The Mobile Radius of Stability TheoremLast modified: Friday, 30-Dec-2011 13:02:34 MST

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 isnotthe 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 methodfor 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 theRadius of Stability model. And what is more, this model is designed expressly for the analysis ofsmall perturbationsin a given nominal value of a parameter. This means that this model is thoroughly unsuitable for the treatment ofsevereuncertainty.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

surprisesassociated with the "economic problem".Obviously, Ben-Haim (2001, 2006, 2010) is very wrong on all fronts.

DEBUNKED!√The reader may wish to read my Official Mobile Debunker of info-gap decision theory.

Also, see my complete list of articles

Moshe's new book! - Sniedovich, M. (2012) Fooled by local robustness,
Risk Analysis, in press.

- Sniedovich, M. (2012) Black swans, new Nostradamuses, voodoo decision theories and the science of decision-making in the face of severe uncertainty,
International Transactions in Operational Research, in press.

- Sniedovich, M. (2011) A classic decision theoretic perspective on worst-case analysis,
Applications of Mathematics,56(5), 499-509.

- Sniedovich, M. (2011) Dynamic programming: introductory concepts, in
Wiley Encyclopedia of Operations Research and Management Science(EORMS), Wiley.

- Caserta, M., Voss, S., Sniedovich, M. (2011) Applying the corridor method to a blocks relocation problem,
OR Spectrum, 33(4), 815-929, 2011.

- Sniedovich, M. (2011)
Dynamic Programming: Foundations and Principles, Second Edition,Taylor & Francis.

- Sniedovich, M. (2010) A bird's view of Info-Gap decision theory,
Journal of Risk Finance, 11(3), 268-283.

- Sniedovich M. (2009) Modeling of robustness against severe uncertainty, pp. 33- 42, Proceedings of the 10th International Symposium on Operational Research, SOR'09, Nova Gorica, Slovenia, September 23-25, 2009.

- Sniedovich M. (2009) A Critique of Info-Gap Robustness Model. In: Martorell et al. (eds),
.Safety, Reliability and Risk Analysis: Theory, Methods and Applications,pp. 2071-2079, Taylor and Francis Group, London.

- Sniedovich M. (2009) A Classical Decision Theoretic Perspective on Worst-Case Analysis, Working Paper No. MS-03-09, Department of Mathematics and Statistics, The University of Melbourne.(PDF File)

- Caserta, M., Voss, S., Sniedovich, M. (2008) The corridor method - A general solution concept with application to the blocks relocation problem. In: A. Bruzzone, F. Longo, Y. Merkuriev, G. Mirabelli and M.A. Piera (eds.),
11th International Workshop on Harbour, Maritime and Multimodal Logistics Modeling and Simulation,DIPTEM, Genova, 89-94.

- Sniedovich, M. (2008) FAQS about Info-Gap Decision Theory, Working Paper No. MS-12-08, Department of Mathematics and Statistics, The University of Melbourne, (PDF File)

- Sniedovich, M. (2008) A Call for the Reassessment of the Use and Promotion of Info-Gap Decision Theory in Australia (PDF File)

- Sniedovich, M. (2008) Info-Gap decision theory and the small applied world of environmental decision-making, Working Paper No. MS-11-08

This is a response to comments made by Mark Burgman on my criticism of Info-Gap (PDF file )

- Sniedovich, M. (2008) A call for the reassessment of Info-Gap decision theory,
Decision Point,24, 10.

- Sniedovich, M. (2008) From Shakespeare to Wald: modeling wors-case analysis in the face of severe uncertainty,
Decision Point,22, 8-9.

- Sniedovich, M. (2008) Wald's Maximin model: a treasure in disguise!,
Journal of Risk Finance,9(3), 287-291.

- Sniedovich, M. (2008) Anatomy of a Misguided Maximin formulation of Info-Gap's Robustness Model (PDF File)

In this paper I explain, again, the misconceptions that Info-Gap proponents seem to have regarding the relationship between Info-Gap's robustness model and Wald's Maximin model.

- Sniedovich. M. (2008) The Mighty Maximin! (PDF File)

This paper is dedicated to the modeling aspects of Maximin and robust optimization.

- Sniedovich, M. (2007) The art and science of modeling decision-making under severe uncertainty,
Decision Making in Manufacturing and Services,1-2, 111-136. (PDF File) .

- Sniedovich, M. (2007) Crystal-Clear Answers to Two FAQs about Info-Gap (PDF File)

In this paper I examine the two fundamental flaws in Info-Gap decision theory, and the flawed attempts to shrug off my criticism of Info-Gap decision theory.

- My reply (PDF File) to Ben-Haim's response to one of my papers. (April 22, 2007)
This is an exciting development!

- Ben-Haim's response confirms my assessment of Info-Gap. It is clear that Info-Gap is fundamentally flawed and therefore unsuitable for decision-making under severe uncertainty.

- Ben-Haim is not familiar with the fundamental concept
point estimate. He does not realize that a function can be a point estimate of another function.So when you read my papers make sure that you do not misinterpret the notion

point estimate. The phrase "A is a point estimate of B" simply means that A is an element of the same topological space that B belongs to. Thus, if B is say a probability density function and A is a point estimate of B, then A is a probability density function belonging to the same (assumed) set (family) of probability density functions.Ben-Haim mistakenly assumes that a point estimate is a point in a Euclidean space and therefore a point estimate cannot be say a function. This is incredible!

- A formal proof that
Info-GapisWald's Maximin Principlein disguise. (December 31, 2006)

This is a very short article entitled Eureka! Info-Gap is Worst Case (maximin) in Disguise! (PDF File)

It shows thatInfo-Gapis not a new theory but rather a simple instance ofWald's famous Maximin Principledating back to 1945, which in turn goes back to von Neumann's work on Maximin problems in the context ofGame Theory(1928).

- A proof that
Info-Gap's uncertainty model is fundamentally flawed. (December 31, 2006)

This is a very short article entitled The Fundamental Flaw in Info-Gap's Uncertainty Model (PDF File) .

It shows that becauseInfo-Gapdeploys a single point estimate under severe uncertainty, there is no reason to believe that the solutions it generates are likely to be robust.

- A math-free explanation of the flaw in Info-Gap. ( December 31, 2006)

This is a very short article entitled The GAP in Info-Gap (PDF File) .

It is a math-free version of the paper above. Read it if you are allergic to math.

- A long essay entitled
What's Wrong with Info-Gap? An Operations Research Perspective(PDF File) (December 31, 2006).

This is a paper that I presented at theASOR Recent Advances in Operations Research(PDF File) mini-conference (December 1, 2006, Melbourne, Australia).

If your organization is promoting Info-Gap, I suggest that you invite me for a seminar at your place. I promise to deliver a lively, informative, entertaining and convincing presentation explaining why it is not a good idea to use — let alone promote — Info-Gap as a decision-making tool.

Here is a list of relevant lectures/seminars on this topic that I gave in the last two years.

- ASOR Recent Advances, 2011, Melbourne, Australia, November 16 2011. Presentation: The Power of the (peer-reviewed) Word. (PDF file).

- Alex Rubinov Memorial Lecture The Art, Science, and Joy of (mathematical) Decision-Making, November 7, 2011, The University of Ballarat. (PDF file).

- Black Swans, Modern Nostradamuses, Voodoo Decision Theories, and the Science of Decision-Making in the Face of Severe Uncertainty (PDF File) .

(Invited tutorial, ALIO/INFORMS Conference, Buenos Aires, Argentina, July 6-9, 2010).

- A Critique of Info-Gap Decision theory: From Voodoo Decision-Making to Voodoo Economics(PDF File) .

(Recent Advances in OR, RMIT, Melbourne, Australia, November 25, 2009)

- Robust decision-making in the face of severe uncertainty(PDF File) .

(GRIPS, Tokyo, Japan, October 16, 2009)

- Decision-making in the face of severe uncertainty(PDF File) .

(KORDS'09 Conference, Vilnius, Lithuania, September 30 -- OCtober 3, 2009)

- Modeling robustness against severe uncertainty (PDF File) .

(SOR'09 Conference, Nova Gorica, Slovenia, September 23-25, 2009)

- How do you recognize a Voodoo decision theory?(PDF File) .

(School of Mathematical and Geospatial Sciences, RMIT, June 26, 2009).

- Black Swans, Modern Nostradamuses, Voodoo Decision Theories, Info-Gaps, and the Science of Decision-Making in the Face of Severe Uncertainty (PDF File) .

(Department of Econometrics and Business Statistics, Monash University, May 8, 2009).

- The Rise and Rise of Voodoo Decision Theory.

ASOR Recent Advances, Deakin University, November 26, 2008. This presentation was based on the pages on my website (voodoo.moshe-online.com).

- Responsible Decision-Making in the face of Severe Uncertainty (PDF File) .

(Singapore Management University, Singapore, September 29, 2008)

- A Critique of Info-Gap's Robustness Model (PDF File) .

(ESREL/SRA 2008 Conference, Valencia, Spain, September 22-25, 2008)

- Robust Decision-Making in the Face of Severe Uncertainty (PDF File) .

(Technion, Haifa, Israel, September 15, 2008)

- The Art and Science of Robust Decision-Making (PDF File) .

(AIRO 2008 Conference, Ischia, Italy, September 8-11, 2008 )

- The Fundamental Flaws in Info-Gap Decision Theory (PDF File) .

(CSIRO, Canberra, July 9, 2008 )

- Responsible Decision-Making in the Face of Severe Uncertainty (PDF File) .

(OR Conference, ADFA, Canberra, July 7-8, 2008 )

- Responsible Decision-Making in the Face of Severe Uncertainty (PDF File) .

(University of Sydney Seminar, May 16, 2008 )

- Decision-Making Under Severe Uncertainty: An Australian, Operational Research Perspective (PDF File) .

(ASOR National Conference, Melbourne, December 3-5, 2007 )

- A Critique of Info-Gap (PDF File) .

(SRA 2007 Conference, Hobart, August 20, 2007)

- What exactly is wrong with Info-Gap? A Decision Theoretic Perspective (PDF File) .

(MS Colloquium, University of Melbourne, August 1, 2007)

- A Formal Look at Info-Gap Theory (PDF File) .

(ORSUM Seminar , University of Melbourne, May 21, 2007)

The Art and Science of Decision-Making Under Severe Uncertainty(PDF File) .

(ACERA seminar, University of Melbourne, May 4, 2007)

- What exactly is Info-Gap? An OR perspective. (PDF File)

ASOR Recent Advances in Operations Researchmini-conference (December 1, 2006, Melbourne, Australia).

Disclaimer:This page, its contents and style, are the responsibility of the author (Moshe Sniedovich) and do not represent the views, policies or opinions of the organizations he is associated/affiliated with.