AN EXPLICIT FORMULA OF THE SHAPLEY VALUE FOR A COOPERATIVE GAME INDUCED FROM THE CONJUGATE POINT
Takeaki Fuchikami, 武暁 淵上, フチカミ, タケアキ, Hidefumi Kawasaki, 英文 川崎, カワサキ, ヒデフミ · Institutional Repositories DataBase (IRDB) · 2005
The conjugate point was introduced by Jacobi to derive a sufficient optimality condition for a variational problem. Recently, the conjugate point was defined for an extremal problem in $ R^n $. The key of the conjugate point is cooperation of variables. Namely, when there exists a conjugate point for a stationary solution $ x in R^n $, we can improve the solution by suitably changing some of the variables. We call such a set of variables a strict conjugate set. This idea leads us to a cooperative game, which we call a conjugate-set game. The Shapley value is an important value in game theory. It evaluates player's contribution in the cooperative game. However, its calculation is usually very hard. The purpose of this paper is to give an explicit formula of the Shapley value for the conjugate-set game induced from the shortest path problem on an ellipsoid.