The Mobile Radius of Stability Theorem
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Some info-gap scholars do a lot of traveling.
Apparently, on the road they do not have easy access to my Maximin Theorem. Just in case you have not encountered this theorem before, I add parenthetically that this theorem states that info-gap's robustness model is a simple instance of Wald's famous Maximin model.
In any event, this lack of easy access may explain why so many info-gap articles continue to misrepresent the true relation between info-gap's robustness model and Wald's famous Maximin model.
In fact, Prof. Yakov Ben-Haim -- the Father of Info-gap decision theory -- continues to insist that info-gap robustness model is not a Maximin model. For example, in his latest book Info-Gap Economics we read this (Ben-Haim 2010, page 9):
Info-gap theory is related to robust-control and min-max methods, but nevertheless different from them. The strategy advocated here is not the amelioration of purportedly worst cases.
This is a significant improvement on Ben-Haim's (2001, 2006) earlier unfounded claims that his theory is fairly new and radically different from all current theories of decision under uncertainty.
Still, the complete lack of understanding as to what Wald's Maximin is all about, and the role and place it has in decision theory –- that pervades the earlier statements is manifested in this statement as well.
The trouble is, however, that unfounded erroneous statements such as these help to sustain the myths about info-gap decision theory that continue to circulate in the info-gap literature. Indeed, such statements continue to sustain those info-gap scholars who are still unwilling to face the fact that info-gap decision theory amounts to much ado about nothing, in fact a reinvention of a wheel, and a square one at that! (see my FAQs about info-gap decision theory).
Be that as it may, to remedy the situation, I created a mobile version of the Maximin Theorem.
Ben-Haim's (2010, p. 9) reference to robust-control prompted me to create another mobile theorem, namely the Mobile Radius of Stability Theorem. This theorem state that info-gap's robustness model is a simple Radius of Stability model.
For those who have not encountered this concept before, note that the Radius of Stability model (circa 1960) is a “bread and butter” paradigm in fields ranging from numerical analysis, applied mathematics, control theory, economics, operations research, to optimization, and beyond. It is the foremost paradigm in these fields for the modeling/analysis of small perturbations in a given nominal value of the parameter of interest. It is certainly the most widely used model of local stability in control theory.
For example, consider the following reference to Hinrichsen and Pritchard (1986a, 1986b) seminal papers in control theory:
Robustness analysis has played a prominent role in the theory of linear systems. In particular the state-state approach via stability radii has received considerable attention, see [HP2], [HP3], and references therein. In this approach a perturbation structure is defined for a realization of the system, and the robustness of the system is identified with the norm of the smallest destabilizing perturbation. In recent years there has been a great deal of work done on extending these results to more general perturbation classes, see, for example, the survey paper [PD], and for recent results on stability radii with respect to real perturbations ...
Paice and Wirth (1998, p.289)
Analysis of the Local Robustness of Stability for Flows.
Mathematics of Control, Signals, and Systems, 11, 289-302.
where HP2 = Hinrichsen and Pritchard (1990), HP3 = Hinrichsen and Pritchard (1992) and PD = Packard and Doyle (1993).
Clearly, this means that Ben-Haim's (2010, p. 9) statement also demonstrates a lack of familiarity with local robustness models used in control theory.
Math-free description
The radius of stability of a system at point p* is the radius of the largest ball centered at p* all of whose elements satisfy pre-determined stability conditions.
The picture is this:
where the rectangle, P, represents the set of all the possible values of the parameter p and the shaded area represents the subset of P whose elements satisfy the stability requirements under consideration.
For our purposes, the most convenient mathematical formulation of the concept "radius of stability" is as follows:
where B(ρ,p*) denotes a ball of radius ρ centered at p*, and P(q) denotes the set of all the values of p that satisfy the stability requirements for system q.
Now, compare this with info-gap's robustness model:
where U(α,û) is a neighborhood (ball) of size α around û, r* is a given number and r(q,u) is a number representing the performance level of system q associated with a given value u of the parameter under consideration.
Even those with very limited mathematical training should be able to figure out that these two models are equivalent. All the same, have a quick look at the formal Radius of Stability Theorem and its proof.
Remark
Note that, the publisher's product description of Ben-Haim's new book Info-Gap Economics reads as follows (See source here):
After every crisis economists and policy analysts ask: can better models help prevent or ameliorate such situations? This book provides an answer. Yes, quantitative models can help if we remember that they are rough approximations to a vastly more complex reality. Models can help if we include realistic but simple representations of uncertainty among our models, and if we retain the pre-eminence of human judgment over the churning of our computers.Info-gap theory is a new method for modeling and managing severe uncertainty. The core of the book presents detailed examples of info-gap analysis of decisions in monetary policy, financial economics, environmental economics for pollution control and climate change, estimation and forecasting. This book is essential reading for economic policy analysts and researchers.
In other words, this description promises a book outlining a new method for the modeling and management of severe uncertainty. But reading the book, you realize that the purported new method is in fact based on a well-established model, known universally as the Radius of Stability model. And what is more, this model is designed expressly for the analysis of small perturbations in a given nominal value of a parameter. This means that this model is thoroughly unsuitable for the treatment of severe uncertainty.
I suggest that you read my review of this book.
Official Mobile Debunker of info-gap decision theory
Based on the above analysis, it is clear that it is very easy to debunk info-gap decision theory.
In fact, the more Ben-Haim attempts to salvage his theory, the easier it is to demonstrate how wrong he is. In his new book Ben-Haim (2010) claims that his robustness model is different from robustness models used in robust-control and the Maximin/Minimax model. Furthermore, he presents info-gap decision theory as a theory that is a response to the challenge posed by surprises associated with the "economic problem".
| DEBUNKED! √ |
The reader may wish to read my Official Mobile Debunker of info-gap decision theory.
