Hans Reijnierse

G-component additive games

According to Maschler, Peleg and Shapley (1972) the bargaining set of aconvex game coincides with its core and the kernel consists of the nucleolus only. In this paper we prove the same properties for Γ-component additive games (=graph restricted games in the sense of Owen (1986)) if Γ is a tree. Furthermore, we give a description of the nucleolus of this type of games which makes it easier accessible for computation

On the core of a collection of coalitions

Abstract. For a collection Ω of subsets of a finite set N we define its core to be equal to the polyhedral cone {x∈IRN: ∑i∈N xi=0 and ∑i∈Sxi≥0 for all S∈Ω}. This note describes several applications of this concept in the field of cooperative game theory. Especially collections Ω are considered with core equal to {0}. This property of a one-point core is proved to be equivalent to the non-degeneracy and balancedness of Ω. Further, the notion of exact cover is discussed and used in a second characterization of collections Ω with core equal to {0}.

Interactive Purchasing Situations

Abstract: This paper introduces a new class of interactive cooperative purchasing situations and provides an explicit alternative characterization of the nucleolus of cooperative games, which offers an alternative to Kohlberg (1971). In our cooperative purchasing situation, the unit price of a commodity depends on the largest order quantity within a cooperating group of players. Due to a decreasing unit price function, players can obtain cost savings by purchasing cooperatively. However, to establish fruitful cooperation a decision has to be made about an adequate allocation of the corresponding cost savings. We analyze Maximum Cooperative Purchasing (MCP)-situation from the perspective of allocation by defining corresponding cooperative MCP-games. It turns out that in an MCP-game all coalitional values can be determined from the values of two-player coalitions. Moreover, it is shown that a decreasing unit price function is a sufficient condition for a non-empty core: the Direct Price solution is both a core element and a marginal vector. It is seen that the nucleolus of an MCP-game can be derived in polynomial time from the Direct Price solution, using a socalled nucleolus determinant. To show this result, the explicit alternative characterization of the nucleolus is used. Using the decomposition of an MCP-game into unanimity games we find an explicit expression for the Shapley value. Interestingly, the Shapley value can be interpreted as a specific tax and subsidize system. Finally, the behavior of the three solution concepts is compared numerically.

On Solving Liability Problems

Abstract: 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.

Convex Congestion Network Problems

This paper analyzes convex congestion network problems.It is shown that for network problems with convex congestion costs, an algorithm based on a shortest path algorithm, can be used to find an optimal network for any coalition. Furthermore an easy way of determining if a given network is optimal is provided.

Cooperative Situations: Representations, Games and Cost Allocations

We apply the procedure to both existing and new classes of cooperative situations: sequencing situations without initial ordering, maintenance problems, minimum cost spanning tree situations, travelling salesman problems, shared taxi problems and travelling repairman problems.