Mixed Levels of Uncertainty in Complex Policy Models

The characterization and treatment of uncertainty poses special challenges when modeling indeterminate or complex coupled systems such as those involved in the interactions between human activity, climate and the ecosystem. Uncertainty about model structure may become as, or more important than, uncertainty about parameter values. When uncertainty grows so large that prediction or optimization no longer makes sense, it may still be possible to use the model as a behavioral test bed to examine the relative robustness of alternative observational and behavioral strategies. When models must be run into portions of their phase space that are not well understood, different submodels may become unreliable at different rates. A common example involves running a time stepped model far into the future. Several strategies can be used to deal with such situations. The probability of model failure can be reported as a function of time. Possible alternative surprises can be assigned probabilities, modeled separately, and combined. Finally, through the use of subjective judgments, one may be able to combine, and over time shift between models, moving from more detailed to progressively simpler order-of-magnitude models, and perhaps ultimately, on to simple bounding analysis.     

A Simple Formula for Calculating the "Mass Density" of a Lognormally Distributed Characteristic: Applications to Risk Analysis

Statements such as "80% of the employees do 20% of the work" or "the richest 1% of society controls 10% of its assets" are commonly used to describe the distribution or concentration of a variable characteristic within a population. Analogous statements can be constructed to reflect the relationship between probability and concentration for unvarying quantities surrounded by uncertainty. Both kinds of statements represent specific usages of a general relationship, the "mass density function," that is not widely exploited in risk analysis and management. This paper derives a simple formula for the mass density function when the uncertainty and/or the variability in a quantity is lognormally distributed; the formula gives the risk analyst an exact, "back-of-the-envelope" method for determining the fraction of the total amount of a quantity contained within any portion of its distribution. For example, if exposures to a toxicant are lognormally distributed with σ_{in x}= 2, 50% of all the exposure is borne by the 2.3% of persons most heavily exposed. Implications of this formula for various issues in risk assessment are explored, including: (1) the marginal benefits of risk reduction; (2) distributional equity and risk perception; (3) accurate confidence intervals for the population mean when a limited set of data is available; (4) the possible biases introduced by the uncritical assumption that extreme "outliers" exist; and (5) the calculation of the value of new information.     

不确定性与企业的对策研究

不确定性是企业不可能避免的一个问题，它一直不断地带给企业机会和危险，它也具有可以确定化的特征。作者探讨并提出了企业在不确定性影响下如何规避可能的损失并获得极大化利益的应对方案。     

E-Capacities and the Ellsberg Paradox

Ellsberg's (1961) famous paradox shows that decision-makers give events with known probabilities a higher weight in their outcome evaluation. In the same article, Ellsberg suggests a preference representation which has intuitive appeal but lacks an axiomatic foundation. Schmeidler (1989) and Gilboa (1987) provide an axiomatisation for expected utility with non-additive probabilities. This paper introduces E-capacities as a representation of beliefs which incorporates objective information about the probability of events. It can be shown that the Choquet integral of an E-capacity is the Ellsberg representation. The paper further explores properties of this representation of beliefs and provides an axiomatisation for them.     

Choices at various levels of uncertainty: An experimental test of the restated diversification theorem

Our “Restated diversification theorem” (Skogh and Wu, 2005) says that risk-averse agents may pool risks efficiently without assignment of subjective probabilities to outcomes, also at genuine uncertainty. It suffices that the agents presume that they face equal risks. Here, the theorem is tested in an experiment where the probability of loss, and the information about this probability, varies. The result supports our theorem. Moreover, it tentatively supports an evolutionary theory of the insurance industry—starting with mutual pooling at uncertainty, turning into insurance priced ex ante when actuarial information is available.     

Dominance and Efficiency in Multicriteria Decision under Uncertainty

This paper proposes several concepts of efficient solutions for multicriteria decision problems under uncertainty. We show how alternative notions of efficiency may be grounded on different decision contexts, depending on what is known about the Decision Maker's (DM) preference structure and probabilistic anticipations. We define efficient sets arising naturally from polar decision contexts. We investigate these sets from the points of view of their relative inclusions and point out some particular subsets which may be especially relevant to some decision situations.     

Backward Induction Is Not Robust: The Parity Problem and the Uncertainty Problem

A cornerstone of game theory is backward induction, whereby players reason backward from the end of a game in extensive form to the beginning in order to determine what choices are rational at each stage of play. Truels, or three-person duels, are used to illustrate how the outcome can depend on (1) the evenness/oddness of the number of rounds (the parity problem) and (2) uncertainty about the endpoint of the game (the uncertainty problem). Since there is no known endpoint in the latter case, an extension of the idea of backward induction is used to determine the possible outcomes. The parity problem highlights the lack of robustness of backward induction, but it poses no conflict between foundational principles. On the other hand, two conflicting views of the future underlie the uncertainty problem, depending on whether the number of rounds is bounded (the players invariably shoot from the start) or unbounded (they may all cooperate and never shoot, despite the fact that the truel will end with certainty and therefore be effectively bounded). Some real-life examples, in which destructive behavior sometimes occurred and sometimes did not, are used to illustrate these differences, and some ethical implications of the analysis are discussed.     

A Rule For Updating Ambiguous Beliefs

When preferences are such that there is no unique additive prior, the issue of which updating rule to use is of extreme importance. This paper presents an axiomatization of the rule which requires updating of all the priors by Bayes rule. The decision maker has conditional preferences over acts. It is assumed that preferences over acts conditional on event E happening, do not depend on lotteries received on E ^c, obey axioms which lead to maxmin expected utility representation with multiple priors, and have common induced preferences over lotteries. The paper shows that when all priors give positive probability to an event E, a certain coherence property between conditional and unconditional preferences is satisfied if and only if the set of subjective probability measures considered by the agent given E is obtained by updating all subjective prior probability measures using Bayes rule.     

Estimating the Ozone Depletion Potential of Methyl Bromide

Ozone depletion potential (ODP) represents the cumulative ozone depletion induced by a particular halocarbon relative to a reference gas (usually trichlorofluoromethane, CFC-11). We focus on ODP estimation for methyl bromide. Previous attempts at its estimation have assumed that components of the ODP equation are lognormally distributed. By considering a wide range of modeling scenarios, we show that this restriction (which is based on computational convenience rather than experimental evidence) has obscured the true uncertainty in the ODP value. Moreover, when publishing point estimates for the ODP value, previous authors have given either mean or median values. We submit that a more appropriate choice for a point estimate is the mode since the distribution of ODP is skewed and since the mode is by definition, the most likely value. For each modeling scenario considered, modal values are given. In general, we find these ODP point estimates are considerably lower than those published elsewhere.     

Propagation of Uncertainty in Risk Assessments: The Need to Distinguish Between Uncertainty Due to Lack of Knowledge and Uncertainty Due to Variability

In quantitative uncertainty analysis, it is essential to define rigorously the endpoint or target of the assessment. Two distinctly different approaches using Monte Carlo methods are discussed: (1) the end point is a fixed but unknown value (e.g., the maximally exposed individual, the average individual, or a specific individual) or (2) the end point is an unknown distribution of values (e.g., the variability of exposures among unspecified individuals in the population). In the first case, values are sampled at random from distributions representing various "degrees of belief" about the unknown "fixed" values of the parameters to produce a distribution of model results. The distribution of model results represents a subjective confidence statement about the true but unknown assessment end point. The important input parameters are those that contribute most to the spread in the distribution of the model results. In the second case, Monte Carlo calculations are performed in two dimensions producing numerous alternative representations of the true but unknown distribution. These alternative distributions permit subject confidence statements to be made from two perspectives: (1) for the individual exposure occurring at a specified fractile of the distribution or (2) for the fractile of the distribution associated with a specified level of individual exposure. The relative importance of input parameters will depend on the fractile or exposure level of interest. The quantification of uncertainty for the simulation of a true but unknown distribution of values represents the state-of-the-art in assessment modeling.