We consider an alternative expression of the Shapley value that reveals a system of compensations: each player receives an equal share of the worth of each coalition he belongs to, and has to compensate an equal share of the worth of any coalition he does not belong to. We give an interpretation in terms of formation of the grand coalition according to an ordering of the players and define the ...

In an earlier paper of the author, the Multiweighted Shapley Values (MWSVs) have been introduced as linear operators on the space of TU games, which satisfy the efficiency and the dummy player axioms. An early dynamic algorithm for computing the Shapley Value is due to late M.Maschler. In the present work, we present a similar algorithm for computing the Multiweighted Shapley Values. For differ...

Uncertain coalitional game deals with situations in which the transferable payoffs are uncertain variables. The uncertain core has been proposed as the solution of uncertain coalitional game. This paper goes further by presenting two definitions of uncertain Shapley value: expected Shapley value and ̨-optimistic Shapley value. Meanwhile, some characterizations of the uncertain Shapley value are...

Optimization theory resolves problems to minimize the total costs when the agents are involved in some conflicts. In this paper, we consider how to allocate the minimized total cost among the agents. To do that, the allocation is required to be fair in a certain sense. We use a game-theoretic point of view, and provide algorithms to compute fair allocations in polynomial time for a certain conf...

The Shapley value, a solution concept from cooperative game theory, has recently been considered for both unrooted and rooted phylogenetic trees. Here, we focus on the Shapley value of unrooted trees and first revisit the so-called split counts of a phylogenetic tree and the Shapley transformation matrix that allows for the calculation of the Shapley value from the edge lengths of a tree. We sh...

Two extensions of the Shapley value are given. First we consider a probabilistic framework in which certain consistent allocation rules such as the Shapley value are characterized. The second generalization of the Shapley value is an extension to the structure of posets by means of a recursive form. In the latter setting, the Shapley value for quasi-concave games is shown to be a core-allocation.

Shapley proved the existence of an ordinal, symmetric and efficient solution for three-player bargaining problems. Ordinality refers to the covariance of the solution with respect to orderpreserving transformations of utilities. The construction of this solution is based on a special feature of the three-player utility space: given a Pareto surface in this space, each utility vector is the idea...

Coalitional game deals with situations that involve cooperations among players, and there are different solution concepts such as the core,the Shapley value and the kernel. In many situations, there is no way to predict the payoff functions except for the expert experiencesand subjective intuitions, which leads to the coalitional game with fuzzy payoffs. Within the framework of credibility theo...