The Inverse Problem Relative to the Shapley Value and the Convexity

Irinel Dragan

University of Texas, Mathematics, Arlington, Texas 76019, USA.

Abstract: In an earlier work, for a given Shapley Value of a cooperative TU game we derived a procedure to find a new game in which the Shapley Value is the same, but it is coalitional rational. Recently, a similar problem has  been solved for other two efficient values, the Egalitarian Allocation and the Egalitarian Nonseparable Contribution. In this paper, by using very similar ideas, we considered the problem:  for a given Shapley Value of a cooperative TU game, find out a new game in which the Shapley Value is the same, but the game is convex. The procedure for finding such a game has been derived, depending on a parameter, and it was proved that in the new game, which is convex, the Shapley Value is unchanged and coalitional rational. As a corollary, it followed that in this new game the Egalitarian Nonseparable Contribution is unchanged and coalitional rational. Numerical examples illustrate the procedure.

Keywords: Inverse Set, Almost Null Family of Game, Convexity Threshold, Egalitarian Nonseparable Contribution.

Pages: 18 – 24 | Full PDF Paper