Decision Making with Imprecise Probabilities

Brian Weatherson

Abstract

Orthodox Bayesian decision theory requires an agent’s beliefs representable by a real-valued function, ideally a probability function. Many theorists have argued this is too restrictive; it can be perfectly reasonable to have indeterminate degrees of belief. So doxastic states are ideally representable by a set of probability functions. One consequence of this is that the expected value of a gamble will be imprecise. This paper looks at the attempts to extend Bayesian decision theory to deal with such cases, and concludes that all proposals advanced thus far have been incoherent. A more modest, but coherent, alternative is proposed. Keywords: Imprecise probabilities, Arrow’s theorem.

1.

Introduction

Orthodox Bayesian decision theory requires agents’ doxastic states to be represented by a probability function, the so-called ‘subjective probabilities’, and their desires to be represented by a real-valued utility function. Once these idealisations are in place, decision theory becomes relatively straightforward. The best choice is the one with the highest expected utility according to the probability function. Because of Newcomb-like problems there is little consensus on how we ought to formalise ‘expected utility according to a probability function’, but in the vast bulk of cases the different approaches will yield equivalent results. The main problem for orthodoxy is that the idealisations made at the start are highly questionable. Many writers have thought that it is no requirement of rationality that agent’s epistemic states be representable by a single probability function. Others have thought that even if this is an ideal, it is so demanding that we cannot expect humans to reach it. One attractive amendment to orthodoxy is to permit agents’s epistemic state to be represented by a set of probability functions. This idea was first suggested by two economists, Gerhard Tintner...

Brian Weatherson

Abstract

Orthodox Bayesian decision theory requires an agent’s beliefs representable by a real-valued function, ideally a probability function. Many theorists have argued this is too restrictive; it can be perfectly reasonable to have indeterminate degrees of belief. So doxastic states are ideally representable by a set of probability functions. One consequence of this is that the expected value of a gamble will be imprecise. This paper looks at the attempts to extend Bayesian decision theory to deal with such cases, and concludes that all proposals advanced thus far have been incoherent. A more modest, but coherent, alternative is proposed. Keywords: Imprecise probabilities, Arrow’s theorem.

1.

Introduction

Orthodox Bayesian decision theory requires agents’ doxastic states to be represented by a probability function, the so-called ‘subjective probabilities’, and their desires to be represented by a real-valued utility function. Once these idealisations are in place, decision theory becomes relatively straightforward. The best choice is the one with the highest expected utility according to the probability function. Because of Newcomb-like problems there is little consensus on how we ought to formalise ‘expected utility according to a probability function’, but in the vast bulk of cases the different approaches will yield equivalent results. The main problem for orthodoxy is that the idealisations made at the start are highly questionable. Many writers have thought that it is no requirement of rationality that agent’s epistemic states be representable by a single probability function. Others have thought that even if this is an ideal, it is so demanding that we cannot expect humans to reach it. One attractive amendment to orthodoxy is to permit agents’s epistemic state to be represented by a set of probability functions. This idea was first suggested by two economists, Gerhard Tintner...

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