alpha-maxmin solutions to fair division problems and the structure of the set of Pareto utility profiles

A simple proof of the equivalence of Pareto optimality plus positiveness and α-maxmin optimality, dispensing with the assumption of closedness of the utility possibility set, is given. Also the structure of the set of Pareto optimal utility profiles is studied. © 2008 Elsevier B.V. All rights reserved. The main results of the recent paper by Sagara ‘‘A characterization of α-maxmin solutions of fair division problems’’ published in Mathematical Social Sciences 55 (2008), 273–280, are concerned with the important issue of fair and efficient division of a measurable space among finitely many individuals. One of these results is concerned with the equivalence of Pareto optimality plus positiveness and α-maxmin optimality, and assumes the closedness of the utility possibility set. We suggest here a simple proof of this equivalence that dispenseswith the closedness assumption. Sagara studies also the structure of the Pareto optimal utility profiles’ set, UP . He shows that if the utility possibility set, U , is closed then UP is homeomorphic with the standard closed simplex,∆, in Rn. Here n is the number of individuals. We give here a short proof of this result andmake further observations on the structure of sets UP and U . ∗ Tel.: +90 312 266 4216; fax: +90 312 266 5140. E-mail address: [email protected]. 0165-4896/$ – see front matter© 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.mathsocsci.2008.11.003 280 F. Hüsseinov / Mathematical Social Sciences 57 (2009) 279–281 First we recall some definitions. A partition scheme of a measurable space (Ω,F ) consists of the non-atomic probability measures μ1, . . . , μn on this space and the set functions f1, . . . , fn that map F into [0, 1], such that fi(∅) = 0 and fi(Ω) = 1, for every i ∈ N = {1, . . . , n}. A partition ofΩ is an ordered n-tuple of disjoint sets in F whose union isΩ . Denote by P the set of all partitions ofΩ . A partition (A1, . . . , An) is positive if μi(Ai) > 0 for all i ∈ N. A partition (A1, . . . , An) isα-maxmin optimal (forα ∈ ∆̊—the interior of∆) if it solves the problem: max{mini∈N α i fi(Ai) | (A1, . . . , An) ∈ P }. A partition (A1, . . . , An) is α-equitable if α −1 i fi(Ai) = α j fj(Aj) for all i, j ∈ N. A set function fi is: (a) μi-continuous from below if B1 ⊂ B2 ⊂ · · · ⊂ B and μi(B \ ∪Bk) = 0 imply fi(Bk)→ fi(B), (b) strictly μi-monotone if A ⊂ B and μi(A) < μi(B) imply fi(A) < fi(B). Property (a) is equivalent to the following two properties: B1 ⊂ B2 ⊂ · · · imply fi(Bk)→ fi(∪Bk), and for A, B ∈ F , μi(A∆B) = 0, where A∆B = (A ∪ B) \ (A ∩ B), implies fi(A) = fi(B). Here fi(A) is theworth of set A to player i andmeasuresμi are technical tools that are used to define possible utilities. The utility possibility set is defined as U = {(x1, . . . , xn) ∈ [0, 1]n : ∃(A1, . . . , An) ∈ P such that xi ≤ fi(Ai), i ∈ N}. The set of all Pareto optimal utility profiles is denoted as UP. The Pareto optimal and the weak Pareto optimal partitions and utility profiles are defined as usual through the concepts of Pareto improvement and strict Pareto improvement. When measures μ1, . . . , μn are absolutely continuous with respect to each other and the assumptions (a) and (b) above are satisfied, a standard argument used in the classical exchangemodels shows that the notions of weak and strong Pareto optimality coincide (Lemma 4.1 in Sagara’s work). Theorem. Let {(Ω,F ), μi, fi, i ∈ N} be a partition scheme such that fi isμi-continuous from below and strictly μi-monotone for each i ∈ N, and let measures μ1, . . . , μn be absolutely continuous with respect to each other. Then the following hold. (i) For α ∈ ∆̊ a partition is α-maxmin optimal if and only if it is Pareto optimal and α-equitable. (ii) A partition is Pareto optimal and positive if and only if it is α-maxmin optimal for some α ∈ ∆̊. Proof. Part (i) is proved in Sagara (2008). We prove (ii). Let partition P = (A1, . . . , An) be Pareto optimal and positive. Setting αi = ∑ k∈N [fk(Ak)] −1 [fi(Ai)]−1 for i ∈ N , we will have α ∈ ∆̊ and α i fi(Ai) = α −1 j fj(Aj) for all i, j ∈ N. (1) Now P is Pareto optimal and α-equitable for α ∈ ∆̊. By part (i) P is α-maxmin optimal. Conversely, let partition P = (A1, . . . , An) be α-maxmin optimal for α ∈ ∆̊. By part (i) then P is Pareto optimal and α-equitable, that is Eq. (1) are satisfied. Now if fj(Aj) = 0 for some j ∈ N , it follows from Eq. (1) that fi(Ai) = 0 for all i ∈ N . From properties of functions fi it follows that μi(Ai) = 0. By absolute continuity of measures μi, i ∈ N we have μi(Aj) = 0 for all i, j ∈ N . Hence μ1(Ω) = 0, which contradicts the assumption μ1(Ω) = 1. Remark. Obviously, for any measures on (Ω,F ) which are absolutely continuous with respect to each other each of properties (a) and (b) is satisfied for one of them if and only if it is satisfied for the other. Therefore the assumption of absolute continuity of measures with respect to each other made in the theorem simplifies the partition scheme to {(Ω,F , μ), fi, i ∈ N},where, for example,μ = μ1. In the remaining part of this note we study the structure of the utility possibility set, U , and Pareto optimal utility profiles’ set, UP . We start with giving a short proof of Sagara’s Theorem 4.1 concerned with the structure of UP. For vectors x, y ∈ Rn we will write x > y if xi > yi for all i ∈ N . Define a mapping h : ∆ → Rn as h(x) = ρ(x)x for x ∈ ∆, where ρ is defined as ρ(x) = sup{r ≥ 0 | rx ∈ U}. Further we will assume that all assumptions of the above theorem are satisfied. F. Hüsseinov / Mathematical Social Sciences 57 (2009) 279–281 281 Proposition 1. If U is closed, then h is a homeomorphism between∆ and UP. Proof. Since U is closed, h(x) ∈ U . Since a weak Pareto optimal utility profile is Pareto optimal h(x) 6∈ UP would imply that there exists u ∈ U such that u > h(x). Then by comprehensiveness of U, h(x) would be a relative interior point of U , which contradicts its definition. Since for each x ∈ U \ UP there exists y ∈ U such that y > x, it follows that U \ UP is a relative open subset of [0, 1]n, and hence UP is a closed set. Since measures μ1, . . . , μn are absolutely continuous with respect to each other U contains a strictly positive vector. Hence ρ(x) > 0 for all x ∈ ∆. It follows easily from this that h is one-to-one. Since∆ is compact to complete the proof it suffices to show that h is continuous. This will follow if we show that ρ is a continuous function. Upper semicontinuity of ρ. Assume xk ∈ ∆, xk → x and lim ρ(xk) > ρ(x). Since ρ is bounded there exists a subsequence {yk} of sequence {xk} such that ρ(yk)→ ρ0 > ρ(x). Closedness of U implies that ρ0x ∈ U . But ρ0 > ρ(x) then would imply that h(x) = ρ(x)x is not a Pareto optimal utility profile. This contradicts the definition of function h. Lower semicontinuity of ρ. Assume xk ∈ ∆, xk → x and lim ρ(xk) < ρ(x). Then there exists a subsequence {zk} of sequence {xk}, such that ρ(zk)→ ρ1 < ρ(x). Thus {ρ(zk)zk} is a sequence in UP with the limit ρ1x not in UP . This contradicts the closedness of UP. Next we show that the utility possibility set U is homeomorphic to D = {x ∈ Rn : xi ≥ 0, i ∈ N and |x| ≤ 1}, where |x| = x1 + · · · + xn, and hence to the standard closed simplex in Rn. Proposition 2. If U is closed then it is homeomorphic to D. Proof. By Proposition 1 mapping h : ∆ → UP is a homeomorphism. Define a mapping H : U → D by setting H(x) =  x |h( x |x| )| for x ∈ U \ {0},