Bayesian Decision Theory and the Representation of Beliefs

Bayesian Decision Theory and the Representation of Beliefs Edi Karni∗ Johns Hopkins University October 21, 2005

Abstract

In this paper I present an axiomatic choice theory for Bayesian decision makers. I use this model to define choice-based subjective probabilities that truly represent Bayesian decision makers’ prior and posterior beliefs. I argue that because of the limitations of the traditional analytical framework, no equivalent results may be obtained for theories that invoke Savage’s (1954) idea of a state space. ∗

I am grateful to Itzhak Gilboa and Robert Nau for their useful comments and to the NSF for financial

support under grant SES-0314249.

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Introduction

Modern theories of decision making under uncertainty characterize a decision maker by a preference relation on a set of alternatives representing courses of action. Uncertainty is captured by the assumption that, for some alternatives, the exact consequence of choosing them is not known at the time the choice is made. With few exceptions, these theories are formulated using the analytical framework of Savage (1954). This framework consists of a set of states, S; a set of consequences, C; and the set, F, of all the mappings from the set of states to the set of consequences. Elements of F are referred to as acts. A decision maker is characterized by a (prior) preference relation, < on F, whose structure, depicted axiomatically, is sufficient for its representation by an expected utility functional, Z

u (f (s)) dπ (s) ,

(1)

S

where u is a utility (real-valued) function defined on the consequences, and π is a (subjective) finitely additive, non-atomic, probability measure on S. States resolve the uncertainty in the sense that, once the true state becomes known, the unique consequence implied by each and every act becomes known. Subsets of S are events. An event is said to obtain if the true state belongs to it. Presumably, implicit in the decision maker’s preference relation is his prior beliefs, that is, a binary relation on the set of events that have the interpretation “at least as likely to obtain as, given the available information.” If a decision maker receives new information pertinent to his assessment of the likely

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realization of alternative events, he is characterized by posterior preference relations reflecting his modified beliefs. That is to say, the change in his preferences is fully accounted for by the change in his beliefs. In this framework a decision maker is considered Bayesian if (a) the representation of his prior preferences involve a unique prior probability measure on the state-space representing his initial, or prior, beliefs; (b) the representations of his posterior, information-dependent, preferences involve unique posterior probability measures on the state-space representing his updated beliefs; and (c) the representations of the posterior preference relations are obtained form the prior preference relation solely by updating the prior probabilities using Bayes rule. This characterization of a Bayesian decision maker is unsatisfactory. In Karni (1996, 2005) I argued that in Savage’s (1954) theory, and all other theories that invoke Savage’s analytical framework, the definition of prior beliefs is predicated on the convention that constant acts are constant utility acts. This convention is neither implied by the axioms nor, more importantly, subject to refutation within Savage’s analytical framework. Without this convention, it is impossible to disentangle the probabilities and marginal utilities; consequently, there is an infinite set of prior probabilities consistent with a decision maker’s prior preferences, and each such prior, when updated according to Bayes rule, gives rise to posteriors that are consistent with the decision maker’s posterior preference relations. To see this, Z let γ be a strictly positive, bounded, real-valued function on S and let E (γ) = γ (s) dπ (s) . S

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Then the prior preference relation, depicted by the representation (1) is also represented by Z uˆ (f (s), s) dˆ π (s) , (2) S

where uˆ (·, s) = u (·) /γ (s) and π ˆ (s) = π (s) γ (s) /E (γ) , for all s ∈ S.1 Consider next an experiment whose outcome is pertinent to the decision maker’s assessment of the likely realization of events. Let X be the set of observations and denote by q (x | s) the conditional probability of the observation x if the true state is s.2 Then, invoking representation (1), the posterior preference relations {