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A Conflict between Finite Additivity and Avoiding Dutch Book

Published online by Cambridge University Press:  01 April 2022

Teddy Seidenfeld  and
Mark J. Schervish
Teddy Seidenfeld
Affiliation:
Department of Philosophy, Washington University, St. Louis
Mark J. Schervish
Affiliation:
Department of Statistics, Carnegie Mellon University, Pittsburgh
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Abstract

For Savage (1954) as for de Finetti (1974), the existence of subjective (personal) probability is a consequence of the normative theory of preference. (De Finetti achieves the reduction of belief to desire with his generalized Dutch-Book argument for previsions.) Both Savage and de Finetti rebel against legislating countable additivity for subjective probability. They require merely that probability be finitely additive. Simultaneously, they insist that their theories of preference are weak, accommodating all but self-defeating desires. In this paper we dispute these claims by showing that the following three cannot simultaneously hold:

(i) Coherent belief is reducible to rational preference, i.e. the generalized Dutch-Book argument fixes standards of coherence.

(ii) Finitely additive probability is coherent.

(iii) Admissible preference structures may be free of consequences, i.e. they may lack prizes whose values are robust against all contingencies.

Type
Research Article
Information
Philosophy of Science , Volume 50 , Issue 3 , September 1983 , pp. 398 - 412
Copyright
Copyright © Philosophy of Science Association 1983

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Footnotes

†

We would like to thank P. C. Fishburn, Jay Kadane, Isaac Levi, Patrick Maher and a referee for their helpful comments. Also, we have benefited from discussions about consequences with E. F. McClennen.

References

de Finetti, B. (1972), Probability, Induction and Statistics. New York: Wiley.Google Scholar
de Finetti, B. (1974), Theory of Probability (2 vols.). New York: Wiley.Google Scholar
Dubins, L. (1975), “Finitely additive conditional probabilities, conglomerability and disintegrations”, Annals of Probability 3: 89–99.10.1214/aop/1176996451CrossRefGoogle Scholar
Fishburn, P. C. (1970), Utility Theory for Decision Making. New York: Wiley.CrossRefGoogle Scholar
Fishburn, P. C. (1981), “Subjective Expected Utility: A Review of Normative Theories”, Theory and Decision 13: 139–199.10.1007/BF00134215CrossRefGoogle Scholar
Heath, D. and Sudderth, W. (1978), “On finitely additive priors, coherence, and extended admissibility”, Annals of Statistics 6: 333–345CrossRefGoogle Scholar
Kadane, J. B., Schervish, M. J., and Seidenfeld, T. (1981), Statistical Implications of Finitely Additive Probability. Technical Report 206. Department of Statistics, Pittsburgh: Carnegie-Mellon University, July.Google Scholar
Kyburg, H. E. (1978), “Subjective probability: considerations, reflections, and problems”, Journal of Philosophical Logic 7: 157–180.10.1007/BF00245926CrossRefGoogle Scholar
Savage, L. J. (1954), Foundations of Statistics. New York: Wiley.Google Scholar
Savage, L. J. (1971), “Elicitation of Personal Probabilities and Expectations”, Journal of the American Statistical Association 66: 783–801.CrossRefGoogle Scholar
Schervish, M. J., Seidenfeld, T., and Kadane, J. B. (1981), The Extent of Non-conglomerability of Finitely Additive Probabilities. Technical Report 198, Department of Statistics, Pittsburgh: Carnegie-Mellon University, March.Google Scholar
Shimony, A. (1955), “Coherence and the axioms of confirmation”, Journal of Symbolic Logic 20: 1–28.10.2307/2268039CrossRefGoogle Scholar