An alternative to the Shapley–Shubik index is called the Banzhaf–Coleman index, which is the…

An alternative to the Shapley–Shubik index is called the Banzhaf–Coleman index, which is the imputation b = (b1, b2,…, bn) defined by

              

where Wi is the set of coalitions that win with player i and lose without player i. We also refer to Wi as the coalitions for which player i is critical for passing a resolution. It is primarily used in weighted voting systems in which player i has wi votes, and a resolution passes if j w j ≥ q, where q is known as the quota. (a) Why is this a reasonable definition as an index of power.

(b) Consider the four player-weighted voting system in which a resolution passes if it receives at least 10 votes. Player 1 has 6 votes, player 2 has 5 votes, player 3 has 4 votes, and player 1 has 2 votes.

(c) Find the Shapley–Shubik index.

(d) Find the Banzhaf–Coleman index.