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The rise and rise of voodoo decision making 
Review 82023 (Posted: June 26 2023)
Reference Marc CiligotTravain. Sur une notion de robustesse. Annales de l'ISUP, 2012, 56(23):3748. Publication type article Year of publication 2012 Downloads hal01334825. Paragraph of special interest Il ne nous semble pas que cette notion de robustesse, même si elle n’est pas sans lien avec la très classique méthode du "pire des cas" (on pourra se référer à [28], [24], [11]) puisse s’y ramener, contrairement à ce qui est affirmé par M. Sniedovich (par exemple dans [41]) , encore moins de façon naturelle. Mais nous n’avons pas de certitude à ce sujet et donc n’entrerons pas dans ce débat ici. S’il s’avère que nous avons tort, cette approche de la robustesse offre au moins un point de vue différent qui, à notre avis, mérite d’être considéré en soi.
English version
It does not seem to us that this notion of robustness, although not without a link to the very classical "worstcase" method (one can refer to [28], [24], [11]), can be reduced to it, contrary to what is asserted by M. Sniedovich (for example in [41]), much less in a natural way. But we have no certainty on this subject and therefore will not enter into this debate here. If it turns out that we are wrong, this approach to robustness at least offers a different perspective that, in our opinion, deserves to be considered in its own right.
Reviewer Moshe Sniedovich IFIG perspective It seems that the author of this article is skeptical about the validity of my claim (e.g. in [41], that infogap robustness model is a maximin model.
[41] = M. Sniedovich. Anatomy of a misguided maximin/minimax formulation of infogap’s robustness model. Working Paper No. MS0408, 2008.
Revisiting
Working Paper No. MS206, 2006
Eureka! InfoGap is WorstCase Analysis (Maximin) in Disguise!
Moshe Sniedovich
Department of Mathematics and Statistics
The University of Melbourne
Melbourne, Australia
1. Introduction
The unusual title that I have chosen for this review is meant to reflect the fact that this review takes us back to 2006, when I launched a campaign whose goal was to contain the spread of infogap decision theory (IGDT) in Australia. The campaign was dormant in recent years but in 2021 I had to revive it because there were signs that some Australian scholars are either unfamiliar with the problematic issues afflicting IGDT and its literature, or prefer to ignore these issues.
One of these issues has to do with the treatment, more precisely mistreatment, in the IGDT literature, of the relationship between IGDT and Wald's famous Maximin paradigm. This issue is important because it raises a very serious question regarding the role and place of IGDT in the state of the art in the area of decisionmaking under severe uncertainty.
The main question that I addressed in Working Paper No. MS206 is this:
Working Paper No. MS206 provided a formal, rigorous (mathematical) proof that IGDT robustsatisficing decision model is a simple maximin model. Since then I have also proved that IGDT's robustness model itself is a simple maximin model, and that the concept "IGDT Robustness" is in fact a reinvention of an old (circa 1960) concept that is known universally as Radius of Stability. I also explained why IGDT is not suitable for the treatment of severe uncertainties of the type it was designed to handle.
To the best of my knowledge, the paper under review here (CiligotTravain 2012) is the only publication todate that raises doubts about the validity of my proofs that IGDT's two core models are maximin models. I take it to mean that the author of this publication has doubt about the validity of my formal proofs, or perhaps that my claims are too good (or bad) to be true. Be it as it may, in this review I provide a straightforward, formal, rigorous proof showing that IGDT's two core models are indeed simple maximin models.
But I also take note of related questions that have been raised recently regarding IGDT and its role and place in decisionmaking under severe uncertainty. For example,
This review focuses on :
These issues are discussed in detail in Review 2. I must say that it is surprising that fundamental questions regarding IGDT are still on the agenda in 2023, more than 20 years after the first book on the theory was published.
2. The Fundamental Theorem of IGDT
The main objective of this review is then to provide a simple (very) detailed, stepbystep, formal, rigorous proof of this very old (almost 20 years!) result:
In the framework of these models:
$$ \begin{align} q & = \text{Decision variable,}\\ \mathcal{Q} & = \text{Decision space: set of all possible values of $q$,}\\ u & = \text{Uncertainty parameter,}\\ \mathfrak{U} &= \text{Uncertainty space: set of all possible values of $u$,}\\ \tilde{u} & = \text{Point estimate of the true value of $u$,}\\ \alpha & = \text{Size ("radius") of neighborhoods around $\tilde{u}$,}\\ \mathscr{U}(\alpha,\tilde{u}) & = \text{Neighborhood of size $\alpha$ around $\tilde{u}$; a subset of $\mathfrak{U}$,}\\ L(q,u) & = \text{Performance level of decision $q$ given $u$,}\\ L_{c} &= \text{Critical performance level,}\\ \hat{\alpha}(q) & = \text{Robustness of decision $q$,}\\ \alpha^{*} & = \text{Robustness of the most robust decision.} \end{align} $$So, by definition, (IGDM) determines the robustness of decision $q$, denoted $\hat{\alpha}(q)$, which is equal to the size ($\alpha$) of the largest neighborhood $\mathscr{U}(\alpha,\tilde{u})$ around $\tilde{u}$ over which decision $q$ satisfies the performance constraint $L(q,u)\le L_{c}$ everywhere in the neighborhood. The (IRSDM) determines the best decision, namely the decision whose robustness ($\hat{\alpha}(q)$) is the largest.
I do hope that the proof provided in this review for this theorem will settle, once and for all, the issue regarding the intimate relationship between IGDT and Wald's maximin paradigm. But based on what I have seen on this front over the last twenty years, I suspect that some IGDT scholars will continue to argue―as usual without proof―that IGDT's two core models are not maximin models.
Having clarified all this, we can return to the main topic of this review.
3. Prototype maximin models
The reference in this review to Abraham Wald in connection with maximin models should not be interpreted as an indication that the mathematician Abraham Wald (19021950) invented this type of models. Maximin/minimax models have been around and used by mathematicians at least since the late 1770s, decades before Abraham Wald developed his statistical decision theory in the late 1930s.
However, it was Abraham Wald who proposed in 1939 that in the face of nonprobabilistic uncertainty in decisionmaking, it is reasonable to minimize the maximum risk. By the mid 1950s this approach to nonprobabilistic uncertainty became very popular in Decision Theory. At present this approach dominates the scene in the field of Robust Optimization.
So the term "Waldtype maximin model" is used here to distinguish between maximin models that have nothing to do with decisionmaking under uncertainty, and maximin models that describe situations involving decisionmaking under nonprobabilistic uncertainty. The later are referred to as Waldtype maximin models.
There are various ways to describe/define what constitutes a maximin model. In this discussion the focus is on maximin models used in decisionmaking under uncertainty, and therefore it is convenient to remind ourselves of the very famous, some would say infamous, rule that is attributed to Abraham Wald, the Father of Wald's maximin paradigm:
This is a slightly modified formulation of John Rawls' phrasing of the rule in his book A Theory of Justice. It reads as follows (Rawls 1971, pp. 152153):
The maximin rule tells us to rank alternatives by their worst possible outcomes: we are to adopt the alternative the worst outcome of which is superior to the worst outcomes of the others.
Remark Wald never formulated his paradigm this way. Such formulations began to appear in the literature in the 1950s, when Wald's paradigm became popular in the then emerging young field of Decision Theory.
This approach to decisionmaking under (nonprobabilistic) uncertainty was proposed by Wald (1939) and by the 1950s became one of the fundamental decision rules in decision theory. In the context of decisionmaking problems, the term "alternatives" means "decisions".
For our discussion here, it is important to note that this rule does not specify what constitutes an "outcome" nor what preference criteria are used, or should be used, to determine what "worse", hence "worst" means in this context. Determining what "outcome" and "worst" mean is the modeler's responsibility, and is a crucial (mathematical) modeling issue. This ambiguity regarding the meaning of "outcome" and "worst" is precisely what makes the Maximin Decision Rule so powerful.
Thus, for the benefit of readers who are not familiar with Wald's maximin paradigm, it is important to point out that maximin models come in various shapes and forms. Here are three prototype models of this type.
The Three Musketeers $$ \begin{align} \large v: &\large = \max_{x\in X}\min_{s\in S(x)} \big\{f(x,s): c(x,s)\in C, \forall s\in S(x)\big\}\tag{Prototype 1}\\ \large v: &\large = \max_{x\in X} \big\{f(x): c(x,s)\in C, \forall s\in S(x)\big\}\tag{Prototype 2}\\ \large v: &\large = \max_{x\in X} \min_{s\in S(x)} \ f(x,s) \tag{Prototype 3} \end{align} $$
In this models $x$ denotes the decision variable and $s$ denotes the (uncertain) state variable. The choice of the format $c(x,s)\in C$ to represent the constraints on $(x,s)$ pairs, is subjective and can be replaced by any other reasonable generic constraint, such as $g(x,s) \le r$, and so on. What is important is the $\forall s\in S(x)$ clause.
Example
In the article A survey of nonlinear robust optimization (Leyffer et al. 2020, p. 343) the following minimax model is used as a general framework for the survey:
$$ \min_{x\in X}\ \big\{f(x): c(x;u) \le 0 , {\color{red} \forall} u\in U(x)\big\} $$where $X$ is the decision space, $U(x)$ is the uncertainty space associated with decision $x$, $f$ is the objective function, and $c$ is the constraint function. Note that, as is the case in Prototype 2, the objective function does not depend on the uncertainty parameter $u$, hence the $\displaystyle \min_{u\in U(x)}$ operation is superfluous. Objective function of this type are popular in the field of Robust Optimization.
What makes this model a minimax model is the (worstcase clause) clause $\forall u\in U(x)$ in the constraint
$$c(x;u) \le 0 \ \ , \ \ \forall u\in U(x)$$In accordance with the Maximin Decision Rule, the $\forall u\in U(x)$ clause requires the worst value of $u$ in $U(x)$ to satisfy the constraint. Since this value is unknown, all values of $u\in U(x)$ are required to satisfy the constraint. Alternatively, this constraint can be rewritten as follows:
$$ 0 \ge \max_{u\in U(x)} c(x;u) $$observing that in this context the worst value of $u\in U(x)$ is the one that maximizes the value of $c(x;u)$ over $U(x)$.
It should be stressed that, methodologically speaking, the above three prototype maximin models are all equivalent in the sense that any one of them can be transformed to the formats of the other two. By inspection,
 The last two prototypes are instances of the first.
 The second prototype lost its $\displaystyle \min_{s\in S(x)}$ operation because this operation is superfluous in cases where the objective function does not depend on parameter $s$.
As an exercise, the reader may wish to show that Prototype 1 can be transformed and take the format of Prototype 3. Hint: define function $F$ as follows:
$$ \begin{align} F(x,s):= \begin{cases} f(x,s) &, \ \ c(x,s) \in C\\ \infty &, \ \ \text{otherwise} \end{cases}\ \ \ , \ \ \ x\in X, s\in S(x) \end{align} $$ and show that $$ w:= \max_{x\in X} \min_{s\in S(x)} \ F(x,s) $$is equivalent to Prototype 1 in the sense that if an optimal solution exists, then $v=w$ and both yield the same solution(s).
Try to figure out what are the implications if it turns out that $w=\infty$.
To give the reader a sense of what maximin models are currently used
4. Proof of The Fundamental Theorem of IGDT
The formal, rigorous proof of The Fundamental Theorem of IGDT provided below is intended for readers who are comfortable with mathematical models of the type displayed above and are familiar with the modeling aspects Wald's maximin paradigm.
Let me reiterate:
I do not expect all proponents of IGDT to change their mind about the theory in view of this yet another proof that the theory's two core models are maximin models. I do hope though that this proof will convince some leading IGDT scholars will that serious misconceptions about Wald's maximin paradigm are circulated in the IGDT literature.5. Conclusions
A formal examination if IGDT reveals the following:
Fact 1: There is no doubt that IGDT's two core models are simple Waldtype maximin models. Many formal, rigorous proofs of this fact have been easily accessible online and in peerreviewed journals since 2007.
 Fact 2: By the same token, there is no doubt that IGDT's key concept, namely InfoGap Robustness Model: $$ \large \hat{\alpha}(q) := \max_{\alpha\ge 0}\ \big\{\alpha: L(q,u) \le L_{c}, \forall u\in \mathscr{U}(\alpha,\tilde{u})\big\}\ \ \ , \ \ \ q\in \mathcal{Q} \tag{IGRM} $$ is a simple Radius of Stability model (circa 1960).
Fact 3: Last but not least, there is no doubt that the strong local orientation of Infogap Robustness Model make the theory unsuitable for the treatment of severe uncertainties of the type stipulated by IGDT.
Bibliographic notes
 Other working papers, articles, lectures, etc. on this topic for the period 20062011 are available online.
 Australian readers might be interested in taking a look at the report entitled Infogap decision theory: a perspective from the Land of the Black Swan that I submitted to the Australian Centre of Excellence for Risk Analysis (ACERA) in 2011.
 A more comprehensive analysis of IGDT can be found in Review 2.
Appendix: Meet some descendants of Prototype 1
Prototype models are very useful in situations where we have to decide whether a given model is, or isn't, a maximin model. Situations like this are encountered quite often in the IGDT literature because some IGDT scholars persist in their effort to convince IGDT users that the theory's two core models are not maximin model.
To dispel those claims, I have shown numerous times, since 2004, and in writing since 2006, that these core models are instances of prototype WaldType maximin models. To no avail.
Hence, another reminder about mathematical objects in general, including models―not necessarily maximin models, is appropriate here.
The admissibility clause simply demands that the instantiation process does not get rid of properties of an object that make it a prototype of the class of objects under consideration. In other words, the admissibility issue simply means that the specification of the instance should not get rid of a property of the prototype that is required for the prototype to be $\dots$ a prototype. For example, consider the following two $2\times 2$ matrices
$$\begin{array}{ccc} M = \begin{bmatrix} a_{1,1} &a_{1,2} \\ a_{2,1}& a_{2,2} \end{bmatrix} &,& P = \begin{bmatrix} 1 &3 \\ 4& 2 \end{bmatrix} \end{array} $$Clearly, $P$ is an instance of $M$, namely it is the instance of $M$ specified by $a_{1,1} = 1; \ a_{1,2} = 3; \ a_{2,1} = 4;\ a_{2,2}= 2.$
On the other hand, in a discussion where $M$ represents a prototype of the class of $2\times 2$ symmetric matrices, it is necessarily assumed that $a_{2,1} = a_{1,2}$, and this property must be preserved by all admissible instances of $M$. Thus, in this context, $P$ is an instance, but not an admissible instance, of $M$. So in this context $P$ cannot be obtained by instantiating $M$ as there is no admissible instantiation of $M$ that yields $P$.
The implication of the Inheritance Theorem for maximin models is then:
The objective of this review being what it is, it is instructive to meet some the descendants, via instantiation, of Prototype 1. As shown above in the proof of The Fundamental Theorem, the two core IGDT models are descendants of this prototype. Recall:
The Three Musketeers $$ \begin{align} \large v: &\large = \max_{x\in X}\min_{s\in S(x)} \big\{f(x,s): c(x,s)\in C, \forall s\in S(x)\big\}\tag{Prototype 1}\\ \large v: &\large = \max_{x\in X} \big\{f(x): c(x,s)\in C, \forall s\in S(x)\big\}\tag{Prototype 2}\\ \large v: &\large = \max_{x\in X} \min_{s\in S(x)} \ f(x,s) \tag{Prototype 3} \end{align} $$
We are ready now to meet the elephant in the IGDT's room.
Yet Another Proof of the Fundamental Theorem of IGDT
By visual inspection,
 Observation 1: Instance 5 is no other than IGDT's robustsatisficing decision model (IRSDM).
 Observation 2: Instance 6 is no other than IGDT's robustness model (IRSDM).
Thus,
Remark.
As an exercise, readers may wish to practice their instantiation skills by proving that IGDT's two core models are admissible instances of Prototype 3 (Hint: incorporate the constraint into the objective function via a proper penalty).