A general power index

I propose a general power index in games. The power of an agent over an outcome is understood as the equilibrium effect on the outcome of variations in the agent’s preferences. I show that the new index, ∆, has the following properties: (i) classic measures of freedom of choice are a special case of the ∆ index in the context of individual choice; (ii) the Banzhaf and Shapley-Shubik indices are special cases of the ∆ index in the context of binary voting games; (iii) in bargaining games, the ∆ index of a player combines his outside option and other parameters; (iv) the ∆ index allows a generalization of the Barry decomposition of success between power and luck to all unidimensional spatial games; (v) the ∆ power of a firm on its price is related to its Lerner index, but also to its technology and to the potential entry of competitors; (vi) in a simple competitive exchange setting, each agent has zero ∆ power on the equilibrium price but a ∆ power density can be defined, which, under some assumptions on preferences, is proportional to the agents’ wealth