Imprecise and Indeterminate Probabilities

Abstract

Abstract Bayesian advocates of expected utility maximizationuse sets of probability distributions to representvery different ideas. Strict Bayesians insist thatprobability judgment is numerically determinateeven though the agent can represent such judgmentsonly in imprecise terms. According to QuasiBayesians rational agents may make indeterminatesubjective probability judgments. Both kinds ofBayesians require that admissible options maximizeexpected utility according to some probabilitydistribution. Quasi Bayesians permit thedistribution to vary with the context of choice.Maximalists allow for choices that do not maximizeexpected utility against any distribution.Maximiners mandate what maximalists allow. Thispaper defends the quasi Bayesian view against strictBayesians on the one hand and maximalists andmaximiners on the other.Keywords. Strict Bayesian, Quasi Bayesian, E -admissibility, E -maximality , Maximizing lowerexpectation. 1 Introduction Suppose that decision-maker X judges that hisavailable options belong to set S. Set S is itself asubset of a set M (Ω) of probability mixtures of afinite subsets of Ω. X's values, goals and beliefscommit him somehow to an evaluation of theelements of M (Ω) as better or worse. Thisevaluation is representable (by us and notnecessarily by X) by a set of weak orderings ofM (Ω) satisfying the requirements imposed by vonNeumann and Morgenstern on the evaluation oflotteries. Each of these weak orderings isrepresentable by a utility function unique up to apositive affine transformation. Consequently wecan define the value structure V [M (Ω)] to be the setof such permissible utility functions or thepermissible von Neumann-Morgenstern preferencesthey represent.For any finite nonempty subset S of M (Ω), V (S) isthe set of restrictions of the members of V [M (Ω)] tothe domain S. This is the value structure for S and itconsists of permissible utility functions for S. Thus,the value structure V [M (Ω)] determines what thevalue structure V (S) would be were S the set ofoptions X judged to be available to him in a givensituation.Let H be a set of propositions such that the decision-maker is sure that exactly one element of H is true.Moreover, if the decision maker X adds anyproposition s asserting that some option in S isgoing to be implemented to what he is certain istrue, the result is consistent with each and everyelement of H .Let O represent possible outcomes of implementingone or another of the available options in S. Thepropositions characterizing such outcomes specifyinformation X cares about according to his goalsand values. The deductive consequences of X'sbody of certainties K and the assumption that s isimplemented while state h in H is true entails thatexactly one consequence in O is true. This is so foreach s and each H .The extended value structure EV (O ) is representableby a set of utility functions defined for elements ofO . Each of these utility functions may be extendedto the set M (O ) of all mixtures of elements of O .Each of the permissible utility functions inEV [M (O )] represents a weak ordering of themembers of M (O ) satisfying von Neumann-Morgenstern requirements.A state of credal probability judgment (credal state)is representable by a set B of permissible probabilitymeasures over a given algebra of propositionsrepresenting the states of nature in H . This set B