A Representation of Partially Ordered Preferences

Abstract

This chapter considers decision-theoretic foundations for robust Bayesian statistics. We modify the approach of Ramsey, deFinetti, Savage and Anscombe, and Aumann in giving axioms for a theory of robust preferences. We establish that preferences which satisfy axioms for robust preferences can be represented by a set of expected utilities. In the presence of two axioms relating to stateindependent utility, robust preferences are represented by a set of probability/utility pairs, where the utilities are almost state-independent (in a sense which we make precise). Our goal is to focus on preference alone and to extract whatever probability and/or utility information is contained in the preference relation when that is merely a partial order. This is in contrast with the usual approach to Bayesian robustness that begins with a class of “priors” or “likelihoods,” and a single loss function, in order to derive preferences from these probability/utility assumptions. I. INTRODUCTION AND OVERVIEW I.I. Robust Bayesian Preferences This essay is about decision-theoretic foundations for robust Bayesian statistics. The fruitful tradition of Ramsey (1931), deFinetti (1937), Savage (1954), and Anscombe and Aumann (1963) seeks to ground Bayesian inference on a normative theory of rational choice. Rather than accept the traditional probability models and loss functions as given, Savage is explicit about the foundations. He axiomatizes a theory of preference using a binary relation over acts, A 1 < A 2 , “act A 1 is not preferred to act A 2 .” Then, he shows that ≲ is represented by a unique personal probability (state-independent) utility pair according to subjective expected utility.

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