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Conceptual thinking about the unknown Uncertainty: expected value, sensitivity analysis, and the value of information Review 1 • Your problem is not as unique as you think it is • You have more data than you think you have • You need less data than you think you need • There is a useful measure that is much simpler than you think it is Review 2 • If it matters, it is detectable/observable • If it can be detected, it can be detected as an amount (or range of possible amounts) • If it can be detected as a range of possible amounts, it can be measured Review 3 • Write down a number • Break it down into pieces (decomposition) • Try different decompositions • Average Clarifying the measurement problem • What is the decision this is supposed to support? • What really is the thing being measured? • Why does this thing matter to the decision? • What do you know about it now? • What is the value of knowing more? Calibration There are two extremes of subjective confidence: Over confidence Under confidence Uncertainty and Risk • Uncertainty = The lack of complete certainty. That is, more than one outcome is possible, so that the true outcome/SON/result/value is not known • Risk = uncertainty involving hazard. That is, some outcomes are bad, involve a loss, or where they are all bad, some are catastrophic EXPECTED VALUE ANALYSIS • Consists of modeling uncertainty as a set of contingencies that are exhaustive and mutually exclusive with specific probabilities of occurrence. • In practice, this means the analyst identifies representative contingencies and assigns probabilities to each of them so that they sum to one. • The probabilities can be based on historically observed frequencies, subjective assessments, or experts (based on information, theory, or both). Calculating the expected value of net benefits • Calculate the net benefits of each contingency • Multiply by that contingency's probability of occurrence. • Sum the weighted benefits E(NB) = SPi (Bi - Ci) Representativeness of contingencies Specification of contingencies A nnualized A nnualized crop value with crop value irrigation without irrigation $ 4 ,5 0 0 ,0 0 0 4 ,5 0 0 ,0 0 0 4 ,5 0 0 ,0 0 0 4 ,0 0 0 ,0 0 0 3 ,0 0 0 ,0 0 0 A nnualized A nnualized cost of dam net benef it & dist ribution system $0 $ 2 0 0 ,0 0 0$4,300,000 2 ,8 0 0 ,0 0 0 2 0 0 ,0 0 0 $1,500,000 3 ,7 0 0 ,0 0 0 2 0 0 ,0 0 0 $600,000 3 ,6 0 0 ,0 0 0 2 0 0 ,0 0 0 $200,000 2 ,8 0 0 ,0 0 0 2 0 0 ,0 0 0 $0 EX(P) V|P 0 .1 0 $430,000 0 .8 0 $480,000 0 .1 0 $0 $910,000 EX(P) 0 .0 5 0 .1 2 0 .6 6 0 .1 2 0 .0 5 1 .0 0 V|P 215000 $180,000 $396,000 $24,000 $0 $815,000 Decision trees and expected NB Decision analysis has two stages. - First, one specifies the logical structure of the decision problem in terms of sequences of decisions and realizations of contingencies using a diagram (called a decision tree) that links an initial decision to final outcomes. - Second, one works backwards from final outcomes to the initial decision, calculating expected values of net benefits across contingencies and pruning dominated branches (ones with lower expected values of net benefits). Vaccine example • Present value of expected net benefits of the vaccination program is simply E(CNV) - E(CV) (i.e., the expected value of the costs when not implementing the program minus the expected costs when implementing the program). Decision tree for vaccination program analysis What’s up for grabs? • Population at risk - Total Population (round to 10K) - Fraction High Risk ? • Infection Rate ? • Mortality Rate ? - Value of Life ? • Time lost to Flu ? - Opportunity Cost of Time ? • Chance of Epidemic - First Year ? - Second Year ? • Total number of people vaccinated • Vaccination Rate • Administrative Costs - Overheads (Fixed) - Dose Price (Variable) • Adverse Reaction Rate ? • Herd Immunity Rate ? • Vaccine Effectiveness Rate? • Discount Rate ? Expected net benefits of vaccinations Expected net benefits of vaccinations Worst; best case analysis • What are our maximum downside risks if we take no action? • What if we do? Histogram of realized net benefits The cost and value of information • Perfect information? • Imperfect information? • Quasi-option value Exogenous learning Endogenous learning