C-diagrams, shifts and solidarity values

In the theory of cooperative games so called dividends of a coalition $S$ are considered, which are defined as $\frac{c_S}{|S|}$. The costs $c_S$ form a c-diagram. On these c-diagrams several types of shifts are defined and analysed. Different solution concepts and their properties are related to shifts. We introduce reward games and fine games as components of a cooperative game. Some solution concepts for applications are analysed in terms of c-diagrams, as well as the solidarity concept. \u

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

On the Shapley value of a minimum cost spanning tree problem

We associate an optimistic coalitional game with each minimum cost spanning tree problem. We define the worth of a coalition as the cost of connection assuming that the rest of the agents are already connected. We define a cost sharing rule as the Shapley value of this optimistic game. We prove that this rule coincides with a rule present in the literature under different names. We also introduce a new characterization using a property of equal contributions.minimum cost spanning tree problems Shapley value

OBLIGATION RULES

We provide a characterization of the obligation rules in the context of minimum cost spanning tree games. We also explore the relation between obligation rules and random order values of the irreducible cost game - it is shown that the later is a subset of the obligation rules. Moreover we provide a necessary and sucient condition on obligation function such that the corresponding obligation rule coincides with a random order value.

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. Finally, we characterize the Shapley value on tree games by five axioms, a counterpart to Shapley's original theorem on the larger class of cooperative games. We also include a brief discussion of the core of tree games.

Realizing efficient outcomes in cost spanning problems

We propose a simple non-cooperative mechanism of network formation in cost spanning tree problems. The only subgame equilibrium payoff is efficient. Moreover, we extend the result to the case of budget restrictions. The equilibrium payoff can them be easily adapted to the framework of Steiner trees.efficiency, cost spanning tree problem, cost allocation, network formation, subgame perfect equilibrium, budget restrictions, Steiner trees

No advantageous merging in minimum cost spanning tree problems

In the context of cost sharing in minimum cost spanning tree problems, we introduce a property called No Advantageous Merging. This property implies that no group of agents can be better off claiming to be a single node. We show that the sharing rule that assigns to each agent his own connection cost (the Bird rule) satisfies this property. Moreover, we provide a characterization of the Bird rule using No Advantageous Merging.Minimum cost spanning tree problems; cost sharing; Bird rule; No Advantageous Merging

Additivity in cost spanning tree problems

We characterize a rule in cost spanning tree problems using an additivity property and some basic properties. If the set of possible agents has at least three agents, these basic properties are symmetry and separability. If the set of possible agents has two agents, we must add positivity. In both characterizations we can replace separability by population monotonicity.cost spanning tree problems additivity characterization