < δ2 ≤ ǫ/2 such for each i ∈ N1 and each j ∈ N2 , z i1 Pi (x j + δ2 , µ j ). Applying our induction hypothesis to the economy with set of agents N2 endowed with A 2 and budget m2 + δ2 we can find an allocation z 2 ≡ (x 2 , µ2 ) that is non-contestable for this economy, and such that for each j ∈ N2 , z 2j Pj z j . Let z δ be the allocation such that for each i ∈ N1 , z iδ ≡ z i1 and for each j ∈ N2 , z δj ≡ z 2j . We claim that z δ ∈ Fm +δ1 +δ2 . By the Decomposition Lemma, for each α ∈ A 2 , xα2 > xα . Thus, for each α ∈ A 2 , xα2 ≤ xα + δ2 . Thus, for each i ∈ N1 and each j ∈ N2 , z iδ Ri z δj . Recall that for

each j ∈ N2 , z 2j Pj z j . By the Decomposition Lemma, for each j ∈ N2 , z j R j z ′j . Since z ′ ∈ Fm +ǫ/2, for each j ∈ N2 and each i ∈ N1 , z ′j R j z i′ . By our choice of δ1 ,

for each α ∈ A 1 , xα1 ≤ xα′ . Thus, for each j ∈ N2 and each i ∈ N1 , z 2j R j z i1 . Thus, for each j ∈ N2 and each i ∈ N1 , z δj Ri z iδ . Since z 2 is non-contestable in the economy with set of agents N2 endowed with A 2 and budget m2 + δ2 , for each pair {i , j } ⊆ N2 , z iδ Ri z δj . Since z 1 is non-contestable in the economy with set of agents N1 endowed with A 1 and budget m1 + δ1 , for each pair {i , j } ⊆ N1 , z iδ Ri z δj . Thus, z δ ∈ Fm +δ1 +δ2 . This proves that the set of δ > 0 such that δ ≤ ǫ and there is an element of Fm +δ that each agent prefers to z , is non-empty. Since the set is bounded, it has a supremum. Let δ∗ be this supremum. Since preferences are continuous, δ∗ is the maximum of this set. Thus, δ∗ = ǫ, for otherwise one can repeat the argument above distributing ǫ − δ∗ from an allocation in Fδ∗ that each agent prefers to z . Theorem 2 implies that value-Maxmin, value-Minmax, money-Maxmin, and money-Minmax rules are budget monotone (see Alkan et al., 1991, Theorem 6).8 We sketch the argument for value-Maxmin rules for completeness. Let u ≡ (u i )i ∈N be a continuous representation of preferences. Let m ∈ R, ǫ > 0, 8

This can also be easily seen from the characterization of value-Maxmin, value-Minmax, money-Maxmin, and money-Minmax allocations by means of the chains of indifferences that connect the agents’ consumptions at the allocation introduced by Alkan (1994) and used to understand incentive properties of non-contestable rules by Velez (2011); Fujinaka and Wakayama (2015); Andersson et al. (2014a,b).

11

z ∈ arg max{u i (z i ) : i ∈ N , z ∈ Fm }, and z ′ ∈ arg max{u i (z i ) : i ∈ N , z ∈ Fm +ǫ }. Let i ∈ N be such that u i (z i ) = min j ∈N u j (z j ). By Theorem 2, max{u i (z i ) : i ∈ N , z ∈ Fm } is an increasing function of m. Thus, u i (z i′ ) > u i (z i ). Let A 1 ≡ {µi : i ∈ N , xµ′ i > xµi } and N1 ≡ {i ∈ N : µi ∈ A 1 }. By the Decomposition Lemma, i ∈ N1 . Thus, all agents who minimize the utility value at z are in N1 . by the Decomposition Lemma, both µ and µ′ are bijections between A 1 and N1 . Thus, each agent in N \N1 prefers her allotment at z than the allotment of each i ∈ N1 at z ′ . Thus, by Theorem 2 (ii), starting from z one can extract, in aggregate, a small amount from the agents in N \ N1 and distribute it among the agents in N1 (Theorem 2 (i)) so that non-contestability is preserved and the minimum utility among the agents is still achieved by an agent in N1 . This would imply that z was not a member of arg max{u i (z i ) : i ∈ N , z ∈ Fm }.

4 4.1

Applications Reconciling intuitive fairness and budget-monotonicity

The following are two examples of rules defined by means of intuitive criteria of fairness that violate budget-monotonicity. Example 1 (A Utilitarian rule may violate budget-monotonicity). Let N ≡ {1, 2} and A ≡ {α, β }. We construct a preference profile, represented by utility functions u 1 and u 2 such that there is no budget-monotone f satisfying that for each m, f (m) ∈ Sm ≡ arg max u 1 (z 1 ) + u 2 (z 2 ). z ∈Fm

1 Let 0 < ǫ < 16 , u 1 (xα , α) ≡ xα for each xα ∈ R, u 1 (xβ , β ) ≡ xβ for each xβ ∈ R, 1 2ǫ xβ + 2(1−2ǫ) for each xβ ≤ −ǫ, and u 2 (xα , α) ≡ xα for each xα ∈ R, u 2 (xβ , β ) ≡ 1−2ǫ

u 2 (xβ , β ) ≡

1−2ǫ 2ǫ x β

+ 1 for each xβ > −ǫ (Fig. 1).

β u2

α

x

2ǫ 1−2ǫ

1 x + 2(1−2ǫ)

− 12

1 2

−ǫ 0

1 2

+ǫ

1 x′

1−2ǫ 2ǫ

x′ +1

Figure 1: u 2 in Example 1. Each point x in the axis that is labeled by an object, say α, represents the bundle (x , α). Preferences are illustrated by means of solid lines that connect bundles that are indifferent for the agent. Dashed lines are auxiliary.

12

Let m ∈ R and z ≡ (x , µ) ∈ Fm . Since z is efficient, µ2 = β . Since z ∈ Fm , a ≤ x2 ≤ m/2 where a solves the equation u 2 (m − a , α) = u 2 (a , β ). For each m one can directly calculate Fm and characterize its image in utility space, i.e., u(Fm ) = {(u 1 (z 1 ), u 2 (z 2 )) : z ∈ Fm } (Fig. 2).

u(Fm ) =

    

seg

Š
Š 2ǫ m 1 1 1 , 1−2ǫ 2 + 2(1−2ǫ) , 2ǫm + 2 ,2ǫm + 2


S m 1−2ǫ m 1 2 , 2ǫ 2 + 1 , m + ǫ, 2 + ǫ

€€

m 2

seg



seg m + ǫ, 12 + ǫ ,(2ǫm + 12 ,2ǫm + 12 )

seg

m 2



m , 1−2ǫ 2ǫ 2 + 1 ,((1 − 2ǫ)m + 2ǫ,(1 − 2ǫ)m + 2ǫ)

u2

m 2


m , 1−2ǫ 2ǫ 2 + 1

if m ≤ −2ǫ if − 2ǫ ≤ m ≤ if m ≥

1 2

(1)

1 2

u2

u 2 = 1−2ǫ 2ǫ u 1 + 1

m 2

u(Fm )


m , 1−2ǫ 2ǫ 2 + 1

u2 = u1 u F1/2 2ǫ 1 u 2 = 1−2ǫ u 1 + 2(1−2ǫ)





m + ǫ, 21 + ǫ

1 1

u1 1 2

€

m 2

2ǫ m 1 , 1−2ǫ 2 + 2(1−2ǫ)

Š

(1 − 2ǫ)m + 2ǫ 2ǫm + 21

(a)


2ǫm + 12 ,2ǫm + 12

u(F−2ǫ )

u1

1 2

(b)

Figure 2: Image of the non-contestable set Fm in utility space, u (Fm ), in Example 1. The explicit expression of these sets for different values of m is given by equation (1). In panel (a) we show u (Fm ), for m ≤ −2ǫ or m ≥ 12 . For these values of m, u (Fm ) is a segment of line. All the segments have equal slope in each of these ranges. Segments for m ≤ −2ǫ are flatter than those for m ≥ 1 1 2 . For −2ǫ ≤ m ≤ 2 , u (Fm ) is the union of two segments, one with the common slope of the segments when m ≤ −2ǫ and the other with the common slope of the segments when m ≥ 12 (Panel (b)). Observe that when m ≤ −2ǫ the utilitarian swf selects the right corner of u (Fm ) and when m ≥ 12 the utilitarian swf selects the left corner of u (Fm ). When the utilitarian swf transitions from selecting the right corner to selecting the left corner of the set, agent 1’s welfare decreases.

Since 0 < ǫ < 14 , the slope of the segments that form Fm for m ≤ −2ǫ is greater than −1 (segments are flatter than the line with slope −1); and the slope of the segments that form Fm for m ≥ 1/2 is lower than −1 (segments are steeper than 13

the line with slope −1). Thus, denoting u(Sm ) ≡ {(u 1 (z 1 ), u 2 (z 2 )) : z ∈ Sm }, we can calculate that:  
1 2ǫm + 21 if m < 0   2ǫm + 2 ,
1 1 u(Sm ) = (0, 1), 2 , 2 if m = 0
  m 1−2ǫ m , + 1 if m > 0 2 2ǫ 2

Let f be such that for each m ∈ R, f (m) ∈ Sm . Then, for each δ ∈ (0, 1/(1 + 4ǫ)), u 1 ( f (−δ)) > u 1 ( f (δ)). Thus, f violates budget-monotonicity.

Example 2 (An Equal-compensation rule may violate budget-monotonicity when there are three agents). Let N ≡ {1, 2, 3} and A ≡ {α, β , γ}. We construct a preference profile R ≡ (Ri )i ∈N such that there is no budget-monotone f such that for each m, § ª ′ ′ f (m) ∈ Em ≡ z ∈ Fm : for each z ∈ Fm , min d i j (z ) ≥ min d i j (z ) . i , j ∈N , i 6= j

i , j ∈N , i 6= j

Let 0 < ǫ < 15 and R a preference profile with numerical representation u = (u i )i ∈N , such that: (1) for each x ∈ R, u 1 (x , α) = u 1 (x , β ) = u 1 (x , γ) = x , (2) for each i ∈ {2, 3}, x ∈ (−∞, −1 − 2ǫ) ∪ (−1, 0), u i (x , α) = u i (x , β ) = u i (x + 6, γ) = x , (3) u 2 (−1 − ǫ, β ) > u 2 (−1 − ǫ, α), (4) u 3 (−1 − ǫ, α) > u 3 (−1 − ǫ, β ), (5) u 2 (ǫ, β ) = u 2 (7, α) = u 2 (7, γ), and (6) u 3 (ǫ, α) = u 3 (7, β ) = u 2 (8, γ) (Fig. 3.)

γ

γ

β

β

γ −1 − ǫ

β

|

R1 α

|

R2

|

|

|

|

|

|

|

|

−1 0

1

2

3

4

5

6

7

α

|

R3

|

|

|

|

|

|

|

|

−1 0

1

2

3

4

5

6

7

−1 − ǫ

α

| |

|

|

|

|

|

|

|

|

−1 0

1

2

3

4

5

6

7

Figure 3: R in Example 2.

Let (x , µ) ∈ F0 . Then, x1 ≥ 0 and µ1 = γ. We claim that if (x , µ) ∈ E0 , then x1 ≥ 2. To prove this, suppose by contradiction that there is z ≡ (x , µ) ∈ E0 such that x1 < 2. Then, there is i ∈ {2, 3} such that xi > −1. Thus, 0 ≥ x2 ≥ −1 and 0 ≥ x3 ≥ −1. Since (x , µ) ∈ F , x2 = x3 > −1. Thus, d 23 (z ) = 0. Now, let z ′ be the allocation defined by z 1′ ≡ (2 + 2ǫ, γ), z 2′ ≡ (−1 − ǫ, β ), and z 3′ ≡ (−1 − ǫ, α). One can verify that z ′ ∈ F0 and mini , j ∈N , i 6= j d i j (z ′ ) > 0. Thus, z 6∈ E0 . This is a contradiction. 14

Let (x , µ) ∈ F1 . Then, x1 ≥ 1/3 and µ1 = γ. σ(1) = γ. We claim that if (x , µ) ∈ E1 , then x1 ≤ 1. To prove this, suppose by contradiction that there is z ≡ (x , µ) ∈ E1 such that x1 > 1. Then, there is i ∈ {2, 3} such that xi < 0. Thus, mini , j ∈N , i 6= j d i j (z ) ≤ 2ǫ. Now, let z ′ be the allocation defined by z 1′ = (1 − 2ǫ, γ), z 2′ ≡ (ǫ, β ), and z 3′ ≡ (ǫ, α). One can verify that z ′ ∈ F1 and mini , j ∈N , i 6= j d i j (z ′ ) = 1 − 3ǫ > 2ǫ. Thus, z 6∈ E1 . This is a contradiction. Thus, f (0) P1 f (1) and f violates budget-monotonicity. Examples 1 and 2 illustrate how non-contestable rules defined on intuitive criteria of fairness may violate budget-monotonicity. It is not necessary to discard all of the recommendations given by a rule because it violates this property. The following lemma states that given two recommendations of a noncontestable rule that induce no violation of budget-monotonicity, one can interpolate them with a non-contestable budget-monotone rule. Lemma 2 (Interpolation of two non-contestable allocations). Let {h , l } ⊆ R be such that h > l , z ∈ Fl , and z ′ ∈ Fh . Suppose that each agent prefers z ′ to z . Then, there is a non-contestable budget-monotone rule f such that f (l ) = z and f (h ) = z ′ . Proof. Let h , l , z , and z ′ as in the statement of the lemma. Let u ≡ (u i )i ∈N be a continuous representation of preferences. Let v ≡ (vi )i ∈N be the representation of preferences given by: for each (xα , α), vi (xα , α) = 2 +

1 (u i (xα , α) − u i (z i′ )). u i (z i′ ) − u i (z i )

Let f be a value-Maxmin non-contestable rule associated with v . For each pair {i , j } ⊆ N , vi (z i ) = v j (z j ) = 1 and vi (z i′ ) = v j (z ′j ) = 2. Thus, we can select f in such a way that f (l ) = z and f (h ) = z ′ . By Theorem 1, f is budget-monotone. It is always possible to recover a weaker form of budget-monotonicity in any rule without losing the rule’s recommendations for the budgets at which the rule is budget-monotone. Theorem 3. Let f be a non-contestable rule and M ⊆ R the set of budgets at which f is budget-monotone. Then, there is a non-contestable rule g that coincides with f on M and satisfies that for each pair {l , h } ⊆ R such that h > l , each agent weakly prefers g (h ) to g (l ). Proof. Let f and M be as in the statement of the theorem. Assume without loss of generality that M 6= ;, for otherwise the result trivially follows from Theorem 1. We define rule g . For each m ∈ M , let g (m) ≡ f (m). Let M be the closure 15

of M . Let m ∈ M \M and {mk }k ∈N a sequence in M that converges to m. Since {mk }k ∈N is convergent, it is bounded. Thus, { f (mk )}k ∈N is bounded. Thus, the set of limit points of { f (mk )}k ∈N is non-empty. Let g (m) ≡ z where z is a limit point of { f (mk )}k ∈N . Let {h , l } ⊆ M be such that h > l . We claim that each agent weakly prefers g (h ) to g (l ). Let {hk }k ∈N and {l k }k ∈N be two sequences in M such that as k → ∞, hk → h , f (hk ) → g (h ), l k → l , and f (l k ) → g (l ). Since h > l , there is k ∈ N such that for each k ≥ K , hk > l k . Since f is budgetmonotone in M , for each k ≥ K , f (hk ) is weakly preferred by each agent to f (l k ). Since preferences are continuous, this property is preserved in the limit and each agent weakly prefers g (h ) to g (l ). If M is bounded below, let g coincide on the set of lower bounds of M with a non-contestable budget-monotone rule that coincides with g at min M (existence of this rule is guaranteed by Lemma 2; see also Corollary 1 below). If M is bounded above, let g coincide on the set of upper bounds of M with a non-contestable budget-monotone rule that coincides with g at max M . Let X ≡ {m ∈ R \ M : inf M < m < sup M }. Since M is closed, X is an open subset of R. Suppose without loss of generality that M is unbounded above and below. Thus, X can be written as the union of pairwise disjoint intervals ∪k ∈K (l k , hk ) where K ⊆ N. Let k ∈ K . We define g on (l k , hk ). We claim that {l k , hk } ⊆ M . Suppose by contradiction, and without loss of generality, that l k ∈ R\M . Thus, there is k ′ ∈ K such that l k ∈ (l k ′ , hk ′ ). Thus, (l k , hk )∩(l k ′ , hk ′ ) 6= ;. This is a contradiction. Let N0 ⊆ N be the set of agents who are indifferent between g (l k ) and g (hk ). For each i ∈ N0 , and each m ∈ (l k , hk ), let g i (m) ≡ g i (l k ). Let α(l k ) be the set of objects received by agents in N0 at g (l k ) and α(hk ) the set of objects received by agents in N0 at g (hk ). By the Decomposition Lemma, α(hk ) = α(l k ). For each i ∈ N \ N0 , let g coincide on (l k , hk ) with a non-contestable budget-monotone rule, for population N \ N0 endowed with objects A \ α(hk ) that interpolates g (l k )|N \N0 and g (hk )|N \N0 . Let m ∈ (l k , hk ) and (x , µ) ≡ g (m). By the Decomposition Lemma for each α ∈ A \ α(hk ), xα is less than the consumption of money of the agent who receives α at g (hk ). Since g (hk ) ∈ F , g (m) ∈ F . It is straightforward to prove that for each pair {l , h } ⊆ R such that h > l , each agent weakly prefers g (h ) to g (l ). Let f be a non-contestable rule and M ⊆ R the set of budgets at which f is budget-monotone. It may be impossible to find a non-contestable budgetmonotone rule that coincides with f on M .9 A satisfactory approximation is 9

An example that shows this is a rule f that assigns the same allotment to an agent, say i , for an interval say [l , h ] but otherwise the welfare of all agents increases with the aggregate budget. Here M = (−∞, l ) ∪ (h , +∞). If g is non-contestable and budget-monotone and coincides with f on M , all agents prefers g (h ) to g ((l + h )/2) and g ((l + h )/2) to g (l ). Since g coincides with f on M , for agent i , g (h ) is no better than f (h ) and g (l ) is no worse than f (l ). Thus, agent i is

16

available, however: In order to recover budget-monotonicity globally, one has to give up the recommendations of f only for a small subset of M (Theorem 4 below). This is surprising. If the set of budgets for which f violates budgetmonotonicity were a single interval, say [a , b ], the problem would be easy to solve. One can rectify f by replacing its values on an interval [a − ǫ, b + ǫ] with those of a budget-monotone rule g ∈ F that interpolates f (a − ǫ) and f (b + ǫ). This strategy is of little help if the complement of M can be an arbitrary set. Suppose for instance that the complement of M were the set of rational numbers. If one tries to enclose in an interval each point at which budgetmonotonicity fails for f and rectify the rule there, one would end up changing the values for the rule for almost every budget. Thus, one would practically lose all the recommendations of f , even though budget-monotonicity fails for an arguably small set of budgets. This is never the case, however. The intuition why this is so is as follows. In order to succeed rectifying f , one has to redefine the rule for a superset of the complement of M that is as close to this set as possible. It turns out that it is enough to redefine the rule for a smaller set, i.e., the complement of the closure of M . Let X be this set. Since X is an open set, it can be written as a countable disjoint union of open intervals. So we can recursively rectify f in these intervals. The result is that we cover X with a family of closed intervals whose union, C , covers very little outside X , and such that for each of these intervals the redefined rule is rectified and interpolates f in its extremes. Moreover, at the extremes of these intervals f is budget-monotone. Surprisingly, no point outside C violates budget-monotonicity for f . This is so because the boundary points of C are points at which f satisfies budgetmonotonicity. Thus, a budget in the complement of C must have budgets in M that are arbitrarily close to its left and must have budgets in M that are arbitrarily close to its right (being in the closure of M only guarantees having points in M arbitrarily close from one side when C is non-empty). At such a budget budget-monotonicity is “inherited” from the points in M that are close to it. Theorem 4. Let f be a non-contestable rule and M ⊆ R the set of budgets at which f is budget-monotone. Then, for each δ > 0 there is a Lebesgue measurable set ∆ ⊆ R whose measure is no greater than δ, and a non-contestable budget-monotone rule that coincides with f on M \ ∆. Proof. Suppose without loss of generality that M 6= ;, for otherwise the result trivially follows from Theorem 1. Let M be the closure of M and X ≡ R \ M . Suppose that X 6= ;. Since M is closed, then X is open. Thus, X can be written as the union of pairwise disjoint intervals ∪k ∈K (l k , hk ) where K ⊆ N (here there indifferent between g (h ) and g (l ). Thus, there is no such a g .

17

may be an interval whose lower limit is −∞ and an interval whose upper limit is +∞). We claim that for each k ∈ K , l k ⊆ M ∪ {−∞} and hk ⊆ M ∪ {+∞}. Suppose by contradiction and without loss of generality that l k ∈ R \ M . Then, there are two intervals in ∪k ∈K (l k , hk ) that are not disjoint. This is a contradiction. We will assume for simplicity in the presentation that K is an infinite set and X = ∪∞ t =1 (l k , hk ). We proceed in two steps. Let λ be the Lebesgue measure on R. Step 1: There is a Lebesgue measurable set C ⊆ R such that X ⊆ C ; λ(C \ X ) < δ; R\C ⊆ M ; there is g ∈ F that coincides with f on R\C and such that for each m ∈ C there is a pair {l , h } ⊆ (M ∩C )∪{−∞, +∞} such that l < h , m ∈ [l , h ] ⊆ C , g (l ) = f (l ) whenever l ∈ M ∩ C , and g (h ) = f (h ) whenever h ∈ M ∩ C ; and for each pair {l , h } ⊆ C such that l < h , each agent prefers g (h ) to g (l ). We construct the set C recursively. Let C0 = ;. We define C1 . Suppose that l 1 ∈ M . Since (l 1 , h1 ) ⊆ X , then either l 1 ∈ M or there is a sequence in M converging to l 1 from the left. Thus, there is l 1′ ∈ (l 1 − δ/22 , l 1 ] ∩ M . By a symmetric argument, if h1 ∈ M , there is h1′ ∈ [h1 , h1 + δ/22 ) ∩ M . Since l 1 < h1 , l 1′ < h1′ . Let C1 ≡ C0 ∪ [l 1′ , h1′ ]. If l 1 = −∞ or h1 = +∞ (since M 6= ;, these two cases cannot hold simultaneously), let C1 ≡ (−∞, h1′ ] or C1 ≡ [l 1′ , +∞), respectively. Let g coincide on C1 \C0 with a non-contestable budget-monotone rule that interpolates f (l 1′ ) when l 1′ ∈ R and f (h1′ ) when h1′ ∈ R. By construction, λ((C1 \C0 )\X ) < δ/2. Let T ≥ 1. Suppose that we have constructed CT satisfying the following prop′ erties. (i) CT is the disjoint union of T ′ ≤ T closed intervals CT = ∪Tt =1 [l t′ , ht′ ];10 (ii) for each t ∈ {1, ..., T ′ }, {l t′ , ht′ } ⊆ M ∪ {−∞, +∞} and l t′ < ht′ ; (iii) for some ′′ T ′′ ≥ T , CT ⊇ ∪Tt =1 (l t , ht ); (iv) for each t ∈ {1, ..., T ′ }, there is t˜ ∈ {1, ..., T ′′ } such that (l t˜ , ht˜ ) ⊆ [l t′ , ht′ ]; (v) for each t ∈ {1, ..., T ′ }, g is defined on [l t′ , ht′ ] and coincides on this interval with a non-contestable budget-monotone rule that interpolates f (l t′ ) when l t′ ∈ R and f (ht′ ) when ht′ ∈ R ; and (vi) λ ((CT \ CT −1 ) \ X ) < δ/2T . Let L ≡ {t > T ′′ : (l t , ht ) 6⊆ CT }. If L = ;, let C ≡ CT . Otherwise, let T ≡ min L. Since L ⊆ N, T is well-defined. Let S ≡ (l T , hT )\CT . Since CT is the finite union of closed sets, it is closed. Thus, S is an open set. We claim that S is an interval. Suppose by means of contradiction that there is a triple {l , m, h } ⊆ R such that {l , h } ⊆ S , m 6∈ S , and l < m < h . Thus, there is t ∈ {1, ..., T ′ }, such that m ∈ [l t′ , ht′ ]. Since {l , h } ⊆ S , l < l t′ < ht′ < h . Thus, there is t˜ ∈ {1, ..., T ′′ } such that (l t˜ , ht˜ ) ⊆ [l t′ , ht′ ] ⊆ S ⊆ (l T , hT ). Since T > T ′′ ≥ t˜ . Thus, T 6= t˜ . This implies that (l t˜ , ht˜ ) ∩ (l T , hT ) = ;. This is a contradiction. Thus, S = (a , b ) where a < b . If a = −∞ or a ∈ M , case that includes a ∈ {h1′ , ..., hT ′ }, let l T′ +1 ≡ a . If a ∈ M \M , there is ǫ < δ/2T +2 such that (a −ǫ, a ]∩M 6= ; and (a −ǫ, a ]∩CT = ;. Let l T′ +1 ∈ (a − ǫ, a ] ∩ M . If b = +∞ or b ∈ M , case that includes b ∈ {l 1′ , ..., l T ′ }, 10

Here we abuse of notation and write [l t′ , ht′ ] when l t′ = −∞ or ht′ = +∞

18

let hT′ +1 ≡ b . If b ∈ M \ M , there is ǫ < δ/2T +2 such that [b , b + ǫ) ∩ M 6= ; and [b , b + ǫ) ∩ CT = ;. Let hT′ +1 ∈ [b , b + ǫ) ∩ M . Let CT +1 ≡ CT ∪ [l T′ +1 , hT′ +1 ]. Let g coincide on CT +1 \ CT with a non-contestable budget-monotone rule that interpolates f (l T′ +1 ) when l T′ +1 ∈ R and f (hT′ +1 ) when hT′ +1 ∈ R. By construction, λ((CT +1 \ CT ) \ X ) < δ/2T +1 . Since CT ∩ [l T′ +1 , hT′ +1 ] ⊆ {l T′ +1 , hT′ +1 }, one can rewrite CT +1 as the disjoint union of T ∗ ≤ T +1 intervals, by joining [l T′ +1 , hT′ +1 ] with any adjacent [l t′ , ht′ ] for t ∈ {1, ..., T ′ }. Since [l T′ +1 , hT′ +1 ] intersects CT at most at its extremes, then this interval is joined with at most two intervals among the disjoint representation of CT . The disjoint representation of CT +1 evidently satisfies (i), (ii), (iii), (v), and (vi) above. We claim that it satisfies (iv). If [l T′ +1 , hT′ +1 ] is joined with an adjacent interval of CT , then since the disjoint representation of CT satisfies (iv), so does the disjoint representation of CT +1 . Suppose now that [l T′ +1 , hT′ +1 ] ∩ CT = ;, i.e., [l T′ +1 , hT′ +1 ] is an interval in the disjoint representation of CT +1 . We claim that (a , b ) = S ≡ (l T , hT ) \ CT = (l T , hT ). Suppose by contradiction that (l T , hT ) ∩ CT 6= ;. Then, {a , b } ∩ CT 6= ;. Suppose without loss of generality that a ∈ CT . Thus, a ∈ R and a ∈ {h1′ , ..., hT ′ }. Thus, [l T′ +1 , hT′ +1 ] ∩ CT 6= ;. This is a contradiction. Thus, (l T , hT ) ⊆ [l T′ +1 , hT′ +1 ]. Recall that if for some t ∈ N, C t ⊇ X , we define C ≡ C t . If for no t ∈ N, C t ⊇ X , let C ≡ ∪∞ t =1 C t . We claim that C ⊇ X . Let m ∈ X . Then there is ′ T ∈ N such that m ∈ (l T , hT ). Thus, m ∈ CT ⊆ C . Let CT ≡ ∪Tt =1 [l t′ , ht′ ] be the disjoint representation of CT satisfying (i)-(vi). Thus, there is t ∈ {1, ..., T ′ } such that m ∈ [l t′ , ht′ ] ⊆ CT ⊆ C . Thus, there is l ≡ l t′ and h ≡ ht′ such that {l , h } ⊆ (M ∩ C ) ∪ {−∞, +∞}, l < h , m ∈ [l , h ] ⊆ C , g (l ) = f (l ) whenever l ∈ M ∩C , and g (h ) = f (h ) whenever h ∈ M ∩C . Since C is the countable union of intervals, it is Lebesgue measurable.PThus, C \ X = ∪∞ \ C t ) \ X is also t =1 (C t +1 P ∞ ∞ Lebesgue measurable and λ(C \ X ) = t =1 λ ((C t +1 \ C t ) \ X ) < t =1 δ/2t = δ. Let {l , h } ⊆ C such that l < h . We claim that each agent prefers g (h ) to g (l ). ′ There is T ∈ N such that {l , h } ⊆ CT ≡ ∪Tt =1 [l t′ , ht′ ]. If l and h belong to the same interval in the disjoint representation of CT , the claim follows because g coincides with a budget-monotone rule in each of these intervals. Suppose that l and h belong to two different intervals l ∈ [l 1′ , h1′ ] and h ∈ [l 2′ , h2′ ]. Since l < h and the intervals are disjoint, h1′ < l 2′ . Thus, {h1′ , l 2′ } ⊆ M . Since g (h1′ ) = f (h1′ ) and g (l 2′ ) = f (l 2′ ), then each agent prefers g (l 2′ ) to g (h1′ ). Since g coincides with a budget-monotone rule in [l 1′ , h1′ ], each agent finds g (h1′ ) at least as good as g (l ). Since g coincides with a budget-monotone rule in [l 2′ , h2′ ], each agent finds g (h ) at least as good as g (l 2′ ). Thus, each agent prefers g (h ) to g (l ). Finally, we claim that R \ C ⊆ M . Let m ∈ R \ C . We claim that there is a sequence in M that converges from the left to m. There are two cases. Case 1. For each n ∈ N there is m ′ ∈ C such that m ′ < m and m − m ′ < 1/n. Let [l n , hn ] ⊆ C such

19

that {l n , hn } ⊆ (M ∩ C ) ∪ {−∞, +∞}, l n < hn , m ′ ∈ [l n , hn ] ⊆ C . Since m ′ 6∈ C , then hn < m and hn ∈ M . Thus, hn ∈ R and m − hn < 1/n. Thus, as n → ∞, hn → m. Case 2. There is ǫ > 0 such that (m − ǫ, m] ⊆ R \ C . Since X ⊆ C , R \ C ⊆ R \ X = M . Thus, (m − ǫ, m] ⊆ M . Thus, there is a sequence in M that converges from the left to m. A symmetric argument shows that there is a sequence in M that converges from the right to m. We claim that m ∈ M . Let {l , h } ⊆ R such that l < m < h . Then, there is an element l ′ of the sequence in M that converges to m from the left such that l < l ′ ≤ m. Since l ′ ∈ M , then each agent prefers f (l ′ ) to f (l ) and each agent finds f (m) at least as good as f (l ′ ). Thus, each agent prefers f (m) to f (l ). The symmetric argument shows that each agent prefers f (h ) to f (m). Thus, m ∈ M . For each m ∈ R \ C , let g (m) ≡ f (m). Step 2. There is a Lebesgue measurable set ∆ ⊆ R whose measure is no greater than δ, and a non-contestable budget-monotone rule g that coincides with f on M \∆. Let ∆ ≡ C \X . Then, C \X is measurable and λ(C \X ) < δ. Since M \∆ = R \ C , then g defined in Step 1 coincides with f in M \ ∆. Since M \ ∆ ⊆ M \ ∆, then g coincides with f in M \ ∆. By definition, g ∈ F . We claim that g is budget-monotone. Let {l , h } ⊆ R such that l < h . If {l , h } ⊆ R \ C ⊆ M , each agent prefers g (h ) = f (h ) to g (l ) = f (l ). If {l , h } ∈ C , by Step 1, each agent prefers g (h ) to g (l ). If l ∈ C and h ∈ R \ C ⊆ M , there is [l ′ , h ′] ⊆ C such that {l ′ , h ′ } ⊆ (M ∩ C ) ∪ {−∞, +∞}, l ′ < h ′ , l ∈ [l ′ , h ′] ⊆ C . Since h 6∈ C , h ′ < h and h ′ ∈ M . Thus, g (h ′ ) = f (h ′ ). Since g (h ) = f (h ) and {h , h ′} ⊆ M , each agent prefers g (h ) to g (h ′ ). Since {h ′ , l } ⊆ C , then each agent finds g (h ′ ) at least as good as g (l ). Thus, each agent prefers g (h ) to g (l ). The symmetric argument shows that if h ∈ C and l ∈ R \ C ⊆ M , each agent prefers g (h ) to g (l ). Now, suppose that X = ;. We show that M = R. Let x ∈ R. Since R = M , there is a sequence in M that converges from the left to m and a sequence in M that converges from the right to m. The same argument that shows m ∈ M in Case 2 above completes the proof.

4.2

Non-conditional existence of non-contestable allocations with a lower bound or an upper bound on consumptions of money: a characterization

Our results concerning existence of budget-monotone rules can be used to further our understanding of the conditions necessary for the existence of noncontestable allocations with a minimum or maximum individual consumption of money constraint. Think for instance of objects in our model as tasks and consumption of money as salaries with a minimum salary constraint. Think also of objects in our model as rooms and consumptions of money as contri20

bution to rent, with the natural constraint that no roommate is paid to occupy a room. Two basic results have been previously obtained concerning this problem. First, for a given preference profile, if budget is large enough, there are noncontestable allocations satisfying a given lower bound in consumptions of money (Svensson, 1983, 1987; Maskin, 1987; Alkan et al., 1991; Velez, 2016). Second, whenever all agents are indifferent between receiving any object with no money, for each negative budget (say the rent in the roomates case) there are noncontestable allocations in which each agent’s consumption of money is no greater than zero (Su, 1999). These two sets of results are not tight, however. That is, one can find profiles of preferences violating the conditions stated in all these results and for which (i) for each positive budget there are non-contestable allocations in which each agent receives a positive consumption of money, and (ii) for each negative budget there are non-contestable allocations in which each agent receives a negative consumption of money. We state necessary and sufficient conditions guaranteeing that (i) and (ii) above are satisfied. Interestingly, these two conditions are equivalent and hold if and only if there is a noncontestable allocation in which each agent receives an object and no money. This last condition is well known to be characterized as follows: for each set of k ≤ n agents, the union of their best objects (i.e., the objects in their best bundles with no money) has at least k objects (Gale, 1960). Our characterization easily follows from a corollary to Theorem 1 that we state first and omit its straightforward proof. Corollary 1 (Interpolation of one non-contestable allocation). Let m ∈ R and z ∈ Fm . Then there is a non-contestable budget-monotone rule f such that f (m) = z . Proposition 1. The following are equivalent: 1. For each m ∈ R such that m < 0, there is (x , µ) ∈ Fm such that x ≪ 0; 2. For each m ∈ R such that m > 0, there is (x , µ) ∈ Fm such that x ≫ 0; 3. There is (0, µ) ∈ F0 ; S 4. For each S ⊆ N , i ∈S {α ∈ A : for each β ∈ A, (0, α) Ri (0, β )} ≥ |S |.

Proof. For each i ∈ N , let Bi ≡ {α ∈ A : for each β ∈ A, (0, α) Ri (0, β )}. From Gale (1960, Theorem 5.2, Pag. 144) there exist a bijection µ : N → A such that for S each i ∈ N , µi ∈ Bi , if and only if for each S ⊆ N , i ∈S Bi ≥ |S |. Thus, statement 4 is equivalent to statement 3. Suppose that there is z ≡ (0, µ) ∈ F0 . By Corollary 1, there is f non-contestable and budget-monotone such that f (0) = (0, µ). 21

Thus, for each m > 0, f (m) ∈ Fm and each agent prefers f (m) to f (0) = (0, µ). By the Decomposition Lemma each agent receives a positive consumption of money at f (m). Thus, statement 3 implies statement 2. By continuity of preferences statement 2 implies statement 3. A similar argument shows that statements 1 and 3 are equivalent.

4.3

Continuity properties of the non-contestable correspondence

A byproduct of our proof of Theorem 1 is that for each non-contestable and budget-monotone rule the welfare of each agent and for each object the consumption of money associated with the object are all continuous functions of the budget. Proposition 2. Let f be non-contestable and budget-monotone. 1. Let w ≡ (wi )i ∈N be a continuous representation of R . Then, for each i ∈ N , wi ◦ fi is continuous. 2. Let α ∈ A. For each m ∈ R, let yα (m) ∈ R be the consumption of money of the agent who receives object α at f (m). Then, yα is continuous. Proof. For statement 1 see the proof that statement 1 implies statements 2 and 4 in Theorem 1. For statement 2 see the proof of that statement 1 implies statements 3 and 5 in Theorem 1. Let u ≡ (u i )i ∈N be a continuous representation of R . Our results allow us to better understand the continuity properties of the correspondence that assigns to each budget its possible utility outcomes, i.e., m ∈ R 7→ u(Fm ) ≡ {(u i (z i ))i ∈N ∈ RN : z ∈ Fm }.11 First, statement 1 in Proposition 2 directly implies that there is always a continuous selection from the correspondence u(F· ). This is an unusual result for a correspondence that is not convex-valued (Michael, 1956). Second, Corollary 1 and Proposition 2 directly imply that u(F· ) is lowerhemicontinuous, i.e., for each m ∈ R, each sequence {mk }k ∈N such that as k → ∞, mk → m, and each v ∈ u(Fm ), there is a sequence {vk }k ∈N such that for each k ∈ N, vk ∈ u(Fmk ) and as k → ∞, vk → v . Finally, statement 2 in Proposition 2 is useful in the study of the generalization of our model in which there are more objects than agents. Here noncontestable allocations may be inefficient (Alkan et al., 1991). However, one can prove that there are non-contestable and efficient allocations for each possible budget as follows. Suppose that there are o objects and o > n. First, introduce 11

The correspondence m ∈ R 7→ Fm may be discontinuous, for an assignment of objects may be sustained in a non-contestable allocation for only a subset of all possible budgets.

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o − n agents who value only money. We refer to the economy with o agents as the extended economy. Since the preferences of the fictitious agents are continuous, for each budget, say m ∈ R, there is an non-contestable allocation in the extended economy with this budget (Alkan et al., 1991; Velez, 2016). Let z be such an allocation. The restriction of this allocation to the “real agents”, i.e., (z i )i ∈N is efficient for the initial economy (Alkan et al., 1991). This, does not prove the existence of a non-contestable and efficient allocation in the original economy for budget m, because the fictitious agents may receive some consumption of money at z . Here is where statement 2 in Proposition 2 comes into play. It directly implies that one can select for each budget a non-contestable allocation in the extended economy in a way that the aggregate consumption of money of the agents in N is a continuous function of the budget. Thus, by a simple fixed point argument, one can show that for each budget, there is a non-contestable and efficient allocation in the economy with o objects and n agents. Moreover, these allocations can be selected so the welfare of each agent is an increasing function of the budget.

5

Discussion

There is a limit to the extent to which one can “non-cooperatively implement” non-contestable rules. First, no non-contestable rule is strategy-proof, i.e., such that each agent has her true preference relation as a dominant strategy in the simultaneous direct revelation game associated with the rule (Alkan et al., 1991; Tadenuma and Thomson, 1995a). This means that each non-contestable rule is manipulable. Not all non-contestable rules are equally manipulable, however. Consider the rules that for a given i ∈ N select for each m an allocation that maximizes agent i ’s welfare in Fm . It is easy to see that each such a rule is valueMaxmin. Thus, these rules are budget-monotone. Interestingly, these rules are the minimally manipulable non-contestable rules with respect to the number of agents who can manipulate the rule profile-wise (Andersson et al., 2014a).12 Second, if one considers the so-called Nash implementation (Maskin, 1999), there is no better news. A game form is a pair (S , g ) where S ≡ S1 × ..Sn is the cross product of the strategy spaces of the agents and g : S → Z m is an outcome function. For each R ∈ R n , the game (S , g , R ) is that in which agents’ strategy spaces are S , the outcome at strategy profile s ∈ S is g (s ), and preferences on these outcomes are R . A game form (S , g ) implements a rule f in Nash equilibria if for each R ∈ R n the unique Nash equilibrium outcome of (S , g , R ) is 12

It is an open question to determine whether the rules that are minimally manipulable in the sense of Andersson et al. (2014b) are budget-monotone.

23

f (R ) (see Jackson (2001) for details of this definition). Tadenuma and Thomson (1995a) show that there is the following limit to the extent to which one can achieve Nash implementation of non-contestable allocations. Let (S , g ) be a game form such that for each R ∈ R n the set of Nash equilibrium outcomes of (S , g , R ) (i) is a subset of F (R ) and (ii) contains all allocations at which all agents are indifferent among all bundles at the allocation. Then, for each R ∈ R n set of Nash equilibrium outcomes of (S , g , R ) contains F (R ). However, this does not imply that each non-contestable rule cannot be implemented in Nash equilibria. The following lemma does. Lemma 3. No non-contestable rule can be implemented in Nash equilibria. Proof. It is well known that the following property is necessary for the existence of a game form that implements a rule f in Nash equilibria (Maskin, 1999). Let R ∈ R n and z ∈ Z . Then, R ′ ∈ R n is obtained by a monotonic transformation of R at z if for each i ∈ N , each xi ∈ R, and each α ∈ A, z i Ri′ (xi , α) implies z i Ri (xi , α). A rule f is monotone if for each pair {R , R ′} ⊆ R n such that R ′ is obtained by a monotonic transformation of R at f (R ), f (R ′ ) = f (R ). We show that no f ∈ F is monotone. Suppose by means of contradiction that f ∈ F is monotone. Let N ≡ {1, ..., n}, A ≡ {α1 , ..., αn }, and m = 0. Let R be the preference profile represented by the following utility functions: for each x ∈ R, u 1 (x , α1 ) ≡ x + n + 1 and for each t ∈ {2, ..., n}, u 1 (x , αt ) ≡ x ; and for each i ∈ N \ {1}, each x ∈ R, and each t ∈ {1, ..., n}, u i (x , α) ≡ x . Let z ≡ (x , µ) ∈ Z m be non-contestable for profile R . Recall that each non-contestable allocation is efficient (Svensson, 1983). Thus, µ1 = α1 . Since for each i ∈ N \ {1}, u 1 (z 1 ) ≥ u 1 (z i ) and u i (z i ) ≥ u i (z 1 ), x1 ∈ [−n, 1]. Let z ≡ (x , µ) be the recommendation of f for profile R with budget 0. Let a ∈ [−n, 0] \ {x1 }. Let R a be the profile represented by the following utility functions: for each i ∈ N , each x ∈ R, u i (x , α1 ) ≡ x + a + a /(n − 1) and for each t ∈ {2, ..., n}, u i (x , αt ) ≡ x . Let z a ≡ (x a , µa ) be the recommendation of f for profile R a with budget 0 (since all agents have the same preference, z a is such that an agent receives object α1 and −a consumption of money, and all other agents receive the other objects and a consumption of money). We claim that µa1 6= α1 . Suppose by contradiction that µa1 = α1 . Then, R is a monotonic transformation of R a at z a . Since f is monotonic, it has to select z a in profile R with budget 0. This contradicts f is single-valued. Suppose without loss of generality that for infinitely many a ∈ [−n, 0] \ {x1 } agent 2 receives object α1 for profile R a with budget 0. Let X ⊆ [−n, 0] \ {x1 } be this set. Let R ′ be the profile in which R2′ ≡ R1 and for each i ∈ N \ {2}, Ri′ ≡ R2 . Then, for each a ∈ X , R ′ is a monotonic transformation of R a at z a . Since f is monotonic, it has to select z a for each a ∈ X , in profile R ′ with budget 0. This contradicts f is single-valued. 24

We can conclude that no non-contestable rule, in particular a budget-monotone one, can be implemented in dominant strategies or in Nash equilibria. This is not the end of the road, however. Given a rule f , it may be possible to construct an extensive game form whose sub-game perfect Nash equilibrium correspondence coincides with f (see Moore and Repullo (1988) for precise definitions of sub-game perfect Nash implementation). It is an interesting open question, beyond the scope of this paper, to systematically study this possibility. It is worth noting that in the two-agent case each equal-compensation rule is budget-monotone (see Lemma 4 below). Moreover, Nicoló and Velez (2016) essentially implement in sub-game perfect equilibria the equal-compensation rules. More precisely, when N ≡ {1, 2}, these authors construct two modified versions of the popular divide-and-choose extensive game form such that for each m ∈ R each of these mechanisms obtains as sub-game perfect Nash equilibrium outcomes the set (here we are also quantifying over all possible preferences profiles R ∈ R n ): § ª ′ ′ E C (m) ≡ z ∈ Fm : for each z ∈ Fm , min d i j (z ) ≥ min d i j (z ) . i , j ∈N , i 6= j

i , j ∈N , i 6= j

Lemma 4. Suppose that there are only two agents. Let {m, m ′ } ⊆ R be such that m ′ > m, z ∈ C (m), and z ′ ∈ C (m ′ ). Then, for each i ∈ N , z i′ Pi z i . Proof. Let m, m ′ , z ≡ (x , µ), and z ′ ≡ (x ′ , µ′ ) be as in the statement of the lemma. Suppose that A ≡ {α, β }, z 1 ≡ (x1 , α), and z 2 ≡ (x2 , β ). We claim that z 1′ P1 z 1 . Suppose by contradiction that z 1 R1 z 1′ . We claim that µ′1 = α. Suppose by contradiction that µ′1 = β . Since z 1 R1 z 1′ and z 1′ R1 z 2′ , (x1 , α) R1 (x2′ , α). Thus, x2′ ≤ x1 . Since z 2 R2 z 1 , z 2 R2 z 2′ . Let z ′′ ∈ Z m ′ be the allocation such that z 1′′ ≡ (x1 + (m ′ − m)/2, α) and z 2′′ ≡ (x2 + (m ′ − m)/2, β ). Thus, z 1′′ P1 z 1 R1 z 1′ and z 2′′ P2 z 2 R2 z 2′ . Thus, z ′ is not efficient. This is a contradiction. Thus, µ′1 = α, µ′2 = β , x1′ ≤ x1 , and x2′ > x2 . Recall that d i j (z ) is the amount of money that one has to add to the bundle of agent j at z so agent i is indifferent between z i and this bundle. Thus, d 12 (z ′ ) < d 12 (z ) and d 21 (z ) < d 21 (z ′ ). Since |N | = 2, and z ∈ E C (m), d 12 (z ) = d 21 (z ) (Tadenuma and Thomson, 1995b). Thus, d 12 (z ′ ) 6= d 21 (z ′ ). This is a contradiction because since |N | = 2 and z ∈ E C (m), d 12 (z ′ ) = d 21 (z ′ ) (Tadenuma and Thomson, 1995b).

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