Size Criterion

This is the most intuitive, but highly impractical, measure of global robustness. According to this criterion, the robustness of a decision/system is the (if necessary, normalized) “size” of the set of “acceptable” outcomes.

More generally, the robustness of a decision/system is the fraction of “acceptable” outcomes associated with the decision/system. That is,

\gamma(q):= \displaystyle \frac{size(A(q))}{size(O(q))}

where

  • O(q) = set of outcomes associated with decision/system q.
  • A(q) = set of acceptable outcomes associated with decision/system q.
  • size(S) = size of set S.

The larger \gamma(q) the more robust q.

It is assumed that size(S)>0 for any non-empty set S and that B \subset C entails that size(B) \le size(c). Note that since A(q) is a subset of O(q), it follows that 0 \le \gamma(q) \le 1.

If the sets under consideration are finite, then we can let size(S)=|S|, where |S| denotes the cardinality of set S. In this case

\gamma(q):= \displaystyle \frac{|A(q)|}{|O(q)|}

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