Shapley–Shubik power index

The Shapley–Shubik power index was formulated by Lloyd Shapley and Martin Shubik in 1954 to measure the powers of players in a voting game. The index often reveals surprising power distribution that is not obvious on the surface.

The constituents of a voting system, such as legislative bodies, executives, shareholders, individual legislators, and so forth, can be viewed as players in an n-player game. Players with the same preferences form coalitions. Any coalition that has enough votes to pass a bill or elect a candidate is called winning, and the others are called losing. Based on Shapley value, Shapley and Shubik concluded that the power of a coalition was not simply proportional to its size.

The power of a coalition (or a player) is measured by the fraction of the possible voting sequences in which that coalition casts the deciding vote, that is, the vote that first guarantees passage or failure.

The power index is normalized between 0 and 1. A power of 0 means that a coalition has no effect at all on the outcome of the game; and a power of 1 means a coalition determines the outcome by its vote. Also the sum of the powers of all the players is always equal to 1.

There are some algorithms for calculating the power index, e.g., dynamic programming techniques, enumeration methods and Monte Carlo methods.

Since Shapley and Shubik have published their paper, several axiomatic approaches have been used to mathematically study the Shapley–Shubik power index, with the anonymity axiom, the null player axiom, the efficiency axiom and the transfer axiom being the most widely used. However, these have been criticised, especially the transfer axiom, which has led to other axioms being proposed as a replacement.