Gerard van der Laan

The average tree solution for cycle-free graph games

In this paper we study cooperative games with limited cooperation possibilities, represented by an undirected cycle-free communication graph. Players in the game can cooperate if and only if they are connected in the graph. We introduce a new single-valued solution concept, the average tree solution. Our solution is characterized by component efficiency and component fairness. The interpretation of component fairness is that deleting a link between two players yields for both resulting components the same average change in payoff, where the average is taken over the players in the component. The average tree solution is always in the core of the restricted game and can be easily computed as the average of n specific marginal vectors, where n is the number of players. We also show that the average tree solution can be generated by a specific distribution of the Harsanyi dividends.

The Average Tree Solution for Cooperative Games with Communication Structure

We study cooperative games with communication structure, represented by an undirected graph. Players in the game are able to cooperate only if they can form a network in the graph. A single-valued solution, the average tree solution, is proposed for this class of games. The average tree solution is defined to be the average of all these payoff vectors. It is shown that if a game has a complete communication structure, then the proposed solution coincides with the Shapley value, and that if the game has a cycle-free communication structure, it is the solution proposed by Herings, van der Laan and Talman in 2008. We introduce the notion of link-convexity, under which the game is shown to have a non-empty core and the average tree solution lies in the core. In general, link-convexity is weaker than convexity. For games with a cycle-free communication structure, link-convexity is even weaker than super-additivity.

Simplicial approximation of unemployment equilibria

Abstract In this paper we consider an exchange economy with fixed prices and quantity rationing. An unemployment equilibrium is an equilibrium with quantity constraints on the supplies only. A simplicial approximation algorithm is used to find an unemployment equilibrium such that at least one commodity is not rationed. Moreover, some insight about the structure of the set of unemployment equilibria is obtained from the sequence of simplices generated by the algorithm.

Harsanyi Power Solutions for Graph-Restricted Games

We consider cooperative transferable utility games, or simply TU-games, with limited communication structure in which players can cooperate if and only if they are connected in the communication graph. Solutions for such graph games can be obtained by applying standard solutions to a modified or restricted game that takes account of the cooperation restrictions. We discuss Harsanyi solutions which distribute dividends such that the dividend shares of players in a coalition are based on power measures for nodes in corresponding communication graphs. We provide axiomatic characterizations of the Harsanyi power solutions on the class of cycle-free graph games and on the class of all graph games. Special attention is given to the Harsanyi degree solution which equals the Shapley value on the class of complete graph games and equals the position value on the class of cycle-free graph games. The Myerson value is the Harsanyi power solution that is based on the equal power measure. Finally, various applications are discussed.

A class of consistent share functions for games in coalition structure

A cooperative game with transferable utility -or simply a TU-game- describes a situation in which players can obtain certain payoffs by cooperation.A value function for these games is a function which assigns to every such a game a distribution of the payoffs over the players in the game.An alternative type of solutions are share functions which assign to every player in a TU-game its share in the payoffs to be distributed.In this paper we consider cooperative games in which the players are organized into an a priori coalition structure being a finite partition of the set of players.We introduce a general method for defining a class of share functions for such games in coalition structure using a multiplication property that states that the share of player i in the total payoff is equal to the share of player i in some internal game within i s a priori coalition, multiplied by the share of this coalition in an external game between the a priori given coalitions.We show that these coalition structure share functions satisfy certain consistency properties.We provide axiomatizations of this class of coalition structure share functions using these consistency and multiplication properties.

Existence of an Equilibrium in a Competitive Economy with Indivisibilities and Money

In this paper we introduce a model of an exchange economy with indivisible goods and money.There are nitely many agents each of whom owns one unit of each of nitely many di erent types of indivisible goods and certain amount of money. Each type of indivisible good is subject to quality di erentiation.We demonstrate that under fairly mild conditions on demand the economy has a price equilibrium.The proof is based on a generalization of the well-known lemma of Knaster, Kuratowski and Mazurkiewicz (KKM) in combinatorial topology.The results in the paper generalize those of Gale in case of just one indivisible good present in the economy. (This abstract was borrowed from another version of this item.) (This abstract was borrowed from another version of this item.)

A fixed point theorem for discontinuous functions

Any function from a non-empty polytope into itself that is locally gross direction preserving is shown to have the fixed point property. Brouwers fixed point theorem for continuous functions is a special case. We discuss the application of the result in the area of non-cooperative game theory. This discussion paper has resulted in a publication in Operations Research Letters . (2008, vol. 36, issue 1, pp.89-93.)(This abstract was borrowed from another version of this item.)(This abstract was borrowed from another version of this item.)

Supply-constrained fixed price equilibria in monetary economies

Abstract In this paper we consider an economy with restrictions on the relative prices of non-money commodities. The non-money commodities are partitioned into two groups, index makers and price following commodities. Then two cases are considered, namely that the relative prices of the index makers are fixed, respectively free. In both cases the money prices are flexible, whereas the relative prices of the price following commodities are tied to the prices of the index makers. The existence of a supply-constrained equilibrium is proved with (i) no rationing on the money commodities (stores of value), and (ii) at least one non-money commodity is not rationed. If prices of the index makers are fixed the result strengthens a theorem of Dehez and Dreze, if the prices are free a theorem of Kurz is strengthened. This paper is not only concerned with these existence results, but also with the question whether supply-constrained equilibria should appear more frequently than demand-constrained equilibria.

A Vector Labeling Method for Solving Discrete Zero Point and Complementarity Problems

In this paper we establish the existence of a discrete zero point of a function from the

The socially stable core in structured transferable utility games

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