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# Civil Systems Planning Benefit/Cost Analysis - PowerPoint PPT Presentation

Civil Systems Planning Benefit/Cost Analysis. Scott Matthews Courses: 12-706 / 19-702/ 73-359 Lecture 16. Admin. Project 1 - avg 85 (high 100) Mid sem grades today - how done?. Recall: Choosing a Car Example. Car Fuel Eff (mpg) Comfort

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### Civil Systems PlanningBenefit/Cost Analysis

Scott Matthews

Courses: 12-706 / 19-702/ 73-359

Lecture 16

• Project 1 - avg 85 (high 100)

• Mid sem grades today - how done?

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• Car Fuel Eff (mpg) Comfort

• Index

• Mercedes 25 10

• Chevrolet 28 3

• Toyota 35 6

• Volvo 30 9

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• Book uses \$ / unit tradeoff

• Our example has no \$ - but same idea

• Assume you’ve thought hard about the car tradeoff and would trade 2 units of C for a unit of F (maybe because you’re a student and need to save money)

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• U(M) = 0.26*1 + 0.74*0 = 0.26

• U(V) = 0.26*(6/7) + 0.74*0.5 = 0.593

• U(T) = 0.26*(3/7) + 0.74*1 = 0.851

• U(H) = 0.26*(4/7) + 0.74*0.6 = 0.593

• Note H isnt really an option - just “checking” that we get same U as for Volvo (as expected)

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• Use hypothetical combinations to determine weights

• Base option = worst on all attributes

• Other options - “swings” one of the attributes from worst to best

• Determine your rank preference, find weights

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• M = \$50,000 V = \$40,000 T = \$20,000 C=\$15,000

• Swing weight table:

• Benchmark 25mpg, \$50k, 3 Comf

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• A is better than B if:

• Pr(Profit > \$z |A) ≥ Pr(Profit > \$z |B), for all possible values of \$z.

• Or (complementarity..)

• Pr(Profit ≤ \$z |A) ≤ Pr(Profit ≤ \$z |B), for all possible values of \$z.

• A FOSD B iff FA(z) ≤ FB(z) for all z

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Stochastic Dominance:Example #1

• CRP below for 2 strategies shows “Accept \$2 Billion” is dominated by the other.

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• Chapter 4 (Risk Profiles) introduced deterministic and stochastic dominance

• We looked at discrete, but similar for continuous

• How do we compare payoff distributions?

• Two concepts:

• A is better than B because A provides unambiguously higher returns than B

• A is better than B because A is unambiguously less risky than B

• If an option Stochastically dominates another, it must have a higher expected value

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• Case 1: A is better than B because A provides unambiguously higher returns than B

• Every expected utility maximizer prefers A to B

• (prefers more to less)

• For every x, the probability of getting at least x is higher under A than under B.

• Say A “first order stochastic dominates B” if:

• Notation: FA(x) is cdf of A, FB(x) is cdf of B.

• FB(x) ≥ FA(x) for all x, with one strict inequality

• or .. for any non-decr. U(x), ∫U(x)dFA(x) ≥ ∫U(x)dFB(x)

• Expected value of A is higher than B

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Source: http://www.nes.ru/~agoriaev/IT05notes.pdf

Option B

FOSD Example

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• How to compare 2 lotteries based on risk

• Given lotteries/distributions w/ same mean

• So we’re looking for a rule by which we can say “B is riskier than A because every risk averse person prefers A to B”

• A ‘SOSD’ B if

• For every non-decreasing (concave) U(x)..

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Option B

SOSD Example

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Area 1

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• As long as criteria are independent (e.g., fun and salary) then

• Then if one alternative SD another on each individual attribute, then it will SD the other when weights/attribute scores combined

• (e.g., marginal and joint prob distributions)

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• What information can we see from just looking at a randomly selected pdf or cdf?

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