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Reviews of publications on Info-Gap decision theory (IGDT)

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Review 8-2023 (Posted: June 26 2023)

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Reference	Marc Ciligot-Travain. Sur une notion de robustesse. Annales de l'ISUP, 2012, 56(2-3):37-48.
Publication type	article
Year of publication	2012
Downloads	hal-01334825.
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 "worst-case" 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
IF-IG perspective	It seems that the author of this article is skeptical about the validity of my claim (e.g. in [41], that info-gap robustness model is a maximin model. [41] = M. Sniedovich. Anatomy of a misguided maximin/minimax formulation of info-gap’s robustness model. Working Paper No. MS-04-08, 2008.

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Revisiting

Working Paper No. MS-2-06, 2006

Eureka! Info-Gap is Worst-Case Analysis (Maximin) in Disguise!

Moshe Sniedovich
Department of Mathematics and Statistics
The University of Melbourne
Melbourne, Australia

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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 info-gap 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 decision-making under severe uncertainty.

The main question that I addressed in Working Paper No. MS-2-06 is this:

Old (2006) Question

Is IGDT a highly specialized application of Wald's Maximin paradigm, or a new theory that is radically different from all current theories for decision-making under uncertainty―as claimed in the main texts on IGDT (Ben-Haim 2001, 2006)?

Working Paper No. MS-2-06 provided a formal, rigorous (mathematical) proof that IGDT robust-satisficing 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 (Ciligot-Travain 2012) is the only publication to-date 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 decision-making under severe uncertainty. For example,

More Recent (2022) Questions

In the recent book entitled Philosophies of Structural Safety and Reliability (Raizer and Elishakoff 2022, p. 115) we read this regarding IGDT:

More recently, Ben-Haim (2001) discusses the information-gap uncertainty:

Info-gap models quantify uncertainty as a size of the gap between what is known and what could be known.

Here, immediately, numerous questions arise on why the information that “could be known” is not presently “known”? Is this due to negligence of the particular designer or of the entire profession which did not think it was necessary to “know” what “could be known,” or the price involved with obtaining the information that “could be known” is too heavy presently, or because this needed information is not released by appropriate firms and companies? It appears that such an analysis, if defined as above, should include an answer to the posed question. Ben-Haim (2001) states:

Info-gap models of uncertainty are particularly useful when data on the uncertainties are quite limited.

One needs to know how “limited” the information should be to utilize info-gap models. The following pertinent questions arise: Why for the data that is available and considered as appropriate for the info-gap models, one cannot use the probabilistic modeling? Why not to utilize in such circumstances the fuzzy sets-based approaches? Is there a difference between the convex modeling of uncertainty (Ben-Haim and Elishakoff 1990) and the information-gap uncertainty? Is the “information-gap uncertainty” a new terminology or a new methodology? As we see there are plenty of questions demanding further research and answers.

This review focuses on :

New (2022) Question

Is the “information-gap uncertainty” a new terminology or a new methodology?

 

Answer 2:

The claim in the IGDT literature is that IGDT's modeling of uncertainty is new, more precisely was new, in the late 1990s when the theory was developed. In Ben-Haim (2001, 2006, p. 11) we read (color is added for emphasis):

In this book we concentrate on the fairly new concept of information-gap uncertainty, whose differences from more classical approaches to uncertainty are real and deep. Despite the power of classical decision theories, in many areas such as engineering, economics, management, medicine and public policy, a need has arisen for a different format for decisions based on severely uncertain evidence.

Regarding the methodology as a whole, in Ben-Haim (2001, 2006, p. xii) we read this (color is added for emphasis):

Info-gap decision theory is radically different from all current theories of decision under uncertainty. The difference originates in the modelling of uncertainty as an information gap rather than as a probability. The need for info-gap modeling and management of uncertainty arises in dealing with severe lack of information and highly unstructured uncertainty.

The fact is, though, that IGDT's two core models are maximin models and its robustness model is a reinvention of the well established Radius of Stability model.

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, step-by-step, formal, rigorous proof of this very old (almost 20 years!) result:

The Fundamental Theorem of IGDT

The two core models of IGDT, namely its robustness model (IGRM) and its robust-satisficing decision model (IRSDM) are simple maximin models. Recall:

$$ \begin{align} \large \hat{\alpha}(q):& \large = \max_{\alpha\ge 0} \ \big\{\alpha: L(q,u) \le L_{c}, \forall u\in \mathscr{U}(\alpha,\tilde{u})\big\} \ \ , \ \ q\in \mathcal{Q}\tag{IGRM}\\ \large \alpha^{*}: & \large = \max_{q\in \mathcal{Q}} \hat{\alpha}(q)\\ & \large = \max_{\substack{q\in \mathcal{Q}\\\alpha\ge 0}}\ \big\{\alpha: L(q,u) \le L_{c}, \forall u\in \mathscr{U}(\alpha,\tilde{u})\big\} \tag{IRSDM} \end{align} $$

Note that in both models $\tilde{u}$ is given, and that in (IGDT) $q$ is also given. In these models the sets $\mathscr{U}(\alpha, \tilde{u}), \alpha\ge 0$ are nested neighborhoods of size ("radius") $\alpha$ around the point estimate $\tilde{u}$ so that

$$ \begin{align} \mathscr{U}(0,\tilde{u}) &=\{\tilde{u}\}\\ \large \mathscr{U}(\alpha, \tilde{u}) & \subseteq \mathscr{U}(\alpha+\delta,\tilde{u}) \ \ , \ \ \forall \alpha\ge 0, \delta\ge 0 \end{align} $$

Assumptions.
With no loss of generality, assume that all the maxima defined above exist and $$\large L(q,\tilde{u}) \le L_{c}\ ,\ \forall q\in \mathcal{Q}$$

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

Abraham Wald
(1902-1950)

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 (1902-1950) 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 non-probabilistic uncertainty in decision-making, it is reasonable to minimize the maximum risk. By the mid 1950s this approach to non-probabilistic uncertainty became very popular in Decision Theory. At present this approach dominates the scene in the field of Robust Optimization.

So the term "Wald-type maximin model" is used here to distinguish between maximin models that have nothing to do with decision-making under uncertainty, and maximin models that describe situations involving decision-making under non-probabilistic uncertainty. The later are referred to as Wald-type 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 decision-making 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:

Maximin Decision Rule

Rank alternatives by their worst-case outcomes and select an alternative whose worst outcome is at least as good as the worst outcomes of the other alternatives.

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. 152-153):

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 decision-making under (non-probabilistic) uncertainty was proposed by Wald (1939) and by the 1950s became one of the fundamental decision rules in decision theory. In the context of decision-making 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 (worst-case 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.

Proof of The Fundamental Theorem of IGDT

In relation to (IRSDM), consider this maximin model, observing that it is a Prototype 1 type model:

$$ \begin{align} \large v & \large = \max_{\substack{q\in \mathcal{Q}\\\alpha\ge 0}}\ \min_{u\in \mathscr{U}(\alpha,\tilde{u})} \ \big\{h(q,\alpha,u): L(q,u) \le L_{c}, \forall u\in \mathscr{U}(\alpha,\tilde{u})\big\} \tag{*} \end{align} $$

where $h$ is a real-valued function of $q\in \mathcal{Q}$, $\alpha\in [0,\infty)$ and $u\in \mathscr{U}(\alpha,\tilde{u})$. Note that for the specific very simple case where

$$ h(q,\alpha,u) = \alpha \ \ \ , \ \ \ q\in \mathcal{Q}, \alpha\in [0,\infty), u\in \mathscr{U}(\alpha,\tilde{u}) $$

this maximin model simplifies to this equivalent maximin model:

$$ \begin{align} \large v & \large = \max_{\substack{q\in \mathcal{Q}\\\alpha\ge 0}}\ \ \min_{u\in \mathscr{U}(\alpha,\tilde{u})}\ \big\{\alpha: L(q,u) \le L_{c}, \forall u\in \mathscr{U}(\alpha,\tilde{u})\big\} \tag{**} \end{align} $$

Since the objective function being minimized here does not depend on $u$, the $\displaystyle \min_{u\in \mathscr{U}(\alpha,\tilde{u})}$ operation is superfluous. Thus, this maximin model can be simplified to this equivalent maximin model:

$$ \begin{align} \large v & \large = \max_{\substack{q\in \mathcal{Q}\\\alpha\ge 0}}\ \big\{\alpha: L(q,u) \le L_{c}, \forall u\in \mathscr{U}(\alpha,\tilde{u})\big\} \tag{***} \end{align} $$

This is no other than (IRSDM). We conclude then that (IRSDM) is a maximin model. Since , (IGRM) is the simple instance of (IRSDM) obtained by letting $\mathcal{Q}$ be a singleton, $\mathcal{Q}=\{q\}$, it follows that (IGRM) is also a maximin model.

Comment:
This proof is based on the fact that any admissible instance of a maximin model is also a maximin model.

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 Wald-type maximin models. Many formal, rigorous proofs of this fact have been easily accessible online and in peer-reviewed journals since 2007.

• Fact 2: By the same token, there is no doubt that IGDT's key concept, namely Info-Gap 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 Info-gap Robustness Model make the theory unsuitable for the treatment of severe uncertainties of the type stipulated by IGDT.

Bottom line questions

• How long will leading IGDT scholars continue to ignore these facts?
• How long will peer reviewed journals will accept for publication articles that ignore these facts?

Bibliographic notes

• Other working papers, articles, lectures, etc. on this topic for the period 2006-2011 are available online.
• Australian readers might be interested in taking a look at the report entitled Info-gap 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.

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Appendix: Meet some descendants of Prototype 1

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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 Wald-Type maximin models. To no avail.

Hence, another reminder about mathematical objects in general, including models―not necessarily maximin models, is appropriate here.

The Inheritance Theorem

If Object D belong to class $\mathscr{D}$ and Object E is an admissible instance of Object D, then Object E also belongs to class $\mathscr{D}$.

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 Inheritance Corollary

If Model A is a maximin model and Model B is an admissible instance of Model A, then Model B is also a maximin model.

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 $$ \begin{align} \large v: &\large = \max_{x\in X}\min_{s\in S(x)} \big\{f(x,s): c(x,s)\in C, \forall s\in S(x)\big\}\tag{Prototype 1}\\ \large v: &\large = \max_{x\in X} \big\{f(x): c(x,s)\in C, \forall s\in S(x)\big\}\tag{Prototype 2}\\ \large v: &\large = \max_{x\in X} \min_{s\in S(x)} \ f(x,s) \tag{Prototype 3} \end{align} $$

 

Instantiation No. 1

Consider the instance of Prototype 1 where the objective function $f$ does not depend on the state variable $s$. In this case Prototype 1 simplifies to this maximin model: $$ \begin{align} v: & = \max_{x\in X}\min_{s\in S(a)} \big\{f(x): c(x,s)\in C, \forall s\in S(x)\big\}\\ & = \max_{x\in X} \big\{f(x): c(x,s)\in C, \forall s\in S(x)\big\}\tag{Instance 1} \\ \end{align} $$

which is no other than Prototype 2.

Verify that the instantiation is admissible.

 

Instantiation No. 2

Consider the instance of Instance 1 specified by $$ \begin{align} X & \longleftarrow \mathcal{Q}\times[0,\infty)\\ x & \longleftarrow (q,\alpha)\\ \end{align} $$ This instantiation yields this maximin model: $$ v:= \max_{\substack{q\in \mathcal{Q}\\ \alpha\ge 0}} \big\{\,f(q,\alpha): c(q,\alpha,s) \in C, \forall s\in S(q,\alpha)\big\}\tag{Instance 2} $$

Verify that the instantiation is admissible.

 

Instantiation No. 3

Consider the instance of Instance 2 specified by $$ \begin{align} S(q,\alpha) & \longleftarrow \mathscr{U}(\alpha,\tilde{u})\\ s & \longleftarrow u\\ \end{align} $$ This instantiation yields this maximin model: $$ v:= \max_{\substack{q\in \mathcal{Q}\\ \alpha\ge 0}} \big\{\,f(q,\alpha): c(q,\alpha,u) \in C, \forall u\in \mathscr{U}(\alpha, \tilde{u})\big\}\tag{Instance 3} $$

Verify that the instantiation is admissible.

 

Instantiation No. 4

Consider the instance of Instance 3 specified by $$ \begin{align} c(x,s) & \longleftarrow L(q,u) \\ C &\longleftarrow (-\infty,L_{c}\,\big] \end{align} $$ This instantiation yields this maximin model: $$ v:= \max_{\substack{q\in \mathcal{Q}\\ \alpha\ge 0}} \big\{f(q,\alpha): L(q,u) \le L_{c}, \forall u\in \mathscr{U}(\alpha, \tilde{u})\big\}\tag{Instance 4} $$

Verify that the instantiation is admissible.

 

Instantiation No. 5

Consider the instance of Instance 4 specified by $$ \begin{align} f(q,\alpha) &\longleftarrow \alpha \end{align} $$ This instantiation yields this maximin model: $$ v:= \max_{\substack{q\in \mathcal{Q}\\ \alpha\ge 0}} \big\{\alpha: L(q,u) \le L_{c}, \forall u\in \mathscr{U}(\alpha,\tilde{u})\big\}\tag{Instance 5} $$

Verify that the instantiation is admissible.

 

Instantiation No. 6

Consider the instance of Instance 5 specified by $$ \begin{align} \mathcal{Q} \longleftarrow \{q\} \end{align} $$ namely the case where $\mathcal{Q}$ is a singleton. Since in this case $$ \max_{\substack{q\in \mathcal{Q}\\ \alpha\ge 0}} \equiv \max_{\alpha\ge 0} $$ this instantiation yields this maximin model: $$ \hat{\alpha}(q) := \max_{ \alpha\ge 0} \big\{\alpha: L(q,u) \le L_{c}, \forall u\in \mathscr{U}(\alpha,\tilde{u}\,)\big\}\tag{Instance 6} $$

Verify that the instantiation is admissible.

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 robust-satisficing decision model (IRSDM).
• Observation 2: Instance 6 is no other than IGDT's robustness model (IRSDM).

Thus,

A Detailed Proof of the Fundamental Theorem of IGDT

The above two observations imply that both (IRDSM) and (IGRM) have been obtained by a sequence of admissible instantiations of Prototype 1. Since Prototype 1 is a maximin model, so are (IRDSM) and (IGRM). It follows then that IGDT's two core models are maximin models.

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).