Power indices for networks, with applications to matching markets

We study situations where agents can form or sever links in a network: what agents can do exactly is described by effectivity functions. A power index assigns to such an effectivity function a number for each agent, measuring the opportunities of that agent. We characterize a class of power indices by four axioms: the Transfer Property, the Dummy Property, Symmetry, and Network Neutrality. As a corollary, we obtain power indices for the case where effectivity functions are induced by preferences of agents about the other agents. Applications include one-to-one, one-to-many, and many-to-many matching markets, as well as roommate problems.