A composite run-to-the-bank rule for multi-issue allocation situations

In this paper, we propose a new extension of the run-to-the-bank rule for bankruptcy situations to the class of multi-issue allocation situations. We show that this rule always yields a core element and that it satisfies self-duality. We characterise our rule by means of a new consistency property, issue-consistency.     

Cost allocation protocols for supply contract design in network situations

The class of Construct and Charge (CC-) rules for minimum cost spanning tree (mcst) situations is considered. CC-rules are defined starting from the notion of charge systems, which specify particular allocation protocols rooted on the Kruskal algorithm for computing an mcst. These protocols can be easily implemented in practical network situations (for instance, in supply transportation networks), are flexible to changes in the network situation and meet the requirement of continuous monitoring by the agents involved. Special charge systems, that we call conservative, lead to a subclass of CC-rules that coincides with the class of obligation rules for mcst situations. The authors thank two anonymous referees both for detailed remarks and for interesting general comments on a previous version of the paper. Stef Tijs and Rodica Branzei are indebted to Daniel Granot for useful discussions on the topics treated in this paper and his hospitality during our research visit at British Columbia University at Vancouver in July–August 2003.     

A characterization of the proportional rule in multi-issue allocation situations

In multi-issue allocation situations, we have to divide a resource among a group of agents. The claim of each agent is a vector specifying the amount claimed by each agent on each issue. We present an axiomatic characterization of the proportional rule.     

New characterizations of the constrained equal awards rule in multi-issue allocation situations

Multi-issue allocation situations are used to study the problem of having to divide an estate among a group of agents. The claim of each agent is a vector specifying the amount claimed by each agent on each issue. We present several axiomatic characterizations of the constrained equal awards rule generalizing some characterizations of these rules in bankruptcy situations.     

The two-stage constrained equal awards and losses rules for multi-issue allocation situations

This paper considers two-stage solutions for multi-issue allocation situations, which are extensions of bankruptcy problems. Characterizations are provided for the two-stage constrained equal awards and constrained equal losses rules, based on the properties of composition up and composition down.     

Multi-stage sequencing situations

This paper introduces the idea of dynamics in cooperative games. The concrete case of multi-stage sequencing situations and the difficulties involved in defining stable cost savings allocations for the games arising from these situations is studied. The MEGS-rule is defined and proven to yield stable allocations. A characterization for the MEGS-rule is given.     

Stochastic dynamo in critical situations

Based on the functional method of consecutive approximations, we consider the problem of magnetic field excitation (stochastic dynamo) by a random velocity field with a finite temporal correlation radius. In critical situations, in the first (diffusion) approximation, the Lyapunov characteristic parameter of the magnetic field energy vanishes. This implies the absence of structure formation (clustering) in realizations of the magnetic field in that approximation. Critical situations occur in problems of magnetic field diffusion in an equilibrium thermal and random pseudoequilibrium and acoustic (in the absence of dissipation) velocity fields. The sign of the Lyapunov characteristic parameter in the second-order approximation determines the possibility of clustering of the magnetic field energy. We show that energy clustering does not occur in a thermal velocity field. In the cases of pseudoequilibrium and acoustic velocity fields, clustering occurs with probability one, i.e., in almost every realization. We evaluate the characteristic time for clustering to be established.     

Benefit sharing in holding situations

A situation is studied where one agent has available an amount of storage facility and the other agents have some goods, part of which can be stored generating benefits. The problem of sharing the benefits produced by full cooperation between agents is tackled in this paper, by introducing a related cooperative game. This game turns out to be a big boss game with interesting theoretical properties. A solution concept, relying on optimal storage plans and associated holding prices, is also introduced, and its relationship with the core of the above holding game is explored in detail. The family of monotonic decreasing bijective mappings, defined on the set of non-negative real numbers, plays an important role in our approach.     

Link formation in cooperative situations

In this paper we study the endogenous formation of cooperation structures or communication graphs between players in a superadditive TU game. For each cooperation structure that is formed, the payoffs to the players are determined by an exogenously given solution. We model the process of cooperation structure formation as a game in strategic form. It is shown that several equilibrium refinements predict the formation of the complete cooperation structure or some structure which is payoff-equivalent to the complete structure. These results are obtained for a large class of solutions for cooperative games with cooperation structures. Received September 1995/Revised version I October 1996/Revised version II April 1997/Final version September 1997     

Partitioning sequencing situations and games

The interaction between sequencing situations and cooperative games starting from the paper of Curiel et al. [Curiel, I., Pederzoli, G., Tijs S., 1989. Sequencing games. European Journal of Operational Research 40, 344–351], has become an established line of research within the theory of operation research games.     

Sequencing interval situations and related games

Uncertainty accompanies almost every situation in real world and it influences our decisions. In sequencing situations it may affect parameters used to determine an optimal order in the queue, and consequently the decision of whether (or not) to rearrange the queue by sharing the realized cost savings. This paper extends the analysis of one-machine sequencing situations and their related cooperative games to a setting with interval data, i.e. when the agents’ costs per unit of time and/or processing time in the system lie in intervals of real numbers obtained by forecasting their values. The question addressed here is: How to determine an optimal order (if the initial order in the queue is not so) and which approach should be used to motivate the agents to adopt the optimal order? This question is an important one that deserves attention both in theory and practice.     

Stability and monotonicity in newsvendor situations

This study considers a supply chain that consists of n retailers, each of them facing a newsvendor problem, and a supplier. Groups of retailers might increase their expected joint profit by joint ordering and inventory centralization. However, we assume that the retailers impose some level of stock that should be dedicated to them. In this situation, we show that the associated cooperative game has a non-empty core. Afterwards, we concentrate on a dynamic situation, where several model cost parameters and the retailers’ dedicated stock levels can change. We investigate how the profit division might be affected by these changes. We focus on four monotonicity properties. We identify several classes of games with retailers, where some of the monotonicity properties hold. Moreover, we show that pairs of cooperative games associated with newsvendor situations do not necessarily satisfy these properties in general, when changes in dedicated stock levels are in concern.     

Inheritance of properties in communication situations

In this paper we consider cooperative games in which the possibilities for cooperation between the players are restricted because communication between the players is restricted. The bilateral communication possibilities are modeled by means of a (communication) graph. We are interested in how the communication restrictions influence the game. In particular, we investigate what conditions on the communication graph guarantee that certain appealing properties of the original game are inherited by the graph-restricted game, the game that arises once the communication restrictions are taken into account. We study inheritance of the following properties: average convexity, inclusion of the Shapley value in the core, inclusion of the Shapley values of a game and all its subgames in the corresponding cores, existence of a population monotonic allocation scheme, and the property that the extended Shapley value is a population monotonic allocation scheme. Received May 1998/Revised version January 2000     

Allocation rules for hypergraph communication situations

This paper considers hypergraph communication situations, where for a group of agents the economic possibilities are described by a coalitional game and the communication possibilities are described by a hypergraph in which the nodes are the agents and the edges are the subgroups of agents who can effect communication. Axiomatic characterizations are provided for two allocation rules, the Myerson value and the position value.     

A value for generalized probabilistic communication situations

We introduce in this work an extension of the model of games with probabilistic graphs arising in Calvo et al. (1999, Math. Soc. Sci. 37, 79), which itself generalizes the one developed by Myerson (1977, Math. of Oper. Res. 2, 225) for games with communications restrictions. In the first of these models, each pair of nodes has a given probability of direct communication. In this paper a more general setting is considered: we suppose that a probability distribution over the set of all possible communication networks among the players is given. A generalization of the Myerson value is defined and characterized in this context.     

Compromise solutions for bankruptcy situations with references

This paper deals with bankruptcy situations in which in addition to the claims, an exogenously given reference point for the allocation of the estate is present. We introduce and analyse two types of compromise solutions and show that they coincide with the τ value of two corresponding TU games. We apply our solutions to a real-life case of allocating university money to degree courses. Research supported by Generalitat Valenciana, through ACOMP06/040, ACOMP07/136 and GV05/189, and by the Government of Spain and FEDER fund under project MTM2005-09184-C02-02. M. Pulido’s research is partially supported by Fundación Séneca of Comunidad Autónoma de la Región de Murcia, through 02911/PI/05. Ruud Hendrickx author acknowledges financial support from the Netherlands Organisation for Scientific Research (NWO).     

Obtaining fast error rates in nonconvex situations

We show that under mild assumptions on the learning problem, one can obtain a fast error rate for every reasonable fixed target function even if the base class is not convex. To that end, we show that in such cases the excess loss class satisfies a Bernstein type condition.