Pivot Player

A pivot player is a player who, in the formation of a coalition, turns a losing coalition into a winning one. This term was introduced by Lloyd Stowell Shapley and Martin Shubik in an important article published in 1954 in which they particularized the Shapley value of a game (Shapley, 1953) in the context of voting games, thus creating the Shapley-Shubik power index. The concept of a pivot needs to be distinguished from the similar but not identical concept of swing, which is the basis for the Banzhaf-Coleman-Penrose index and many others.

Let N={1,2,…,n} be a set of players. A cooperative game v on N is said to be expressed in characteristic function form if a function v (the characteristic function) is given that assigns a value v(S) to every coalition S ⫅N under the condition v(Ø) = 0. A game v is simple if v(S) = 0 or v(S) = 1 for all coalitions S ⫅N. In the first case, S is losing; in the second case, winning.

Consider a generic ordering (i.e., permutation) of the players of N. Suppose that player i is in the jth position of this ordering. Suppose that the coalition S is composed of all players who are from the first to the jth position in such ordering, is winning, while the same coalition without player i is losing. Then the player i is the pivot for that permutation. The Shapley—Shubik Index assigns to each player the quotient between the number of times he or she is a pivot in all possible permutations, and the number n! of all possible permutations.

The concept of pivot player should be distinguished from that of swing, used in other power measures, in particular such as the Penrose-Banzhaf-Coleman measure. A swing for the ith member of N is a pair of coalitions (S, S\{i}) such that S is a winning coalition and S\{i} is not winning. Essentially, the difference between a pivot player and a swing lies in the fact that identification of a pivot takes into account the ordering in which a player becomes crucial for a coalition, whereas identification of a swing takes into account [Page 480]only the coalition itself. Put otherwise, the pivot is based on permutations, while the swing is based on combinations.

Following the paper by Shapley and Shubik, various other versions of pivot players have been specified, such as the pivot of queue, the pivot of roll call, and the pivot of ternary roll call, introduced by Dan S. Felsenthal and Moshé Machover in 1998.