Decision Making under Interval Uncertainty

Vladik Kreinovich
Department of Computer Science
University of Texas at El Paso
El Paso, TX 79968, USA
vladik@utep.edu

To make a decision, we must:
* find out the user's preference, and
* help the user select an alternative which is the best -- according to
  these preferences.
A general way to describe user preferences is via the notion of
utility: we select a very bad alternative B and a very good
alternative G; utility u(A) of an alternative A if then defined as the
probability p for which A is equivalent to the lottery in which we get
G with probability p, and B otherwise. One can prove that utility is
determined uniquely modulo linear re-scaling (corresponding to
different choices of G and B), and that the utility of a decision with
probabilistic consequences is equal to the expected utility of these
consequences.

Once the utility function u(d) is elicited, we select the decision d
with the largest utility u(d). Interval techniques can help in finding
the optimizing decision.

Often, we do not know the exact probability distribution, so we need
to extract, from the sample, the characteristics of a distribution
which are most appropriate for decision making. We show that, under
reasonable assumptions, we should select moments and cumulative
distribution function (cdf). Based on a finite sample, we can only
find bounds on these characteristics, so we need to deal with bounds
(intervals) on moments and bounds on cdf (a.k.a. p-boxes).

Once we know intervals [u(d)] of possible values of utility, which
decision shall we select? We can simply select a decision d which
may} be optimal, but there are usually many such decisions;
which of them should be select? It is reasonable to assume that
this selection should not depend on linear re-scaling of utility;
under this assumption, we get Hurwicz optimism-pessimism criterion
alpha * max u(d) + (1 - alpha) * min u(d) --> max.
The next question is how to select alpha: interestingly, e.g.,
too optimistic values (alpha > 0.5) do not lead to good decisions.

In some situations, it is difficult to elicit even interval-valued
utilities. In many such situations, there are reasonable symmetries
which can be used to make a decision. We show how this method works on
the example of selecting a location for a meteorological tower.

Finally, while optimization problems are ubiquitous, sometimes, we
need to go beyond optimization: e.g., we need to make sure that the
system is controllable for all disturbances within a given range.
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