DETERMINING THE NUCLEOLUS OF COMPROMISE STABLE GAMES

The main goal is to illustrate that the so-called indirect function of a cooperative game in characteristic function form is applicable to determine the nucleolus for a subclass of coalitional games called compromise stable transferable utility (TU) games. In accordance with the Fenchel–Moreau theory on conjugate functions, the indirect function is known as the dual representation of the characteristic function of the coalitional game. The key feature of a compromise stable TU game is the coincidence of its core with a box prescribed by certain upper and lower core bounds. For the purpose of the determination of the nucleolus, we benefit from the interrelationship between the indirect function and the prekernel of coalitional TU games. The class of compromise stable TU games contains the subclasses of clan games, big boss games and $1$- and $2$-convex $n$-person TU games. As an adjunct, this paper reports the indirect function of clan games for the purpose of determining their nucleolus.

PROJECT MANAGEMENT GAMES

This paper studies situations in which companies can cooperate in order to decrease the earliest completion time of a project that consists of several tasks. This is beneficial for the client who wants the project to be completed as early as possible. The client is willing to pay more for an earlier completion time. The total payoff must be allocated among the companies that cooperate. Cooperative game theory is used to model this situation. Conditions for the core of the game to be nonempty are derived. We study a class of project management games for which necessary and sufficient conditions for the nonemptiness of the core can be derived. We will show that a subset of the set of balanced project management games can be partitioned into a class of 1-convex games and a class of big boss games. Expressions for the extreme points of the core, the τ-value, the nucleolus, and the Shapley-value of games in these two classes are derived.

SHAPLEY VALUE IN A MODEL OF INFORMATION TRANSFERAL

In this paper we analyze the value of the information in a cooperative model. There is an agent (the innovator), having relevant information which can be sold to some potential buyers. The n potential users of the information share a market. The expected utility of each of them can be improved by obtaining the information. The whole situation is modelled as a (n + 1)–person cooperative game. We study the properties of the characteristic function of this game. It fulfills a weak version of the superadditivity property, namely 0-monotonicity. The game is proved to be monotonic. We compute the Shapley value and we prove it is an imputation for the game. We compare the Shapley value with the equilibrium studied in a noncooperative model by Quintas (1995). We also study some limit cases when the potential buyers are completely informed or uninformed. It includes Big Boss Games (Muto et al. (1988)) and other limit cases. We conclude that in a cooperative environment the buyers avoid being exploited by the innovator. Conversely the innovator obtains a higher utility in a noncooperative game.

ON COOPERATIVE GAMES RELATED TO MARKET SITUATIONS AND AUCTIONS

We consider a market situation with two corners. One corner consists of a single seller with one object, and the other corner consists of potential buyers who all want the object. We suppose that the valuations of the object for the different buyers are known by all of them. Then two types of cooperative games, which we call market games and ring games, corresponding to such market situations are considered. Market games are related to special total big boss games, while ring games are related to special convex games, the peer group games. It turns out that there exists a duality relation between the market game and the ring game arising from the same two-corner market situation. For both classes of games relevant solution concepts are studied.