On the Relation between Robust and Bayesian Decision Making

Define the following sequence of transforming functions T^k(∙):³

T^k (Z) = e^kL

Since T^k(∙) is increasingly convex as k increases, a Bayesian with objective
T^k (Z) will become increasingly risk averse in terms of the coefficient of absolute
risk aversion. As a result, the value of the transformed loss T^k(Z) increases
disproportionately with the size of the loss Z.

Intuitively, this implies that the largest of all losses Z associated with some
action x obtains increasing relative weight. This should move the solution to the
Bayesian decision problem closer and closer to the robust solution. Proposition
1 below confirms this intuition:

Proposition 1 Let xk denote the solution to the transformed Bayesian decision
problem (6) with prior probabilities pi > 0 (i = l,...n). Let x^t_r denote the
solution to the robust decision problem (f). Then

Iim ∣∣^χk ^- <11 = ⁰
k→∞

The proof of proposition 1 can be found in the appendix. Proposition 1
shows that robust decisions can be interpreted as decisions of a Bayesian with
an infinite degree of risk-aversion and arbitrary strictly positive priors over the
domain to which the robust decision maker cannot assign prior probabilities.
In Bayesian terms the desire for robustness represents a choice of a particular
objective function, which has the property that optimal decisions are robust to
the assignment of prior probabilities.

The next subsection illustrates proposition 1 using a univariate example.

3.1 An Example

Consider the following simple loss function, which has been considered amongst
others by Brainard (1967) and Onatski (2000):

Z(x, s) = (sx — π*)²

The variable π* denotes an inflation target pursued by the central bank while
sx denotes the inflation rate that results when the decision maker chooses policy
³The particular sequence T^k is just chosen for convenience and other sequences might give
the same result.

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