Even and odd marginal worth vectors, Owen’s multilinear extension and convex games

In this paper the authors characterize convex games by means of Owen’s multilinear extension and the marginal worth vectors associated with even or odd permutations. Therefore they have obtained a refinement of the classic theorem; Shapley, Ichiishi in order to characterize the convexity of a game by its marginal worth vectors. The authors also give new expressions for the marginal worth vectors in relation to unanimity coordinates and the first partial derivatives of Owen’s multilinear extension. A sufficient condition for the convexity is given and also one application of the integer part of a convex game.