On the Relation between Robust and Bayesian Decision Making

4 Extension

While the loss function considered so far assumed a finite dimensional deci-
sion vector, macroeconomists tend to use infinite horizon models with infinite
dimensional decision vectors. In this section we show that the results of the
previous section extend in a natural way to the infinite horizon problems with
discounting.

Consider the following loss function

∞
L(x,s) = ^ β^tl(xt,s)
¢=0

where x_t ∈ Rⁿ denotes the period t decision, the vector x = (x0, x'_r,... )' the
stacked period decisions, and β < 1 a discount factor. The period loss function
l(∙,s) is assumed to be strictly convex and twice continuously differentiable for
all s. The period decision x_t must be chosen from a compact and convex set of
feasible decisions Ω_t that might depend on past decisions. Furthermore, there
is a compact set Ω_x C Rⁿ such that Ω_t C Ω_x for all t.

The robust decision maker minimizes

∞

min max) β^t l(x_t,s)
{x_t∖x_t∈Ω_t} s∈Ω, ¢=0

To construct the transformed Bayesian problem it might seem natural at first
to transform the period loss function l(∙, ∙) to preserve the time separability of
the objective function, e.g. to let the Bayesian minimize

However, the solution to this problem will not necessarily converge to the so-
lution of the robust decision problem as к increases without bound. This is
the case because a marginal change of some decision might have its strongest
impact for a state s⅛ that differs from the worst-case state s* associated with
the robust decision. When, in addition, the sign of the utility change for s⅛ is
opposite to the sign of the utility change for s*, then the Bayesian decisions for
(8) fails to converge as к → ∞. This is illustrated in the following example.

Convergence will be slower, the less weight is attached to the worst state associated with the
robust decision.

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