Remarks on the integer Talmud solution for integer bankruptcy problems

In Fragnelli et al. (TOP 22:892–933, 2014; TOP 24:88–130, 2016), 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 extension to our setting of the constrained equal losses solution and of the constrained equal awards solution. Here, we analyze the possibilities of extending the Talmud solution to the integer situation, starting from the existing approaches for the non-integer case; some of these approaches are compatible with the non-integer claims, but in order to comply with as much as possible of the approaches it is necessary to switch to integer claims.     

A coalitional procedure leading to a family of bankruptcy rules

We provide a general coalitional procedure that characterizes a family of rules for bankruptcy problems inspired by the Talmud.     

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.     

A characterization of a family of consensus rules for committee elections

A committee election rule is proposed that is based on a measure of vote concentration. Relationships are established between properties of the committee selection function, and properties of the function used to measure vote concentration.     

Bankruptcy and the per capita nucleolus: The claim-and-right rules family

This article describes the per capita nucleolus for bankruptcy games as a bankruptcy rule. This rule, called the clights rule, is based on the well-known constrained equal awards principle and it takes into account a vector of clights, a new term which is a blend of claims and rights. These clights only depend on the vector of claims while the height of the estate determines whether the clights should be interpreted as modified claims or as rights. It is shown that both the clights rule and the Aumann–Maschler rule can be captured within the family of so-called claim-and-right rules.     

Bilateral comparisons and consistent fair division rules in the context of bankruptcy problems

We analyze the problem of extending a given bilateral principle of justice to a consistentn-creditor bankruptcy rule. Based on the bilateral principle, we build a family of binary relations on the set of creditors in order to make bilateral comparisons between them. We find that the possibility of extending a specific bilateral principle of justice in a consistent way is closely related to the quasi-transitivity of the binary relations mentioned above.     

Equivalence of axioms for bankruptcy problems

The bankruptcy problem is concerned with how to divide the net worth of the bankrupt firm among its creditors. In this paper, we investigate the logical relations between various axioms in the context of bankruptcy. Those axioms are: population-and-resource monotonicity, consistency, converse consistency, agreement, and separability. In most axiomatic models, they are not directly related. However, we show that they are equivalent on the class of bankruptcy problems under minor additional requirements. Received: April 1998/Revised version: January 1999     

The pseudo-average rule: bankruptcy,cost allocation and bargaining

A division rule for claims problems, also known as bankruptcy or rationing problems, based on the pseudo-average solution is studied (for 2-person problems). This solution was introduced in Moulin (Jpn Econ Rev 46:303–332, 1995) for discrete cost allocation problems. Using the asymptotic approach, we obtain a division rule for claims problems. We characterize the division rule axiomatically and show that it coincides with the rule associated to the equal area bargaining solution (this is not true for n = 3). Moreover, following Moulin and Shenker (J Econ Theor 64:178–201, 1994), we show that its associated solution for continuous homogeneous goods is precisely the continuous pseudo-average solution.     

Using Bayesian networks for bankruptcy prediction: Some methodological issues

This study provides operational guidance for building naïve Bayes Bayesian network (BN) models for bankruptcy prediction. First, we suggest a heuristic method that guides the selection of bankruptcy predictors. Based on the correlations and partial correlations among variables, the method aims at eliminating redundant and less relevant variables. A naïve Bayes model is developed using the proposed heuristic method and is found to perform well based on a 10-fold validation analysis. The developed naïve Bayes model consists of eight first-order variables, six of which are continuous. We also provide guidance on building a cascaded model by selecting second-order variables to compensate for missing values of first-order variables. Second, we analyze whether the number of states into which the six continuous variables are discretized has an impact on the model’s performance. Our results show that the model’s performance is the best when the number of states for discretization is either two or three. Starting from four states, the performance starts to deteriorate, probably due to over-fitting. Finally, we experiment whether modeling continuous variables with continuous distributions instead of discretizing them can improve the model’s performance. Our finding suggests that this is not true. One possible reason is that continuous distributions tested by the study do not represent well the underlying distributions of empirical data. Finally, the results of this study could also be applicable to business decision-making contexts other than bankruptcy prediction.     

A Semilattice of Degrees of Computable Metrics

Siberian Mathematical Journal - Under study is the ordering $ {mathcal{CM}}_{c}({mathbf{X}}) $ of $ c $ -degrees of computable metrics on a Polish space  $ {mathbf{X}} $ with...     

A characterization of Kruskal sharing rules for minimum cost spanning tree problems

In Tijs et al. (Eur J Oper Res 175:121–134, 2006) a new family of cost allocation rules is introduced in the context of cost spanning tree problems. In this paper we provide the first characterization of this family by means of population monotonicity and a property of additivity.     

Market discipline of subordinated debt in banking: The case of costly bankruptcy

This paper extends the contingent-claim valuation framework of subordinated debt by explicitly incorporating bankruptcy cost in the model. The model is then used to investigate the role of subordinated debt in alleviating the moral hazard problem in banking. The incorporation of bankruptcy cost in the framework of the analysis provides new evidence about subordinated debt. The reaction of subordinated debt prices can completely remove risk-shifting incentives of equityholders only when bankruptcy costs are accounted for. The extent of subordinated debt’s discipline is shown to depend critically on the relative magnitude of subordinated debt, senior debt and bankruptcy costs.     

A Separation Property of the Observer-Based Stabilizing Controller for Linear Delay Systems

Siberian Mathematical Journal - This paper is concerned with the observer-based controller for the stabilization of linear systems with multiple delays. We design a state observer for the...     

Avoider-Enforcer: The rules of the game

An Avoider-Enforcer game is played by two players, called Avoider and Enforcer, on a hypergraph F⊆X². The players claim previously unoccupied elements of the board X in turns. Enforcer wins if Avoider claims all vertices of some element of F, otherwise Avoider wins. In a more general version of the game a bias b is introduced to level up the players' chances of winning; Avoider claims one element of the board in each of his moves, while Enforcer responds by claiming b elements. This traditional set of rules for Avoider-Enforcer games is known to have a shortcoming: it is not bias monotone.We relax the traditional rules in a rather natural way to obtain bias monotonicity. We analyze this new set of rules and compare it with the traditional ones to conclude some surprising results. In particular, we show that under the new rules the threshold bias for both the connectivity and Hamiltonicity games, played on the edge set of the complete graph Kn, is asymptotically equal to n/logn. This coincides with the asymptotic threshold bias of the same game played by two “random” players.     

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.     

Skew-product decompositions of Brownian motions on manifolds: A probabilistic aspect of the Lichnerowicz-Szabo theorem

We show that the only compact simply connected manifolds for which the radial part of Brownian motion enjoys the Markov property are compact two points homogeneous spaces, i.e. rank one symmetric spaces.     

Mixed succession rules: The commutative case

We begin a systematic study of the enumerative combinatorics of mixed succession rules, i.e. succession rules such that, in the associated generating tree, nodes are allowed to produce sons at several different levels according to different production rules. Here we deal with a specific case, namely that of two different production rules whose rule operators commute. In this situation, we are able to give a general formula expressing the sequence associated with the mixed succession rule in terms of the sequences associated with the component production rules. We end by providing examples illustrating our approach.     

The ultratriangular form for prime-power lattice rules

A recent development in the theory of lattice rules has been the introduction of the unique ultratriangular D-Z form for prime-power rules. It is known that any lattice rule may be decomposed into its Sylow p-components. These components are prime-power rules, each of which has a unique ultratriangular form. By reassembling these ultratriangular forms in a defined way, it is possible to obtain a canonical form for any lattice rule. A special case occurs when the ultratriangular forms of each of the Sylow p-components have a consistent set of column indices. In this case, it is possible to obtain a unique canonical D-Z form. Given the column indices and the invariants for an ultratriangular form, we may obtain a formula for the number of ultratriangular forms, and hence the number of prime-power lattice rules, having these column indices and invariants.     

A Note on the Jones Polynomials of 3-Braid Links

Siberian Mathematical Journal - The braid group on $ n $ strands plays a central role in knot theory and low dimensional topology. 3-braids were classified, up to conjugacy, into normal forms....