Endowment additivity and the weighted proportional rules for adjudicating conflicting claims

We propose and study a new axiom, restricted endowment additivity, for the problem of adjudicating conflicting claims. This axiom requires that awards be additively decomposable with respect to the endowment whenever no agent’s claim is filled. For two-claimant problems, restricted endowment additivity essentially characterizes weighted extensions of the proportional rule. With additional agents, however, the axiom is satisfied by a great variety of rules. Further imposing versions of continuity and consistency, we characterize a new family of rules which generalize the proportional rule. Defined by a priority relation and a weighting function, each rule aims, as nearly as possible, to assign awards within each priority class in proportion to these weights. We also identify important subfamilies and obtain new characterizations of the constrained equal awards and proportional rules based on restricted endowment additivity

Two families of rules for the adjudication of conflicting claims

We define two families of rules to adjudicate conflicting claims. The first family contains the constrained equal awards, constrained equal losses, Talmud, and minimal overlap rules. The second family, which also contains the constrained equal awards and constrained equal losses rules, is obtained from the first one by exchanging, for each problem, how well agents with relatively larger claims are treated as compared to agents with relatively smaller claims. In each case, we identify the subfamily of consistent rules.claims problems, constrained equal awards rule, constrained equal losses rule, Talmud rule, minimal overlap rule, ICI rules, CIC rules, consistency.

Refining the Lorenz‐ranking of rules for claims problems on restricted domains

The comparison of the central rules for claims problems, according to the Lorenz order, has been studied not only on the entire set of problems but also on some restricted domains. We provide new characterizations of the adjusted proportional rule as being Lorenz‐maximal or Lorenz‐minimal within a class of rules on the half‐domains. Using this result, we rank the adjusted proportional, the minimal overlap, and the average‐of‐awards rules by analyzing whether or not these rules satisfy progressivity and regressivity on the half‐domains. We also find that the adjusted proportional rule violates two well‐known claim monotonicity properties.Agencia Estatal de Investigación | Ref. PID2019‐106281GB‐I00Agencia Estatal de Investigación | Ref. PID2021‐124030NB‐C33Xunta de Galicia | Ref. ED481A 2021/325Universidade de Vigo/CISU

Problemes de repartiment: regles i jocs cooperatius

Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2020, Director: F. Javier Martínez de Albéniz[en] In this work we study claims problems, which happen when there is a certain estate to be distributed and the sum of the claims of all agents is greater than the estate. We study it from an axiomatic point of view, studying the properties of those problems and their distribution rules. We structure the work in 4 parts. Firstly, we present the claims problems and the different distribution rules, paying attention on the 4 most important rules. Besides that, we define some more rules that are not so important, but worth mentioning. The next part is the most important of this work, the characterization of the 4 most important sharing rules. We begin the chapter defining some properties that will later use to make the characterization of them. In the third part, we look at claims problems from a different point of view, from cooperative game theory. We study the relations between cooperative games and claims problems, specifically between distribution rules and point solutions for that kind of games. Finally, we discuss two recent papers where claims problems are used to solve current problems in society. The first tells us about the distribution of the health budget in Catalonia. The second studies the distribution of the permission of CO 2 emissions amongst the different world regions