Nucleoli as maximizers of collective satisfaction functions |
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Authors: | Peter Sudhölter Bezalel Peleg |
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Institution: | (1) Institute of Mathematical Economics, University of Bielefeld, Postfach 100131, D-33501 Bielefeld, Germany (e-mail: psudhoelter@wiwi.uni-bielefeld.de), DE;(2) Center for Rationality and Interactive Decision Theory, The Hebrew University of Jerusalem, Feldman Building, Givat-Ram, 91904 Jerusalem, Israel (e-mail: pelegba@math.huji.ac.il), IL |
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Abstract: | Two preimputations of a given TU game can be compared via the Lorenz order applied to the vectors of satisfactions. One preimputation
is `socially more desirable' than the other, if its corresponding vector of satisfactions Lorenz dominates the satisfaction
vector with respect to the second preimputation. It is shown that the prenucleolus, the anti-prenucleolus, and the modified
nucleolus are maximal in this Lorenz order. Here the modified nucleolus is the unique preimputation which lexicographically
minimizes the envies between the coalitions, i.e. the differences of excesses. Recently Sudh?lter developed this solution
concept. Properties of the set of all undominated preimputations, the maximal satisfaction solution, are discussed. A function
on the set of preimputations is called collective satisfaction function if it respects the Lorenz order. We prove that both
classical nucleoli are unique minimizers of certain `weighted Gini inequality indices', which are derived from some collective
satisfaction functions. For the (pre)nucleolus the function proposed by Kohlberg, who characterized the nucleolus as a solution
of a single minimization problem, can be chosen. Finally, a collective satisfaction function is defined such that the modified
nucleolus is its unique maximizer.
Received: 18 October 1996 / Accepted: 31 January 1997 |
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