On the Relation between Robust and Bayesian Decision Making

Example 2 Let the optimal robust decision be given by x^,_r' = [x^,_r^l₀,x^t_r^l₁,... )
and the state S{ (i = 1, 2) that maximizes the loss for this and any neighboring
decisions is given by si. Next consider the decision

x' = _x* ₊ (d', O, O, O ... )

which is equal to x^t_r, except for the first period. Suppose that altering the decision
from x^t_r to x causes the loss in period zero to increase by γ₁ > O units in state
sɪ. This causes x to be suboptimal for the robust decision maker.

Next, consider a Bayesian decision maker with objective (8) who considers
a deviation from x^t_r to x. The change ∆⅛ in the first period loss is given by

where γ₂ = l(x*_r0 + d,sfi) - l(x^t_r0,s2).  Suppose γ₂ < O and l(x*_r0,s2) >

l(x^t_r0,sι) + Y1 > O, which cannot be excluded, then

Iim ∆_k = -∞
k→∞

which indicates that a Bayesian with objective function (8) will prefer x to x_r∙
for all sufficiently large k.

To obtain a convergence result similar to the one in section 3 one has to
define the transformed loss function as

T^k(T(x, s)) = e^k(∑∞ ^e‘^z(^_t,sf)⁾                          (₁₀)

and let the Bayesian minimize

I

min V e^k(∑ ∑o        .

{≈_t∣≈_t∈Ω_t} ²-'

i=1

where p_i are prior probabilities.

Proposition 3 below shows that, as k increases without bound, the Bayesian
solution to problem (11) converges to the robust solution in terms of the follow-
ing vector norm:

∞

ll^xl∣β = V ^βtx't^xt

¢=0

10

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