The Shapley Value of Phylogenetic Trees

Every weighted tree corresponds naturally to a cooperative game that we call a "tree game"; it assigns to each subset of leaves the sum of the weights of the minimal subtree spanned by those leaves. In the context of phylogenetic trees, the leaves are species and this assignment captures the diversity present in the coalition of species considered. We consider the Shapley value of tree games and suggest a biological interpretation. We determine the linear transformation M that shows the dependence of the Shapley value on the edge weights of the tree, and we also compute a null space basis of M. Both depend on the "split counts" of the tree. Finally, we characterize the Shapley value on tree games by four axioms, a counterpart to Shapley's original theorem on the larger class of cooperative games.Comment: References added, and a section (calculating the Shapley value of a tree game from its subtrees) was removed for length reasons (request of referee) and may appear in another paper. 16 pages; related work at http://www.math.hmc.edu/~su/papers.html. Journal of Mathematical Biology, to appear. The original article is available at http://www.springerlink.co

Power indices expressed in terms of minimal winning coalitions

A voting situation is given by a set of voters and the rules of legislation that determine minimal requirements for a group of voters to pass a motion. A priori measures of voting power, such as the Shapley-Shubik index and the Banzhaf value, show the influence of the individual players. We used to calculate them by looking at marginal contributions in a simple game consisting of winning and losing coalitions derived from the rules of the legislation. We introduce a new way to calculate these measures directly from the set of minimal winning coalitions. This new approach logically appealing as it writes measures as functions of the rules of the legislation. For certain classes of games that arise naturally in applications the logical shortcut drastically simplifies calculations. The technique generalises directly to all semivalues. Keywords. Shapley-Shubik index, Banzhaf index, semivalue, minimal winning coalition, Möbius transform.Shapley-Shubik index, Banzhaf index, semivalue, minimal winning coalition, Möbius transform.

Marginal contributions and externalities in the value

For games in partition function form, we explore the implications of distinguishing between the concepts of intrinsic marginal contributions and externalities. If one requires efficiency for the grand coalition, we provide several results concerning extensions of the Shapley value. Using the axioms of efficiency, anonymity, marginality and monotonicity, we provide upper and lower bounds to players' payoffs when affected by external effects, and a characterization of an ''externality-free'' value. If the grand coalition does not form, we characterize a payoff configuration on the basis of the principle of balanced contributions. We also analyze a game of coalition formation that yields sharp prediction

Weighted Banzhaf power and interaction indexes through weighted approximations of games

The Banzhaf power index was introduced in cooperative game theory to measure the real power of players in a game. The Banzhaf interaction index was then proposed to measure the interaction degree inside coalitions of players. It was shown that the power and interaction indexes can be obtained as solutions of a standard least squares approximation problem for pseudo-Boolean functions. Considering certain weighted versions of this approximation problem, we define a class of weighted interaction indexes that generalize the Banzhaf interaction index. We show that these indexes define a subclass of the family of probabilistic interaction indexes and study their most important properties. Finally, we give an interpretation of the Banzhaf and Shapley interaction indexes as centers of mass of this subclass of interaction indexes

Steady Marginality: A Uniform Approach to Shapley Value for Games with Externalities

The Shapley value is one of the most important solution concepts in cooperative game theory. In coalitional games without externalities, it allows to compute a unique payoff division that meets certain desirable fairness axioms. However, in many realistic applications where externalities are present, Shapley's axioms fail to indicate such a unique division. Consequently, there are many extensions of Shapley value to the environment with externalities proposed in the literature built upon additional axioms. Two important such extensions are "externality-free" value by Pham Do and Norde and value that "absorbed all externalities" by McQuillin. They are good reference points in a space of potential payoff divisions for coalitional games with externalities as they limit the space at two opposite extremes. In a recent, important publication, De Clippel and Serrano presented a marginality-based axiomatization of the value by Pham Do Norde. In this paper, we propose a dual approach to marginality which allows us to derive the value of McQuillin. Thus, we close the picture outlined by De Clippel and Serrano

Characterization of the Shapley-Shubik Power Index Without the Efficiency Axiom

We show that the Shapley-Shubik power index on the domain of simple (voting) games can be uniquely characterized without the e¢ ciency axiom. In our axiomatization, the efficiency is replaced by the following weaker require- ment that we term the gain-loss axiom: any gain in power by a player implies a loss for someone else (the axiom does not specify the extent of the loss). The rest of our axioms are standard: transfer (which is the version of additivity adapted for simple games), symmetry or equal treatment, and dummySimple Games, Shapley-Shubik Power Index, Effciency Axiom