Indirect control and power in mutual control structures

In a mutual control structure (mcs) agents exercise control over each other. Typical examples occur in the area of corporate governance: firms and investment companies exercise mutual control, in particular by owning each others' stocks. We represent such situations in two equivalent ways: by a function assigning to each coalition the set of controlled players, and by a simple game structure in which for each player a simple game describes who controls that player. These concepts are similar to authority distributions and command games in Hu and Shapley, 2003a and Hu and Shapley, 2003b. An mcs is invariant if it incorporates all indirect control relations. We axiomatically develop a class of power indices for invariant mcs. We impose four axioms with a plausible interpretation in this framework, which together characterize a broad class of power indices based on dividends resulting both from exercising and from undergoing control. Extra conditions can further refine this broad class.