On the Relation between Robust and Bayesian Decision Making

1 Introduction

In recent years robust or maxmin decision theory has been put forward as an al-
ternative to standard Bayesian decision theory in macroeconomics (e.g. Hansen
and Sargent (2000), (2001)).¹

The key idea behind robust decision theory is that agents might face un-
certainty that they cannot quantify in terms of prior probabilities because ’too
little is known’ to do so.

Without prior probabilities Bayesian decisions are not defined. Robust de-
cision theory fills this gap by postulating that any action is evaluated according
to the worst outcome that it can generate among the uncertain states to which
prior probabilities cannot be assigned, see Gilboa and Schmeidler (1989) for an
axiomatization.

A key motivation for introducing robust decision makers into macroeconomic
models is that such models can explain behavior that seems not to be rational
from a Bayesian perspective and thereby improve the descriptive performance of
otherwise standard macroeconomic models. Hansen et al. (1999), for example,
show that a slight preference for robustness can explain a substantial part of
the observed equity premiums.

Despite its increasing popularity in applied macroeconomics (e.g. Onatski
and Stock (2000), Tetlow and von zur Muehlen (2001)), the relation of robust
decision theory to standard Bayesian decision theory seems to have received
little attention. At the same time, it seems important to understand the links
between the two problems since they might inform us in which ways robust
decision makers may alter and improve the descriptive performance of macroe-
conomic models. Moreover, possible links are potentially informative about how
to compute robust decisions in applications.

The present paper shows that robust decision problems can be interpreted
in terms of the limit of a sequence of Bayesian decision problems. For a simple
class of robust decision problems, I show that there is a sequence of Bayesian

¹I use the term robust decision theory synonymous to the term ’maxmin decision theory’,
as put forward by Gilboa and Schmeidler (1989). Hansen et. al. (2002) have shown how these
two classes of problems can be linked.

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