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An application of nonstandard analysis to game theory

  • Eugene Wesley (a1)

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In this article an application of extended fields of real numbers to the proof of theorems in the theory of cooperative games will be presented. The proofs set forth below involve the use of A. Robinson's theory of nonstandard analysis and are metamathematical in character. Alternative proofs utilizing ordinary topological methods can in fact be carried out quite briefly. However, attempts to apply nonstandard analysis to game theory are relatively novel. For this reason these results may be of interest not only insofar as they present new information on the theory of the kernel of a cooperative game, but also in that they serve to demonstrate the possibility of effectively exploiting nonstandard analysis as a tool for future investigation in this area. It may well turn out that nonstandard analysis could serve as the most natural vehicle through which concepts defined for games with a finite number of players might be extended to games with a continuum of players.

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[1]Davis, M. and Maschler, M., The kernel of a cooperative game, Naval Research Logistics Quarterly, vol. 12 (1965), pp. 223259.
[2]Maschler, M. and Peleg, B., A characterization, existence proof, and dimension bounds for the kernel of a game, Pacific Journal of Mathematics, vol. 18 (1966) pp. 289328.
[3]Peleg, B., Existence theorem for the bargaining set, Essays in mathematical economics in honor of Oskar Morgenstern, Shubik, M., ed., Princeton University Press, Princeton, N.J., 1967, pp. 5356. See also Bulletin of the American Mathematical Society, vol. 69 (1963), pp. 109–110.
[4]Robinson, A., Complex function theory over non-Archimedean fields, Technical (Scientific) Note No. 30, Contract No. AF 61 (052)-182, Hebrew University, Jerusalem, Israel, 1962.
[5]Robinson, A., Introduction to model theory and to the meta-mathematics of algebra, University of California, Los Angeles, North-Holland Publishing Company, Amsterdam, 1963.
[6]Robinson, A., Nonstandard analysis, North-Holland Publishing Company, Amsterdam, 1966.

An application of nonstandard analysis to game theory

  • Eugene Wesley (a1)

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