Marco Slikker

Social and economic networks in cooperative game theory

Preface. Part I: Social and Economic Networks in Cooperative Situations. 1. Games and Networks. 2. Restricted Cooperation in Games. 3. Inheritance of Properties in Communication Situations. 4. Variants on the Basic Model. Part II: Network Formation. 5. Noncooperative Games. 6. A Network-Formation Model in Extensive Form. 7. A Network-Formation Model in Strategic Form. 8. Network Formation with Costs for Establishing Links. 9. A One-Stage Model of Network Formation and Payoff Division. 10. Network Formation and Potential Games. 11. Network Formation and Reward Functions. References. Notations. Index.

Cooperation between multiple news-vendors with transshipments

We study a situation with n retailers, each of them facing a news-vendor problem, i.e., selling to customers over a finite period of time (product with a short life cycle, such as fashion). Groups of retailers might improve their expected joint profit by coordinating their orders, followed by transshipments after demand realization is known. We analyze these situations by defining a cooperative game, called a general news-vendor game, for such a situation with n retailers. We concentrate on whether it makes sense to cooperate by studying properties of general news-vendor games. Our main result states that general news-vendor games have non-empty cores.

Network formation models with costs for establishing links

Abstract. In this paper we study endogenous formation of communication networks in situations where the economic possibilities of groups of players can be described by a cooperative game. We concentrate on the influence that the existence of costs for establishing communication links has on the communication networks that are formed. The starting points in this paper are two game-theoretic models of the formation of communication links that were studied in the literature fairly recently, the extensive-form model by Aumann and Myerson (1988) and the strategic-form model that was studied by Dutta et al. (1998). We follow their analyses as closely as possible and use an extension of the Myerson value to determine the payoffs to the players in communication situations when forming links is not costless. We find that it is possible that as the costs of establishing links increase, more links are formed.

A one-stage model of link formation and payoff division

In this paper we introduce a strategic form model in which cooperation structures and divisions of the payoffs are determined simultaneously. We analyze the cooperation structures and payoff divisions that result according to several equilibrium concepts. We find that essentially no cycles will result and that a player need not profit from a central position in a cooperation structure.

On a New Class of Parallel Sequencing Situations and Related Games

This paper considers a special class of sequencing situations with two parallel machines in which each agent has precisely two jobs to be processed, one on each machine. The costs of an agent depend linearly on the final completion time of his jobs. We describe a procedure that provides an optimal processing order of the jobs for some particular classes. Furthermore, we study cooperative games arising from these sequencing situations. Our main result will be on the balancedness of these games.

Link Monotonic Allocation Schemes

A network is a graph where the nodes represent players and the links represent bilateral interaction between the players. A reward game assigns a value to every network on a fixed set of players. An allocation scheme specifies how to distribute the worth of every network among the players. This allocation scheme is link monotonic if extending the network does not decrease the payoff of any player. We characterize the class of reward games that have a link monotonic allocation scheme. Two allocation schemes for reward games are studied, the Myerson allocation scheme and the position allocation scheme, which are both based on allocation rules for communication situations. We introduce two notions of convexity in the setting of reward games and with these notions of convexity we characterize the classes of reward games where the Myerson allocation scheme and the position allocation scheme are link monotonic. As a by-product we find a characterization of the Myerson value and the position value on the class of reward games using potentials.

An Iterative Procedure for Evaluating Digraph Competitions

A competition which is based on the results of (partial) pairwise comparisons can be modelled by means of a directed graph. Given initial weights on the nodes in such digraph competitions, we view the measurement of the importance (i.e., the cardinal ranking) of the nodes as an allocation problem where we redistribute the initial weights on the basis of insights from cooperative game theory. After describing the resulting procedure of redistributing the initial weights, an iterative process is described that repeats this procedure: at each step the allocation obtained in the previous step determines the new input weights. Existence and uniqueness of the limit is established for arbitrary digraphs. Applications to the evaluation of, e.g., sport competitions and paired comparison experiments are discussed.

Characterizations of a multi-choice value

Abstract. A multi-choice game is a generalization of a cooperative game in which each player has several activity levels. We study the extended Shapley value as proposed by Derks and Peters (1993). Van den Nouweland (1993) provided a characterization that is an extension of Youngs (1985) characterization of the Shapley value. Here we provide several other characterizations, one of which is the analogue of Shapleys (1953) original characterization. The three other characterizations are inspired by Myersons (1980) characterization of the Shapley value using balanced contributions.

Bidding for surplus in network allocation problems

In this paper we study non-cooperative foundations of network allocation rules. We focus on three allocation rules: the Myerson value, the position value and the component-wise egalitarian solution. For any of these three rules we provide a characterization based on component efficiency and some balanced contribution property. Additionally, we present three mechanisms whose equilibrium payoffs are well defined and coincide with the three rules under consideration if the underlying value function is monotonic. Non-monotonic value functions are shown to deal with allocation rules applied to monotonic covers. The mechanisms are inspired by the implementation of the Shapley value by Perez-Castrillo and Wettstein [Bidding for the surplus: a non-cooperative approach to the Shapley value, J. Econ. Theory 100 (2) (2001) 274–294].

The egalitarian solution for convex games: some characterizations

The egalitarian solution for TU-games as introduced by Dutta and Ray [Dutta, B., Ray, D., 1989. A concept of egalitarianism under participation constraints, Econometrica 57, 615–635] is studied. Five characterizations of the restriction of this solution to the class of convex games are given. They all involve a stability property due to the concept of the equal division core from Selten [Selten, R., 1972. Equal share analysis of characteristic function experiments. In: Sauermann, H. (Ed.), Beitrage zur experimentellen Wirtschaftsforschung, Vol. 3, J.C.B. Mohr, Tubingen, pp. 130–165] and all but the third characterization involve a property restricting maximum payoffs. The first two characterizations use in addition efficiency and the reduced game properties of Hart and Mas-Colell [Hart, S., Mas-Colell, A., 1989. Potential, value and consistency. Econometrica 57, 589–614] and Davis and Maschler [Davis, M., Maschler, M., 1965. The kernel of a cooperative game. Naval Research Logistics Quarterly 12, 223–259], respectively. The fourth and fifth characterization only need in addition weak variants of the reduced game properties mentioned above. The third characterization involves besides the stability condition, efficiency and a new consistency property.