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ing such values on the basis offixed outcome threshold(cf.Frohwein and Lambert[2000]).However,it should be noted that agencies of-ten regulate on the basis of an extreme percentile
F 1()(e.g.,95th percentile)rather than the probabil-ity of exceeding some fixed outcome threshold.There-fore,it is plausible to also consider the conditional ex-pected value on the basis of a fixed probability
threshold.It is not claimed that this measure of therisk of extreme events can,by itself,capture all facets ofrisk—no measure can.However,the conditional ex-pected value under consideration here can,possibly inconjunction with other measures of risk,provide helpful
information to the decision maker.For example,a man-ager may be concerned about the expected perfor-mance of his worst of 10 employees(0.9),or an en-vironmental scientist about the expected contaminationmeasured in the worst of 100 soil samples(0.99).
The companion paper(Frohwein and Lambert 2000)provides references on the use of conditional expectedvalues as a measure of the risk of extreme events.
The following section discusses the problemswith averaging out and folding back conditional ex-pected values,defined by a fixed nonexceedanceprobability,in decision trees.Next,the develop-ments to overcome these difficulties are outlined.Asrequired by the proposed approach,approximate ex-pressions for the conditional expected values arethen derived and conditions that enable the sequen-tial optimization of the conditional expected valueare established.Then,the optimization process issummarized and depicted in a flowchart.After the
key results have been reiterated,an example(con-tamination remediation)is provided to illustrate theapplication of the proposed method.Finally,conclud-ing remarks highlight the general importance of theresults for risk analysis.
2.PROBLEMS WITH AVERAGING OUT AND
FOLDING BACK CONDITIONAL
EXPECTED VALUES IN DECISION TREES
The conditional expected value of the outcome,
conditioned on the outcome magnitude falling in the
upper 100(1)percent of possible outcome mag-
nitudes,as a function of the chosen policy s,can be ex-
pressed as
f4,(s)E[X|X F 1(;s)],(1)
where X is a random variable,F 1(;s)denotes the in-verse of the cumulative probability distribution of X,given policy s,and is the decision maker’s nonex-ceedance probability of concern.The notation“f4,”for the conditional expected value follows previouspapers on the topic of conditional expected values asmeasure of the risk of extreme events(Asbeck andHaimes 1984,Haimes et al.1990).Averaging out andfolding back the conditional expected values f4,can-not be accomplished in the same manner as for un-conditional expected values,i.e.,
E[X|X F 1()]p1 E[X1|X1 F 1()]...
pn E[Xn|Xn F 1()],(2)where pi denotes the probability of obtaining random
variable Xi and where the pi’s sum to 1.Haimes et al.(1990)first identified this difficulty,which is ascribed to the nonseparability and non-monotonicity of conditional expected values.Uncon-ditional expected values,on the other hand,are sep-arable and monotonic.For a mathematical definitionof separability and monotonicity see,e.g.,Li(1990).Frohwein and Lambert(2000)show that the con-ditional expected value f4,,conditioned on the out-come exceeding threshold,is second-order separa-ble(Li 1990,Li and Haimes 1990,1991)because itcan be expressed and optimized in terms of the par-tial expected value f4,*and the exceedance probabil-
ity