The Shapley Value of Coalitions to Other Coalitions

The Shapley value for an n-person game is decomposed into a 2^n x 2^n value matrix giving the value of every coalition to every other coalition. The cell φ(ν, N) in the symmetric matrix is positive, zero, or negative, dependent on whether row coalition ν is beneficial, neutral, or unbeneficial to column coalition J. This enables viewing the values of coalitions from multiple perspectives. The n x 1 Shapley vector, replicated in the bottom row and right column of the 2^n x 2^n matrix, follows from summing the elements in all columns or all rows in the n x n player value matrix replicated in the upper left part of the 2^n x 2^n matrix. A proposition is developed, illustrated with an example, revealing desirable matrix properties, and applicable for weighted Shapley values. For example, the Shapley value of a coalition to another coalition equals the sum of the Shapley values of each player in the first coalition to each player in the second coalition.

Kjell Hausken