On the Relation between Robust and Bayesian Decision Making

When Ω_s contains the states of the world in a primitive sense, then equation
(3) implies that the decision maker has a preference for global robustness, as an
action x is evaluated with respect to all possible outcomes. For the case where
Ω_s indexes a set of probability distributions (economic models), equation (3)
allows for a preference for local robustness, as the decision maker seeks only to
be robust with respect to models contained in Ω_s.

It is useful to rewrite the robust decision problem as follows:

min R(x∖    with

c Ω

I

R(x) ≡ ∑L(x,s_i)I(x,s_i)                      (4)

i=l

where I(x,Si) is an indicator function that is equal to one if Si is a maximizer
of L(x,s_i) and that is equal to zero otherwise.² Rewriting the robust objective
in this way helps to highlight the relation to the Bayesian problem (2).

The indicator functions appearing in (4) look almost like ’prior probabilities’
of the robust decision maker. These ’robust priors’ put all probability weight on
the worst state associated with a given decision x. Since this worst state may
shift with x, the ’prior’ of the robust decision maker may shift with the chosen
decision. This is a major difference to Bayesian priors.

Given the previous observation, there exists an immediate equivalence be-
tween robust and Bayesian decisions, as pointed out by Chamberlain (2000)
and Hansen et al. (2002): if the Bayesian’s priors put all probability weight on
the worst state associated with the robust decision, then the optimal Bayesian
decision is identical to the robust decision. Note, however, that these priors
need not be rational from a Bayesian perspective.

Instead of choosing the Bayesian’s priors, this paper seeks to choose an
objective function for the Bayesian problem to achieve an equivalence between
the optimal Bayesian and robust decisions that holds (almost) independent from
the prior probabilities assigned by the Bayesian. This is done in the next section.

²If there are several maximizers I define the indicator function to be 1 only for the state
with the lowest index i.

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