Power Indices based on Ordinal Games

Shapley (1953) and Banzhaf (1965) solved multiperson cooperative games by assessing a value (power index) to each player of a game. Shapley value can be interpreted as giving each player his average marginal contribution to all coalitions of players. Banzhaf power index related to a player i corresponds to the probability that a coalition wins when the player i joins randomly the coalition. Given a game v : 2N → IR (i.e. a real valued set function on N), where N is a set of players, the Shapley power index related to the player i ∈ N is given by