Info-gap decision theory

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Info-gap decision theory is a non-probabilistic decision theory seeking to optimize robustness to failure, or opportunity of windfall. This differs from classical decision theory, which typically maximizes the expected utility.

In many fields, including engineering, economics, management, biological conservation, medicine, homeland security, and more, analysts use models and data to evaluate and formulate decisions. An info-gap is the disparity between what is known and what needs to be known in order to make a reliable and responsible decision. Info-gaps are Knightian uncertainties: a lack of knowledge, an incompleteness of understanding. Info-gaps are non-probabilistic and cannot be insured against or modelled probabilistically. A common info-gap, though not the only kind, is uncertainty in the shape of a probability distribution. Another common info-gap is uncertainty in the functional form of a property of the system, such as friction force in engineering, or the Phillips curve in economics.

[edit] Info-gap models

Info-gaps are quantified by info-gap models of uncertainty. An info-gap model is an unbounded family of nested sets all sharing a common structure. 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.

A common example of an info-gap model is the fractional error model. The best estimate of an uncertain function $u(x)\!\,$ is ${\tilde{u}}(x)$ , 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:

$\mathcal{U}(\alpha, {\tilde{u}}) = \left \{ u(x): \ |u(x) - {\tilde{u}}(x) | \le \alpha {\tilde{u}}(x) \right \} , \ \ \ \alpha \ge 0$

At any horizon of uncertainty $α$ , the set $\mathcal{U}(\alpha, {\tilde{u}})$ contains all functions $u(x)\!\,$ whose fractional deviation from ${\tilde{u}}(x)$ is no greater than $α$ . 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 $\mathcal{U}(\alpha, {\tilde{u}})$ is nested if $\alpha < \alpha^\prime$ implies that:

$\mathcal{U}(\alpha, {\tilde{u}}) \ \subset \ \mathcal{U}(\alpha^\prime, {\tilde{u}})$

Contraction. The info-gap model $\mathcal{U}(0,{\tilde{u}})$ is a singleton set containing its center point:

$\mathcal{U}(0,{\tilde{u}}) = \{ {\tilde{u}} \}$

The nesting axiom imposes the property of "clustering" which is characteristic of info-gap uncertainty. Furthermore, the nesting axiom implies that the uncertainty sets $\mathcal{U}(\alpha, u)$ become more inclusive as $α$ grows, thus endowing $α$ with its meaning as an horizon of uncertainty. The contraction axiom implies that, at horizon of uncertainty zero, the estimate ${\tilde{u}}$ is correct.

[edit] Robustness and opportuneness

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.

The robustness function expresses the greatest level of uncertainty at which failure cannot occur; the opportuneness function is the least level of uncertainty which entails the possibility of sweeping success. The robustness and opportuneness functions address, respectively, the pernicious and propitious facets of uncertainty.

Let $q$ be a decision vector of parameters such as design variables, time of initiation, model parameters or operational options. We can verbally express the robustness and opportuneness functions as the maximum or minimum of a set of values of the uncertainty parameter $α$ of an info-gap model:

${\hat{\alpha}}(q) = \max \{ \alpha: \ \mbox{minimal requirements are always satisfied}\}$	(robustness)	(1)
${\hat{\beta}}(q) = \min \{ \alpha: \ \mbox{sweeping success is possible}\}$	(opportuneness)	(2)

We can "read" eq. (1) as follows. The robustness ${\hat{\alpha}}(q)$ of decision vector $q$ is the greatest value of the horizon of uncertainty $α$ for which specified minimal requirements are always satisfied. ${\hat{\alpha}}(q)$ expresses robustness — the degree of resistance to uncertainty and immunity against failure — so a large value of ${\hat{\alpha}}(q)$ is desirable. Eq. (2) states that the opportuneness ${\hat{\beta}}(q)$ is the least level of uncertainty $α$ which must be tolerated in order to enable the possibility of sweeping success as a result of decisions $q$ . ${\hat{\beta}}(q)$ is the immunity against windfall reward, so a small value of ${\hat{\beta}}(q)$ is desirable. A small value of ${\hat{\beta}}(q)$ reflects the opportune situation that great reward is possible even in the presence of little ambient uncertainty. The immunity functions ${\hat{\alpha}}(q)$ and ${\hat{\beta}}(q)$ are complementary and are defined in an anti-symmetric sense. Thus "bigger is better" for ${\hat{\alpha}}(q)$ while "big is bad" for ${\hat{\beta}}(q)$ . The immunity functions — robustness and opportuneness — are the basic decision functions in info-gap decision theory.

The robustness function involves a maximization, but not of the performance or outcome of the decision. The greatest tolerable uncertainty is found at which decision $q$ satisfices the performance at a critical survival-level. One may select an action $q$ according to its robustness ${\hat{\alpha}}(q)$ , whereby the robustness function underlies a satisficing decision algorithm which maximizes the immunity to pernicious uncertainty.

The opportuneness function in eq. (2) involves a minimization, however not, as might be expected, of the damage which can accrue from unknown adverse events. The least horizon of uncertainty is sought at which decision $q$ enables (but does not necessarily guarantee) large windfall gain. Unlike the robustness function, the opportuneness function does not satisfice, it "windfalls". When ${\hat{\beta}}(q)$ is used to choose an action $q$ , one is "windfalling" by optimizing the opportunity from propitious uncertainty in an attempt to enable highly ambitious goals or rewards.

Given a scalar reward function $R (q, u)$ , depending on the decision vector $q$ and the info-gap-uncertain function $u$ , the minimal requirement in eq. (1) is that the reward $R (q, u)$ be no less than a critical value $r c$ . Likewise, the sweeping success in eq. (2) is attainment of a "wildest dream" level of reward $r w$ which is much greater than $r c$ . Usually neither of these threshold values, $r c$ and $r w$ , is chosen irrevocably before performing the decision analysis. Rather, these parameters enable the decision maker to explore a range of options. In any case the windfall reward $r w$ is greater, usually much greater, than the critical reward $r c$ :

r w > r c

The robustness and opportuneness functions of eqs. (1) and (2) can now be expressed more explicitly:

${\hat{\alpha}}(q, {r_{\rm c}}) = \max \left \{ \alpha: \ \left ( \min_{u \in \mathcal{U}(\alpha, \tilde{u})} R(q,u) \right ) \ge {r_{\rm c}} \right \}$	(3)
${\hat{\beta}}(q, {r_{\rm w}}) = \min \left \{ \alpha: \ \left ( \max_{u \in \mathcal{U}(\alpha, \tilde{u})} R(q,u) \right ) \ge {r_{\rm w}} \right \}$	(4)

${\hat{\alpha}}(q, {r_{\rm c}})$ is the greatest level of uncertainty consistent with guaranteed reward no less than the critical reward $r c$ , while ${\hat{\beta}}(q, {r_{\rm w}})$ is the least level of uncertainty which must be accepted in order to facilitate (but not guarantee) windfall as great as $r w$ . The complementary or anti-symmetric structure of the immunity functions is evident from eqs. (3) and (4).

These definitions can be modified to handle multi-criterion reward functions.

The robustness function generates robust-satisficing preferences on the options. A decision maker will usually prefer a decision option $q\,\!$ over an alternative $q^\prime$ if the robustness of $q\,\!$ is greater than the robustness of $q^\prime$ at the same value of critical reward $r c$ . That is:

$q > _{\rm r} q^\prime$ if ${\hat{\alpha}}(q, {r_{\rm c}}) > {\hat{\alpha}}(q^\prime, {r_{\rm c}})$

(5)

Let $\mathcal{Q}$ be the set of all available or feasible decision vectors $q$ . A robust-satisficing decision is one which maximizes the robustness on the set $\mathcal{Q}$ of available $q$ -vectors and satisfices the performance at the critical level $r c$ :

${\hat{q}_{{\rm c}}}({r_{\rm c}}) = \arg \max_{q \in \mathcal{Q}} {\hat{\alpha}}(q, {r_{\rm c}})$

Usually, though not invariably, the robust-satisficing action ${\hat{q}_{{\rm c}}}({r_{\rm c}})$ depends on the critical reward $r c$ .

The opportuneness function generates opportune-windfalling preferences on the options. A decision maker who chooses to focus on windfall opportunity will prefer a decision $q$ over an alternative $q^\prime$ if $q$ is more opportune than $q^\prime$ at the same level of reward $r w$ . Formally:

$q > _{\rm o} q^\prime$ if ${\hat{\beta}}(q, {r_{\rm w}}) < {\hat{\beta}}(q^\prime, {r_{\rm w}})$

(6)

The opportune-windfalling decision, ${\hat{q}_{{\rm w}}}({r_{\rm w}})$ , minimizes the opportuneness function on the set of available decisions:

${\hat{q}_{{\rm w}}}({r_{\rm w}}) = \arg \min_{q \in \mathcal{Q}} {\hat{\beta}}(q, {r_{\rm w}})$

The two preference rankings, eqs. (5) and (6), as well as the corresponding the optimal decisions ${\hat{q}_{{\rm c}}}({r_{\rm c}})$ and ${\hat{q}_{{\rm w}}}({r_{\rm w}})$ , may be different.

The robustness and opportuneness functions have many properties which are important for decision analysis. Robustness and opportuneness both trade-off against aspiration for outcome: robustness and opportuneness deteriorate as the decision maker's aspirations increase. Robustness is zero for model-best anticipated outcomes. Robustness curves of alternative decisions may cross, implying reversal of preference depending on aspiration. Robustness may be either sympathetic or antagonistic to opportuneness: a change in decision which enhances robustness may either enhance or diminish opportuneness. Various theorems have also been proven which show how the probability of success is enhanced by enhancing the info-gap robustness, without of course knowing the underlying probability distribution.

[edit] References

The basic text for info-gap theory:

Yakov Ben-Haim, Info-Gap Decision Theory: Decisions Under Severe Uncertainty, Academic Press, 2nd edition, Sep. 2006. ISBN 0-12-373552-1.

An application to engineering structures:

Y. Kanno and I. Takewaki, Robustness analysis of trusses with separable load and structural uncertainties, International Journal of Solids and Structures, Volume 43, Issue 9, May 2006, pp.2646--2669.

An application to neural nets:

S.G. Pierce, K. Worden and G. Manson, 2006, A novel information-gap technique to assess reliability of neural network-based damage detection Journal of Sound and Vibration, 293: Issues 1--2, pp.96--111.

Applications to biological conservation:

A. Moilanen, and B.A. Wintle, 2006, Uncertainty analysis favours selection of spatially aggregated reserve structures. Biological Conservation, Volume 129, Issue 3, May 2006, Pages 427--434.

Helen M. Regan, Yakov Ben-Haim, Bill Langford, Will G. Wilson, Per Lundberg, Sandy J. Andelman, Mark A. Burgman, 2005, Robust decision making under severe uncertainty for conservation management, Ecological Applications, vol.15(4): 1471--1477.

An application to mathematical biology:

Yohay Carmel and Yakov Ben-Haim, 2005, Info-gap robust-satisficing model of foraging behavior: Do foragers optimize or satisfice?, American Naturalist, 166: 633-641.

An application to homeland security:

L. Joe Moffitt, John K. Stranlund, and Barry C. Field, 2005, Inspections to Avert Terrorism: Robustness Under Severe Uncertainty, Journal of Homeland Security and Emergency Management, Vol. 2: No. 3. http://www.bepress.com/jhsem/vol2/iss3/3

An application to financial economics:

Yakov Ben-Haim, 2005, Value at risk with Info-gap uncertainty, Journal of Risk Finance, vol. 6, #5, pp.388-403.

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Category: Decision theory

Info-gap decision theory

From Wikipedia, the free encyclopedia

[edit] Info-gap models

[edit] Robustness and opportuneness

[edit] References

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