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 and . 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.