Abstract
In this paper we characterize convex games by means of Owen's multilinear extension and the marginal worth vectors associated with even or odd permutations.
Therefore we have obtained a refinement of the classic theorem; Shapley (1971), Ichiishi (1981) in order to characterize the convexity of a game by its marginal worth vectors.
We 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 to the integer part of a convex game.
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The authors are grateful to A. Mas-Colell, J. E. Martínez-Legaz and E. Hendor for their stimulating and informative comments. We are very much indebted to G. Owen for his special and useful seminar given in Barcelona two years ago. Errors and shortcomings are the sole responsability of the authors. Comments and suggestions will be welcome. A part of this research has been supported by the University of Barcelona. In addition Carles Rafels' work was supposed by a Spanish research grant, DGICYT projet PB 92-0615. Thank you to University of Barcelona and Polytechnic University of Catalonia for the support to communicate this research in the International Conference on Game Theory in Stony Brook (NY) last July (1993).
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Rafels, C., Ybern, N. Even and odd marginal worth vectors, Owen's multilinear extension and convex games. Int J Game Theory 24, 113–126 (1995). https://doi.org/10.1007/BF01240037
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DOI: https://doi.org/10.1007/BF01240037