User talk:Sniedo/Info-Gap

Background, working assumptions, and a look ahead

Decision under severe uncertainty is a formidable task. It should therefore be expected that the development of methodologies capable of handling this task would be an even greater undertaking. Indeed, over the past sixty years an enormous effort has gone into the development of such methodologies. Yet, for all the knowledge and expertise that have accrued in this area of decision theory, no fully satisfactory general methodology is available to date.

Now, info-gap was designed expressly for this purpose: it is portrayed as a methodology for solving decision problems that are subject to severe uncertainty. And what is more, its aim is to seek solutions that are robust.

Thus, to have a clear picture of info-gap's modus operandi and its role and place in decision theory and robust optimization, it is imperative to examine it within this context. In other words, it is necessary to establish info-gap's relation to classical decision theory and robust optimization. To this end, the following questions must be addressed:

What are the characteristics of decision problems that are subject to severe uncertainty?
What difficulties arise in the modelling and solution of such problems?
What type of robustness is sought?
How does info-gap theory address these issues?
In what way is info-gap decision theory similar to and/or different from other theories for decision under uncertainty?

In this regard, two important points must be elucidated at the outset:

Considering the severity of the uncertainty that info-gap was designed to tackle, it is essential to clarify the difficulties posed by severe uncertainty.
Since info-gap is a non-probabilistic method that seeks to maximize robustness to uncertainty, it is imperative to compare it to the single most important "non-probabilistic" model in classical decision theory, namely Wald's Maximin paradigm (Wald 1945, 1950). After all, this paradigm has dominated the scene in classical decision theory for well over sixty years now.

So, first, let us get clear on the assumptions that undrlie severe uncertainty.

Working assumptions

The uncertainty associated with a decision problem is captured in the info-gap framework by three basic constructs:

A parameter $\displaystyle u$ whose true value is subject to severe uncertainty.
A region of uncertainty $\displaystyle {\mathfrak {U}}\$ where the true value of $\displaystyle u\$ lies.
An estimate $\ \displaystyle {\tilde {u}}\$ of the true value of $\displaystyle u\$ .

It should be pointed out, though, that as such these constructs are generic meaning that they can be employed to model situations where the uncertainty is not severe but mild, indeed very mild. So it is vital to be clear that to express the severity of the uncertainty these three constructs are given specific definitions in the Info-gap framework.

Working Assumptions

The region of uncertainty $\displaystyle {\mathfrak {U}}\$ is relatively large.
In fact, Ben-Haim (2006, p. 210) indicates that in the context of info-gap theory most of the commonly encountered regions of uncertainty are unbounded.

The estimate $\displaystyle {\tilde {u}}\$ is a poor approximation of the true value of $\displaystyle \ u\$ .
That is, the estimate is a poor indication of the true value of $\displaystyle \ u\$ (Ben-Haim, 2006, p. 280) and is likely to be substantially wrong (Ben-Haim, 2006, p. 280).

In the picture $\displaystyle u^{\circ }\$ represents the true (unknown) value of $\ \displaystyle u\$ .
The point to note here is that conditions of severe uncertainty entail that the estimate $\displaystyle {\tilde {u}}\$ can -- relatively speaking -- be distant from the true value $\displaystyle u^{\circ }\$ . This is particularly pertinant for methodologies, like info-gap, that seek robustness to uncertainty. Indeed, assuming otherwise would -- methodologically speaking -- be tantamount to engaging in whishful thinking.

In short, the situations that info-gap is designed to take on are demanding in the extreme. Hence, the challenge that one faces conceptually, methodologically and technically is considerable. It is essential therefore to examine whether info-gap robustness analysis succeeds in this task, and whether the tools that it deploys in this effort are different from those made available by Wald's (1945) Maximin paradigm especially for robust optimization.

So let us take a quick look at this stalwart of classical decision theory and robust optimization.

Wald's Maximin paradigm

The basic idea of the famous Maximin paradigm can be expressed in plain language as follows:

Maximin Rule
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 outcome of the others.
Rawls (1971, p. 152)

Thus, according to the Maximin paradigm, in the framework of decision-making under severe uncertainty, the robustness of an alternative is a measure of how well this alternative can cope with the worst uncertain outcome that it can generate. Needless to say, this attitude towards severe uncertainty often leads to the selection of highly conservative alternatives. This is precisely the reason that Wald's Maximin paradigm is not always a satisfactory methodology for decision-making under severe uncertainty (Tintner 1952).

As indicated in the overview, info-gap's robustness model is a Maximin model in disguise. More specifically, it is a simple instance of Wald's Maximin model where:

The region of uncertainty associated with an alternative decision is an immediate neighbourhood of the estimate $\displaystyle {\tilde {u}}\$ .
The uncertain outcomes of an alternative are determined by a characteristic function of the performance requirement under consideration.

Thus, aside from the conservatism issue, the far more serious issue of validity must be addressed. The point here is that the validity of the results generated by info-gap's robustness analysis are crucially contingent on the quality of the estimate $\displaystyle {\tilde {u}}\$ , which, according to info-gap's own working assumptions, is poor and likely to be substantially wrong (Ben-Haim, 2006, p. 280-281).

And there is yet another reason why the intimate relation to Maximin is crucial to this discussion. This has to do with the portrayal of Info-gap's role and place in decision theory vis-a-vis other methodologies and its contribution to the state of the art in this field.

Role and place in decision theory

Info-gap is emphatic about its advancement of the state of the art in decision theory vis-a-vis existing methodologies in this area (color is used here 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.
Ben-Haim (2006, p.xii)

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.
Ben-Haim (2006, p. 11)

These strong claims must be substantiated. In particular, a clear-cut, unequivocal answer must be given to the question: in what way is info-gap's generic robustness model different, indeed radically different, from Maximin?

Subsequent sections of this article describe various aspects of info-gap decision theory and its applications, how it attempts to cope with the working assumptions cited above and its intimate relationship with Wald's classical Maximin paradigm and worst-case analysis.