Geodesic
The lexicogfraphic prekernel
Matematičeskaâ teoriâ igr i eë priloženiâ, Tome 5 (2013) no. 3, pp. 88-114.

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The lexicographic prekernel of a cooperative game with transferable utilities (TU) is a subset of the payoff vectors lexicographically minimizing the vector of maximal surpluses of one player over another one. This solution is non-empty for every TU game, it is efficient, is contained both in the prekernel and in the least core, and may not contain the prenucleolus [9]. A combinatorial characterization of the lexicographic prekernel being a weak analog of the known characterization of the prenucleolus by Kohlberg [4] with the help of balanced collections of coalitions is given. The difference consists in sets of vectors to be lexicographic minimized: the prenucleolus deals with excess vectors, and the lexicographi prekernel deals with vectors of maximal surpluses. It is shown that finding the lexicographic prekernel comes to solving a finite set (not more than the number of players) of optimization and of combinatorial problems.
Keywords: cooperative game, prenucleolus, lexicographic prekernel.
Mots-clés : solution, prekernel 
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Elena B. Yanovskaya. The lexicogfraphic prekernel. Matematičeskaâ teoriâ igr i eë priloženiâ, Tome 5 (2013) no. 3, pp. 88-114. http://geodesic.mathdoc.fr/item/MGTA_2013_5_3_a4/

[1] Davis M., Maschler M., “The kernel of a cooperative game”, Naval Research Logistics Quarterly, 12 (1965), 223–259 | DOI | MR | Zbl

[2] Faigle U., Kern W., Kuipers J., “Computing an element in the lexicographic kernel of a game”, Math. Methods of Operations Research, 63 (2006), 427–433 | DOI | MR | Zbl

[3] Katsev I., Yanovskaya E., “The restricted prenucleolus”, Mathematical Social Sciences, 66:1 (2013), 56–65 | DOI | MR | Zbl

[4] Kohlberg E., “On the nucleolus of a characteristic function game”, SIAM Journal of Applied Mathematics, 20 (1971), 62–66 | DOI | MR | Zbl

[5] Orshan G., Sudhölter P., Reconfirming the prenucleolus, Working paper 325, Bielefeld University, Center for Mathematical Economics, 2001

[6] Orshan G., Sudhölter P., “The positive core of a cooperative game”, International Journal of Game Theory, 39 (2010), 113–136 | DOI | MR | Zbl

[7] Peleg B., “On the reduced game property and its converse”, International Journal of Game Theory, 15 (1986), 187–200 ; “A correction”, International Journal of Game Theory, 16 (1987), 209 | DOI | MR | Zbl | MR

[8] Peleg B., Sudhölter P., Introduction to the Theory of Cooperative Games, Theory and Decision Library. Series C, 34, Kluwer Academic Publishers, 2003 | DOI | MR

[9] Yarom M., “The lexicographic kernel of a cooperative game”, Mathematics of Operations Research, 6 (1981), 66–100 | DOI | MR