Info-gap decision theory

Info gap decision theory seeks to optimize robustness to failure under severe uncertainty, in particular applying sensitivity analysis of the stability radius type to perturbations in the value of a given estimate of the parameter of interest. It has some connections with Wald's maximin model; some authors distinguish them, others consider them instances of the same principle. It has been developed by Yakov Ben Haim, and has found many applications and described as a theory for decision making under "severe uncertainty". It has been criticized as unsuited for this purpose, and alternatives proposed, including such classical approaches as robust optimization.

In contrast to probabilistic decision theory, info gap analysis does not use probability distributions: it measures the deviation of errors (differences between the parameter and the estimate), but not the probability of outcomes – in particular, the estimate is in no sense more or less likely than other points, as info gap does not use probability. Info gap, by not using probability distributions, is robust in that it is not sensitive to assumptions on probabilities of outcomes. However, the model of uncertainty does include a notion of "closer" and "more distant" outcomes, and thus includes some assumptions, and is not as robust as simply considering all possible outcomes, as in minimax. Further, it considers a fixed universe so it is not robust to unexpected (not modeled) events. The connection to minimax analysis has occasioned some controversy: (Ben Haim 1999, pp. 271–2) argues that info gap's robustness analysis, while similar in some ways, is not minimax worst case analysis, as it does not evaluate decisions over all possible outcomes, while (Sniedovich, 2007) argues that the robustness analysis can be seen as an example of maximin (not minimax). This is discussed in criticism, below, and elaborated in the classical decision theory perspective.

It is common to make decisions under uncertainty. What can be done to make good (or at least the best possible) decisions under conditions of uncertainty? Info gap robustness analysis evaluates each feasible decision by asking: how much deviation from an estimate of a parameter value, function, or set, is permitted and yet "guarantee" acceptable performance? In everyday terms, the "robustness" of a decision is set by the size of deviation from an estimate that still leads to performance within requirements when using that decision. It is sometimes difficult to judge how much robustness is needed or sufficient. However, according to info gap theory, the ranking of feasible decisions in terms of their degree of robustness is independent of such judgments. Info gap theory also proposes an opportuneness function which evaluates the potential for windfall outcomes resulting from favorable uncertainty.

Suppose you are a project manager, supervising two teams: orange and white. Some revenue at the end of the year will be achieved. You have a limited timescale, and you aim to decide how to space these resources between the orange and white, so that the total revenues are large.

The actual revenue may be different. For uncertainty level we can define an envelope. Lower uncertainty would correspond to a smaller envelope. These envelopes are called info gap models of uncertainty, since they describe one's understanding of the uncertainty surrounding the revenue functions. We can find a model for the total revenue. Figure 5 shows the info gap model of the total revenue.

High revenues would typically earn a project manager the senior management's respect, but if the total revenues are below a certain threshold, it will cost said project manager's job. We will define such a threshold as a critical revenue, since total revenues beneath the critical revenue will be considered as failure. This is shown in Figure 6. If the uncertainty will increase, the envelope of uncertainty will become more inclusive, to include instances of the total revenue function that, for the specific allocation, yields a revenue smaller than the critical revenue. The robustness measures the immunity of a decision to failure. A robust satisficer is a decision maker that prefers choices with higher robustness. If, for some allocation , the correlation between the critical revenue and the robustness is illustrated, the result is a graph somewhat similar to that in Figure 7. This graph, called robustness curve of allocation , has two important features, that are common to (most) robustness curves:

The curve is non increasing. This captures the notion that when higher requirements (higher critical revenue) are in place, failure to meet the target is more likely (lower robustness). This is the tradeoff between quality and robustness. At the nominal revenue, that is, when the critical revenue equals the revenue under the nominal model (the estimate of the revenue functions), the robustness is zero. This is since a slight deviation from the estimate may decrease the total revenue. The decision depends on the value of failure.

As well as the threat of losing your job, the senior management offers you a carrot: if the revenues are higher than some revenue, you will be rewarded. If the uncertainty will decrease, the envelope of uncertainty will become less inclusive, to exclude all instances of the total revenue function that, for the specific allocation, yields a revenue higher than the windfall revenue. If, for some allocation , we will illustrate the correlation between the windfall revenue and the robustness, we will have a graph somewhat similar to Figure 10. This graph, called opportuneness curve of allocation , has two important features, that are common to (most) opportuneness curves:

The curve is non decreasing. This captures the notion that when we have higher requirements (higher windfall revenue), we are more immune to failure (higher opportuneness, which is less desirable). That is, we need a more substantial deviation from the estimate in order to achieve our ambitious goal. This is the tradeoff between quality and opportuneness. At the nominal revenue, that is, when the critical revenue equals the revenue under the nominal model (our estimate of the revenue functions), the opportuneness is zero. This is since no deviation from the estimate is needed in order to achieve the windfall revenue.

Info gaps are quantified by info gap models of uncertainty. An info gap model is an unbounded family of nested sets. For example, a frequently encountered example is a family of nested ellipsoids all having the same shape. The structure of the sets in an info gap model derives from the information about the uncertainty. In general terms, the structure of an info gap model of uncertainty is chosen to define the smallest or strictest family of sets whose elements are consistent with the prior information. Since there is, usually, no known worst case, the family of sets may be unbounded. A common example of an info gap model is the fractional error model. The best estimate of an uncertain function is , but the fractional error of this estimate is unknown. The following unbounded family of nested sets of functions is a fractional error info gap model:

At any horizon of uncertainty , the set contains all functions whose fractional deviation from is no greater than However, the horizon of uncertainty is unknown, so the info gap model is an unbounded family of sets, and there is no worst case or greatest deviation. There are many other types of info gap models of uncertainty. All info gap models obey two basic axioms:

Nesting. The info gap model is nested if implies that:

Contraction. The info gap model is a singleton set containing its center point:

The nesting axiom imposes the property of "clustering" which is characteristic of info gap uncertainty. Furthermore, the nesting axiom implies that the uncertainty sets become more inclusive as grows, thus endowing with its meaning as a horizon of uncertainty. The contraction axiom implies that, at horizon of uncertainty zero, the estimate is correct. Recall that the uncertain element may be a parameter, vector, function or set. The info gap model is then an unbounded family of nested sets of parameters, vectors, functions or sets.

Uncertainty may be either pernicious or propitious. That is, uncertain variations may be either adverse or favorable. Adversity entails the possibility of failure, while favorability is the opportunity for sweeping success. Info gap decision theory is based on quantifying these two aspects of uncertainty, and choosing an action which addresses one or the other or both of them simultaneously. The pernicious and propitious aspects of uncertainty are quantified by two "immunity functions": the robustness function expresses the immunity to failure, while the opportuneness function expresses the immunity to windfall gain.

Based on these function, one can then decided on a course of action by optimizing for uncertainty: choose the decision which is most robust (can withstand the greatest uncertainty; "satisficing"), or choose the decision which requires the least uncertainty to achieve a windfall. Formally, optimizing for robustness or optimizing for opportuneness yields a preference relation on the set of decisions, and the decision rule is the "optimize with respect to this preference". In the below, let be the set of all available or feasible decision vectors .

A typical engineering application is the vibration analysis of a cracked beam, where the location, size, shape and orientation of the crack is unknown and greatly influence the vibration dynamics. uses the fact that the robustness curves of different alternatives can intersect.

Sensitivity analysis – how sensitive conclusions are to input assumptions – can be performed independently of a model of uncertainty: most simply, one may take two different assumed values for an input and compares the conclusions. From this perspective, info gap can be seen as a technique of sensitivity analysis, though by no means the only.

The robust optimization literature provides methods and techniques that take a global approach to robustness analysis. These methods directly address decision under severe uncertainty, and have been used for this purpose for more than thirty years now. Wald's Maximin model is the main instrument used by these methods. The principal difference between the Maximin model employed by info gap and the various Maximin models employed by robust optimization methods is in the manner in which the total region of uncertainty is incorporated in the robustness model. Info gap takes a local approach that concentrates on the immediate neighborhood of the estimate. In sharp contrast, robust optimization methods set out to incorporate in the analysis the entire region of uncertainty, or at least an adequate representation thereof. In fact, some of these methods do not even use an estimate.

Classical decision theory, offers two approaches to decision making under severe uncertainty, namely maximin and Laplaces' principle of insufficient reason (assume all outcomes equally likely); these may be considered alternative solutions to the problem info gap addresses. Further, as discussed at decision theory: alternatives to probability theory, probabilists, particularly Bayesians probabilists, argue that optimal decision rules (formally, admissible decision rules) can always be derived by probabilistic methods (this is the statement of the complete class theorems), and thus that non probabilistic methods such as info gap are unnecessary and do not yield new or better decision rules.

As attested by the rich literature on robust optimization, maximin provides a wide range of methods for decision making in the face of severe uncertainty. Indeed, as discussed in criticism of info gap decision theory, info gap's robustness model can be interpreted as an instance of the general maximin model.

As for Laplaces' principle of insufficient reason, in this context it is convenient to view it as an instance of Bayesian analysis. The essence of the Bayesian analysis is applying probabilities for different possible realizations of the uncertain parameters. In the case of Knightian (non probabilistic) uncertainty, these probabilities represent the decision maker's "degree of belief" in a specific realization. In our example, suppose there are only five possible realizations of the uncertain revenue to allocation function. The decision maker believes that the estimated function is the most likely, and that the likelihood decreases as the difference from the estimate increases. Figure 11 exemplifies such a probability distribution. Now, for any allocation, one can construct a probability distribution of the revenue, based on his prior beliefs. The decision maker can then choose the allocation with the highest expected revenue, with the lowest probability for an unacceptable revenue, etc. The most problematic step of this analysis is the choice of the realizations probabilities. When there is an extensive and relevant past experience, an expert may use this experience to construct a probability distribution. But even with extensive past experience, when some parameters change, the expert may only be able to estimate that is more likely than , but will not be able to reliably quantify this difference. Furthermore, when conditions change drastically, or when there is no past experience at all, it may prove to be difficult even estimating whether is more likely than
Nevertheless, methodologically speaking, this difficulty is not as problematic as basing the analysis of a problem subject to severe uncertainty on a single point estimate and its immediate neighborhood, as done by info gap. And what is more, contrary to info gap, this approach is global, rather than local. Still, it must be stressed that Bayesian analysis does not expressly concern itself with the question of robustness. Bayesian analysis raises the issue of learning from experience and adjusting probabilities accordingly. In other words, decision is not a one stop process, but profits from a sequence of decisions and observations.