Counter Examples

The fundamental flaw in info-gap decision theory is so obvious that it is straightforward to construct counterexamples to its central proposition which is that … info-gap decision theory seeks decisions that are robust against a severe uncertainty of the type that it postulates.

The point is that info-gap’s robustness model is a model of local robustness of the radius of stability type. This means that, by construction/definition, info-gap’s robustness model seeks to measure the robustness of decisions to small perturbations in a nominal value of the parameter of interest. It does not seek to measure the robustness of decisions against large variations in the given nominal value. The implication therefore is that it does not seek to measure the robustness of decisions against the variation of the parameter over the uncertainty space under consideration.

So, all it takes to come up with a convincing counterexample is to draw a picture displaying the acceptable regions of two decisions:

  • One that is robust globally (over the uncertainty space) but is fragile locally (in the neighborhood of the point estimate)
  • One that is fragile globally (over the uncertainty space) but is robust locally (in the neighborhood of the point estimate)

For example, consider this:

where robustness is sought with respect to the constraint r^{*}\le r(x,u) over the uncertainty space U. The shaded areas represent the points in U that satisfy this constraint for the respective values of decision x.

Clearly, decision x' is far more robust than decision x'' globally over the uncertainty space \mathbf{U}. Also, according to the precepts of info-gap decision theory, the info-gap robustness of decision x'' is clearly much higher (larger ) than the info-gap robustness of decision x'.

It goes without saying that it is as easy to construct counterexamples that are far more extreme.

More …

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