Appendix -- Flaw of Averages.pdf - Richard de Neufville

de Neufville + Scholtes D R A F T October 14, 2008value of the system. In practice, f(x) for a system design not a simple algebraic expression; it istypically a computer model, ranging from a spreadsheet valuation to a set of complex, interlinkedcomputer models.Put in simple language, the mathematical proposition means that the answer you get from arealistic description of the effect of uncertain parameters generally differs – often greatly -- fromthe answer you get from using best or most likely estimates of uncertain parameters.This proposition may seem counter-intuitive. A natural reasoning might be that:• if one uses an average value of an uncertain parameter,• then the effects of its upside value will counter-balance the effects of the downside value.False!The difficulty is that the effects of the upside and downside values of the parameter will in generalnot cancel each other out. Mathematically speaking, this is because our models of the system,f(x), are non-linear. In plain English, the upside and downside effects do not cancel out becauseactual systems behave irregularly.The asymmetric behavior of systems, where the upside and the downside are not equal, occursin three different ways:• the system response to changes is non-linear;• the system response involves some discontinuity; or• management rationally imposes a discontinuity.The following examples illustrate these conditions.The system response is non-linear:The cost of achieving any outcome generally varies with its level or quantity. Typically, there areinitial costs that have to be covered, which means that the costs/unit are high for small levels ofoutput, and lower for higher levels. Even greater output may require special efforts – such ashigher wages for overtime or the use of less productive materials. Put another way, the systemmay show economies of scale over some range, and increased marginal costs and diseconomiesof scale elsewhere. The overall picture is thus a supply curve similar to Figure X .When the costs – or indeed any other major aspect of the project – do not vary linearly, the Flawof Averages applies. The box on the supply of Electric Power illustrates the effect.Appendix: Flaw of Averages Page 3 of 6

de Neufville + Scholtes D R A F T October 14, 2008The Supply of Electric PowerConsider a regional utility whose production comes from a mix of low-cost hydropower andexpensive oil-fired thermal plants. Its average cost per kilowatt-hour (kwh) will be lower if Greenpolicies reduce demand, and higher if economic growth drives up consumption, as below.Level of UseAverage CostProbabilityMegawatt-hours$ / kilowatt-hour1200 0.3 0.081000 0.4 0.05800 0.3 0.04What is the utility’s margin of profitability when it is obliged to sell power to customers at a fixedprice of $0.06 / kwh?If we focus on the most likely or the average forecast, that is, a consumption of 1 Megawatt-hour,then the utility has a margin of $ 0.01/kwh (= 0.06 – 0.050) for an overall profit of $10,000.If however we consider the actual range of possibilities, then we can see that the high costsincurred when demand is high, boosted by the high volumes of demands, lead to losses that arenot compensated on average by the possible higher profitability if demand and prices are low. Inthis example, the average value is thus actually $3,200 compared to the $10,000 estimateLevel of UseCost Margin Overall ExpectedProbabilityMegawatt-hrs$ / kwh $ / kwh Profit Profit1200 0.3 0.08 - 0.02 - $24,000 - 7,2001000 0.4 0.05 + 0.01 $10,000 4,000800 0.3 0.04 + 0.02 $16,000 6,400Total 3,200A focus on the most likely or average conditions thus leads to an entirely misleading assessmentof performance. This is an example of the Flaw of Averages.Appendix: Flaw of Averages Page 5 of 6

de Neufville + Scholtes D R A F T October 14, 2008Valuation of an Oil FieldConsider the problem of valuing a 1M barrel oil field that could be acquired in 6 months. It hasknown extraction costs of $75/bbl, and the best estimate of the price in 6 months is $80/bbl.However, the price is equally likely to remain at $80/bbl, drop to $70/bbl or rise to $90/bbl .If we focus on the average future price, the value of the field is $5 M (= 1M (80 – 75) )What is the value of the field if we recognize the price uncertainty? An instinctive reaction is thatthis value must be lower because the project is more risky. However, when you do thecalculation scenario-by-scenario you find that if the price is $10/bbl higher, the value increases by$10M to $15M. However, if the price is $10/bbl lower, the value does not drop by $10M to -$5Mas implied by a loss of $5/bbl when production costs exceed the market price. This is becausemanagement has the flexibility not to pump, and to eliminate the lost so the value of the field isactually 0 when the price is low. The net result is the actual average value of the field would be$6.67M, higher that is $5M estimate derived on the basis of the average oil price of $80/bbl.The field is worth more on average, not less. This manifestation of the Flaw of Averagesillustrates why it worthwhile taking flexibility into account in design. If management is notcommitted yourself to pumping, it can avoid the downside if the low oil price occurs while stillmaintaining the upsides. This flexibility increases the average value of the project compared to aninflexible alternative.1 Provide reference2 Provide references3 The undergraduate engineering curricula at leading European and Asia universities commonlyhave no room for economic or social studies. This is the case for example at the University ofCambridge, the Technical University of Delft, the ETH???, and the Tokyo Institute of Technology.In the United States, the accreditation boards require engineering undergraduates to take aquarter of their credits in some form of liberal arts. However, this material is rarely integrated intothe professional curriculum.Appendix: Flaw of Averages Page 6 of 6