Rationing problems: Extensions and Multi-issue analysis

Abstract

This thesis concerns itself with the study of situations in which a group of agents lay separate claims to a scarce resource. These situations are as old as the discipline of economics itself; indeed, a number of ancient documents including the works of Aristotle, the essays of Maimonides and the Babylonian Talmud address this very old problem.

Despite the ancient nature of these texts, these situations were not tackled formally until the early eighties by O'Neill (1982), who provided an extremely simple mathematical model to explain a wide variety of economic problems, including, among others, the assignment of taxes, bankruptcy, the distribution of emergency supplies and cost-sharing of a public good. In general we can refer to these situations as problems of adjudicating conflicting claims or (standard) rationing problems.

The present study seeks to enrich rationing problems from different perspectives:

* In Chapter 2 we introduce an extension of the standard rationing model, in which agents are not only identified by their respective claims to some amount of a scarce resource, but also by some exogenous ex-ante conditions (initial stock of resource or net worth of agents, for instance), other than claims. The essence of this chapter is that those agents who have less (with a worse ex-ante condition) should somehow be given some priority over those who have more (with a better ex-ante condition). Within this framework, we define a generalization of the constrained equal awards rule and provide two different characterizations of this generalized rule. Finally, we use the corresponding dual properties to characterize a generalization of the constrained equal losses rule.

* In Chapter 3 we present a variant of the multi-issue rationing model, where agents stake their claim for several issues. In this variant, the amount of resource available for each issue is constrained to a quantity fixed a priori according to exogenous criteria. The aim is to distribute the amount corresponding to each issue while taking into account the allocation for the remaining issues (issue-allocation interdependence). We name these problems constrained multi-issue allocation situations (CMIA). In order to solve these problems, we first reinterpret some single-issue (standard) egalitarian rationing rules as a minimization program based on the idea of finding a feasible allocation that lies as close as possible to a specific reference point. We extend this family of egalitarian rules to the CMIA framework. Specifically, we extend the constrained equal awards rule, the constrained equal losses rule and the reverse Talmud rule to the multi-issue rationing setting, which are found to be particular cases of a family of rules, namely the extended α-egalitarian family. This family is analysed and characterized by using consistency principles (over agents and over issues) and a property based on the Lorenz-dominance criterion.

* Finally, in Chapter 4 we consider how to solve a rationing problem in which the resource cannot be directly assigned to agents. We propose a two-stage procedure in which the resource is first allocated to groups of agents and then divided among their members. We name these situations decentralized rationing problems. Within this framework, we define extensions of the constrained equal awards, the constrained equal losses and the proportional rules. We show that the first two rules do not preserve certain essential properties and prove the conditions under which both rules do preserve those properties. We characterize the extension of the proportional rule as the only solution that satisfies individual equal treatment of equals. Furthermore, we prove that the proportional rule is the only solution that assigns the same allocation regardless of

whether the resource is distributed directly to agents or in a decentralized manner (with agents grouped). Finally, we analyse a strategic game based on decentralized rationing problems in which agents can move freely across groups to submit their claims.