On the Relation between Robust and Bayesian Decision Making

While the state s entering the loss function (1) may be interpreted literally
as a state of nature, it is more interesting to interpret s as indexing probability
distributions over random events. The latter implies that each s is associated
with a different economic model about the underlying stochastic process, which
is an interpretation more in line with the recent literature on robust control in
macroeconomics.

First, consider a Bayesian decision maker. Based on Savage’s axioms such a
decision maker can construct subjective prior probabilities pi (г = 1,... n) that
describe the likelihood with which the decision maker believes that state s⅛ will
realize.

Given these priors a Bayesian acts to

I

min E [L(x, s)] = min ∖ L(x, Si)pi                  (2)

.r∈Ω^                   x∈Ω_x <

i=l

Next, consider what has been called a robust decision maker who cannot
assign meaningful priors to the realization of the state s. The inability to assign
prior probabilities might be due to a failure of some of Savage’s axioms, e.g.
if there is no random variable with uniform distribution that allows for the
calibration of probabilities.

Uncertainty that cannot be quantified in terms of subjective probabilities has
been called Knightian uncertainty in the literature. The existence of Knightian
uncertainty opens many possible ways for modeling the decision problem. One
intuitive way, suggested by Blinder (1998), is to simply average over the states
of the world. The resulting decision problem would be equivalent to a Bayesian
decision problem with pi = ∣ (г = 1,... ,1 ).

The most widely advocated method to model decisions in the presence of
Knightian uncertainty is to let the decision maker choose the action x that
minimizes the maximum possible loss associated with x. In mathematical terms

min max L(x,s)                           (3)

x∈Ω_x s∈Ω_s

Let x* denote the solution to the minimization part of problem (3). An ax-
iomatic formulation for such a decision theory has been given by Gilboa and
Schmeidler (1989).

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