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On the Uniqueness of Subjective Probabilities

Edi Karni and David Schmeidler
Economic Theory
Vol. 3, No. 2 (Apr., 1993), pp. 267-277
Published by: Springer
Stable URL: http://www.jstor.org/stable/25054697
Page Count: 11
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On the Uniqueness of Subjective Probabilities
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Abstract

The purpose of this paper is twofold: First, within the framework of Savage (1954), we suggest axiomatic foundations for the representation of event-dependent preference relations over acts. This representation has the form of expectation of event-dependent utility with respect to non-unique subjective probabilities on the set of states. Second, we give an economic-theoretic motivation for selecting a unique probability distribution as an appropriate concept of "subjective probabilities." However, unlike in Savage's theory, this notion of subjective probabilities does not necessarily represent the decisions-maker's belief regarding the likelihood of events. Our approach involves a departure from Savage's postulate P4, which guarantees the completeness of Savage's likelihood relation on the set of all events. Instead, we assume the existence of a finite partition of the set of states, $\{S_{1},\ldots ,S_{n}\}$, such that, for events within each element of this partition P4 is satisfied. This weakening of Savage's axioms suffices for the existence of an expected event-dependent utility representation, but not for the uniqueness of the subjective probabilities. In many economic problems involving decision-making under uncertainty the existence of a unique probability is presumed and, in fact, is essential for the statement of the result. An example is Arrow's (1965) finding that all risk averse decision-makers will invest in a risky asset provided its expected rate of return exceeds that of an alternative risk-free asset. We show that a unique probability distribution can be chosen so as to render such results meaningful. Namely, any risk averse decision-maker will hold a positive position in the risky asset if and only if its expected rate of return with respect to the chosen probability exceeds that of the riskless asset.

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