On the Relation between Robust and Bayesian Decision Making

decision problems with ever increasing risk aversion that has the property that
the associated optimal decisions converge to the optimal robust decision.

Convergence is robust to the precise assignment of prior probabilities by the
Bayesian as long as strictly positive probability is assigned to all states over
which the robust decision faces Unquantifiable uncertainty. This suggests the
following Bayesian interpretation of robust decision theory: it represents the
choice of a particular objective function that has the property that optimal
Bayesian decisions are insensitive (or robust) to many different priors.

These results hold not only when the desire for robustness is of a global
nature but also if desired robustness is locally restricted to some small set of
perturbations around a reference model.

Besides ever increasing risk-aversion, the sequence of Bayesian decision prob-
lems has a second interesting property: utility fails to be time separable even
if the objective function of the robust decision maker (seemingly) displays such
time separability. This property emerges because the worst case evaluated by
the robust decision maker depends on the full decision vector and not only on
the decision of a single period.

The next section introduces the decision problem and describes the robust
and Bayesian approach to its solution. Section 3 derives the convergence result
which is illustrated in section 3.1 with the help of a simple example. Section
4 extends the setup to infinite dimensional decision problems with discounting.
An appendix collects the proofs.

2 Bayesian and Robust Decision Problems

Consider a decision maker whose objective can be described by a simple loss
function that depends on a decision vector x ∈ Rⁿ and an unknown state of the
world s:

L⁽x,s)                                   ⁽¹⁾

T(∙, s) is assumed to be twice continuously differentiable and strictly convex for
all s. The state of the world s is assumed to belong to some finite and known
set Ω_s = {sι,... si} and the decision is assumed to belong to a compact and
convex set Ω_x C Rⁿ of feasible decisions.

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