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 x^t_r denote the
solution to the robust decision problem (7). Then

Iim ∣∣4 ^{- x}rllβ = 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 ∖T^k(L(x,s))l       .                ,∣∞∞ _fl⅛,z    x

---______ = _kβt y^l(x_t,s_i)e^k^_h=o^β i(^ħ,s_i))p_i      (13)
∂Xt                     ‘ .

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

∂E[T ^fc(L(x,s))]
______∂x_t_______
∂E[T ^k(L(x,s))]
∂x_t+j

The limit of (14) for k → ∞ depends on the states s that maximize 4∞=oβ^hl(x_h,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(x^t_r,s₁ )

and let

Ω_max = {i∖L(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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