×

zbMATH — the first resource for mathematics

The sigma-core of a cooperative game. (English) Zbl 0737.90081
The \(\sigma\)-core of a cooperative game with side-payments is the set of \(\sigma\)-additive elements of the game core. A simple proof of Schmeidler’s theorem on the \(\sigma\)-core and core equality conditions are given for exact games. For general monotone games stronger conditions are proved through the conditions of \(\sigma\)-continuity of conjugate game functions. The conditions imply that the function forms a capacity in the sense of Choquet. The results known for capacities are translated into a general \(\sigma\)-core theorem, which in particular gives a necessary and sufficient condition for the non-emptiness of the \(\sigma\)- core.

MSC:
91A12 Cooperative games
PDF BibTeX XML Cite
Full Text: DOI EuDML
References:
[1] Adamski W.: Capacitylike set functions and upper envelopes of measures. Math. Ann.229, 237–244 (1977) · Zbl 0341.28002
[2] Anger B., Lembke J.: Infinitely subadditive capacities as upper envelopes of measures. Z. Wahrscheinlichkeitstheorie verw. Geb.68, 403–414 (1985) · Zbl 0553.28002
[3] Bierlein D.: Über die Fortsetzung von Wahrscheinlichkeitsfeldern. Z. Wahrscheinlichkeitstheorie verw. Geb.1, 28–46 (1962) · Zbl 0109.10601
[4] Bondareva O. N.: Some applications of linear programming methods to the theory of cooperative games. Problemy Kibernet.10, 119–139 (1963) [Russian] · Zbl 1013.91501
[5] Delbaen F.: Convex games and extreme points. J. Math. Anal. Appl.45, 210–233 (1974) · Zbl 0337.90084
[6] Dellacherie C.: Ensembles Analytiques, Capacités, Mesures de Hausdorff. LNM295, Berlin-Heidelberg-New York 1972 · Zbl 0259.31001
[7] Dellacherie C., Meyer P.: Probabilities and Potential. North Holland Publishing Company, Amsterdam-New York-Oxford 1978 · Zbl 0494.60001
[8] Dunford N., Schwartz J. T.: Linear Operators, Part I (fourth printing). Inter-science Publishers Inc. New York 1967
[9] Huber P. J., Strassen V.: Minimax tests and the Neymann-Pearson lemma for capacities. Ann. Statist.1, 251–263;2, 223–224 (1973) · Zbl 0259.62008
[10] Kannai Y.: Countably additive measures in cores of games. J. Math. Anal. Appl.27, 227–240 (1969) · Zbl 0181.46902
[11] Kindler J.: A Mazur-Orlicz type theorem for submodular set functions. J. Math. Anal. Appl.120, 533–546 (1986) · Zbl 0605.28004
[12] Kindler J.: The sigma-core of convex games and the problem of measure extension. Manuscripta Math.66, 97–108 (1989) · Zbl 0682.28005
[13] König H.: Über das von Neumannsche Minimax-Theorem. Arch. Math19, 482–487 (1968) · Zbl 0179.21001
[14] Schmeidler D.: Cores of exact games I. J. Math. Anal. Appl.40, 214–225 (1972) · Zbl 0243.90071
[15] Shapley, L.S.: On balanced sets and cores. Naval Res. Logist. Quart.14, 453–460 (1967)
[16] Terkelsen F.: Some minimax theorems. Math. Scand.31, 405–413 (1972) · Zbl 0259.90042
[17] Topsøe F.: Compactness in spaces of measures. Studia Math.36, 195–212 (1970) · Zbl 0201.06202
[18] Ulam, S.M.: Zur Maßtheorie in der allgemeinen Mengenlehre. Fund. Math.16, 141–150 (1930) · JFM 56.0920.04
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.