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. We also include a brief discussion of the core of tree games.
Abstract. We develop a procedure for implementing an e‰cient and envy-free allocation of m objects among n individuals with the possibility of monetary side-payments, assuming that players have quasi-linear utility functions. The procedure eliminates envy by compensating envious players. It is fully descriptive and says explicitly which compensations should be made, and in what order. Moreover, it is simple enough to be carried out without computer support. We formally characterize the properties of the procedure, show how it establishes envy-freeness with minimal resources, and demonstrate its application to a wide class of fair-division problems.
We establish axioms under which a bargaining solution can be found by the maximization of the CES function and is unique up to specifications of the distribution and elasticity parameters. This solution is referred to as the CES solution which includes the Nash and egalitarian solutions as special cases. Next, we consider a normalization of the CES function and establish axioms, under which a bargaining solution can be found by the maximization of the normalized CES function and is unique up to specifications of the distribution and substitution parameters. We refer to this solution as the normalized CES solution which includes the Nash and Kalai-Smorodinsky solutions as special cases. Our paper contributes to bargaining theory by establishing unified characterizations of existing as well as a great variety of new bargaining solutions.
We consider the class of proper monotonic simple games and study coalition formation when an exogenous weight vector and a solution concept are combined to guide the distribution power within winning coalitions. These distributions induce players' preferences over coalitions in a hedonic game. We formalize the notion of semistrict core stability, which is stronger than the standard core concept but weaker than the strict core notion and derive two characterization results for the semistrict core, dependent on conditions we impose on the solution concept. It turns out that a bounded power condition, which connects exogenous weights and the solution, is crucial. It generalizes a condition termed "absence of the paradox of smaller coalitions" that was previously used to derive core existence results.JEL Classification: D72, C71
We model the process of coalition formation in the 16th German Bundestag as a hedonic coalition formation game. In order to induce players' preferences in the game we apply the Shapley value of the simple game describing all winning coalitions in the Bundestag. Using different stability notions for hedonic games we prove that the "most" stable government is formed by the Union Parties together with the Social Democratic Party. JEL Classification: D72, C71.
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