On the Relation between Robust and Bayesian Decision Making



Proposition 3 Let x*k denote the solution to the transformed Bayesian decision
problem (11) with prior probabilities
pi > 0 (i = l,...n). Let xtr denote the
solution to the robust decision problem (7). Then

Iim ∣∣4 - xrllβ = 0
κ→∞

The proof of proposition 3 is identical to the one of proposition 1 with the
exception that one has to substitute expressions involving the standard vector
norm by the ’discounted’ norm (12).

As a final remark, I want to stress that the transformed Bayesian utility
function fails to be time separable.6 Marginal utility for the Bayesian problem
is given by

E Tk(L(x,s))l       .                ,fl⅛,z    x

---______ = kβt y^l(xt,si)ek^h=oβ i(^ħ,si))pi      (13)
∂Xt                     ‘ .

Since this expression does not converge for k → ∞, consider the ratio of marginal
utilities instead:

E[T fc(L(x,s))]
______∂xt_______
E[T k(L(x,s))]
xt+j

(14)


The limit of (14) for k → ∞ depends on the states s that maximize 4=oβhl(xh,s),
where the latter expression is the term showing up in the exponent of (13). Since
the states that maximize this expression depend on the whole decision vector
x,
a decision change in some period other than
t or t + j may well alter this ratio.

5 Appendix

This section proves proposition 1. Rename states s such that at x*

T(x*, si) T(x), s2) ... L(xtr,s1 )

and let

Ωmax = {iL(x*r,si) = L(x*,sι)}

I first prove the following auxiliary result:

6The subsequent arguments assume a non-atomistic decision maker who takes into account
that the maximizing states are a function of his/her own decision
x. See section 6 in Hansen
et al. (2002) for further discussion.

11



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