- Email: [email protected]
Fairness and implementability in allocation of indivisible objects with monetary compensations
Journal of Mathematical Economics 43 (2007) 549–563
Fairness and implementability in allocation of indivisible objects with monetary compensations Toyotaka Sakai Faculty of Economics and IGSSS, Yokohama National University, Yokohama 240-8501, Japan Received 4 January 2005; received in revised form 19 March 2006; accepted 11 July 2006 Available online 20 February 2007
Abstract This paper introduces a general framework for the fair allocation of indivisible objects when each agent can consume at most one (e.g., houses, jobs, queuing positions) and monetary compensations are possible. This framework enables us to deal with identical objects and monotonicity of preferences in ranking objects. We show that the no-envy solution is the only solution satisfying equal treatment of equals, Maskin monotonicity, and a mild continuity property. The same axiomatization holds if the continuity property is replaced by a neutrality property. © 2007 Elsevier B.V. All rights reserved. JEL classification: C78; D63; D71 Keywords: Indivisible goods; Queuing; Generalized queuing; No-envy; Maskin monotonicity; Nash implementation; Walrasian social choice
1. Introduction This paper introduces a general framework for the fair allocation of indivisible objects when each agent can consume at most one (e.g., houses, jobs, positions) and monetary compensations are possible. By allowing identical objects and monotonicity of preferences in ranking objects, this framework unifies various classes of problems that have been studied separately so far. Examples include problems of allocating different objects (e.g., Svensson, 1983; Alkan et al., 1991), identical objects (e.g., Tadenuma and Thomson, 1991), single object (e.g., Tadenuma and Thomson, 1993, 1995), queuing positions (e.g., Maniquet, 2003; Chun, 2006a,b). E-mail address: [email protected]. URL: http://www.geocities.jp/toyotaka sakai/index.html. 0304-4068/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.jmateco.2006.07.006
550
T. Sakai / Journal of Mathematical Economics 43 (2007) 549–563
Our purpose is to identify solutions that satisfy certain fairness and implementability properties. As a fairness property, we consider equal treatment of equals, which states that two agents with the same preference should receive indifferent consumption bundles. As an implementability property, we consider Maskin monotonicity, which is necessary and (in this model, when there are at least three agents) sufficient for a solution to be Nash implementable (Maskin, 1999).1 The no-envy solution is the solution that associates with each preference profile the set of allocations at which no agent prefers the consumption bundle of any other to his own (Foley, 1967). Geanakoplos and Nalebuff (1988) show that, in classical exchange economies with divisible goods, any solution satisfying equal treatment of equals and Maskin monotonicity is a subsolution of the no-envy solution. Fleurbaey and Maniquet (1997) show that, in abstract economic environments, the same implication holds if a preference domain satisfies a richness condition. Since we can show that our preference domain satisfies the condition, any solution satisfying equal treatment of equals and Maskin monotonicity is a subsolution of the no-envy solution in this environment. We study how solutions satisfying the two properties differ from the no-envy solution. Our main theorem shows that there is no difference if we focus on solutions satisfying a mild continuity property. In other words, the no-envy solution is the only solution satisfying equal treatment of equals, Maskin monotonicity, and the continuity property. Furthermore, the same axiomatization holds if the continuity property is replaced by a neutrality property introduced by Tadenuma and Thomson (1991). Since the continuity and neutrality properties are so weak that they are satisfied by all standard solutions, our theorem suggests that the no-envy solution is essentially the only solution satisfying equal treatment of equals and Maskin monotonicity. Svensson (1983, Theorem 4) shows that the Walrasian solution from equal income coincides with the no-envy solution. Since the Walrasian solution from equal income satisfies the axioms characterizing the no-envy solution, the equivalence is in fact obtained as a corollary to our theorem. It is known that the no-envy solution satisfies many desirable properties in economies with indivisible objects with money. For example, this solution satisfies Pareto efficiency, (Svensson, 1983; Alkan et al., 1991), a welfare lower bound property (Bevia, 1996), and a consistency property for population changes (Tadenuma and Thomson, 1991). 2 On the other hand, our theorem justifies the no-envy solution from the perspectives of fairness and implementability. This paper is organized as follows. Section 2 introduces the model. Section 3 presents the main results. Section 4 concludes. Appendix A offers proofs and remarks on continuity properties. 2. The model 2.1. Basic notion We extend the model developed by Alkan et al. (1991) so as to deal with identical objects and monotonicity of preferences in ranking objects. 3 Let N ≡ {1, 2, . . . , n} and O ≡ {a1 , a2 , . . . , an }
1
As an implementability property, one may wish to consider “strategy-proofness" (no one can gain by misrepresenting his preference). However, it is known that this property is incompatible with various fairness properties including equal treatment of equals (e.g., Schummer, 2000; Svensson and Larsson, 2002; Ohseto, 2001; Bochet and Sakai, 2007). 2 Tadenuma and Thomson (1991, Theorem 1) show that, in a variable population model with different objects, if a subsolution of the no-envy solution satisfies the consistency property and a neutrality property (same us ours), then it in fact coincides with the no-envy solution. 3 Earlier works of economies with indivisible objects and money include Kaneko (1983), Svensson (1983), and Quinzii (1984).
T. Sakai / Journal of Mathematical Economics 43 (2007) 549–563
551
be finite sets of agents and objects, respectively, where |N| = |O|. We allow some objects to be “null” to describe situations where there are fewer “real” objects than agents. Let A ≡ {A1 , A2 , . . . , AL } be a partition of O, i.e., ∪L l=1 Al = O and for each l, l with l = l , Al ∩ Al = ∅. This partition identifies which objects are identical: two objects are identical if and only if they are in the same component Al of A. When there is no confusion, we identify each Al with an object contained in Al . We assume that there is at least one non-null object to make the problem interesting, that is, |A| ≥ 2. There is also an amount M ∈ R of a single perfectly divisible good called money. A feasible allocation, or simply an allocation, is a pair: z ≡ (zi )i∈N ≡ (σ(i), mi )i∈N ≡ (σ, m), where σ : N → A is a function such that for each l ∈ {1, 2, . . . , L}, |{i ∈ N : σ(i) = Al }| = |Al |, and m ∈ RN is such that i∈N mi = M. Here σ(i) is the indivisible object that i receives, and mi is agent i’s consumption of money. When (σ, m) is such that for each i, j ∈ N, σ(i) = σ(j) implies mi = mj , we may write mσ(i) instead of mi . Let Z be the set of allocations. A partial order is an irreflexive and transitive binary relation on A.4 We use a partial order to deal with monotonicity of preferences in ranking objects. For example, if A ∈ A denotes a larger house and B ∈ A denotes a smaller house, then it is natural to assume that A B. A preference for agent i ∈ N is a transitive and complete binary relation Ri over his consumption space A × R that satisfies: • Money monotonicity: for each (A, mi ) ∈ A × R and each mi > mi , (A, mi )Pi (A, mi ), • Finiteness: for each (A, mi ) ∈ A × R and each B ∈ A, there is mi ∈ R such that (A, mi )Ii (B, mi ). • -object monotonicity: for each A, B ∈ A and each mi ∈ R, if A B, then (A, mi )Pi (B, mi ).5 Let R be the set of preferences. A profile is an n-tuple of preferences R ≡ (R1 , R2 , . . . , Rn ) ∈ RN . A preference Ri ∈ R is quasi-linear if there exists a “valuation vector” v(Ri ) ≡ (v(Ri , A1 ), v(Ri , A2 ), . . . , v(Ri , AL )) ∈ RL such that v(Ri , A1 ) = 0 and Ri is represented as U(Ri , A, mi ) = v(Ri , A) + mi for (A, mi ) ∈ A × R.6 Let Q ⊆ R be the subset of quasi-linear preferences. We call (A, ) a primitive pair and fix it throughout the paper. This notion is a new device introduced by this study. Various classes of problems discussed in the literature can be described using this notion: • Different objects: Let A ≡ {{a1 }, {a2 }, . . . , {an }} and = ∅. In this class, there are as many different real objects as agents, and no object monotonicity is imposed. • Different objects and null objects, a general case: Let A ≡ {A1 , A2 , . . . , AL } be an arbitrary partition of A. Let us interpret AL as a null object, and A1 , . . . , AL−1 as desirable real objects. Then, for each l ≤ L − 1, let Al AL .7 Several interesting subclasses can be identified: (i) when all real objects are different (e.g., Tadenuma and Thomson, 1993, 1995), we let |Al | = 1 4 5 6 7
Irreflexivity: for each A ∈ A, AⱭA; transitivity: for each A, B, C ∈ A, if A B and B C, then A C. Note that, when = ∅ (i.e., for no A, B ∈ A, A B), this condition is vacuously satisfied. Note that v(Ri , A1 ) = 0 is just a normalization and this restriction does not lose any generality. We can also consider the case where some real objects, say Al , are “bads” by instead assuming that AL Al .
552
T. Sakai / Journal of Mathematical Economics 43 (2007) 549–563
for each l ≤ L − 1; (ii) when all real objects are identical and the number of real objects is at least two (e.g., Tadenuma and Thomson, 1991), we let L = 2 and |A1 | ≥ 2; (iii) when there is only one real object (e.g., Tadenuma and Thomson, 1993, 1995), we let L = 2 and |A1 | = 1. • Queuing: Let A ≡ {{a1 }, {a2 }, . . . , {an }} and be such that {a1 } {a2 } · · · {an }. Agents stand to receive a service, and no two agents can be served simultaneously. All agents prefer being the first to being the second, and being the second to being the third, and so on. Maniquet (2003) and Chun (2006a,b) study this class when preferences are linear.8 • Generalized queuing: Let A ≡ {A1 , A2 , . . . , AL } be an arbitrary partition of A and be such that A1 A2 · · · AL . In this queuing problem, |Al |-agents are served in the lth position. We also allow the case where there are fewer positions than agents, that is, AL denotes a null object, and so |AL |-agents cannot be served. 2.2. No-envy and efficiency A solution is a non-empty valued correspondence ψ from RN to Z. We introduce two familiar solutions. The no-envy notion states that no agent prefers the consumption bundle of any other to his own (Tinbergen, 1953; Foley, 1967). An allocation z ∈ Z is envy-free for R ∈ RN if for each i, j ∈ N, zi Ri zj . Given R ∈ RN , let F (R) be the set of envy-free allocations for R. The nonemptiness of F (R) is shown by Alkan et al. (1991, Theorem 2). We call F the no-envy solution. This solution is fundamental in the theory of fair allocation.9 We also study a standard efficiency notion. An allocation z ∈ Z is Pareto efficient for R ∈ RN if there exists no z ∈ Z such that for each i ∈ N, zi Ri zi and for some j ∈ N, zj Pj zj . Given R ∈ RN , let P(R) be the set of Pareto efficient allocations for R. We call P the Pareto solution. It is known that for each R ∈ RN , F (R) ⊆ P(R) (Svensson, 1983, Theorem 2; Alkan et al., 1991, Theorem 1). 2.3. Axioms We are interested in solutions satisfying the following basic fairness requirement: two agents whose preferences are the same should receive indifferent consumption bundles: Equal treatment of equals: For each R ∈ RN , each z ∈ ψ(R), and each i, j ∈ N, if Ri = Rj , then zi Ii zj (and so zi Ij zj ). Given R ∈ RN and z ∈ Z, a profile R ∈ RN is a Maskin monotonic transformation of R at z, if for each i ∈ N, {w ∈ Z : zi Ri wi } ⊆ {w ∈ Z : zi Ri wi }. Let MT (R, z) be the set of Maskin monotonic transformations of R at z. General results by Maskin (1999) imply in our context that the following property is necessary and (when |N| ≥ 3) sufficient for a solution to be implementable in Nash equilibrium.10 8 9 10
In queuing problems, a preference Ri is linear if it is represented as U(Ri , {al }, mi ) = γi (n − l) + mi with γi > 0. Thomson (2005b) offers a survey of the literature. For Maskin’s results and the definition of Nash implementation, see Jackson (2001).
T. Sakai / Journal of Mathematical Economics 43 (2007) 549–563
553
Maskin monotonicity: For each R ∈ RN and each z ∈ ψ(R), if R ∈ MT (R, z), then z ∈ ψ(R ). Next, we consider the requirement that a solution should be robust to small perturbations of preferences. If a sequence of quasi-linear preferences {Rνi } is such that {v(Rνi )} → v(Ri ), then it is natural to consider that the sequence {Rνi } also converges to Ri . In this case, we write {Rνi } → Ri . Restricted continuity: Let {Rν }, R be a sequence of profiles and a profile in QN , respectively. Let {zν }, z be a sequence of allocations and an allocation in Z, respectively. Suppose that: (i) for each i ∈ N, {v(Rνi )} → v(Ri ), (ii) for each ν ∈ N, zν ∈ ψ(Rν ), (iiii) mν → m and there is ν ∈ N such that for each ν ≥ ν , σ ν = σ. Then, z ∈ ψ(R). Restricted continuity is only defined for quasi-linear preferences. However, we can also consider continuity properties for general preferences by introducing a natural topology on the space of preferences. This topic is briefly examined in Appendix B. However, restricted continuity is quite weak as a continuity requirement and this approach has the advantage of not requiring any complicated mathematical notion. Given two allocations z, z ∈ Z and a profile R ∈ RN , we write z R z if for each i ∈ N, zi Ii zi , and there is a bijection π : N → N such that for each i ∈ N, zπ(i) = zi . The next mild neutrality property states that if two allocations are equivalent in welfare and one is obtained from the other by a permutation, then one is chosen if and only if the other is also chosen (Tadenuma and Thomson, 1991). Neutrality: For each R ∈ RN , each z ∈ ψ(R), and each z ∈ Z, if z R z , then z ∈ ψ(R). Restricted continuity and neutrality are so weak that they are satisfied by all standard solutions in this literature. 3. Main results 3.1. Axiomatizations of the no-envy solution We present axiomatizations of the no-envy solution. Theorem 1. The no-envy solution is the only solution satisfying equal treatment of equals, Maskin monotonicity, and restricted continuity. The same axiomatization holds if restricted continuity is replaced by neutrality. The proof of Theorem 1 is relegated to Appendix A. We here outline the strategy of the proof. We first show that RN satisfies a certain richness condition (Lemmas 1 and 2). This fact and the results by Geanakoplos and Nalebuff (1988), Fleurbaey and Maniquet (1997) imply that if a solution satisfies equal treatment of equals and Maskin monotonicity, then it is a subsolution of the no-envy solution (Lemma 3). We next show that if a subsolution of the no-envy solution satisfies Maskin monotonicity and at least one of restricted continuity and neutrality, then it coincides with the no-envy solution (Lemmas 4–6). These lemmas complete the proof. 3.2. Tightness of Theorem 1 Clearly, equal treatment of equals and Maskin monotonicity are indispensable to obtain the axiomatizations. The next example shows that restricted continuity and neutrality are also indispensable.
554
T. Sakai / Journal of Mathematical Economics 43 (2007) 549–563
Example 1. Let z ∈ Z. For each R ∈ RN , let ψ(R) ≡ F (R) \ {z }. It is obvious that, if ψ is welldefined, ψ satisfies equal treatment of equals and Maskin monotonicity, but Theorem 1 implies that ψ violates neutrality and restricted continuity. It remains to show that ψ is well-defined, i.e., for each R ∈ RN , ψ(R) = ∅. To prove this, it suffices to show that for each R ∈ RN , |ψ(R)| ≥ 2. Here, we only show this for the simple case where N ≡ {1, 2}, A ≡ {A, B}, and ≡ ∅. A proof for general cases is somewhat complicated but depends on a similar idea.11 Let R ∈ RN and z ∈ F (R). Without loss of generality, assume z = ((A, mA ), (B, mB )). If for each i ∈ N, z1 Ii z2 , then (z2 , z1 ) ∈ F (R), so |F (R)| ≥ 2. If z1 P1 z2 , then for a sufficiently small ε > 0, (A, mA − ε)P1 (B, mB + ε) and (B, mB + ε)P2 (A, mA − ε). Therefore, ((A, mA − ε), (B, mB + ε)) ∈ F (R), so |F (R)| ≥ 2. The case z2 P2 z1 is parallel to the previous case. Thus, in either case, |F (R)| ≥ 2. The axiomatizations do not hold on the domain of quasi-linear preferences. The main reason is that equal treatment of equals and Maskin monotonicity do not imply no-envy on such a smaller domain. The following is an example of a solution on QN that satisfies F ψ P and the axioms in Theorem 1: ¯ ∈ QN and z¯ ≡ (σ, ¯ m) ¯ ∈ Z be such that σ¯ is the unique maximizer of Example 2. Let R ¯ z¯ ) in v(R , σ(i)) over σ and for each i, j ∈ N, z¯ i P¯ i z¯ j . There are many such a pair (R, i i∈N general, but here we fix one. Define the solution ψ as follows: ¯ z¯ ), P(R) if R ∈ MT(R, ψ(R) ≡ F (R) otherwise. Obviously, F ψ P. Also, it is easy to see that ψ satisfies restricted continuity and neutrality. ¯ z¯ ), there are no i, j ∈ N such that Ri = Rj . Hence, if By construction, for each R ∈ MT(R, ¯ z¯ ), and so ψ(R) = F (R). Thus, R ∈ QN is such that for some i, j ∈ N, Ri = Rj , then R ∈ / MT(R, ψ satisfies equal treatment of equals. It remains to show that ψ is Maskin monotonic. Let R ∈ QN , z ∈ ψ(R), and R ∈ MT(R, z). ¯ z¯ ). Then z ∈ ψ(R) = P(R). Since z ∈ P(R) and R ∈ MT(R, ¯ z¯ ), by Consider the case R ∈ MT(R, quasi-linearity, z and z¯ share the same assignment function. Therefore, by quasi-linearity, R ∈ ¯ z¯ ) together imply R ∈ MT(R, ¯ z¯ ). Hence, ψ(R ) = P(R ). Since P is MT(R, z) and R ∈ MT(R, ¯ z¯ ). Then / MT(R, Maskin monotonic, by z ∈ P(R), z ∈ P(R ) = ψ(R ). Next consider the case R ∈ z ∈ ψ(R) = F (R). Since F is Maskin monotonic, z ∈ F (R ). Since F (R ) ⊆ ψ(R ), z ∈ ψ(R ). Hence, in either case, z ∈ ψ(R ). Thus, ψ is Maskin monotonic. We remark that the above example is constructed so as to apply to the smaller domain of linear preferences. This implies that in queuing problems by Maniquet (2003) and Chun (2006a,b), the set of our axioms does not characterize the no-envy solution. 3.3. Walrasian social choice Theorem 1 can be applied to study Walrasian social choice in this economy. An allocation z ≡ (σ, m) ∈ Z is a Walrasian allocation from equal income for R ∈ RN if there exist p ∈ RA +
11
The proof is available from the author upon request.
T. Sakai / Journal of Mathematical Economics 43 (2007) 549–563
555
and K ∈ R such that for each i ∈ N, (σ(i), mi ) maximizes Ri over the set: {(A, mi ) ∈ A × R : pA + mi ≤ K}. Given R ∈ RN , let W(R) be the set of Walrasian allocations from equal income for R. We call W the Walrasian solution from equal income (Varian, 1974). The equivalence of the Walrasian solution from equal income and the no-envy solution is established by Svensson (1983, Theorem 4).12 Since the Walrasian solution from equal income satisfies the axioms listed in Theorem 1, the equivalence can be in fact obtained as a corollary to Theorem 1. Corollary 1. The Walrasian solution from equal income coincides with the no-envy solution. Theorem 1 and Corollary 1 together imply the following axiomatizations of the Walrasian solution from equal income. Corollary 2. The Walrasian solution from equal income is the only solution satisfying equal treatment of equals, Maskin monotonicity, and restricted continuity. The same axiomatization holds if restricted continuity is replaced by neutrality. Let us compare Corollary 2 and results characterizing the same solution in other environments. In economies with divisible goods, Thomson (1979, Theorem 1 .D2) shows that, when preferences satisfy a boundary condition,13 if a subsolution of the Pareto efficient solution satisfies equal treatment of equals, “Nash implementability by a convex mechanism”, and “continuity for linear preferences”, then it is contained in the Walrasian solution from equal income.14 “Nash implementability by a convex mechanism” requires a solution to be implementable in Nash equilibrium by a mechanism whose outcome function satisfies a certain convexity condition. Therefore, it is stronger than the standard Nash implementability notion and hence implies Maskin monotonicity. “Continuity for linear preferences” is an exact counterpart of our restricted continuity to linear preferences. In economies with divisible goods, Nagahisa and Suh (1995, Theorem 2) show that, when preferences satisfy the boundary condition and twice differentiability, the Walrasian solution from equal income is the only subsolution of the Pareto efficient solution that satisfies equal treatment of equals and “local independence”. “Local independence” states that if an allocation is chosen at a preference profile and the preferences change to retain their marginal rates of substitutions at the allocation, then the allocation should be reselected (see, Nagahisa, 1991). Since Maskin monotonic transformations are examples of such a change, “local independence” is stronger than Maskin monotonicity. The most notable difference between Corollary 2 and the results by Thomson and Nagahisa– Suh is the absence of Pareto efficiency in the former. However, in the present model, equal treatment of equals and Maskin monotonicity imply no-envy, which in turn implies Pareto efficiency.
12 Svensson imposes different conditions on preferences as ours. However, the equivalence result does not depend on the conditions. For a given envy-free allocation z, Svensson constructs a price vector p ∈ RA as follows: for each i ∈ N and each A ∈ A, let m(i, z, A) ∈ R be such that zi Ii (A, m(i, z, A)). Then let K ≥ maxi,A m(i, z, A). For each A ∈ A, let pA ≡ K − m(i, z, A) where σ(i) = A. Then one can easily check that p is a price vector that supports z when K is the equal income. 13 The boundary condition: for each x ∈ RL , {y ∈ RL : xI y} ⊆ RL . Without this condition, the Walrasian solution i ++ + ++ from equal income is not Maskin monotonic (see, for example, Jackson, 2001). 14 Thomson’s theorem is based on Hurwicz’s (1979, Theorem 1) characterization of the Walrasian solution in classical private ownership economies.
556
T. Sakai / Journal of Mathematical Economics 43 (2007) 549–563
Therefore, equal treatment of equals and Maskin monotonicity imply Pareto efficiency. Taking this fact into account, the axioms in Corollary 2 parallel the axioms in the results by Thomson and Nagahisa–Suh. Remark 1. Thomson and Nagahisa–Suh use no-envy instead of equal treatment of equals. The relaxation of no-envy to equal treatment of equals is obtained from observations by Geanakoplos and Nalebuff (1988), Fleurbaey and Maniquet (1997). 3.4. Further results on Maskin monotonic solutions Tadenuma and Thomson (1995) study the following independence property. Complete indifference: For each R ∈ RN and each z ∈ Z, if for each i, j ∈ N, zi Ii zj , then z ∈ ψ(R). Tadenuma and Thomson show that, when there is only one real object, any solution satisfying Maskin monotonicity and complete indifference contains the no-envy solution. As Thomson (2005b, Chapter 10.8) suggests, this result holds if there are more than one real object. We remark that this result still holds even if monotonicity of preferences in ranking objects is further imposed: Proposition 1. If a solution satisfies Maskin monotonicity and complete indifference, then it is a supersolution of the no-envy solution. If the solution also satisfies equal treatment of equals, then it in fact coincides with the no-envy solution. Proof. The first part can be shown by the same way as in the result by Tadenuma and Thomson, so omitted. The second part follows from the first part and the relation that any solution satisfying equal treatment of equals and Maskin monotonicity is contained in the no-envy solution (Lemma 3). An allocation z ∈ Z is egalitarian-equivalent for R ∈ RN if there exists z0 ∈ A × R such that for each i ∈ N, zi Ii z0 (Pazner and Schmeidler, 1978). Let E(R) be the set of egalitarian-equivalent allocations. We call E the egalitarian-equivalent solution. We first explain that, when = ∅, no ψ ⊆ E is Maskin monotonic. Note that any ψ ⊆ E satisfies equal treatment of equals. Since equal treatment of equals and Maskin monotonicity imply no-envy, if ψ ⊆ E is Maskin monotonic, then ψ ⊆ F . However, Thomson (1990) shows that, when |A| ≥ 3 and = ∅, there is R ∈ RN such that E(R) ∩ F (R) = ∅. These arguments imply that, when |A| ≥ 3 and = ∅, no ψ ⊆ E is Maskin mootonic. Also, Ohseto (2004, Theorem 5) shows that, when |A| = 2 and = ∅, no ψ ⊆ E is Maskin mootonic.15 Thus, we can conclude that if = ∅, no ψ ⊆ E is Maskin monotonic. We next slightly generalize this result: Even if = ∅, no ψ ⊆ E is Maskin monotonic. Proposition 2. No subsolution of the egalitarian-equivalent solution is Maskin monotonic. Proof. Suppose, by contradiction, that there exists ψ ⊆ E that is Maskin monotonic. Let R ∈ RN be such that for each i, j ∈ N, Ri = Rj . Let z ≡ (σ, m) ∈ ψ(R). Then for each i, j ∈ N, zi Ii zj . Let ε > 0. For each i ∈ N, let Rεi be such that zi Iiε (σ(j), mj + ε) for each j ∈ N with σ(j) = σ(i). 15 Ohseto shows his result on preference domains such that the number of common preferences of agents is at least three. We do not deal with such finitely restricted preference domains.
T. Sakai / Journal of Mathematical Economics 43 (2007) 549–563
557
Let Rε ≡ (Rεi )i∈N . When ε is sufficiently small, Rε ∈ RN . By Maskin monotonicity, z ∈ ψ(Rε ). However, z is not egalitarian-equivalent for Rε . Ohseto (2004, Footnote 9) says that egalitarian-equivalence and Maskin monotonicity seem to be incompatible in general. Proposition 2 confirms that this view is true in our general model for indivisible objects allocation. 4. Concluding remarks Our theorem states that, for a quite large class of problems, the no-envy solution is essentially the only Nash implementable solution satisfying equal treatment of equals. Hence, the next involves designing a “natural” mechanism that implements the no-envy solution.16 Such a mechanism is ¯ designed by Azacis (2004) when preferences are quasi-linear and extending his mechanism to nonquasi-linear preferences is of interest. Svensson (1991) constructs a price-reporting mechanism that implements the Walrasian solution in a market model. Since the no-envy solution coincides with the Walrasian solution from equal income in the present model, the idea of his mechanism may help design a mechanism implementing the no-envy solution in our model. Saijo et al. (1996) and Thomson (2005a) design mechanisms implementing the no-envy solution in economies with infinitely divisible goods.17 Designing counterparts of their mechanisms is also still open in our model. Acknowledgements This paper is based on Chapter 3 of my dissertation submitted to the University of Rochester. I am grateful to my advisor William Thomson for his continuous encouragement and support. I also thank an anonymous referee for his/her constructive and detailed referee report and Olivier Bochet, John Duggan, Norio Takeoka as well as participants of the seminars at Kobe University, Kyoto University, Tsukuba University, University of Namur, University of Rochester, Yokohama National University, and the Seventh International Meeting of the Society for Social Choice and Welfare at Osaka University for their helpful suggestions. Appendix A. Proof of Theorem 1 We first show that the space of preferences we consider satisfies a certain richness condition, called “monotonic closedness”. This condition implies that any solution satisfying equal treatment of equals and Maskin monotonicity is a subsolution of the no-envy solution. Then, we show that if a subsolution of the no-envy solution satisfies Maskin monotonicity and either restricted continuity or neutrality, then it in fact coincides with the no-envy solution. Since we identify each member of the partition, Al ∈ A, with an object contained in Al , al ∈ Al , we write al to denote Al when there is no risk of confusion. L An L-tuple of consumption bundles (al , mal )L l=1 ∈ ×l=1 ({al } × R) is -consistent if for each l, l such that l l , mal < mal . Given two objects a, b ∈ A, a is -connected to b if there are objects b1 , b2 , . . . , bk ∈ A such that a = b1 , b = bk , and for each k = 1, 2, . . . , k − 1, either
16 17
For critique to “unnatural” mechanisms, see Saijo et al. (1996) and Jackson (2001). Thomson’s mechanism is designed for more general economic environments.
558
T. Sakai / Journal of Mathematical Economics 43 (2007) 549–563
bk bk +1 or bk +1 bk . Given objects a, b ∈ A such that a b, we write a 0 b if there exists no c ∈ A such that a c and c b. L Lemma 1. If two -consistent tuples (al , mal )L l=1 , (al , mal )l=1 are such that for each l, ml > ml , then there exists Ri ∈ R such that
(a1 , ma1 )Ii (a2 , ma2 )Ii · · · Ii (aL , maL ) and (a1 , ma1 )Ii (a2 , ma2 )Ii · · · Ii (aL , maL ). Proof. Let X ≡ {X1 , X2 , . . . , XK } be the partition of A such that (i) for each k and each a, b ∈ Xk , a is -connected to b; (ii) for each k, k with k = k , each a ∈ Xk , and each b ∈ Xk , a is not connected to b. Note that X is uniquely determined. ¯ i: Step 1 Defining R Substep 1-1 Let k ∈ {1, 2, . . . , K}. For each a, b ∈ Xk such that a 0 b, let ha,b : [ma , ma ] → [mb , mb ] be a strictly increasing bijection such that for each ma ∈ [ma , ma ], ma < ha,b (ma ). Such a bijection exists since ma < mb and ma < mb . Then, for each ma ∈ [ma , ma ], let (a, ma )I¯i (b, ha,b (ma )). Substep 1-2 For each k, fix bk ∈ Xk . Let k, k be such that k < k . Let hbk ,bk : [m bk , m bk ] → [m bk , m bk ] be a strictly increasing bijection, and for each mbk ∈ [m bk , mbk ], let (bk , mbk )I¯i (bk , hbk ,bk (mbk )). Substep 1-3 For each a ∈ A and each ma , ma ∈ R with ma < ma , let (a, ma )P¯ i (a, ma ). Substep 1-4 For each ε > 0, let (a1 , ma1 − ε)I¯i (a2 , ma2 − ε)I¯i · · · I¯i (aL , maL − ε) and (a1 , ma1 + ε)I¯i (a2 , ma2 + ε)I¯i · · · I¯i (aL , maL + ε). ¯ i , i.e., for each (a, ma ), (b, mb ) ∈ A × R, Step 2 Defining Ri : Let Ri be the transitive closure of R (a, ma )Ri (b, mb ) if and only if there are bundles: (b1 , mb1 ), (b2 , mb2 ), . . . , (bK , mbK ) such that (i) (a, ma ) = (b1 , mb1 ), (b, mb ) = (bk , mbk ), and (ii) for each k ≤ K − 1: ¯ i (bk+1 , mbk+1 ), (bk , mbk )R where (a, ma )Pi (b, mb ) holding if and only if for some k ≤ K − 1, (bk , mbk )P¯ i (bk+1 , mbk+1 ). Step 3 Concluding: Obviously, Ri is transitive and complete. By Substep 1-3, Ri satisfies money monotonicity. By Substeps 1-1, -2 and -4, Ri satisfies finiteness. By Substep 1-1, Ri satisfies -object monotonicity. Hence, Ri ∈ R.
T. Sakai / Journal of Mathematical Economics 43 (2007) 549–563
559
Geanakoplos and Nalebuff (1988) essentially show that in economies with divisible goods, if a preference space satisfies certain conditions, then equal treatment of equals and Maskin monotonicity together imply no-envy.18 Fleurbaey and Maniquet (1997) show that in an abstract social choice environment, if a preference space satisfies the following condition, the same relation holds. Monotonic closedness: For each i, j ∈ N, each Ri , Rj ∈ R, and each z ∈ Z with zj Pi zi , there is Ri ∈ R such that (1) zj Pi zi , (2) {w ∈ Z : zi Ri wi } ⊆ {w ∈ Z : zi Ri wi }, (3) {w ∈ Z : zj Rj wj } ⊆ {w ∈ Z : zj Ri wj }. Lemma 2. The preference space R is monotonically closed. Proof. Let i, j ∈ N, Ri , Rj ∈ R, and z ∈ Z be such that zj Pi zi . Write zi = (a, ma ) and zj = (b, mb ). For each c ∈ A, let m(i, c), m(j, c) ∈ R be such that (a, ma )Ii (c, m(i, c))
and
(b, mb )Ij (c, m(j, c)).
Consider the tuple: ((b, mb ), {(c, max{m(i, c), m(j, c)})}c=b ) ∈ ×L l=1 ({al } × R). Given c, d ∈ A, if c d, then m(i, c) < m(i, d) and m(j, c) < m(j, d), hence max{m(i, c), m(j, c)} < max{m(i, d), m(j, d)}. Thus, this tuple is -consistent. Since the set of -consistent tuples is open, there is a sufficiently small ε > 0 such that the tuple: ((b, mb ), {(c, ε + max{m(i, c), m(j, c)})}c=b ) ∈ ×L l=1 ({al } × R) is -consistent. Clearly, the tuple ((a, ma ), {(c, m(i, c))}c=a ) ∈ ×L l=1 ({al } × R) is -consistent as well. Therefore, by Lemma 1, there exists Ri ∈ R such that both tuples: ((a, ma ), {(c, m(i, c))}c=a ) ∈ ×L l=1 ({al } × R) and ((b, mb ), {(c, ε + max{m(i, c), m(j, c)})}c=b ) ∈ ×L l=1 ({al } × R) constitute indifference curves. Clearly, Ri satisfies the desired properties.
Lemma 3. If a solution satisfies equal treatment of equals and Maskin monotonicity, then it is a subsolution of the no-envy solution.
18 They also impose Pareto efficiency and state the result for two agents. However, as Moulin (1993), Fleurbaey and Maniquet (1997) point out, Pareto efficiency is redundant and their result holds for an arbitrary number of agents.
560
T. Sakai / Journal of Mathematical Economics 43 (2007) 549–563
Proof. The proof below is due to Geanakoplos and Nalebuff (1988, Proposition 4) and Fleurbaey and Maniquet (1997, Theorem 1). Suppose, by contradiction, that there are R ∈ RN and z ∈ ψ(R) such that for some i, j ∈ N, zj Pi zi . By monotonic closedness, there are Ri , Rj ∈ R such that Ri = Rj , zj Pi zi , {w ∈ Z : zi Ri wi } ⊆ {w ∈ Z : zi Ri wi } and {w ∈ Z : zj Rj wj } ⊆ {w ∈ Z : zj Rj wj }. By Maskin monotonicity, z ∈ ψ(Ri , Rj , RN\{i,j} ). However, since Ri = Rj and zj Pi zi , this contradicts equal treatment of equals. Lemma 4. If a subsolution of the no-envy solution satisfies Maskin monotonicity and restricted continuity, then it coincides with the no-envy solution. Proof. Let ψ be a subsolution of the no-envy solution satisfying Maskin monotonicity and restricted continuity. Let R ∈ RN and z ∈ F (R). Since z ∈ F (R), for each i, j ∈ N, if σ(i) = σ(j), then mi = mj . Hence, for each a ∈ A, we can write mσ(i) instead of mi when a = σ(i). Let R be the profile of quasi-linear preferences such that for each i, j ∈ N, Ri = Rj and zi Ii zj . By construction, R ∈ QN . For each ε > 0, each i ∈ N, and each ν ∈ N, let Rνi be the quasi-linear preference such that for each j = i with σ(j) = σ(i), ε ν σ(i), mσ(i) − I zj . ν i Given ν ∈ N, let Rν ≡ (Rν1 , Rν2 , . . . , Rνn ) and zν ≡ (σ ν , mν ) ∈ ψ(Rν ). When ε is sufficiently small, for each ν ∈ N, Rν ∈ RN . Hereafter, let ε > 0 be such a small number. Since zν ∈ F (Rν ), for each i, j ∈ N, if σ ν (i) = σ ν (j), then mνi = mνj . Hence, for each ν ∈ N and each a ∈ A, we can write mνσ(i) instead of mνi when a = σ(i). Claim 1. For each i ∈ N, σ ν (i) = σ(i). Suppose, by contradiction, that for some i, j ∈ N, σ ν (i) = σ(j) = σ(i). Without loss of generality, assume σ ν (j) = σ(j). Case 1 (mνσ ν (i) ≥ mσ ν (i) ). Let δ ≡ mνσ ν (i) − mσ ν (i) ≥ 0. By definition of Rν , for each c = σ(i), σ ν (i), ε zνi = (σ ν (i), mσ ν (i) + δ)Iiν σ(i), mσ(i) − + δ Iiν (c, mc + δ), ν ε ε ν ν ν zi = (σ (i), mσ ν (i) + δ)Ij σ(i), mσ(i) + + δ Ijν c, mc + + δ . ν ν ν ν ν ν Since zj Rj zi and σ (j) = σ(j), by (A.2),
(A.1) (A.2)
ε + δ. ν Then by (A.1), zνj Piν zνi , a contradiction to zν ∈ ψ(Rν ). mνσ ν (j) ≥ mσ ν (j) +
Case 2 (mνσ ν (i) < mσ ν (i) ). Then there exists k ∈ N such that mνσ ν (k) > mσ ν (k) . By definition of Rν , zνk = (σ ν (k), mνσ ν (k) )Piν (σ ν (k), mσ ν (k) )Rνi (σ ν (i), mσ ν (i) )Piν (σ ν (i), mνσ ν (i) ) = zνi , where (σ ν (k), mσ ν (k) )Piν (σ ν (i), mσ ν (i) ) holding iff σ ν (k) = σ(i). Thus, agent i envies agent k, a contradiction to zν ∈ ψ(Rν ).
T. Sakai / Journal of Mathematical Economics 43 (2007) 549–563
561
Claim 2. For each i ∈ N, mνσ(i) ≤ mσ(i) + (ε/ν). Suppose, by contradiction, that for some i ∈ N, mνσ(i) > mσ(i) + (ε/ν). Then, there is j ∈ N such that mνσ(j) < mσ(j) . Since zνj Rνj zνi , we have σ(j) = σ(i). By definition of Rν , ε ν I (σ(j), mσ(j) )Pjν (σ(j), mνσ(j) ) = zνj . zνi = (σ(i), mνσ(i) )Pjν σ(i), mσ(i) + ν j Therefore, agent j envies agent i, a contradiction to zν ∈ ψ(Rν ). Claim 3. For each i ∈ N, mσ(i) − νε ≤ mνσ(i) . Suppose, by contradiction, that for some i ∈ N, mσ(i) − νε > mνσ(i) . Then, there is j ∈ N such that mνσ(j) > mσ(j) . Since zνi Rνi zνj , we have σ(j) = σ(i). By definition of Rν , ε ν Pi (σ(i), mνσ(i) ) = zνi . zνj = (σ(j), mνσ(j) )Piν (σ(j), mσ(j) )Iiν σ(i), mσ(i) − ν Therefore, agent i envies agent j, a contradiction to zν ∈ ψ(Rν ). Thus, we have established that for each ν ∈ N and each i ∈ N, mσ(i) − (ε/ν) ≤ mνσ(i) ≤ mσ(i) + (ε/ν). This implies that the sequence {zν } converges to z. Clearly, the sequence {Rν } also converges to R . Thus, by restricted continuity, z ∈ ψ(R ). Since R ∈ MT(R , z), by Maskin monotonicity, z ∈ ψ(R). Thus, ψ(R) = F (R). The following is a weaker version of a decomposition lemma in Alkan et al. (1991, Lemma 3). Lemma 5. Let R ∈ RN and z, z ∈ F (R). If for each i, j ∈ N, zi Ii zi , then z R z . Proof. Let R ∈ RN and z ≡ (σ, m), z ≡ (σ , m ) ∈ F (R) be such that for each i ∈ N, zi Ii zi . Let a ∈ A. It suffices to show that ma = ma . Suppose that ma > ma . Let i ∈ N be such that σ (i) = a. Then, since z ∈ F (R), zi Ri (a, ma )Pi (a, ma ) = (σ (i), m σ (i) ) = zi , a contradiction. Next, suppose that ma < ma . Let i ∈ N be such that σ(i) = a. Then, since z ∈ F (R), zi Ri (a, ma )Pi (a, ma ) = (σ(i), mσ(i) ) = zi , a contradiction. Thus, ma = ma .
Lemma 6. If a subsolution of the no-envy solution satisfies Maskin monotonicity and neutrality, then it coincides with the no-envy solution. Proof. Let ψ be a subsolution of the no-envy solution satisfying Maskin monotonicity and neutrality. Let R ∈ RN and z ∈ F (R). Let R be the profile of quasi-linear preferences such that for each i, j ∈ N, Ri = Rj and zi I i zj . By construction, R ∈ QN . Let w ∈ ψ(R ). Let us show w R z. Since z ∈ P(R ), if there is i ∈ N with wi Pi zi , then there is j ∈ N with zj Pj wj , and then wi Pi zi Ii zj Pi wj , which contradicts equal treatment of equals. Hence, for each i ∈ N, zi Ii wi . Since z, w ∈ F (R ), by Lemma 5, z R w. By neutrality, z ∈ ψ(R ). Since R ∈ MT(R , z), by Maskin monotonicity, z ∈ ψ(R). Proof of Theorem 1. We omit the easy proof that the no-envy solution satisfies equal treatment of equals, Maskin monotonicity, restricted continuity, and neutrality. Let ψ be a solution satisfying equal treatment of equals and Maskin monotonicity. By Lemma 3, it is a subsolution of the no-envy solution. If ψ further satisfies restricted continuity, then by Lemma 4, it coincides with the no-envy solution. On the other hand, if ψ satisfies neutrality, then by Lemma 6, it coincides with the no-envy solution.
562
T. Sakai / Journal of Mathematical Economics 43 (2007) 549–563
Appendix B. Continuity properties We briefly mention relations between restricted continuity and other topological properties. In the present model, the Kannai topology is the smallest topology such that the set: {((a, mi ), (b, mi ), Ri ) ∈ (A × R) × (A × R) × R : (a, mi )Pi (b, mi )} is open with respect to the product topology. Here, this topology coincides with other standard topologies such as the closed convergence topology (e.g., Hildenbrand, 1974; Redekop, 1993). We refer to Kannai (1970) and Redekop (1993) for interpretations and properties of this topology. We list several facts on topological properties with respect to the Kannai topology.19 • • • • •
A solution ψ satisfies restricted continuity if and only if the subgraph of ψ on QN is closed. If a solution has a closed graph, then it satisfies restricted continuity. If a closed-valued solution is upper hemicontinuous, then its graph is closed. The no-envy solution is compact-valued and upper hemicontinuous, so its graph is closed. The no-envy solution is the only closed-valued solution satisfying equal treatment of equals, Maskin monotonicity, and upper hemicontinuity. Alternately, it is the only solution satisfying equal treatment of equals and Maskin monotonicity whose graph is closed.
Therefore, we could proceed the present study based on Kannai continuity properties. However, we chose to use restricted continuity because of its mildness and simplicity. References Alkan, A., Demange, G., Gale, D., 1991. Fair allocation of indivisible goods and criteria of justice. Econometrica 59, 1023–1039. ¯ Azacis, H., 2004. Double Implementation in a Market for Indivisible Goods with a Price Constraint. Universitat Aut`onoma de Barcelona, Mimeo. Bevia, C., 1996. Identical preferences lower bound solution and consistency in economies with indivisible goods. Social Choice and Welfare 13, 113–126. Bochet, O., Sakai, T., 2007. Strategic manipulations of multi-valued solutions in economies with indivisibilities. Mathematical Social Sciences 53, 53–68. Chun, Y., 2006a. A pessimistic approach to the queueing problem. Mathematical Social Sciences 51, 171–181. Chun, Y., 2006b. No-envy in queueing problems. Economic Theory 29, 151–162. Fleurbaey, M., Maniquet, F., 1997. Implementability and horizontal equity imply no-envy. Econometrica 65, 1215–1219. Foley, D., 1967. Resource allocation and the public sector. Yale Economic Essays 7, 45–98. Geanakoplos, J., Nalebuff, B., 1988. On a fundamental conflict between equity and efficiency. Yale University/Princeton University, Mimeo. Hildenbrand, W., 1974. Core and Equilibria of a Large Economy. Princeton University Press, Princeton, NJ. Hurwicz, L., 1979. On allocations attainable through Nash equilibria. In: Laffont, J.-J. (Ed.), Aggregation and Revealed Preferences. North-Holland Publishing Co. Jackson, M., 2001. A crash course in implementation theory. Social Choice and Welfare 18, 655–708. Kaneko, M., 1983. Housing markets with indivisibilities. Journal of Urban Economics 13, 22–50. Kannai, Y., 1970. Continuity properties of the core of a market. Econometrica 38, 791–815. Maniquet, F., 2003. A characterization of the Shapley value in queueing problems. Journal of Economic Theory 109, 90–103. Maskin, E., 1999. Nash equilibrium and welfare optimality. Review of Economic Studies 66, 23–38.
19
Proofs are available from the author upon request.
T. Sakai / Journal of Mathematical Economics 43 (2007) 549–563
563
Moulin, H., 1993. On the fair and coalitionproof allocation of private goods. In: Binmore, K., Kirman, A., Tani, P. (Eds.), Frontiers of Game Theory. The MIT Press, Cambridge, MA. Nagahisa, R.-I., 1991. A local independence condition for characterization of Walrasian allocation rule. Journal of Economic Theory 54, 106–123. Nagahisa, R.-I., Suh, S.-C., 1995. A characterization of the Walras rule. Social Choice and Welfare 12, 335–352 (Reprinted in: Walker, D., 2001. The Legacy of Leon Walras. Edward Elgar Publishing). Ohseto, S., 2001. Strategy-proof and efficient allocation of an indivisible good on finitely restricted domains. International Journal of Game Theory 29, 365–374. Ohseto, S., 2004. Implementing egalitarian-equivalent allocation of indivisible goods on restricted domains. Economic Theory 23, 659–670. Pazner, E., Schmeidler, D., 1978. Egalitarian equivalent allocations: a new concept of economic equity. Quarterly Journal of Economics 92, 671–683. Quinzii, M., 1984. Core and competitive equilibria with indivisibilities. International Journal of Game Theory 13, 41–60. Redekop, J., 1993. The questionnaire topology on some spaces of economic preferences. Journal of Mathematical Economics 22, 479–494. Saijo, T., Tatamitani, Y., Yamato, T., 1996. Toward natural implementation. International Economic Review 37, 941–980. Schummer, J., 2000. Eliciting preferences to assign positions and compensation. Games and Economic Behavior 30, 293–318. Svensson, L.-G., 1983. Large indivisibilities: an analysis with respect to price equilibrium and fairness. Econometrica 51, 939–954. Svensson, L.-G., 1991. Nash implementation of competitive equilibria in a model with indivisible goods. Econometrica 59, 869–877. Svensson, L.-G., Larsson, B., 2002. Strategy-proof and nonbossy allocation of indivisible goods and money. Economic Theory 20, 483–502. Tadenuma, K., Thomson, W., 1991. No-envy and consistency in economies with indivisible goods. Econometrica 59, 1755–1767. Tadenuma, K., Thomson, W., 1993. The fair allocation of an indivisible good when monetary compensations are possible. Mathematical Social Sciences 25, 117–132. Tadenuma, K., Thomson, W., 1995. Games of fair division. Games and Economic Behavior 9, 191–204. Thomson, W., 1979. Comment: on allocations attainable through Nash equilibria. In: Laffont, J.-J. (Ed.), Aggregation and Revealed Preferences. North-Holland Publishing Company. Thomson, W., 1990. On the non-existence of envy-free and egalitarian-equivalent allocations in economies with indivisibilities. Economics Letters 34, 227–229. Thomson, W., 2005a. Divide and permute. Games and Economic Behavior 52, 186–200. Thomson, W., 2005b. Theory of Fair Allocation. University of Rochester, Mimeo. Tinbergen, J., 1953. Redelijke Inkomensverdeling, 2nd ed. N.V. Gulden Press. Varian, H., 1974. Equity, envy, and efficiency. Journal of Economic Theory 9, 63–91.
Fairness and implementability in allocation of indivisible objects with monetary compensations Toyotaka Sakai Faculty of Economics and IGSSS, Yokohama National University, Yokohama 240-8501, Japan Received 4 January 2005; received in revised form 19 March 2006; accepted 11 July 2006 Available online 20 February 2007
Abstract This paper introduces a general framework for the fair allocation of indivisible objects when each agent can consume at most one (e.g., houses, jobs, queuing positions) and monetary compensations are possible. This framework enables us to deal with identical objects and monotonicity of preferences in ranking objects. We show that the no-envy solution is the only solution satisfying equal treatment of equals, Maskin monotonicity, and a mild continuity property. The same axiomatization holds if the continuity property is replaced by a neutrality property. © 2007 Elsevier B.V. All rights reserved. JEL classification: C78; D63; D71 Keywords: Indivisible goods; Queuing; Generalized queuing; No-envy; Maskin monotonicity; Nash implementation; Walrasian social choice
1. Introduction This paper introduces a general framework for the fair allocation of indivisible objects when each agent can consume at most one (e.g., houses, jobs, positions) and monetary compensations are possible. By allowing identical objects and monotonicity of preferences in ranking objects, this framework unifies various classes of problems that have been studied separately so far. Examples include problems of allocating different objects (e.g., Svensson, 1983; Alkan et al., 1991), identical objects (e.g., Tadenuma and Thomson, 1991), single object (e.g., Tadenuma and Thomson, 1993, 1995), queuing positions (e.g., Maniquet, 2003; Chun, 2006a,b). E-mail address: [email protected]. URL: http://www.geocities.jp/toyotaka sakai/index.html. 0304-4068/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.jmateco.2006.07.006
550
T. Sakai / Journal of Mathematical Economics 43 (2007) 549–563
Our purpose is to identify solutions that satisfy certain fairness and implementability properties. As a fairness property, we consider equal treatment of equals, which states that two agents with the same preference should receive indifferent consumption bundles. As an implementability property, we consider Maskin monotonicity, which is necessary and (in this model, when there are at least three agents) sufficient for a solution to be Nash implementable (Maskin, 1999).1 The no-envy solution is the solution that associates with each preference profile the set of allocations at which no agent prefers the consumption bundle of any other to his own (Foley, 1967). Geanakoplos and Nalebuff (1988) show that, in classical exchange economies with divisible goods, any solution satisfying equal treatment of equals and Maskin monotonicity is a subsolution of the no-envy solution. Fleurbaey and Maniquet (1997) show that, in abstract economic environments, the same implication holds if a preference domain satisfies a richness condition. Since we can show that our preference domain satisfies the condition, any solution satisfying equal treatment of equals and Maskin monotonicity is a subsolution of the no-envy solution in this environment. We study how solutions satisfying the two properties differ from the no-envy solution. Our main theorem shows that there is no difference if we focus on solutions satisfying a mild continuity property. In other words, the no-envy solution is the only solution satisfying equal treatment of equals, Maskin monotonicity, and the continuity property. Furthermore, the same axiomatization holds if the continuity property is replaced by a neutrality property introduced by Tadenuma and Thomson (1991). Since the continuity and neutrality properties are so weak that they are satisfied by all standard solutions, our theorem suggests that the no-envy solution is essentially the only solution satisfying equal treatment of equals and Maskin monotonicity. Svensson (1983, Theorem 4) shows that the Walrasian solution from equal income coincides with the no-envy solution. Since the Walrasian solution from equal income satisfies the axioms characterizing the no-envy solution, the equivalence is in fact obtained as a corollary to our theorem. It is known that the no-envy solution satisfies many desirable properties in economies with indivisible objects with money. For example, this solution satisfies Pareto efficiency, (Svensson, 1983; Alkan et al., 1991), a welfare lower bound property (Bevia, 1996), and a consistency property for population changes (Tadenuma and Thomson, 1991). 2 On the other hand, our theorem justifies the no-envy solution from the perspectives of fairness and implementability. This paper is organized as follows. Section 2 introduces the model. Section 3 presents the main results. Section 4 concludes. Appendix A offers proofs and remarks on continuity properties. 2. The model 2.1. Basic notion We extend the model developed by Alkan et al. (1991) so as to deal with identical objects and monotonicity of preferences in ranking objects. 3 Let N ≡ {1, 2, . . . , n} and O ≡ {a1 , a2 , . . . , an }
1
As an implementability property, one may wish to consider “strategy-proofness" (no one can gain by misrepresenting his preference). However, it is known that this property is incompatible with various fairness properties including equal treatment of equals (e.g., Schummer, 2000; Svensson and Larsson, 2002; Ohseto, 2001; Bochet and Sakai, 2007). 2 Tadenuma and Thomson (1991, Theorem 1) show that, in a variable population model with different objects, if a subsolution of the no-envy solution satisfies the consistency property and a neutrality property (same us ours), then it in fact coincides with the no-envy solution. 3 Earlier works of economies with indivisible objects and money include Kaneko (1983), Svensson (1983), and Quinzii (1984).
T. Sakai / Journal of Mathematical Economics 43 (2007) 549–563
551
be finite sets of agents and objects, respectively, where |N| = |O|. We allow some objects to be “null” to describe situations where there are fewer “real” objects than agents. Let A ≡ {A1 , A2 , . . . , AL } be a partition of O, i.e., ∪L l=1 Al = O and for each l, l with l = l , Al ∩ Al = ∅. This partition identifies which objects are identical: two objects are identical if and only if they are in the same component Al of A. When there is no confusion, we identify each Al with an object contained in Al . We assume that there is at least one non-null object to make the problem interesting, that is, |A| ≥ 2. There is also an amount M ∈ R of a single perfectly divisible good called money. A feasible allocation, or simply an allocation, is a pair: z ≡ (zi )i∈N ≡ (σ(i), mi )i∈N ≡ (σ, m), where σ : N → A is a function such that for each l ∈ {1, 2, . . . , L}, |{i ∈ N : σ(i) = Al }| = |Al |, and m ∈ RN is such that i∈N mi = M. Here σ(i) is the indivisible object that i receives, and mi is agent i’s consumption of money. When (σ, m) is such that for each i, j ∈ N, σ(i) = σ(j) implies mi = mj , we may write mσ(i) instead of mi . Let Z be the set of allocations. A partial order is an irreflexive and transitive binary relation on A.4 We use a partial order to deal with monotonicity of preferences in ranking objects. For example, if A ∈ A denotes a larger house and B ∈ A denotes a smaller house, then it is natural to assume that A B. A preference for agent i ∈ N is a transitive and complete binary relation Ri over his consumption space A × R that satisfies: • Money monotonicity: for each (A, mi ) ∈ A × R and each mi > mi , (A, mi )Pi (A, mi ), • Finiteness: for each (A, mi ) ∈ A × R and each B ∈ A, there is mi ∈ R such that (A, mi )Ii (B, mi ). • -object monotonicity: for each A, B ∈ A and each mi ∈ R, if A B, then (A, mi )Pi (B, mi ).5 Let R be the set of preferences. A profile is an n-tuple of preferences R ≡ (R1 , R2 , . . . , Rn ) ∈ RN . A preference Ri ∈ R is quasi-linear if there exists a “valuation vector” v(Ri ) ≡ (v(Ri , A1 ), v(Ri , A2 ), . . . , v(Ri , AL )) ∈ RL such that v(Ri , A1 ) = 0 and Ri is represented as U(Ri , A, mi ) = v(Ri , A) + mi for (A, mi ) ∈ A × R.6 Let Q ⊆ R be the subset of quasi-linear preferences. We call (A, ) a primitive pair and fix it throughout the paper. This notion is a new device introduced by this study. Various classes of problems discussed in the literature can be described using this notion: • Different objects: Let A ≡ {{a1 }, {a2 }, . . . , {an }} and = ∅. In this class, there are as many different real objects as agents, and no object monotonicity is imposed. • Different objects and null objects, a general case: Let A ≡ {A1 , A2 , . . . , AL } be an arbitrary partition of A. Let us interpret AL as a null object, and A1 , . . . , AL−1 as desirable real objects. Then, for each l ≤ L − 1, let Al AL .7 Several interesting subclasses can be identified: (i) when all real objects are different (e.g., Tadenuma and Thomson, 1993, 1995), we let |Al | = 1 4 5 6 7
Irreflexivity: for each A ∈ A, AⱭA; transitivity: for each A, B, C ∈ A, if A B and B C, then A C. Note that, when = ∅ (i.e., for no A, B ∈ A, A B), this condition is vacuously satisfied. Note that v(Ri , A1 ) = 0 is just a normalization and this restriction does not lose any generality. We can also consider the case where some real objects, say Al , are “bads” by instead assuming that AL Al .
552
T. Sakai / Journal of Mathematical Economics 43 (2007) 549–563
for each l ≤ L − 1; (ii) when all real objects are identical and the number of real objects is at least two (e.g., Tadenuma and Thomson, 1991), we let L = 2 and |A1 | ≥ 2; (iii) when there is only one real object (e.g., Tadenuma and Thomson, 1993, 1995), we let L = 2 and |A1 | = 1. • Queuing: Let A ≡ {{a1 }, {a2 }, . . . , {an }} and be such that {a1 } {a2 } · · · {an }. Agents stand to receive a service, and no two agents can be served simultaneously. All agents prefer being the first to being the second, and being the second to being the third, and so on. Maniquet (2003) and Chun (2006a,b) study this class when preferences are linear.8 • Generalized queuing: Let A ≡ {A1 , A2 , . . . , AL } be an arbitrary partition of A and be such that A1 A2 · · · AL . In this queuing problem, |Al |-agents are served in the lth position. We also allow the case where there are fewer positions than agents, that is, AL denotes a null object, and so |AL |-agents cannot be served. 2.2. No-envy and efficiency A solution is a non-empty valued correspondence ψ from RN to Z. We introduce two familiar solutions. The no-envy notion states that no agent prefers the consumption bundle of any other to his own (Tinbergen, 1953; Foley, 1967). An allocation z ∈ Z is envy-free for R ∈ RN if for each i, j ∈ N, zi Ri zj . Given R ∈ RN , let F (R) be the set of envy-free allocations for R. The nonemptiness of F (R) is shown by Alkan et al. (1991, Theorem 2). We call F the no-envy solution. This solution is fundamental in the theory of fair allocation.9 We also study a standard efficiency notion. An allocation z ∈ Z is Pareto efficient for R ∈ RN if there exists no z ∈ Z such that for each i ∈ N, zi Ri zi and for some j ∈ N, zj Pj zj . Given R ∈ RN , let P(R) be the set of Pareto efficient allocations for R. We call P the Pareto solution. It is known that for each R ∈ RN , F (R) ⊆ P(R) (Svensson, 1983, Theorem 2; Alkan et al., 1991, Theorem 1). 2.3. Axioms We are interested in solutions satisfying the following basic fairness requirement: two agents whose preferences are the same should receive indifferent consumption bundles: Equal treatment of equals: For each R ∈ RN , each z ∈ ψ(R), and each i, j ∈ N, if Ri = Rj , then zi Ii zj (and so zi Ij zj ). Given R ∈ RN and z ∈ Z, a profile R ∈ RN is a Maskin monotonic transformation of R at z, if for each i ∈ N, {w ∈ Z : zi Ri wi } ⊆ {w ∈ Z : zi Ri wi }. Let MT (R, z) be the set of Maskin monotonic transformations of R at z. General results by Maskin (1999) imply in our context that the following property is necessary and (when |N| ≥ 3) sufficient for a solution to be implementable in Nash equilibrium.10 8 9 10
In queuing problems, a preference Ri is linear if it is represented as U(Ri , {al }, mi ) = γi (n − l) + mi with γi > 0. Thomson (2005b) offers a survey of the literature. For Maskin’s results and the definition of Nash implementation, see Jackson (2001).
T. Sakai / Journal of Mathematical Economics 43 (2007) 549–563
553
Maskin monotonicity: For each R ∈ RN and each z ∈ ψ(R), if R ∈ MT (R, z), then z ∈ ψ(R ). Next, we consider the requirement that a solution should be robust to small perturbations of preferences. If a sequence of quasi-linear preferences {Rνi } is such that {v(Rνi )} → v(Ri ), then it is natural to consider that the sequence {Rνi } also converges to Ri . In this case, we write {Rνi } → Ri . Restricted continuity: Let {Rν }, R be a sequence of profiles and a profile in QN , respectively. Let {zν }, z be a sequence of allocations and an allocation in Z, respectively. Suppose that: (i) for each i ∈ N, {v(Rνi )} → v(Ri ), (ii) for each ν ∈ N, zν ∈ ψ(Rν ), (iiii) mν → m and there is ν ∈ N such that for each ν ≥ ν , σ ν = σ. Then, z ∈ ψ(R). Restricted continuity is only defined for quasi-linear preferences. However, we can also consider continuity properties for general preferences by introducing a natural topology on the space of preferences. This topic is briefly examined in Appendix B. However, restricted continuity is quite weak as a continuity requirement and this approach has the advantage of not requiring any complicated mathematical notion. Given two allocations z, z ∈ Z and a profile R ∈ RN , we write z R z if for each i ∈ N, zi Ii zi , and there is a bijection π : N → N such that for each i ∈ N, zπ(i) = zi . The next mild neutrality property states that if two allocations are equivalent in welfare and one is obtained from the other by a permutation, then one is chosen if and only if the other is also chosen (Tadenuma and Thomson, 1991). Neutrality: For each R ∈ RN , each z ∈ ψ(R), and each z ∈ Z, if z R z , then z ∈ ψ(R). Restricted continuity and neutrality are so weak that they are satisfied by all standard solutions in this literature. 3. Main results 3.1. Axiomatizations of the no-envy solution We present axiomatizations of the no-envy solution. Theorem 1. The no-envy solution is the only solution satisfying equal treatment of equals, Maskin monotonicity, and restricted continuity. The same axiomatization holds if restricted continuity is replaced by neutrality. The proof of Theorem 1 is relegated to Appendix A. We here outline the strategy of the proof. We first show that RN satisfies a certain richness condition (Lemmas 1 and 2). This fact and the results by Geanakoplos and Nalebuff (1988), Fleurbaey and Maniquet (1997) imply that if a solution satisfies equal treatment of equals and Maskin monotonicity, then it is a subsolution of the no-envy solution (Lemma 3). We next show that if a subsolution of the no-envy solution satisfies Maskin monotonicity and at least one of restricted continuity and neutrality, then it coincides with the no-envy solution (Lemmas 4–6). These lemmas complete the proof. 3.2. Tightness of Theorem 1 Clearly, equal treatment of equals and Maskin monotonicity are indispensable to obtain the axiomatizations. The next example shows that restricted continuity and neutrality are also indispensable.
554
T. Sakai / Journal of Mathematical Economics 43 (2007) 549–563
Example 1. Let z ∈ Z. For each R ∈ RN , let ψ(R) ≡ F (R) \ {z }. It is obvious that, if ψ is welldefined, ψ satisfies equal treatment of equals and Maskin monotonicity, but Theorem 1 implies that ψ violates neutrality and restricted continuity. It remains to show that ψ is well-defined, i.e., for each R ∈ RN , ψ(R) = ∅. To prove this, it suffices to show that for each R ∈ RN , |ψ(R)| ≥ 2. Here, we only show this for the simple case where N ≡ {1, 2}, A ≡ {A, B}, and ≡ ∅. A proof for general cases is somewhat complicated but depends on a similar idea.11 Let R ∈ RN and z ∈ F (R). Without loss of generality, assume z = ((A, mA ), (B, mB )). If for each i ∈ N, z1 Ii z2 , then (z2 , z1 ) ∈ F (R), so |F (R)| ≥ 2. If z1 P1 z2 , then for a sufficiently small ε > 0, (A, mA − ε)P1 (B, mB + ε) and (B, mB + ε)P2 (A, mA − ε). Therefore, ((A, mA − ε), (B, mB + ε)) ∈ F (R), so |F (R)| ≥ 2. The case z2 P2 z1 is parallel to the previous case. Thus, in either case, |F (R)| ≥ 2. The axiomatizations do not hold on the domain of quasi-linear preferences. The main reason is that equal treatment of equals and Maskin monotonicity do not imply no-envy on such a smaller domain. The following is an example of a solution on QN that satisfies F ψ P and the axioms in Theorem 1: ¯ ∈ QN and z¯ ≡ (σ, ¯ m) ¯ ∈ Z be such that σ¯ is the unique maximizer of Example 2. Let R ¯ z¯ ) in v(R , σ(i)) over σ and for each i, j ∈ N, z¯ i P¯ i z¯ j . There are many such a pair (R, i i∈N general, but here we fix one. Define the solution ψ as follows: ¯ z¯ ), P(R) if R ∈ MT(R, ψ(R) ≡ F (R) otherwise. Obviously, F ψ P. Also, it is easy to see that ψ satisfies restricted continuity and neutrality. ¯ z¯ ), there are no i, j ∈ N such that Ri = Rj . Hence, if By construction, for each R ∈ MT(R, ¯ z¯ ), and so ψ(R) = F (R). Thus, R ∈ QN is such that for some i, j ∈ N, Ri = Rj , then R ∈ / MT(R, ψ satisfies equal treatment of equals. It remains to show that ψ is Maskin monotonic. Let R ∈ QN , z ∈ ψ(R), and R ∈ MT(R, z). ¯ z¯ ). Then z ∈ ψ(R) = P(R). Since z ∈ P(R) and R ∈ MT(R, ¯ z¯ ), by Consider the case R ∈ MT(R, quasi-linearity, z and z¯ share the same assignment function. Therefore, by quasi-linearity, R ∈ ¯ z¯ ) together imply R ∈ MT(R, ¯ z¯ ). Hence, ψ(R ) = P(R ). Since P is MT(R, z) and R ∈ MT(R, ¯ z¯ ). Then / MT(R, Maskin monotonic, by z ∈ P(R), z ∈ P(R ) = ψ(R ). Next consider the case R ∈ z ∈ ψ(R) = F (R). Since F is Maskin monotonic, z ∈ F (R ). Since F (R ) ⊆ ψ(R ), z ∈ ψ(R ). Hence, in either case, z ∈ ψ(R ). Thus, ψ is Maskin monotonic. We remark that the above example is constructed so as to apply to the smaller domain of linear preferences. This implies that in queuing problems by Maniquet (2003) and Chun (2006a,b), the set of our axioms does not characterize the no-envy solution. 3.3. Walrasian social choice Theorem 1 can be applied to study Walrasian social choice in this economy. An allocation z ≡ (σ, m) ∈ Z is a Walrasian allocation from equal income for R ∈ RN if there exist p ∈ RA +
11
The proof is available from the author upon request.
T. Sakai / Journal of Mathematical Economics 43 (2007) 549–563
555
and K ∈ R such that for each i ∈ N, (σ(i), mi ) maximizes Ri over the set: {(A, mi ) ∈ A × R : pA + mi ≤ K}. Given R ∈ RN , let W(R) be the set of Walrasian allocations from equal income for R. We call W the Walrasian solution from equal income (Varian, 1974). The equivalence of the Walrasian solution from equal income and the no-envy solution is established by Svensson (1983, Theorem 4).12 Since the Walrasian solution from equal income satisfies the axioms listed in Theorem 1, the equivalence can be in fact obtained as a corollary to Theorem 1. Corollary 1. The Walrasian solution from equal income coincides with the no-envy solution. Theorem 1 and Corollary 1 together imply the following axiomatizations of the Walrasian solution from equal income. Corollary 2. The Walrasian solution from equal income is the only solution satisfying equal treatment of equals, Maskin monotonicity, and restricted continuity. The same axiomatization holds if restricted continuity is replaced by neutrality. Let us compare Corollary 2 and results characterizing the same solution in other environments. In economies with divisible goods, Thomson (1979, Theorem 1 .D2) shows that, when preferences satisfy a boundary condition,13 if a subsolution of the Pareto efficient solution satisfies equal treatment of equals, “Nash implementability by a convex mechanism”, and “continuity for linear preferences”, then it is contained in the Walrasian solution from equal income.14 “Nash implementability by a convex mechanism” requires a solution to be implementable in Nash equilibrium by a mechanism whose outcome function satisfies a certain convexity condition. Therefore, it is stronger than the standard Nash implementability notion and hence implies Maskin monotonicity. “Continuity for linear preferences” is an exact counterpart of our restricted continuity to linear preferences. In economies with divisible goods, Nagahisa and Suh (1995, Theorem 2) show that, when preferences satisfy the boundary condition and twice differentiability, the Walrasian solution from equal income is the only subsolution of the Pareto efficient solution that satisfies equal treatment of equals and “local independence”. “Local independence” states that if an allocation is chosen at a preference profile and the preferences change to retain their marginal rates of substitutions at the allocation, then the allocation should be reselected (see, Nagahisa, 1991). Since Maskin monotonic transformations are examples of such a change, “local independence” is stronger than Maskin monotonicity. The most notable difference between Corollary 2 and the results by Thomson and Nagahisa– Suh is the absence of Pareto efficiency in the former. However, in the present model, equal treatment of equals and Maskin monotonicity imply no-envy, which in turn implies Pareto efficiency.
12 Svensson imposes different conditions on preferences as ours. However, the equivalence result does not depend on the conditions. For a given envy-free allocation z, Svensson constructs a price vector p ∈ RA as follows: for each i ∈ N and each A ∈ A, let m(i, z, A) ∈ R be such that zi Ii (A, m(i, z, A)). Then let K ≥ maxi,A m(i, z, A). For each A ∈ A, let pA ≡ K − m(i, z, A) where σ(i) = A. Then one can easily check that p is a price vector that supports z when K is the equal income. 13 The boundary condition: for each x ∈ RL , {y ∈ RL : xI y} ⊆ RL . Without this condition, the Walrasian solution i ++ + ++ from equal income is not Maskin monotonic (see, for example, Jackson, 2001). 14 Thomson’s theorem is based on Hurwicz’s (1979, Theorem 1) characterization of the Walrasian solution in classical private ownership economies.
556
T. Sakai / Journal of Mathematical Economics 43 (2007) 549–563
Therefore, equal treatment of equals and Maskin monotonicity imply Pareto efficiency. Taking this fact into account, the axioms in Corollary 2 parallel the axioms in the results by Thomson and Nagahisa–Suh. Remark 1. Thomson and Nagahisa–Suh use no-envy instead of equal treatment of equals. The relaxation of no-envy to equal treatment of equals is obtained from observations by Geanakoplos and Nalebuff (1988), Fleurbaey and Maniquet (1997). 3.4. Further results on Maskin monotonic solutions Tadenuma and Thomson (1995) study the following independence property. Complete indifference: For each R ∈ RN and each z ∈ Z, if for each i, j ∈ N, zi Ii zj , then z ∈ ψ(R). Tadenuma and Thomson show that, when there is only one real object, any solution satisfying Maskin monotonicity and complete indifference contains the no-envy solution. As Thomson (2005b, Chapter 10.8) suggests, this result holds if there are more than one real object. We remark that this result still holds even if monotonicity of preferences in ranking objects is further imposed: Proposition 1. If a solution satisfies Maskin monotonicity and complete indifference, then it is a supersolution of the no-envy solution. If the solution also satisfies equal treatment of equals, then it in fact coincides with the no-envy solution. Proof. The first part can be shown by the same way as in the result by Tadenuma and Thomson, so omitted. The second part follows from the first part and the relation that any solution satisfying equal treatment of equals and Maskin monotonicity is contained in the no-envy solution (Lemma 3). An allocation z ∈ Z is egalitarian-equivalent for R ∈ RN if there exists z0 ∈ A × R such that for each i ∈ N, zi Ii z0 (Pazner and Schmeidler, 1978). Let E(R) be the set of egalitarian-equivalent allocations. We call E the egalitarian-equivalent solution. We first explain that, when = ∅, no ψ ⊆ E is Maskin monotonic. Note that any ψ ⊆ E satisfies equal treatment of equals. Since equal treatment of equals and Maskin monotonicity imply no-envy, if ψ ⊆ E is Maskin monotonic, then ψ ⊆ F . However, Thomson (1990) shows that, when |A| ≥ 3 and = ∅, there is R ∈ RN such that E(R) ∩ F (R) = ∅. These arguments imply that, when |A| ≥ 3 and = ∅, no ψ ⊆ E is Maskin mootonic. Also, Ohseto (2004, Theorem 5) shows that, when |A| = 2 and = ∅, no ψ ⊆ E is Maskin mootonic.15 Thus, we can conclude that if = ∅, no ψ ⊆ E is Maskin monotonic. We next slightly generalize this result: Even if = ∅, no ψ ⊆ E is Maskin monotonic. Proposition 2. No subsolution of the egalitarian-equivalent solution is Maskin monotonic. Proof. Suppose, by contradiction, that there exists ψ ⊆ E that is Maskin monotonic. Let R ∈ RN be such that for each i, j ∈ N, Ri = Rj . Let z ≡ (σ, m) ∈ ψ(R). Then for each i, j ∈ N, zi Ii zj . Let ε > 0. For each i ∈ N, let Rεi be such that zi Iiε (σ(j), mj + ε) for each j ∈ N with σ(j) = σ(i). 15 Ohseto shows his result on preference domains such that the number of common preferences of agents is at least three. We do not deal with such finitely restricted preference domains.
T. Sakai / Journal of Mathematical Economics 43 (2007) 549–563
557
Let Rε ≡ (Rεi )i∈N . When ε is sufficiently small, Rε ∈ RN . By Maskin monotonicity, z ∈ ψ(Rε ). However, z is not egalitarian-equivalent for Rε . Ohseto (2004, Footnote 9) says that egalitarian-equivalence and Maskin monotonicity seem to be incompatible in general. Proposition 2 confirms that this view is true in our general model for indivisible objects allocation. 4. Concluding remarks Our theorem states that, for a quite large class of problems, the no-envy solution is essentially the only Nash implementable solution satisfying equal treatment of equals. Hence, the next involves designing a “natural” mechanism that implements the no-envy solution.16 Such a mechanism is ¯ designed by Azacis (2004) when preferences are quasi-linear and extending his mechanism to nonquasi-linear preferences is of interest. Svensson (1991) constructs a price-reporting mechanism that implements the Walrasian solution in a market model. Since the no-envy solution coincides with the Walrasian solution from equal income in the present model, the idea of his mechanism may help design a mechanism implementing the no-envy solution in our model. Saijo et al. (1996) and Thomson (2005a) design mechanisms implementing the no-envy solution in economies with infinitely divisible goods.17 Designing counterparts of their mechanisms is also still open in our model. Acknowledgements This paper is based on Chapter 3 of my dissertation submitted to the University of Rochester. I am grateful to my advisor William Thomson for his continuous encouragement and support. I also thank an anonymous referee for his/her constructive and detailed referee report and Olivier Bochet, John Duggan, Norio Takeoka as well as participants of the seminars at Kobe University, Kyoto University, Tsukuba University, University of Namur, University of Rochester, Yokohama National University, and the Seventh International Meeting of the Society for Social Choice and Welfare at Osaka University for their helpful suggestions. Appendix A. Proof of Theorem 1 We first show that the space of preferences we consider satisfies a certain richness condition, called “monotonic closedness”. This condition implies that any solution satisfying equal treatment of equals and Maskin monotonicity is a subsolution of the no-envy solution. Then, we show that if a subsolution of the no-envy solution satisfies Maskin monotonicity and either restricted continuity or neutrality, then it in fact coincides with the no-envy solution. Since we identify each member of the partition, Al ∈ A, with an object contained in Al , al ∈ Al , we write al to denote Al when there is no risk of confusion. L An L-tuple of consumption bundles (al , mal )L l=1 ∈ ×l=1 ({al } × R) is -consistent if for each l, l such that l l , mal < mal . Given two objects a, b ∈ A, a is -connected to b if there are objects b1 , b2 , . . . , bk ∈ A such that a = b1 , b = bk , and for each k = 1, 2, . . . , k − 1, either
16 17
For critique to “unnatural” mechanisms, see Saijo et al. (1996) and Jackson (2001). Thomson’s mechanism is designed for more general economic environments.
558
T. Sakai / Journal of Mathematical Economics 43 (2007) 549–563
bk bk +1 or bk +1 bk . Given objects a, b ∈ A such that a b, we write a 0 b if there exists no c ∈ A such that a c and c b. L Lemma 1. If two -consistent tuples (al , mal )L l=1 , (al , mal )l=1 are such that for each l, ml > ml , then there exists Ri ∈ R such that
(a1 , ma1 )Ii (a2 , ma2 )Ii · · · Ii (aL , maL ) and (a1 , ma1 )Ii (a2 , ma2 )Ii · · · Ii (aL , maL ). Proof. Let X ≡ {X1 , X2 , . . . , XK } be the partition of A such that (i) for each k and each a, b ∈ Xk , a is -connected to b; (ii) for each k, k with k = k , each a ∈ Xk , and each b ∈ Xk , a is not connected to b. Note that X is uniquely determined. ¯ i: Step 1 Defining R Substep 1-1 Let k ∈ {1, 2, . . . , K}. For each a, b ∈ Xk such that a 0 b, let ha,b : [ma , ma ] → [mb , mb ] be a strictly increasing bijection such that for each ma ∈ [ma , ma ], ma < ha,b (ma ). Such a bijection exists since ma < mb and ma < mb . Then, for each ma ∈ [ma , ma ], let (a, ma )I¯i (b, ha,b (ma )). Substep 1-2 For each k, fix bk ∈ Xk . Let k, k be such that k < k . Let hbk ,bk : [m bk , m bk ] → [m bk , m bk ] be a strictly increasing bijection, and for each mbk ∈ [m bk , mbk ], let (bk , mbk )I¯i (bk , hbk ,bk (mbk )). Substep 1-3 For each a ∈ A and each ma , ma ∈ R with ma < ma , let (a, ma )P¯ i (a, ma ). Substep 1-4 For each ε > 0, let (a1 , ma1 − ε)I¯i (a2 , ma2 − ε)I¯i · · · I¯i (aL , maL − ε) and (a1 , ma1 + ε)I¯i (a2 , ma2 + ε)I¯i · · · I¯i (aL , maL + ε). ¯ i , i.e., for each (a, ma ), (b, mb ) ∈ A × R, Step 2 Defining Ri : Let Ri be the transitive closure of R (a, ma )Ri (b, mb ) if and only if there are bundles: (b1 , mb1 ), (b2 , mb2 ), . . . , (bK , mbK ) such that (i) (a, ma ) = (b1 , mb1 ), (b, mb ) = (bk , mbk ), and (ii) for each k ≤ K − 1: ¯ i (bk+1 , mbk+1 ), (bk , mbk )R where (a, ma )Pi (b, mb ) holding if and only if for some k ≤ K − 1, (bk , mbk )P¯ i (bk+1 , mbk+1 ). Step 3 Concluding: Obviously, Ri is transitive and complete. By Substep 1-3, Ri satisfies money monotonicity. By Substeps 1-1, -2 and -4, Ri satisfies finiteness. By Substep 1-1, Ri satisfies -object monotonicity. Hence, Ri ∈ R.
T. Sakai / Journal of Mathematical Economics 43 (2007) 549–563
559
Geanakoplos and Nalebuff (1988) essentially show that in economies with divisible goods, if a preference space satisfies certain conditions, then equal treatment of equals and Maskin monotonicity together imply no-envy.18 Fleurbaey and Maniquet (1997) show that in an abstract social choice environment, if a preference space satisfies the following condition, the same relation holds. Monotonic closedness: For each i, j ∈ N, each Ri , Rj ∈ R, and each z ∈ Z with zj Pi zi , there is Ri ∈ R such that (1) zj Pi zi , (2) {w ∈ Z : zi Ri wi } ⊆ {w ∈ Z : zi Ri wi }, (3) {w ∈ Z : zj Rj wj } ⊆ {w ∈ Z : zj Ri wj }. Lemma 2. The preference space R is monotonically closed. Proof. Let i, j ∈ N, Ri , Rj ∈ R, and z ∈ Z be such that zj Pi zi . Write zi = (a, ma ) and zj = (b, mb ). For each c ∈ A, let m(i, c), m(j, c) ∈ R be such that (a, ma )Ii (c, m(i, c))
and
(b, mb )Ij (c, m(j, c)).
Consider the tuple: ((b, mb ), {(c, max{m(i, c), m(j, c)})}c=b ) ∈ ×L l=1 ({al } × R). Given c, d ∈ A, if c d, then m(i, c) < m(i, d) and m(j, c) < m(j, d), hence max{m(i, c), m(j, c)} < max{m(i, d), m(j, d)}. Thus, this tuple is -consistent. Since the set of -consistent tuples is open, there is a sufficiently small ε > 0 such that the tuple: ((b, mb ), {(c, ε + max{m(i, c), m(j, c)})}c=b ) ∈ ×L l=1 ({al } × R) is -consistent. Clearly, the tuple ((a, ma ), {(c, m(i, c))}c=a ) ∈ ×L l=1 ({al } × R) is -consistent as well. Therefore, by Lemma 1, there exists Ri ∈ R such that both tuples: ((a, ma ), {(c, m(i, c))}c=a ) ∈ ×L l=1 ({al } × R) and ((b, mb ), {(c, ε + max{m(i, c), m(j, c)})}c=b ) ∈ ×L l=1 ({al } × R) constitute indifference curves. Clearly, Ri satisfies the desired properties.
Lemma 3. If a solution satisfies equal treatment of equals and Maskin monotonicity, then it is a subsolution of the no-envy solution.
18 They also impose Pareto efficiency and state the result for two agents. However, as Moulin (1993), Fleurbaey and Maniquet (1997) point out, Pareto efficiency is redundant and their result holds for an arbitrary number of agents.
560
T. Sakai / Journal of Mathematical Economics 43 (2007) 549–563
Proof. The proof below is due to Geanakoplos and Nalebuff (1988, Proposition 4) and Fleurbaey and Maniquet (1997, Theorem 1). Suppose, by contradiction, that there are R ∈ RN and z ∈ ψ(R) such that for some i, j ∈ N, zj Pi zi . By monotonic closedness, there are Ri , Rj ∈ R such that Ri = Rj , zj Pi zi , {w ∈ Z : zi Ri wi } ⊆ {w ∈ Z : zi Ri wi } and {w ∈ Z : zj Rj wj } ⊆ {w ∈ Z : zj Rj wj }. By Maskin monotonicity, z ∈ ψ(Ri , Rj , RN\{i,j} ). However, since Ri = Rj and zj Pi zi , this contradicts equal treatment of equals. Lemma 4. If a subsolution of the no-envy solution satisfies Maskin monotonicity and restricted continuity, then it coincides with the no-envy solution. Proof. Let ψ be a subsolution of the no-envy solution satisfying Maskin monotonicity and restricted continuity. Let R ∈ RN and z ∈ F (R). Since z ∈ F (R), for each i, j ∈ N, if σ(i) = σ(j), then mi = mj . Hence, for each a ∈ A, we can write mσ(i) instead of mi when a = σ(i). Let R be the profile of quasi-linear preferences such that for each i, j ∈ N, Ri = Rj and zi Ii zj . By construction, R ∈ QN . For each ε > 0, each i ∈ N, and each ν ∈ N, let Rνi be the quasi-linear preference such that for each j = i with σ(j) = σ(i), ε ν σ(i), mσ(i) − I zj . ν i Given ν ∈ N, let Rν ≡ (Rν1 , Rν2 , . . . , Rνn ) and zν ≡ (σ ν , mν ) ∈ ψ(Rν ). When ε is sufficiently small, for each ν ∈ N, Rν ∈ RN . Hereafter, let ε > 0 be such a small number. Since zν ∈ F (Rν ), for each i, j ∈ N, if σ ν (i) = σ ν (j), then mνi = mνj . Hence, for each ν ∈ N and each a ∈ A, we can write mνσ(i) instead of mνi when a = σ(i). Claim 1. For each i ∈ N, σ ν (i) = σ(i). Suppose, by contradiction, that for some i, j ∈ N, σ ν (i) = σ(j) = σ(i). Without loss of generality, assume σ ν (j) = σ(j). Case 1 (mνσ ν (i) ≥ mσ ν (i) ). Let δ ≡ mνσ ν (i) − mσ ν (i) ≥ 0. By definition of Rν , for each c = σ(i), σ ν (i), ε zνi = (σ ν (i), mσ ν (i) + δ)Iiν σ(i), mσ(i) − + δ Iiν (c, mc + δ), ν ε ε ν ν ν zi = (σ (i), mσ ν (i) + δ)Ij σ(i), mσ(i) + + δ Ijν c, mc + + δ . ν ν ν ν ν ν Since zj Rj zi and σ (j) = σ(j), by (A.2),
(A.1) (A.2)
ε + δ. ν Then by (A.1), zνj Piν zνi , a contradiction to zν ∈ ψ(Rν ). mνσ ν (j) ≥ mσ ν (j) +
Case 2 (mνσ ν (i) < mσ ν (i) ). Then there exists k ∈ N such that mνσ ν (k) > mσ ν (k) . By definition of Rν , zνk = (σ ν (k), mνσ ν (k) )Piν (σ ν (k), mσ ν (k) )Rνi (σ ν (i), mσ ν (i) )Piν (σ ν (i), mνσ ν (i) ) = zνi , where (σ ν (k), mσ ν (k) )Piν (σ ν (i), mσ ν (i) ) holding iff σ ν (k) = σ(i). Thus, agent i envies agent k, a contradiction to zν ∈ ψ(Rν ).
T. Sakai / Journal of Mathematical Economics 43 (2007) 549–563
561
Claim 2. For each i ∈ N, mνσ(i) ≤ mσ(i) + (ε/ν). Suppose, by contradiction, that for some i ∈ N, mνσ(i) > mσ(i) + (ε/ν). Then, there is j ∈ N such that mνσ(j) < mσ(j) . Since zνj Rνj zνi , we have σ(j) = σ(i). By definition of Rν , ε ν I (σ(j), mσ(j) )Pjν (σ(j), mνσ(j) ) = zνj . zνi = (σ(i), mνσ(i) )Pjν σ(i), mσ(i) + ν j Therefore, agent j envies agent i, a contradiction to zν ∈ ψ(Rν ). Claim 3. For each i ∈ N, mσ(i) − νε ≤ mνσ(i) . Suppose, by contradiction, that for some i ∈ N, mσ(i) − νε > mνσ(i) . Then, there is j ∈ N such that mνσ(j) > mσ(j) . Since zνi Rνi zνj , we have σ(j) = σ(i). By definition of Rν , ε ν Pi (σ(i), mνσ(i) ) = zνi . zνj = (σ(j), mνσ(j) )Piν (σ(j), mσ(j) )Iiν σ(i), mσ(i) − ν Therefore, agent i envies agent j, a contradiction to zν ∈ ψ(Rν ). Thus, we have established that for each ν ∈ N and each i ∈ N, mσ(i) − (ε/ν) ≤ mνσ(i) ≤ mσ(i) + (ε/ν). This implies that the sequence {zν } converges to z. Clearly, the sequence {Rν } also converges to R . Thus, by restricted continuity, z ∈ ψ(R ). Since R ∈ MT(R , z), by Maskin monotonicity, z ∈ ψ(R). Thus, ψ(R) = F (R). The following is a weaker version of a decomposition lemma in Alkan et al. (1991, Lemma 3). Lemma 5. Let R ∈ RN and z, z ∈ F (R). If for each i, j ∈ N, zi Ii zi , then z R z . Proof. Let R ∈ RN and z ≡ (σ, m), z ≡ (σ , m ) ∈ F (R) be such that for each i ∈ N, zi Ii zi . Let a ∈ A. It suffices to show that ma = ma . Suppose that ma > ma . Let i ∈ N be such that σ (i) = a. Then, since z ∈ F (R), zi Ri (a, ma )Pi (a, ma ) = (σ (i), m σ (i) ) = zi , a contradiction. Next, suppose that ma < ma . Let i ∈ N be such that σ(i) = a. Then, since z ∈ F (R), zi Ri (a, ma )Pi (a, ma ) = (σ(i), mσ(i) ) = zi , a contradiction. Thus, ma = ma .
Lemma 6. If a subsolution of the no-envy solution satisfies Maskin monotonicity and neutrality, then it coincides with the no-envy solution. Proof. Let ψ be a subsolution of the no-envy solution satisfying Maskin monotonicity and neutrality. Let R ∈ RN and z ∈ F (R). Let R be the profile of quasi-linear preferences such that for each i, j ∈ N, Ri = Rj and zi I i zj . By construction, R ∈ QN . Let w ∈ ψ(R ). Let us show w R z. Since z ∈ P(R ), if there is i ∈ N with wi Pi zi , then there is j ∈ N with zj Pj wj , and then wi Pi zi Ii zj Pi wj , which contradicts equal treatment of equals. Hence, for each i ∈ N, zi Ii wi . Since z, w ∈ F (R ), by Lemma 5, z R w. By neutrality, z ∈ ψ(R ). Since R ∈ MT(R , z), by Maskin monotonicity, z ∈ ψ(R). Proof of Theorem 1. We omit the easy proof that the no-envy solution satisfies equal treatment of equals, Maskin monotonicity, restricted continuity, and neutrality. Let ψ be a solution satisfying equal treatment of equals and Maskin monotonicity. By Lemma 3, it is a subsolution of the no-envy solution. If ψ further satisfies restricted continuity, then by Lemma 4, it coincides with the no-envy solution. On the other hand, if ψ satisfies neutrality, then by Lemma 6, it coincides with the no-envy solution.
562
T. Sakai / Journal of Mathematical Economics 43 (2007) 549–563
Appendix B. Continuity properties We briefly mention relations between restricted continuity and other topological properties. In the present model, the Kannai topology is the smallest topology such that the set: {((a, mi ), (b, mi ), Ri ) ∈ (A × R) × (A × R) × R : (a, mi )Pi (b, mi )} is open with respect to the product topology. Here, this topology coincides with other standard topologies such as the closed convergence topology (e.g., Hildenbrand, 1974; Redekop, 1993). We refer to Kannai (1970) and Redekop (1993) for interpretations and properties of this topology. We list several facts on topological properties with respect to the Kannai topology.19 • • • • •
A solution ψ satisfies restricted continuity if and only if the subgraph of ψ on QN is closed. If a solution has a closed graph, then it satisfies restricted continuity. If a closed-valued solution is upper hemicontinuous, then its graph is closed. The no-envy solution is compact-valued and upper hemicontinuous, so its graph is closed. The no-envy solution is the only closed-valued solution satisfying equal treatment of equals, Maskin monotonicity, and upper hemicontinuity. Alternately, it is the only solution satisfying equal treatment of equals and Maskin monotonicity whose graph is closed.
Therefore, we could proceed the present study based on Kannai continuity properties. However, we chose to use restricted continuity because of its mildness and simplicity. References Alkan, A., Demange, G., Gale, D., 1991. Fair allocation of indivisible goods and criteria of justice. Econometrica 59, 1023–1039. ¯ Azacis, H., 2004. Double Implementation in a Market for Indivisible Goods with a Price Constraint. Universitat Aut`onoma de Barcelona, Mimeo. Bevia, C., 1996. Identical preferences lower bound solution and consistency in economies with indivisible goods. Social Choice and Welfare 13, 113–126. Bochet, O., Sakai, T., 2007. Strategic manipulations of multi-valued solutions in economies with indivisibilities. Mathematical Social Sciences 53, 53–68. Chun, Y., 2006a. A pessimistic approach to the queueing problem. Mathematical Social Sciences 51, 171–181. Chun, Y., 2006b. No-envy in queueing problems. Economic Theory 29, 151–162. Fleurbaey, M., Maniquet, F., 1997. Implementability and horizontal equity imply no-envy. Econometrica 65, 1215–1219. Foley, D., 1967. Resource allocation and the public sector. Yale Economic Essays 7, 45–98. Geanakoplos, J., Nalebuff, B., 1988. On a fundamental conflict between equity and efficiency. Yale University/Princeton University, Mimeo. Hildenbrand, W., 1974. Core and Equilibria of a Large Economy. Princeton University Press, Princeton, NJ. Hurwicz, L., 1979. On allocations attainable through Nash equilibria. In: Laffont, J.-J. (Ed.), Aggregation and Revealed Preferences. North-Holland Publishing Co. Jackson, M., 2001. A crash course in implementation theory. Social Choice and Welfare 18, 655–708. Kaneko, M., 1983. Housing markets with indivisibilities. Journal of Urban Economics 13, 22–50. Kannai, Y., 1970. Continuity properties of the core of a market. Econometrica 38, 791–815. Maniquet, F., 2003. A characterization of the Shapley value in queueing problems. Journal of Economic Theory 109, 90–103. Maskin, E., 1999. Nash equilibrium and welfare optimality. Review of Economic Studies 66, 23–38.
19
Proofs are available from the author upon request.
T. Sakai / Journal of Mathematical Economics 43 (2007) 549–563
563
Moulin, H., 1993. On the fair and coalitionproof allocation of private goods. In: Binmore, K., Kirman, A., Tani, P. (Eds.), Frontiers of Game Theory. The MIT Press, Cambridge, MA. Nagahisa, R.-I., 1991. A local independence condition for characterization of Walrasian allocation rule. Journal of Economic Theory 54, 106–123. Nagahisa, R.-I., Suh, S.-C., 1995. A characterization of the Walras rule. Social Choice and Welfare 12, 335–352 (Reprinted in: Walker, D., 2001. The Legacy of Leon Walras. Edward Elgar Publishing). Ohseto, S., 2001. Strategy-proof and efficient allocation of an indivisible good on finitely restricted domains. International Journal of Game Theory 29, 365–374. Ohseto, S., 2004. Implementing egalitarian-equivalent allocation of indivisible goods on restricted domains. Economic Theory 23, 659–670. Pazner, E., Schmeidler, D., 1978. Egalitarian equivalent allocations: a new concept of economic equity. Quarterly Journal of Economics 92, 671–683. Quinzii, M., 1984. Core and competitive equilibria with indivisibilities. International Journal of Game Theory 13, 41–60. Redekop, J., 1993. The questionnaire topology on some spaces of economic preferences. Journal of Mathematical Economics 22, 479–494. Saijo, T., Tatamitani, Y., Yamato, T., 1996. Toward natural implementation. International Economic Review 37, 941–980. Schummer, J., 2000. Eliciting preferences to assign positions and compensation. Games and Economic Behavior 30, 293–318. Svensson, L.-G., 1983. Large indivisibilities: an analysis with respect to price equilibrium and fairness. Econometrica 51, 939–954. Svensson, L.-G., 1991. Nash implementation of competitive equilibria in a model with indivisible goods. Econometrica 59, 869–877. Svensson, L.-G., Larsson, B., 2002. Strategy-proof and nonbossy allocation of indivisible goods and money. Economic Theory 20, 483–502. Tadenuma, K., Thomson, W., 1991. No-envy and consistency in economies with indivisible goods. Econometrica 59, 1755–1767. Tadenuma, K., Thomson, W., 1993. The fair allocation of an indivisible good when monetary compensations are possible. Mathematical Social Sciences 25, 117–132. Tadenuma, K., Thomson, W., 1995. Games of fair division. Games and Economic Behavior 9, 191–204. Thomson, W., 1979. Comment: on allocations attainable through Nash equilibria. In: Laffont, J.-J. (Ed.), Aggregation and Revealed Preferences. North-Holland Publishing Company. Thomson, W., 1990. On the non-existence of envy-free and egalitarian-equivalent allocations in economies with indivisibilities. Economics Letters 34, 227–229. Thomson, W., 2005a. Divide and permute. Games and Economic Behavior 52, 186–200. Thomson, W., 2005b. Theory of Fair Allocation. University of Rochester, Mimeo. Tinbergen, J., 1953. Redelijke Inkomensverdeling, 2nd ed. N.V. Gulden Press. Varian, H., 1974. Equity, envy, and efficiency. Journal of Economic Theory 9, 63–91.