Non-Archimedean Subjective Probabilities in Decision Theory and Games
AbstractDecember 7, 1997 To allow conditioning on counterfactual events, zero probabilities can be replaced by infinitesimal probabilities that range over a non-Archimedean ordered field. This paper considers a suitable minimal field that is a complete metric space. Axioms similar to those in Anscombe and Aumann (1963) and in Blume, Brandenburger and Dekel (1991) are used to characterize preferences which: (i) reveal unique non-Archimedean subjective probabilities within the field; and (ii) can be represented by the non-Archimedean subjective expected value of any real-valued von Neumann--Morgenstern utility function in a unique cardinal equivalence class, using the natural ordering of the field. Keywords: Non-Archimedean probabilities, subjective expected utility, Anscombe--Aumann axioms, lexicographic expected utility, conditional probability systems, reduction of compound lotteries.
Download InfoIf you experience problems downloading a file, check if you have the proper application to view it first. In case of further problems read the IDEAS help page. Note that these files are not on the IDEAS site. Please be patient as the files may be large.
Bibliographic InfoPaper provided by Stanford University, Department of Economics in its series Working Papers with number 97038.
Date of creation: 07 Dec 1997
Date of revision:
Contact details of provider:
Postal: Ralph Landau Economics Building, Stanford, CA 94305-6072
Web page: http://www-econ.stanford.edu/econ/workp/
More information through EDIRC
Non-Archimedean probabilities; subjective expected utility; Anscombe--Aumann axioms; lexicographic expected utility; conditional probability systems; reduction of compound lotteries;
Other versions of this item:
- Hammond, Peter J., 1999. "Non-Archimedean subjective probabilities in decision theory and games," Mathematical Social Sciences, Elsevier, vol. 38(2), pages 139-156, September.
Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.:
- Reinhard Selten, 1973. "A Simple Model of Imperfect Competition, where 4 are Few and 6 are Many," Working Papers 008, Bielefeld University, Center for Mathematical Economics.
- McLennan, Andrew, 1989. "The Space of Conditional Systems is a Ball," International Journal of Game Theory, Springer, vol. 18(2), pages 125-39.
- David Kreps & Robert Wilson, 1998.
Levine's Working Paper Archive
237, David K. Levine.
- Hammond, P.J. & , ., 1987. "Consequentialist foundations for expected utility," CORE Discussion Papers 1987016, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
- Anderson, Robert M., 1991. "Non-standard analysis with applications to economics," Handbook of Mathematical Economics, in: W. Hildenbrand & H. Sonnenschein (ed.), Handbook of Mathematical Economics, edition 1, volume 4, chapter 39, pages 2145-2208 Elsevier.
- Myerson, Roger B, 1986.
"Multistage Games with Communication,"
Econometric Society, vol. 54(2), pages 323-58, March.
- McLennan, Andrew, 1989. "Consistent Conditional Systems in Noncooperative Game Theory," International Journal of Game Theory, Springer, vol. 18(2), pages 141-74.
- Roger B. Myerson, 1977.
"Refinements of the Nash Equilibrium Concept,"
295, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
- Blume, Lawrence & Brandenburger, Adam & Dekel, Eddie, 1991. "Lexicographic Probabilities and Equilibrium Refinements," Econometrica, Econometric Society, vol. 59(1), pages 81-98, January.
- F J Anscombe & R J Aumann, 2000. "A Definition of Subjective Probability," Levine's Working Paper Archive 7591, David K. Levine.
- Rajan, Uday, 1998. "Trembles in the Bayesian Foundations of Solution Concepts of Games," Journal of Economic Theory, Elsevier, vol. 82(1), pages 248-266, September.
- Blume, Lawrence & Brandenburger, Adam & Dekel, Eddie, 1991. "Lexicographic Probabilities and Choice under Uncertainty," Econometrica, Econometric Society, vol. 59(1), pages 61-79, January.
- Karni, Edi & Schmeidler, David, 1991. "Utility theory with uncertainty," Handbook of Mathematical Economics, in: W. Hildenbrand & H. Sonnenschein (ed.), Handbook of Mathematical Economics, edition 1, volume 4, chapter 33, pages 1763-1831 Elsevier.
- Lavalle, Irving H & Fishburn, Peter C, 1992. " State-Independent Subjective Expected Lexicographic Utility," Journal of Risk and Uncertainty, Springer, vol. 5(3), pages 217-40, July.
- Antonio Quesada, 2003. "Negative results in the theory of games with lexicographic utilities," Economics Bulletin, AccessEcon, vol. 3(20), pages 1-7.
- repec:ebl:ecbull:v:3:y:2003:i:20:p:1-7 is not listed on IDEAS
For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (Thomas Krichel).
If references are entirely missing, you can add them using this form.