Dutch-Book Arguments against using Conditional Probabilities for Conditional Bets

Affiliation(s)

School of Historical and Philosophical Studies, University of Melbourne, Parkville, Victoria, Australia.

School of Historical and Philosophical Studies, University of Melbourne, Parkville, Victoria, Australia.

ABSTRACT

We consider here an important family of conditional bets, those that proceed to settlement if and only if some agreed evidence is received that a condition has been met. Despite an opinion widespread in the literature, we observe that when the evidence is strong enough to generate certainty as to whether the condition has been met or not, using traditional conditional probabilities for such bets will NOT preserve a gambler from having a synchronic Dutch Book imposed upon him. On the contrary (I show) the gambler can be Dutch-Booked if his betting ratios ever depart from a rather different probability, one that involves the probability of the agreed evidence being provided. We note furthermore that this same alternative probability assessment is necessary if the evidence is weaker (*i.e.* if it fails to provide knowledge whether or not the condition has been met.) By contrast, some of the (rather different) probability assessments proposed by Jeffrey, precisely for such situations, still expose the gambler to a Dutch-Book.

We consider here an important family of conditional bets, those that proceed to settlement if and only if some agreed evidence is received that a condition has been met. Despite an opinion widespread in the literature, we observe that when the evidence is strong enough to generate certainty as to whether the condition has been met or not, using traditional conditional probabilities for such bets will NOT preserve a gambler from having a synchronic Dutch Book imposed upon him. On the contrary (I show) the gambler can be Dutch-Booked if his betting ratios ever depart from a rather different probability, one that involves the probability of the agreed evidence being provided. We note furthermore that this same alternative probability assessment is necessary if the evidence is weaker (

KEYWORDS

Probability; Conditioning; Dutch-Book; Conditional Probability; Bayesianism; Jeffrey Conditioning

Probability; Conditioning; Dutch-Book; Conditional Probability; Bayesianism; Jeffrey Conditioning

Cite this paper

Hutchison, K. (2012). Dutch-Book Arguments against using Conditional Probabilities for Conditional Bets.*Open Journal of Philosophy, 2,* 195-201. doi: 10.4236/ojpp.2012.23030.

Hutchison, K. (2012). Dutch-Book Arguments against using Conditional Probabilities for Conditional Bets.

References

[1] De Finetti, B. (1974-1975). Theory of probability: A critical introductory treatment. (2 vols.) A. Machí, & A. Smith (Trans.), New York: Wiley.

[2] Gillies, D. (2000). Philosophical theories of probability. London: Routledge.

[3] Hacking, I. (1967). Slightly more realistic personal probability. Philosophy of Science, 34, 311-325. doi:10.1086/288169

[4] Harman, G. (1983). Problems with probabilistic semantics. In A. Orenstein & R. Stern (Ed.), Developments in semantics (pp. 242-245). New York: Haven.

[5] Howson, C. (1977). Bayesian rules of updating. Erkenntnis, 45, 195-208.

[6] Howson, C., & Urbach, P. (2006). Scientific reasoning: The Bayesian approach (3rd ed.). Chicago, IL: Open Court.

[7] Hutchison, K. (1999). What are conditional probabilities conditional upon? British Journal for the Philosophy of Science, 50, 665-695. doi:10.1093/bjps/50.4.665

[8] Jeffrey, R. (1983). The logic of decision (2nd ed.). Chicago, IL: University of Chicago Press.

[9] Skyrms, B. (1987). Dynamic coherence and probability kinematics. Philosophy of Science, 54, 1-20. doi:10.1086/289350

[10] Talbott, W. (2008). Bayesian epistemology (revision of Mar 26, 2008). Stanford Encyclopedia of Philosophy. URL (last checked 9 Jun 2012) http://plato.stanford.edu/entries/epistemology-bayesian/

[11] Teller, P. (1973). Conditionalization and observation. Synthese, 26, 218-258. doi:10.1007/BF00873264

[12] Weatherson, B. (2003). From classical to intuitionistic probability, Notre Dame Journal of Formal Logic, 44, 111-123. doi:10.1305/ndjfl/1082637807.

[1] De Finetti, B. (1974-1975). Theory of probability: A critical introductory treatment. (2 vols.) A. Machí, & A. Smith (Trans.), New York: Wiley.

[2] Gillies, D. (2000). Philosophical theories of probability. London: Routledge.

[3] Hacking, I. (1967). Slightly more realistic personal probability. Philosophy of Science, 34, 311-325. doi:10.1086/288169

[4] Harman, G. (1983). Problems with probabilistic semantics. In A. Orenstein & R. Stern (Ed.), Developments in semantics (pp. 242-245). New York: Haven.

[5] Howson, C. (1977). Bayesian rules of updating. Erkenntnis, 45, 195-208.

[6] Howson, C., & Urbach, P. (2006). Scientific reasoning: The Bayesian approach (3rd ed.). Chicago, IL: Open Court.

[7] Hutchison, K. (1999). What are conditional probabilities conditional upon? British Journal for the Philosophy of Science, 50, 665-695. doi:10.1093/bjps/50.4.665

[8] Jeffrey, R. (1983). The logic of decision (2nd ed.). Chicago, IL: University of Chicago Press.

[9] Skyrms, B. (1987). Dynamic coherence and probability kinematics. Philosophy of Science, 54, 1-20. doi:10.1086/289350

[10] Talbott, W. (2008). Bayesian epistemology (revision of Mar 26, 2008). Stanford Encyclopedia of Philosophy. URL (last checked 9 Jun 2012) http://plato.stanford.edu/entries/epistemology-bayesian/

[11] Teller, P. (1973). Conditionalization and observation. Synthese, 26, 218-258. doi:10.1007/BF00873264

[12] Weatherson, B. (2003). From classical to intuitionistic probability, Notre Dame Journal of Formal Logic, 44, 111-123. doi:10.1305/ndjfl/1082637807.