Computation and comparison of division rules to adjudicate conflicting claims

Resumen

The contents of this thesis, entitled Computation and comparison of division rules to adjudicate conflicting claims, are the result of the work carried out by Iago Núñez Lugilde during the studies corresponding to the Programa de Doctorado en Estadística e Investigación Operativa por la Universidad de A Coruña, la Universidad de Santiago de Compostela y la Universidad de Vigo. This thesis was supervised by Professor Estela Sánchez Rodríguez (departamento de Estatística e Investigación Operativa, Universidade de Vigo), with the collaboration of Professors Miguel Ángel Mirás Calvo and Carmen Quinteiro Sandomingo (departamento de Matemáticas, Universidade de Vigo) and Professor Arantza Estévez Fernández (Operations Analytics department, Vrije Universiteit Amsterdam), who was in charge of the required stay to obtain the international mention. The thesis focuses on the study of conflicting claims problems. In Chapter 1, we incorporate the average-of-awards rule into the Lorenz ranking of division rules. Furthermore, to compare the different distribution methods, we define new coefficients, based on the Gini index, intended to measure the discrepancy between the allocationsmade by any two rules for a given conflicting claims problem. In Chapter 2, we refine the relationships among division rules with the Lorenz-order. We study the differences in the behaviour of the rules depending on the amount of resource to be distributed compared to the overall claim. We provide a new characterisation of the adjusted proportional rule as being Lorenz-maximal or Lorenz-minimal within a class of rules. Using these results, we rank this rule, along with the minimal overlap and the average-of-awards rules by analysing whether or not they satisfy, among others, the properties of progressivity or regressivity. In Chapter 3, we develop an algorithm to compute the average-of-awards rule. This algorithm provides a new interpretation of the rule as a point of fairness between stable and utopia imputations. We illustrate the developed analysis by studying a real example in which the worlds most polluting regions must share CO2 emissions during the decade from 2020 to 2030 to curb climate change. The set of awards vectors of a conflicting claims problem with n agents is a non-empty convex polytope obtained from the intersection of an n-rectangle with a hyperplane in Rn. These polytopes also arise naturally in the study of other set solutions related to cooperative game theory. In Chapter 4, we analyse in detail the geometry of this type of polytopes. We develop explicit expressions to compute their exact volume and their centroid, and we show that they are #P-complete problems, establishing an analogy with the computation of the random arrival rule. In Chapter 5, we study situations in which the allocation of a resource must be repeated respecting certain priority conditions imposed by an allocation used before. We define two cooperative games, using conflicting claims problems as a tool, and we study their properties. We present an applied example where the allocation of a university¿s funds is carried out taking into account priority conditions based on some objectives imposed on the different groups. Finally, we use priority games to analyse the cooperation restricted by a hierarchical structure. In Chapter 6, we introduce the R package, ClaimsProblems. We do a thorough review of all its functions and their use for any conflicting claims problem. All the functions related to this type of problems used throughout the thesis are programmed in the package and explained in this chapter.