Jos A. M. Potters

Cores and related solution concepts for multi-choice games

A multi-choice game is a generalization of a cooperative game in which each player has several activity levels. Cooperative games form a subclass of the class of multi-choice games.This paper extends some solution concepts for cooperative games to multi-choice games. In particular, the notions of core, dominance core and Weber set are extended. Relations between cores and dominance cores and between cores and Weber sets are extensively studied. A class of flow games is introduced and relations with non-negative games with non-empty cores are investigated.

Sequencing and Cooperation

In machine scheduling the first problem is to find a timetable that is optimal with respect to some efficiency criterion. If the jobs come from different clients the solution of the optimization problem is not the end of the story. In addition, we have to decide how the minimal total cost must be distributed among the parties involved. In this note, cost allocation problems will be considered to arise from one-machine scheduling problems with an additive and weakly increasing cost function. We will show that the cooperative games related to these cost allocation problems have a nonempty core. Furthermore, we give a rule that assigns a core element of the associated cost saving game to each scheduling problem of this kind and an initial order of the jobs.

Permutation games: Another class of totally balanced games

SummaryA class of cooperative games in characteristic function form arising from certain sequencing problems and assignment problems, is introduced. It is shown that games of this class are totally balanced. In the proof of this fact we use the Birkhoff-von Neumann theorem on doubly stochastic matrices and the Bondareva-Shapley theorem on balanced games. It turns out that this class of permutation games coincides with the class of totally balanced games if the number of players is smaller than four. For larger games the class of permutation games is a nonconvex subset of the convex cone of totally balanced games.ZusammenfassungWir führen eine Klasse von kooperativen Spielen in charakteristischer Funktionsform ein, die bei gewissen Folgeproblemen und Zuordnungsproblemen auftreten. Wir zeigen, daß diese Spiele vollständig balanciert sind. Zum Beweis verwenden wir den Satz von Birkhoff-von Neumann über doppelt stochastische Matrizen und den Satz von Bondareva-Shaplex über balancierte Spiele. Es zeigt sich, daß diese Klasse von Permutationsspielen mit der Klasse von vollständig balancierten Spiele übereinstimmt, falls die Zahl der Spieler kleiner als vier ist. Für größere Spiele ist die Klasse der Permutationsspiele eine nichtkonvexe Teilmenge des konvexen Kegels der vollständig balancierten Spiele.

Airport problems and consistent allocation rules

Abstract A class of single valued rules for airport problems is considered. The common properties of these rules are efficiency, reasonableness and a weak form of consistency. These solutions are automatically members of the core for the associated airport game. Every weighted Shapley value, the nucleolus, and the modified nucleolus turn out to belong to this class of rules. The τ -value, however, does not to belong to this class. As a side result we prove that, for airport games, the modified nucleolus and the prenucleolus of the dual game coincide. Furthermore, we investigate monotonicity properties of the rules and axiomatize the Shapley value, nucleolus, and modified nucleolus on the class of airport games.

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

Flow-shops with a dominant machine

Abstract Flow-shop problems with a dominant machine are introduced. We show that for this kind of problems we can restrict our search for an optimal schedule to permutation schedules if the optimality criterion is regular. Furthermore, we give an expression for the completion times with respect to a semi-active permutation schedule and we give fast algorithms for the weighted completion times and the maximal lateness criterion.

Prosperity properties of TU-games

Abstract. An important open problem in the theory of TU-games is to determine whether a game has a stable core (Von Neumann-Morgenstern solution (1944)). This seems to be a rather difficult combinatorial problem. There are many sufficient conditions for core-stability. Convexity is probably the best known of these properties. Other properties implying stability of the core are subconvexity and largeness of the core (two properties introduced by Sharkey (1982)) and a property that we have baptized extendability and is introduced by Kikuta and Shapley (1986). These last three properties have a feature in common: if we start with an arbitrary TU-game and increase only the value of the grand coalition, these properties arise at some moment and are kept if we go on with increasing the value of the grand coalition. We call such properties prosperity properties. In this paper we investigate the relations between several prosperity properties and their relation with core-stability. By counter examples we show that all the prosperity properties we consider are different.

Computing the nucleolus by solving a prolonged simplex algorithm

This paper describes a fast algorithm to find the nucleolus of any game with a nonempty imputation set. It is based on the algorithm scheme of Maschler et al. Maschler, M., J. Potters, S. Tijs. 1992. The general nucleolus and the reduced game property. Internal. J. Game Theory21 83--106. for the general nucleolus.

Minimality of Consistent Solutions for Strategic Games, in Particular for Potential Games

SummarySolutions defined on classes of strategic games, satisfying One-Person Rationality (OPR), Non-emptiness (NEM) and Consistency (CONS) are considered. The main question to be answered is whether these conditions characterize the Nash Equilibrium solutionNE for the given class of games. Depending on the structure of the class of games positive as well as negative answers are obtained. A graph-theoretical framework will be developed to express sufficient conditions for a positive or a negative answer. For the class of (finite) strategic games with at least one Nash equilibrium the answer is positive. For several classes of potential games the answer is negative.

On finding an envy-free Pareto-optimal division

This paper describes an algorithm to find an (α-)envy-free Pareto-optimal division in the case of a finite number of homogeneous infinitely divisible goods and linear utility functions. It is used to find an allocation in the classical cake division problem that is almost Pareto-optimal and α-envy-free.