Yakov Ben-Haim

ROBUST DECISION-MAKING UNDER SEVERE UNCERTAINTY FOR CONSERVATION MANAGEMENT

In conservation biology it is necessary to make management decisions for endangered and threatened species under severe uncertainty. Failure to acknowledge and treat uncertainty can lead to poor decisions. To illustrate the importance of considering uncertainty, we reanalyze a decision problem for the Sumatran rhino, Dicerorhinus sumatrensis, using information-gap theory to propagate uncertainties and to rank management options. Rather than requiring information about the extent of parameter uncertainty at the outset, information-gap theory addresses the question of how much uncertainty can be tolerated before our decision would change. It assesses the robustness of decisions in the face of severe uncertainty. We show that different management decisions may result when uncertainty in utilities and probabilities are considered in decision-making problems. We highlight the importance of a full assessment of uncertainty in conservation management decisions to avoid, as much as possible, undesirable outcomes.

A non-probabilistic concept of reliability

Abstract Uncertainty can be modelled either probabilistically or non-probabilistically. The former option leads to the concept of reliability as the probability of no-failure. In this paper non-probabilistic convex models of uncertainty are used to formulate reliability in terms of acceptable system performance given uncertain operating environment or uncertain geometrical imperfections. It is shown that probabilistic reliability can be very sensitive to small inaccuracy in the probabilistic model. Consequently, the non-probabilistic concept of reliability is useful when insufficient information is available for verifying a probabilistic model. In addition, a theorem is presented showing that analogous convex and probabilistic models of input uncertainty can lead to very different predictions of the range of output variation.

A non-probabilistic measure of reliability of linear systems based on expansion of convex models⋆

Abstract In this paper we develop a rigorous quantitative alternative to the probabilistic theory of reliability. The intuitive concept of reliability which underlies our analysis is that a system is reliable if it is robust with respect to uncertainty. That is, a system is reliable if it can tolerate a large amount of uncertainty before failure can occur. Conversely, a system is unreliable if it is fragile with respect to uncertainty. We model uncertainty with non-probabilistic convex models, and measure the amount of uncertainty with the expansion parameters of the convex models. The measure of reliability developed here is the amount of uncertainty the system can tolerate before failure. We consider linear dynamic systems with uncertain inputs and uncertain output failure states. The reliability of these systems hinges on the disjointness of convex sets. Using the hyperplane separation theorem for convex sets, we introduce a concept of modal reliability.

Uncertainty, Probability and Information-gaps

Abstract This paper discusses two main ideas. First, we focus on info-gap uncertainty, as distinct from probability. Info-gap theory is especially suited for modelling and managing uncertainty in system models: we invest all our knowledge in formulating the best possible model; this leaves the modeller with very faulty and fragmentary information about the variation of reality around that optimal model. Second, we examine the interdependence between uncertainty modelling and decision-making. Good uncertainty modelling requires contact with the end-use, namely, with the decision-making application of the uncertainty model. The most important avenue of uncertainty-propagation is from initial data- and model-uncertainties into uncertainty in the decision-domain. Two questions arise. Is the decision robust to the initial uncertainties? Is the decision prone to opportune windfall success? We apply info-gap robustness and opportunity functions to the analysis of representation and propagation of uncertainty in several of the Sandia Challenge Problems.

Decision making in an uncertain world: information-gap modeling in water resources management

Information-gap (info-gap) modeling is put forth as a basic approach for enhancing decision making under uncertainty, especially when there is a high level of uncertainty and little information is available. The great need for having realistic techniques for describing severe uncertainty can be illustrated in water resource management by pointing out the wide range of uncertainties present in sustainable development when taking into account hydrological, socioeconomic, political, and other considerations. Some illustrative systems problems in watershed management are utilized to explain how info-gap modeling can be employed in practice.

Info‐Gap Robust‐Satisficing Model of Foraging Behavior: Do Foragers Optimize or Satisfice?

In this note we compare two mathematical models of foraging that reflect two competing theories of animal behavior: optimizing and robust satisficing. The optimal‐foraging model is based on the marginal value theorem (MVT). The robust‐satisficing model developed here is an application of info‐gap decision theory. The info‐gap robust‐satisficing model relates to the same circumstances described by the MVT. We show how these two alternatives translate into specific predictions that at some points are quite disparate. We test these alternative predictions against available data collected in numerous field studies with a large number of species from diverse taxonomic groups. We show that a large majority of studies appear to support the robust‐satisficing model and reject the optimal‐foraging model.

Robust rationality and decisions under severe uncertainty

Abstract This paper develops a prescriptive approach to decision-making with severely uncertain information, and explores risk-taking behavior, based on non-probabilistic set-models of information-gap uncertainty. Info-gap models are well suited for representing uncertainty arising from severe lack of information, and lead naturally to a decision strategy which maximizes the decision-makers immunity to uncertainty, while also achieving no less than a specified minimum reward. We prove a “gamblers theorem” which quantifies the trade-off between reward and immunity to uncertainty. This trade-off forces the decision-maker to gamble, but without employing a probabilistic framework. We present a complementary theorem expressing the trade-off between immunity and windfall reward, and a further result characterizing the antagonism between robustness to failure and opportunity for success. Next, we develop a measure of risk-sensitivity based on the idea of immunity to uncertainty, without any probabilistic underpinning and without the assumptions of von Neumann–Morgenstern utility theory. We prove a theorem which establishes the relation between a decision-makers aversion to uncertainty and the information which is available to him. Our final theorem establishes conditions in which the magnitude of the decision-makers commitment of resources will increase with his fondness for risk.

SET-MODELS OF INFORMATION-GAP UNCERTAINTY : AXIOMS AND AN INFERENCE SCHEME

Abstract The sparsity and complexity of information in many technological situations has led to the development of new methods for quantifying uncertain evidence, and new schemes of inference from uncertain data. This paper deals with set-models of information-gap uncertainty which employ geometrical rather than measure-theoretic tools, and which are radically different from both probability and fuzzy-logic possibility models. The first goal of this paper is the construction of an axiomatic basis for info-gap models of uncertainty. The result is completely different from Kolmogorovs axiomatization of probability. Once we establish an axiomatically distinct framework for uncertainty, we arrive at a new possibility for inference and decision from uncertain evidence. The development of an inference scheme from info-gap models of uncertainty is the second goal of this paper. This inference scheme is illustrated with two examples: a logical riddle and a mechanical engineering design decision.

Value‐at‐risk with info‐gap uncertainty

Purpose – To study the effect of Knightian uncertainty – as opposed to statistical estimation error – in the evaluation of value-at-risk (VaR) of financial investments. To develop methods for augmenting existing VaR estimates to account for Knightian uncertainty. Design/methodology/approach – The value at risk of a financial investment is assessed as the quantile of an estimated probability distribution of the returns. Estimating a VaR from historical data entails two distinct sorts of uncertainty: probabilistic uncertainty in the estimation of a probability density function (PDF) from historical data, and non-probabilistic Knightian info-gaps in the future size and shape of the lower tail of the PDF. A PDF is estimated from historical data, while a VaR is used to predict future risk. Knightian uncertainty arises from the structural changes, surprises, etc., which occur in the future and therefore are not manifested in historical data. This paper concentrates entirely on Knightian uncertainty and does not consider the statistical problem of estimating a PDF. Info-gap decision theory is used to study the robustness of a VaR to Knightian uncertainty in the distribution. Findings – It is shown that VaRs, based on estimated PDFs, have no robustness to Knightian errors in the PDF. An info-gap safety factor is derived that multiplies the estimated VaR in order to obtain a revised VaR with specified robustness to Knightian error in the PDF. A robustness premium is defined as a supplement to the incremental VaR for comparing portfolios. Practical implications – The revised VaR and incremental VaR augment existing tools for evaluating financial risk. Originality/value – Info-gap theory, which underlies this paper, is a non-probabilistic quantification of uncertainty that is very suitable for representing Knightian uncertainty. This enables one to assess the robustness to future surprises, as distinct from existing statistical techniques for assessing estimation error resulting from randomness of historical data.

The graph model for conflict resolution with information-gap uncertainty in preferences

Information-gap models, for formally modeling the uncertainty of preferences of decision makers involved in a conflict, are devised for employment with the graph model for conflict resolution. These information-gap models are designed for handling a variety of situations for expressing severe preference-uncertainty of a decision maker, including both transitive and intransitive preferences among the states or possible scenarios in a conflict. Applications of these decision technologies to the game of chicken and the Cuban Missile Crisis of 1962 illustrate how the information-gap models can be conveniently utilized in practice and how strategic insights can be gained through rigorous examination of the robustness of equilibrium solutions to uncertainty in preferences. It is also shown that uncertainty-analyses can lead to modification of a decision makers prior preferences.