Bankruptcy Problems Research Articles

Bankruptcy problems with two references: an impartial compromise

Pulido et al. (Annals Oper Res 158:133–141, 2008) present an extension of the classical bankruptcy problem (O’Neill in Math Social Sci 2:345–371, 1982) where the involved agents have, apart from the claims vector, an additional reference vector. To analyze this extended problem, they propose the extreme and the diagonal approaches, both of them restricted to the case in which the reference vector is lower than the claims vector. We note that if the claims and the reference vectors are interchanged, the allocation proposed by the extreme approach varies. Therefore, by introducing the idea of impartiality, in the current approach, we propose an extension of their model in which no relation is assumed between the claims and reference vectors.

A precedence constraint value revisited

This paper discusses games where cooperation is restricted by a hierarchical structure. The model assumes that there is a hierarchy between certain coalitions given by a partition. Face games (Gonzalez-Diaz and Sanchez-Rodriguez in Games Econ Behav 62:100–105, 2008) play a central role in the definition of the proposed hierarchical game. It turns out that the Shapley value of the hierarchical game coincides with the Faigle and Kern precedence constraint value (Faigle and Kern in Int J Game Theory 21:249–266, 1992) and with some particular weighted Shapley value (Kalai and Samet in Int J Game Theory 16:205–222, 1987). Two new characterizations of this hierarchical value are given. Finally, an application of the model is given for bankruptcy problems.

Bankruptcy problems with non-integer claims: definition and characterizations of the ICEA Solution

In Fragnelli et al. (TOP 22:892–933, 2014), we considered a bankruptcy problem with the additional constraint that the estate has to be assigned in integer unities, allowing for non-integer claims. We dealt with the question of existence and uniqueness of a solution and proposed the “box method” that is strongly oriented towards the constrained equal losses solution; uniqueness may be guaranteed by introducing a ranking on the claimants. Here, we introduce a solution oriented towards the constrained equal awards solution and give three characterizations and a simple method for determining the solution.

Two characterizations of the random arrival rule

The main focus of this note is on the random arrival rule. We adopt two characterizations from transferable utility games theory and reinterpret them in the framework of bankruptcy problems. One is that the random arrival rule can be formulated as the vector of marginal compensations of a potential function. The other one is that the random arrival rule is the only rule satisfying balanced compensations.

Bankruptcy Problems of Heat Supply Companies

Bankruptcy Problems of Heat Supply Companies

Bankruptcy Problem Allocations and the Core of Convex Games

A well-known result related to bankruptcy problems establishes that a vector is a bankruptcy allocation if and only if it belongs to the core of the associated O’Neill’s bankruptcy game. In this paper we show that this game is precisely the unique TU-game based on convex functions that satisfies the previous result. In addition, given a bankruptcy problem, we show a way for constructing bankruptcy games such that the set of bankruptcy allocations is a subset of their core or their core is a subset of the set of bankruptcy allocations. Also, we show how these results can be applied for finding new bankruptcy solutions.

A noncooperative approach to bankruptcy problems with an endogenous estate

We introduce a new class of bankruptcy problems in which the value of the estate is endogenous and depends on agents’ investment decisions. There are two investment alternatives: investing in a company (risky asset) and depositing money into a savings account (risk-free asset). Bankruptcy is possible only for the risky asset. We define a game between agents each of which aims to maximize his expected payoff by choosing an investment alternative and a company management which aims to maximize profits by choosing a bankruptcy rule. Our agents are differentiated by their incomes. We consider three most prominent bankruptcy rules in our base model: the proportional rule, the constrained equal awards rule and the constrained equal losses rule. We show that only the proportional rule is a part of any pure strategy subgame perfect Nash equilibrium. This result is robust to changes in income distribution in the economy and can be extended to a larger set of bankruptcy rules and multiple types. However, extension to multiple company framework with competition leads to equilibria where the noncooperative support for the proportional rule disappears.

A new approach for bounding awards in bankruptcy problems

The solution for the contested garment problem, proposed in the Babylonic Talmud, suggests that each agent should receive at least some part of the resources whenever the claim exceeds the available amount. In this context, we propose a new method to define lower bounds on awards, an idea that has underlied the theoretical analysis of bankruptcy problems from its beginning (O’Neill, Math Soc Sci 2:345–371, 1982) to present day (Dominguez and Thomson, Econ Theory 28:283–307, 2006). Specifically, starting from the fact that a society establishes its own set of commonly accepted equity principles, our proposal ensures to each agent the smallest amount she gets according to all the admissible rules. We analyze its recursive application for different sets of equity principles.

Integer solutions to bankruptcy problems with non-integer claims

In this paper, we consider a bankruptcy problem with the additional constraint that the estate has to be assigned in integer unities; differently from the approaches by other authors, we allow for non-integer claims. This situation poses new questions on the existence and the uniqueness of the solution. We propose a new solution method that leads to the “box solution”, for which uniqueness may be guaranteed introducing a ranking on the claimants. Some examples complete the paper.

On Ibn Ezra's Procedure and Shapley Value

We examine ibn Ezra's procedure (Rabinovitch 1973; O'Neill 1982) historically used to solve the Rights Arbitration problem in the general framework of bankruptcy problems. When the greatest claim is larger than or equal to the estate, the procedure is a maximal game (Aumann 2010). However, when the greatest claim is smaller than the estate, the axioms of efficiency (the whole estate is distributed) and satiation are difficult to satisfy simultaneously. We discuss both axioms to show that their importance and necessity are radically different. From then, for the part of the estate not covered by the greatest claim, we examine four possible procedures: the minimal overlap rule, Alcalde et al.'s (2005) iterative generalization, dictator game, and unanimity game. They pose many problems mainly because: (i) only the greatest claimant remains in the game; (ii) playing two different subgames is unrealistic; (iii) some claimants may receive more than they claim, which violates the satiation axiom; (iv) under the minimal overlap rule players simultaneously play a unanimity game and a dictator game; (v) Alcalde et al.'s generalization is is efficient but converges very slowly. This is why we propose an original and different rule, the series of nested unanimity games (SNUG). We show that it is guarantees efficiency and satiation for all claimants and compared to the other rules, SNUG is attractively homogenous --- only unanimity games are played.

How to divide a cake when people have different metabolism?

This paper deals with bankruptcy problems in which the players have different utility functions defined in terms of the quantity of allocated resources. We tackle this kind of situation by means of a game without transferable utility and provide two characterizations of the CEA-rule in this context.

Computation of the random arrival rule for bankruptcy problems

Bankruptcy problems are a fundamental class of fair division problems in microeconomics. Among the various solution concepts proposed for the problem, the random arrival rule is one of the most prominent. In this paper, we conduct a computational analysis of the rule. It is shown that the allocation returned by the rule is #P-complete to compute. The general complexity result is complemented by a pseudo-polynomial-time dynamic programming algorithm for the random arrival rule.

On Solving Liability Problems

This paper introduces liability problems, as a generalization of bankruptcy problems, where every agent not only owns a certain amount of cash money, but also has outstanding claims and debts towards the other agents. Assuming that the agents want to cash their claims, we will analyze liability rules which prescribe how the total available amount of cash should be allocated among the agents. In particular we focus on bankruptcy rule based bilateral transfer schemes. Existence of these schemes is established and it is seen that within the class of hierarchical liability problems, such a transfer scheme is unique. Although in general a bankruptcy rule based bilateral transfer scheme need not be unique, we show that the resulting bankruptcy rule based transfer allocation is. This leads to the definition of bankruptcy rule based liability rules. For hierarchical liability problems an alternative characterization of such liability rules is provided. Moreover it is shown that the axiomatic characterization of the Aumann-Maschler bankruptcy rule on the basis of consistency can be extended to the corresponding liability rule. We conclude with a discussion of alternative approaches to solve liability problems.

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).

Characterization of the Reverse Talmud bankruptcy rule by Exemption and Exclusion properties

The Reverse Talmud rule for bankruptcy problems applies the Constrained Equal Awards rule (CEA rule) for ‘large’ amounts of the estate, while it applies the Constrained Equal Losses rule (CEL rule) for ‘small’ amounts of the estate. The CEA rule, respectively CEL rule, can be axiomatized using the Exemption and Exclusion properties respectively. There is no rule that satisfies both these two properties. In this paper we axiomatize the Reverse Talmud rule by using compatible weaker versions of the Exemption and Exclusion properties.

Coalitional manipulation on networks

Abstract We study allocation rules that are robust to coalitional manipulation by transferring, merging, or splitting individual characteristics among coalition partners (e.g. merging or splitting claims in bankruptcy problems). Coalition formation is restricted by an exogenous network (a non-directed graph) so that only connected subsets of agents can form a coalition. We offer a full characterization of non-manipulable rules without any assumption on the network structure. This result yields a variety of useful corollaries for specific networks such as complete network, trees, and networks without a “bridge”, and corollaries for specialized models dealing with bankruptcy, surplus sharing, cost sharing, income redistribution, social choice with transferable utility, etc.

A new analysis of a simple model of fair allocation

In a recent article, Fragnelli and Gagliardo [Fragnelli, V., Gagliardo, S., (2012). Cooperative models for allocating an object. Economics Letters 117, 227–229] propose several procedures to solve a basic problem of fair allocation. We scrutinize their proposal and contextualize it into recent developments of the literature on bankruptcy problems. Our analysis supports two of the procedures they propose; namely, the Shapley and Talmud rules.

Stochastic bankruptcy games

We study bankruptcy problems where the estate and the claims have stochastic values and we allow these values to be correlated. We associate a transferable utility game with uncertainty to a stochastic bankruptcy problem and use the Weak Sequential Core as a solution concept for such games. We test the stability of a number of well known division rules in this stochastic setting and find that all of them are unstable, with the exception of the stochastic extension of the Constrained Equal Awards rule which leads to a Weak Sequential Core element.

The minimal overlap rule: restrictions on mergers for creditors’ consensus

As it is known, there is no rule satisfying additivity in the complete domain of bankruptcy problems. This paper proposes a notion of partial additivity in this context, to be called μ-additivity. We find out that this property, together with two quite compelling axioms, equal treatment of equals and continuity, identify the minimal overlap rule, introduced by O’Neill (Math. Soc. Sci. 2:345–371, 1982).

Leximin rules for bankruptcy problems under uncertainty

We model bankruptcy problems under uncertainty under the assumption that there are several possible states of nature, each of which is identified with a different bankruptcy problem. For this multi-dimensional extension of classic bankruptcy problems, we consider situations in which agents exhibit at the same time additive preferences and leximin preferences on their possible results. We propose division rules which combine different rationality principles and guarantee efficiency with respect to leximin preferences.