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The consistent Shapley value for hyperplane games. (English) Zbl 0682.90105
Summary: A new value is defined for n-person hyperplane games, i.e., non- sidepayment cooperative games, such that for each coalition, the Pareto optimal set is linear. This is a generalization of the Shapley value for side-payment games.
It is shown that this value is consistent in the sense that the payoff in a given game is related to payoffs in reduced games (obtained by excluding some players) in such a way that corrections demanded by coalitions of a fixed size are cancelled out. Moreover, this is the only consistent value which satisfies Pareto optimality (for the grand coalition), symmetry and covariancy with respect to utility changes of scales. It can be reached by players who start from an arbitrary Pareto optimal payoff vector and make successive adjustments.

MSC:
91A12 Cooperative games
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[1] Aumann RJ and Maschler M (1985) ?Game Theoretic Analysis of a Bankruptcy Problem from the Talmud?. Journal of Economic Theory 36, pp. 195-213 · Zbl 0578.90100 · doi:10.1016/0022-0531(85)90102-4
[2] Hart S and Mas-Collel A (1989) ?Potential, Value and Consistency?. To appear in Econometrica. · Zbl 0675.90103
[3] Lensberg T (1988) ?Stability and the Nash Solution.? Journal of Economic Theory 45, pp. 330-341. · Zbl 0657.90106 · doi:10.1016/0022-0531(88)90273-6
[4] Orshan G ?The Consistent Shapley Value in Hyperplane Games from a Global Standpoint?. M. Sc. Thesis, Dept. of Mathematics, The Hebrew University of Jerusalem, November 1987. · Zbl 0764.90101
[5] Peleg B (1986) ?On the Reduced Game Property and Its Converse.? International Journal of Game Theory 15, pp. 187-200. · Zbl 0629.90099 · doi:10.1007/BF01769258
[6] Peleg B (1985) ?An Axiomatization of the Core of Coperative Games Without Side Payments.? Journal of Mathematical Economics 14, pp. 203-214. · Zbl 0581.90102 · doi:10.1016/0304-4068(85)90020-5
[7] Shapley LS (1953) ?A Value forn-Person Games.? Contributions to the Theory of Games II, Annals of Mathematics Study 28, ed. Tucker AW and Kuhn HW, pp. 307-317.
[8] Sobolev AI (1975) ?The Characterization of Optimality Principles in Cooperative Games by Functional Equations.? Mathematical Methods in the Social Sciences (Russian language), ed. N.N. Vorobjev, Vilnius, pp. 94-151.
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