Veto players, the kernel of the Shapley value and its characterization

In this article, we provide a new basis for the kernel of the Shapley value (Shapley, 1953), which is used to construct a new axiom of invariance, and to provide a new axiomatic characterization of the Shapley value. This characterization only invokes marginalistic principles, and does not rely on classical axioms such as symmetry, efficiency or linearity. Moreover, our approach reveals a new instructive role played by veto players

New axiomatizations of the Shapley interaction index for bi-capacities

International audienceBi-capacities are a natural generalization of capacities (or fuzzy measures) in a context of decision making where underlying scales are bipolar. They are able to capture a wide variety of decision behaviours. After a short presentation of the basis structure, we introduce the Shapley value and the interaction index for capacities. Afterwards, the case of bi-capacities is studied with new axiomatizations of the interaction index

Shapley Value Based Pricing for Auctions and Exchanges

This is the author accepted manuscript. The final version is available from Elsevier via the DOI in this record.This paper explores how the Shapley value can be used as the basis of a payment rule for auctions and exchanges. The standard Shapley value is modified so that losing bidders do not make or receive any payments. The new rule, called the balanced winner contribution (BWC) rule, satisfies a variation of Myerson’s balanced contribution property. The payment rule is fair in the sense that, with respect to reported values, the members of every pair of traders make equal contributions to each other’s share of the gains from trade. BWC payments can be used in single-item auctions and more complex auctions and exchanges with multiple items and package bidding. A series of examples is presented to illustrate how the BWC rule works and how the payments compare to those based on competitive prices, the core, and the Vickrey-Clarke-Groves mechanis

Pure bargaining problems with a coalition structure

The final publication is available at Springer via http://dx.doi.org/10.1007/s41412-016-0007-2We consider here pure bargaining problems endowed with a coalition structure such that each union is given its own utility. In this context we use the Shapley rule in order to assess the main options available to the agents: individual behavior, cooperative behavior, isolated unions behavior, and bargaining unions behavior. The latter two respectively recall the treatment given by Aumann–Drèze and Owen to cooperative games with a coalition structure. A numerical example illustrates the procedure. We provide criteria to compare any pair of behaviors for each agent, introduce and axiomatically characterize a modified Shapley rule, and determine its natural domain, that is, the set of problems where the bargaining unions behavior is the best option for all agents.Peer ReviewedPostprint (author's final draft

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

Exact Algorithms for Solving Stochastic Games

Shapley's discounted stochastic games, Everett's recursive games and Gillette's undiscounted stochastic games are classical models of game theory describing two-player zero-sum games of potentially infinite duration. We describe algorithms for exactly solving these games

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

1-concave basis for TU games

The first stage of research, twenty years ago, on the subclass of 1-convex TU games dealt with its characterization through some regular core structure. Appealing abstract and practical examples of 1-convex games were missing until now. Both drawbacks are solved. On the one hand, a generating set for the cone of 1-concave cost games is introduced with clear affinities to the unanimity games taking into account the complementary transformation on coalitions. The dividends within this new game representation are used to characterize the 1-concavity constraint as well as to investigate the core property of the Shapley value for cost games. We present a simple formula to compute the nucleolus and the τ-value within the class of 1-convex/1-concave games and show that in a 1-convex/1-concave game there is an explicit relation between the nucleolus and the Shapley value. On the other hand, an appealing practical example of 1-concave cost game has cropped up not long ago in Sales’s Ph.D study of Catalan university library consortium for subscription to journals issued by Kluwer publishing house, the so-called library cost game which turn out to be decomposable into the abstract 1-concave cost games of the generating set mentioned above