The Shapley value of coalitions to other coalitions

The Shapley value for ann-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 phi(IJ)(v,N) in the symmetric matrix is positive, zero, or negative, dependent on whether row coalitionIis beneficial, neutral, or unbeneficial to column coalitionJ. This enables viewing the values of coalitions from multiple perspectives. Then 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 then x nplayer 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.