Utilitarian resource allocation

Transcription

1 Utilitarian resource allocation Albin Erlanson and Karol Szwagrzak March 19, 2014 Extended Abstract Keywords: Claims; Indivisible objects; Consistency; Resource-monotonicity JEL classification: D70, D63, D61 A standard claims problem consists of n persons with individual claims c i on an endowment E. The sum of the claims is greater than the endowment and we have to decide who gets what. This model was introduced by O Neill (1982) and it was the start of a research field. The question addressed is how to adjudicate these conflicting claims. 1 One interpretation of the model is that a firm goes bankrupt and the assets of the firm must be divided among the claimants. Another interpretation is when a public project needs to be financed. We can then think of a claim as persons income and the question is how much each person should pay. There are many other scenarios which fit this simple but useful model. In this paper we enrich the basic model by allowing for multiple firms. Thus, each person has a list of claims, one for each firm. Similarly to the standard claims problem there is not enough of the endowment to meet the claims, and the question is how to divide the endowment of each firm among the claimants. We will call each firm a resource. These resources can be either divisible or indivisible. Preferences are monotone for each agent. Hence, there is no need to incorporate preferences into the model, and a multidimensional claims problem is a resource allocation problem without any preference information. In Theorem 1 we show that utilitarian resource allocation is a consequence of imposing two basic principles of distributive justice: consistency and resource-monotonicity, Bonn University and Lund University. aerlanso@uni-bonn.de University of Southern Denmark. karol.s.flores@gmail.com 1 See Thomson (2003) for a survey on the extensive literature on claims problems. 1

2 together with an independence of irrelevant alternatives property. Note that a utilitarian rule need not to treat all agents symmetric. These rules are an example of asymmetric rules for the claims problem, and they are a generalization of the class of symmetric parametric rules (Young, 1987). In a recent paper Stovall (2014) characterizes the class of asymmetric parametric rules for the standard claims problem with one endowment. Our paper is a contribution to this recent interest in asymmetric rules for claims problems. For our second result, contained in Theorem 2, we assume that all resources are divisible. We replace the property of independence of irrelevant alternatives by continuity and symmetry and obtain a narrower class of utilitarian rules, generalizing the class of parametric rules (Young, 1987) to the allocation of multiple divisible resources. In our third and last result, Theorem 3 we show that a rule satisfying consistency, claims-truncation invariance and composition is a weighted rule. This is also a utilitarian rule. But it is fully specified by a list of weights, one for each agent and each resource. Multidimensional claims problems We now introduce the elements of a multidimensional claims problem. 2 A number of divisible and indivisible resources are to be allocated among a group of claimants drawn form the finite set A. Let N denote the subsets of A. The resource kinds that are available in indivisible units are indexed by I while those available in divisible units are indexed by D. Let K denote the union of I and D. Let C R D + Z I + denote the space of possible resource profiles. For every group of agents N N, a (multidimensional) claims problem is the pair (C, E), where C C N and E C are such that, for each resource kind k K, i N Ck i E k. For each N N, let P N denote the claims problems involving the agents in N. An allocation for the claims problem (C, E) P N is a profile z C N such that N z i = E and, for each i N, z i C i. Let Z(C, E) denote the collection of all allocations for claims problem (C, E). An allocation rule, or simply a rule, is a function ϕ recommending allocations for all possible claims problems: for each N N and each (C, E) P N, ϕ(c, E) Z(C, E). 2 The basic mathematical notation is as follows: Let {Y i } i I be a family of sets Y i indexed by I. Let Y I i I Y i. For each y Y I and each J I, we denote by y J the projection of y onto Y J. If x, y R I, then x y means that, for each i I, x i y i. For each i I, e i R I denotes the ith standard basis vector, the vector with a one in the ith coordinate and zeros elsewhere. 2

3 Axioms We look for rules that satisfies desirable axioms. Below is the list of axioms that we consider. Resource monotonicity: For each pair (C, E), ( C, Ē) PN such that C = C and Ē E, ϕ(c, E) ϕ( C, Ē). Consistency: For each pair N, N N such that N N, each (C, E) P N, and each i N, ϕ i (C N, i N ϕ i (C, E)) = ϕ i (C, E). For each (C, E) P N let C E denote the profile in C N such that, for each k K, (C E) k = (min{c k i, E k }) i N. Claims-truncation invariance: For each (C, E) P N, ϕ(c E, E) = ϕ(c, E). Composition: For each (C, E) P N, each E C such that E E, ϕ(c, E) = x implies ϕ(c, E) = x + ϕ(c x, E E). Independence of irrelevant alternatives (IIA): For each pair (C, E), ( C, Ē) P N such that Z(C, E) Z( C, Ē) and ϕ(c, E) Z( C, Ē), ϕ(c, E) = ϕ( C, Ē). The following properties are only meaningful when all resources are perfectly divisible, i.e. when I =. Symmetry: For each (C, E) P N, each pair i, j N, C i = C j implies ϕ j (C, E) = ϕ j (C, E). Continuity: The rule is continuous in each of its arguments. Rules Let U consist of all profiles U {Ui k : i A, k K} such that Ui k : R + R is strictly concave and continuous. A rule ϕ is utilitarian if there is a U U such that, for each N N and each (C, E) P N, ϕ(c, E) = arg max { k K i N U i k (zi k ) : z Z(C, E) }. (1) 3

4 Let ϕ U denote the utilitarian rule specified by U U. When resources are indivisible we need to ensure that the solution to the optimization problem in (1) is integer valued. This is done by specifying additional structure on the selection of functions from U. Let I denote the profile of functions (U i ) i A such that each U i : R R is strictly concave and for each e R + and c R A + arg max { i A U i(z i ) : 0 z c } is a profile of integers. (2) Let U denote the profile of functions in U such that, for each k I, (U k i ) i A is in I. A rule ϕ is utilitarian if there is U U such that ϕ = ϕ U. Note that this definition refines the previous one. When there are only divisible resources they coincide. Results Before showing our results we characterize the structure of the profiles of functions in U. In order to do this we introduce a class of matrices and reformulate the optimization program in (1) as an integer linear program. Let M denote the class of matrices m {m ik R + : i A; k R + } such that: (i) for each i A, m i1 > m i2 > ; and (ii) all non-zero entries in matrix m are distinct. Lemma 1 (Linear approximation). If (U i ) i A I (m M), c R A, then there is m M ((U i ) A I) such that, for each e R +, if x = arg max { i A U i(z i ) : i A z i = e, 0 z c } y = arg max { i A ci k=1 m ikz ik : i A then, for each i A, x i = c i k=1 y ik and y {0, 1} A c i. and ci k=1 z ik = e, 0 z ik 1 } Except of providing an explicit description of utilitarian rules for the case of indivisible resources Lemma 1 is also used to prove that the utilitarian rule are well defined. Now we are ready to state the main results of the paper. Theorem 1. A rule is consistent, resource-monotonic, and satisfies IIA if and only if it is utilitarian. In our next result we characterizes a subclass of the utilitarian rules: the parametric rules. A rule ϕ is parametric if, for each k K, there is a function H k : R + R + R such that, for each c R +, H k (c, ) is strictly concave, for each N N and each (C, E) C N C, 4

5 ϕ(c, E) = arg max{ k K i N Hk (Ci k, zi k ) : z Z(C, E)}. The parametric rules for the standard claims problem were characterized in Young (1987) on the basis of symmetry, pairwise consistency and continuity. Theorem 2. Suppose that resources are divisible. A rule is consistent, continuous, and symmetric if and only if it is parametric. Theorem 2 generalizes Young s result to a multidimensional setting. Suppose now that all resources are indivisible. We will introduce a class of weighted rules. First we need some notation to define the rules. Let W denote the collection of all profiles w such that i. for each i A and k K, w k i > 0; and ii. for each k K and each pair i, j A, w k i w k j. A rule ϕ is weighted if there is a profile of weights w W such that and, for each N N and each (C, E) P N, ϕ(c, E) = arg max{ k I i N wk i zi k : z Z(C, E)}. Now we can state our last result. Theorem 3. Suppose that resources are indivisible. A rule is consistent, claimstruncation invariant, and satisfies composition if and only if it is weighted. References O Neill, B. (1982). A problem of rights arbitration from the Talmud. Mathematical Social Siences 2, Stovall, J. E. (2014). Asymmetric parametric division rules. Games and Economic Behavior Volume 84 (4), Thomson, W. (2003). Axiomatic and game-theoretic analysis of bankruptcy and taxation problems: a survey. Mathematical Social Siences 45 (3), Young, P. (1987). On dividing an amount according to claims or liabilities. Mathematics of Operations Research 12 (3),