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1 Introduction

Bayesian decision theory and its generalizations provide a powerful set of
tools for analyzing problems involving state-contingent uncertainty. In prob-
lems of this class, decision-makers begin with a complete specification of
uncertainty in terms of a state space (a set of possible states of the world).
The ultimate problem is choose between acts, represented as mappings from
a the state space to a set of possible outcomes. In many applications, there is
an intermediate stage in which the decision-maker may obtain information in
the form of a signal about the state of the world, represented by a refinement
of the state space. That is, any possible value of the signal means that the
true state of the world must lie in some subset of the state space.

In the standard Bayesian model, the decision-maker is endowed with a
prior probability distribution over the states of the world. The standard rules
of probability enable the derivation of a probability for any given event, and
a probability distribution for any random variable. Given the observation of
a signal, Bayesian updating involves the use of conditional probabilities to
derive a posterior distribution.

It has been widely recognised since the work of Ellsberg (1961) that
decision-makers do not always think in terms of well-defined probabilities.
Rather, some events may be regarded as being, in some sense, ambiguous.
A wide range of definitions of ambiguity and proposals for modelling am-
biguous preferences have been put forward. The most influential has been
the multiple priors approach of Gilboa and Schmeidler (1989, 1994). In this
approach, uncertainty is represented by a convex set of probability distribu-
tions. Ambiguity-averse individuals choose to evaluate acts on the basis of
the least favorable probability distribution (the maxmin EU model) but a
range of other decision criteria are possible.

It is possible to apply Bayesian updating to multiple priors in the ob-
vious fashion, by deriving a posterior distribution for each element of the
set However this approach raises a number of issues. First, it is possible in
some circumstances to observe a signal that has probability zero for some
prior distributions under consideration (this cannot occur in the standard
Bayesian setting). More generally, as Halpern notes (2003, p84) an observed
signal will, in general be more probable under some priors than others, and is
therefore informative about the weight that should be placed on alternative
priors. The procedure of updating priors separately makes no use of this
information.

A more fundamental difficulty with state-contingent models of decision-
making under uncertainty is that decision-makers do not possess a complete
state-contingent description of the uncertainty they face. Life is full of sur-

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