Axiomatizations of two types of Shapley values for games on union closed systems

A situation in which a finite set of players can obtain certain payoffs by cooperation can be described by a cooperative game with transferable utility, or simply a TU-game. A (single-valued) solution for TU-games assigns a payoff distribution to every TU-game. A well-known solution is the Shapley value. In the literature various models of games with restricted cooperation can be found. So, instead of allowing all subsets of the player set N to form, it is assumed that the set of feasible coalitions is a subset of the power set of N. In this paper, we consider such sets of feasible coalitions that are closed under union, i.e. for any two feasible coalitions also their union is feasible. We consider and axiomatize two solutions or rules for these games that generalize the Shapley value: one is obtained as the conjunctive permission value using a corresponding superior graph, the other is defined as the Shapley value of a modified game similar as the Myerson value for games with limited communication. © 2010 The Author(s)

Proportionate flow shop games

htmlabstractIn a proportionate flow shop problem several jobs have to be processed through a fixed sequence of machines and the processing time of each job is equal on all machines. By identifying jobs with agents, whose costs linearly depend on the completion time of their jobs, and assuming an initial processing order on the jobs, we face two problems: the first one is how to obtain an optimal order that minimizes the total processing cost, the second one is how to allocate the cost savings obtained by ordering the jobs optimally. In this paper we focus on the allocation problem. PFS games are defined as cooperative games associated to proportionate flow shop problems. It is seen that PFS games have a nonempty core. Moreover, it is shown that PFS games are convex if the jobs are initially ordered in decreasing urgency. For this case an explicit game independent expression for the Shapley value is provid

Component efficient solutions in line-graph games with applications

Recently, applications of cooperative game theory to economic allocation problems have gained popularity. We investigate a class of cooperative games that generalizes some economic applications with a similar structure. These are the so-called line-graph games being cooperative TU-games in which the players are linearly ordered. Examples of situations that can be modeled like this are sequencing situations and water distribution problems. We define four properties with respect to deleting edges that each selects a unique component efficient solution on the class of line-graph games. We interpret these solutions and properties in terms of dividend distributions, and apply them to economic situations. © 2006 Springer-Verlag