Joachim Rosenmüller

L. P. -games with Sufficiently Many Players

L.P.-games aren-person cooperative games arising from a linear program as studied by Owen and Billera/Raanan. Employing a version of nondegeneracy of games developed by Weidner and the author, we show that methods from the Geometry of Numbers are the suitable tool in order to obtain lower bounds for the number of players of each type that ensure that the core and the competitive equilibrium coincide.

The nucleolus of homogeneous games with steps

Abstract Homogeneous games were introduced by von Neumann and Morgenstern in the constant-sum case. Peleg studied the kernel and the nucleolus within this framework. However, for the general nonconstant-sum case Ostmann invented the unique minimal representation, Rosenmuller gave a second characterization and Sudholter discovered the “incidence vector”. Based on these results Peleg and Rosenmuller treated several solution concepts for “games without steps”. The present paper treats the case of games “with steps”. It is shown that with a suitable version of a “truncated game” the nucleolus of a game is essentially the one obtained by truncating behind the “largest step”. As the truncated version has “no steps”, the case “with steps” is reduced to the one “without steps”, which is treated in the paper by Peleg and Rosenmuller.

The least core, nucleolus, and kernel of homogeneous weighted majority games☆

Abstract Homogeneous weighted majority games were already introduced by von Neumann and Morgenstern; they discussed uniqueness of the representation and the “main simple solution” for constant-sum games. For the same class of games Peleg studied the kernel and the nucleolus. Ostmann and Rosenmuller described the nature of representations of general homogeneous weighted majority games; see also Sudholter. The present paper starts out to close the gap: for the general homogeneous weighted majority game “without steps,” we discuss the least core, the nucleolus, and the kernel and show their close relationship (coincidence) with the unique minimal representation.

Weighted majority games and the matrix of homogeneity

If a weighted majority game (not necessarily constant sum or super additive) is described by the weights (voting strength) of the players involved and a majority level, then it is desirable to know whether the game is in addition homogeneous. The paper provides a recursive procedure defining a test for homogeneity. This procedure involves the computation of a number theoretical function, the “matrix of homogeneity”. If this matrix is knownall majority levels with respect to which the given set of weights represents a homogeneous simple weighted majority game are known at once.ZusammenfassungEin gewichtetes Majoritätsspiel (nicht notwendig superadditiv oder mit Konstantsummeneigenschaft) kann beschrieben werden durch die Gewichte (die Abstimmungsstärke) der beteiligten Spieler und durch eine Mehrheitsschranke. Wenn eine solche Darstellung gegeben ist, so ist es wünschenswert, das Spiel zusätzlich auf die Eigenschaft der Homogenität hin zu untersuchen. Die vorliegende Arbeit stellt eine rekursive Prozedur zur Verfügung, die einen Test auf Homogenität einer vorgelegten Darstellung eines Spieles impliziert. Die Prozedur ermöglicht die Berechnung einer zahlentechnischen Funktion, der „Homogenitätsmatrix”. Wenn diese Matrix bekannt ist, kann man bei einem vorgegebenen Gewichtssatz alle Majoritätsschranken angeben, bezüglich welcher die vorgelegten Gewichte ein homogenes einfaches gewichtetes Majoritätsspiel darstellen.

A characterization of vNM-stable sets for linear production games

Abstract. We discuss linear production games or market games with a continuum of players which are represented as minima of finitely many nonatomic measures.¶Within this context we consider vNM-Stable Sets according to von Neumann and Morgenstern. We classify or characterize all solutions of this type which are convex polyhedra, i.e., which are the convex hull of finitely many imputations. Specifically, in each convex polyhedral vNM-Stable Set (and not only in the symmetric ones), the different types of traders must organize themselves into cartels. The vNM-Stable Set is then the convex hull of the utility distributions of the cartels.¶Using the results from the continuum, we obtain a similar characterization also for finite glove market games.

Computing the Minkowski sum of prisms

AbstractWithin this paper we study the Minkowski sum of prisms (“Cephoids”) in a finite dimensional vector space. For a vector

Implementing the modified LH algorithm

a \in \mathbb{R}^n

The canonical extensive form of a game form: symmetries

with positive components we write