Additive adjudication of conflicting claims

In a “claims problem” (O’Neill 1982), a group of individuals have claims on a resource but its endowment is not sufficient to honour all of the claims. We examine the following question: If a claims problem can be decomposed into smaller claims problems, can the solutions of these smaller problems be added to obtain the solution of the original problem? A natural condition for this decomposition is that the solution to each of the smaller problems is non-degenerate, assigning positive awards to each claimant. We identify the only consistent and endowment monotonic adjudication rules satisfying this property; they are generalizations of the canonical “constrained equal losses rule” sorting claimants into priority classes and distributing the amount available to each class using a weighted constrained equal losses rule. The constrained equal losses rule is the only symmetric rule in this family of rules

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

A Characterization of a Family of Rules for the Adjudication of Conflicting Claims

We consider the problem of adjudicating conflicting claims, and characterize the family of rules satisfying four standard invariance requirements, homogeneity, two composition properties, and consistency. It takes as point of departure the characterization of the family of two-claimant rules satisfying the first three requirements, and describes the restrictions imposed by consistency on this family and the further implications of this requirement for problems with three or more claimants. The proof, which is an alternative to Moulin's original proof (Econometrica, 2000), is based on a general method of constructing consistent extensions of two-claimant rules (Thomson, 2001), which exploits geometric properties of paths of awards, seen in their entirety.claims problems, consistent extensions, proportional rule, constrained equal awards rule, constrained equal losses rule.

On the Existence of Consistent Rules to Adjudicate Conflicting Claims: A Constructive Geometric Approach

For the problem of adjudicating conflicting claims, a rule is consistent if the choice it makes for each problem is always in agreement with the choice it makes for each "reduced problem" obtained by imagining that some claimants leave with their awards and reassessing the situation from the viewpoint of the remaining claimants. We develop a general technique to determine whether a given two-claimant rule admits a consistent extension to general populations, and to identify this extension if it exists. We apply the technique to a succession of examples. One application is to a one-parameter family of rules that offer a compromise between the constrained equal awards and constrained equal losses rules. We show that a consistent extension of such a rule exists only if all the weight is placed on the former or all the weight is placed on the latter. Another application is to a family of rules that provide a compromise between the constrained equal awards and proportional rules, and a dual family that provide a compromise between the constrained equal losses and proportional rules. In each case, we identify the restrictions implied by consistency.claims problems, consistent extensions, proportional rule, constrained equal awards rule, constrained equal losses rule.

Strong composition down. Characterizations of new and classical bankruptcy rules.

This paper is devoted to the study of claims problems. We identify the family of rules that satisfy strong composition down (robustness with respect to reevaluations of the estate) and consistency (robustness with respect to changes in the set of agents) together. Such a family is the Þxed path rules, which is a generalization of the weighted constrained equal awards rules. In addition, once strong composition down and consistency are combined with homogeneity only the weighted constrained equal awards rules survive. We also prove that the constrained equal awards rule is the only rule that satisÞes strong composition down, consistency and equal treatment of equals together.strong composition down, Þxed path rules, constrained equal awards rule, weighted constrained equal awards rules

On the investment implications of bankruptcy laws

Axiomatic analysis of bankruptcy problems reveals three major principles: (i) proportionality (PRO), (ii) equal awards (EA), and (iii) equal losses (EL). However, most real life bankruptcy procedures implement only the proportionality principle. We construct a noncooperative investment game to explore whether the explanation lies in the alternative implications of these principles on investment behavior. Our results are as follows (i) EL always induces higher total investment than PRO which in turn induces higher total investment than EA; (ii) PRO always induces higher egalitarian social welfare than both EA and EL in interior equilibria; (iii) PRO induces higher utilitarian social welfare than EL in interior equilibria but its relation to EA depends on the parameter values (however, a numerical analysis shows that on a large part of the parameter space, PRO induces higher utilitarian social welfare than EA)

The Minimal Overlap Rule: Restrictions on Mergers for Creditors' Consensus

This paper proposes a notion of partial Additivity in bankruptcy, -Additivity. We show that this property, together with Anonymity and Continuity, identifies the Minimal Overlap rule, introduced by O'Neill (1982).Bankruptcy Problems; Additivity; Minimal Overlap Rule

Centralized Clearing Mechanisms in Financial Networks:A Programming Approach

We consider financial networks where agents are linked to each other with financial contracts. A centralized clearing mechanism collects the initial endowments, the liabilities and the division rules of the agents and determines the payments to be made. A division rule specifies how the assets of the agents should be rationed, the four most common ones being the proportional, the priority, the constrained equal awards, and the constrained equal losses division rules. Since payments made depend on payments received, we are looking for solutions to a system of equations. The set of solutions is known to have a lattice structure, leading to the existence of a least and a greatest clearing payment matrix. Previous research has shown how decentralized clearing selects the least clearing payment matrix. We present a centralized approach towards clearing in order to select the greatest clearing payment matrix. To do so, we formulate the determination of the greatest clearing payment matrix as a programming problem. When agents use proportional division rules, this programming problem corresponds to a linear programming problem. We show that for the other common division rules, it can be written as an integer linear programming problem

Uniqueness of Clearing Payment Matrices in Financial Networks

We study bankruptcy problems in financial networks in the presence of general bankruptcy laws. The set of clearing payment matrices is shown to be a lattice, which guarantees the existence of a greatest and a least clearing payment. Multiplicity of clearing payment matrices is both a theoretical and a practical concern. We present a new condition for uniqueness that generalizes all the existing conditions proposed in the literature. Our condition depends on the decomposition of the financial network into strongly connected components. A strongly connected component which contains more than one agent is called a cycle and the involved agents are called cyclical agents. If there is a cycle without successors, then one of the agents in such a cycle should have a positive endowment. The division rule used by a cyclical agent with a positive endowment should be positive monotonic and the rule used by a cyclical agent with a zero endowment should be strictly monotonic. Since division rules involving priorities are not positive monotonic, uniqueness of the clearing payment matrix is a much bigger concern for such division rules than for proportional ones. We also show how uniqueness of clearing payment matrices is related to continuity of bankruptcy rules

Solutions for cooperative games with restricted coalition formation and almost core allocations

The thesis focuses on cooperative games with transferable utility and incorporates two topics: Solutions for TU-games with restricted coalition formation and the stability of the grand coalition. Chapters 3 is devoted to a new solution for cooperative games with coalition structures, called the α-egalitarian Owen value, and this coalitional value is characterized by three approaches. Firstly, we provide two axiomatizations by introducing the α-indemnificatory null player axiom, and the (intra) coalitional quasi-balanced contributions axiom. Secondly, we characterize the coalitional value by introducing an α-guarantee potential function. Finally, the coalitional value is implemented by a punishment-reward bidding mechanism. In Chapter 4, we continue to work with TU-games restricted by coalition structures and propose a coalitional value called the two-step Shapley-solidarity value. A procedural interpretation is provided for this coalitional value, and we introduce a new axiom called the coalitional A-null player axiom to axiomatize the value based on additivity. Moreover, two other axiomatizations on the basis of quasi-balanced contributions for the grand coalition are also provided. In Chapter 5, we focus on cooperative games with communication structures and provide efficient extensions of the Myerson value (Myerson, 1977). The idea lies in introducing the Shapley payoffs of the underlying game as players' claims to derive a graph-induced bankruptcy problem. Then, two efficient extensions of the Myerson value are achieved through bankruptcy rules, including the CEA rule and the CEL rule (Aumann &amp; Maschler, 1985).Moreover, corresponding axiomatizations are also provided.Chapter 6 proceeds with studying the stability of the grand coalition for cost TU-games by addressing an optimization problem to maximize the total shareable cost over what we called the almost core. We analyze the computational complexity of this optimization problem, in relation to the computational complexity of related problems for the core. In particular, we consider a special class of games, i.e., the minimum cost spanning tree games. We show that maximizing the total shareable costs over the (non-negative) almost core is NP-hard for mcst games, and we provide a tight 2-approximation algorithm for this almost core optimization problem with the additional non-negative constraint. <br/