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Civil Systems PlanningBenefit/Cost Analysis

Scott MatthewsCourses: 12-706 / 19-702/ 73-359Lecture 16

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Admin

Project 1 - avg 85 (high 100)Mid sem grades today - how done?

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Recall: Choosing a Car Example

Car Fuel Eff (mpg) Comfort

IndexMercedes 25 10Chevrolet 28 3Toyota 35 6Volvo 30 9

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“Pricing out”

Book uses $ / unit tradeoffOur example has no $ - but same idea“Pricing out” simply means knowing

your willingness to make tradeoffsAssume 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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With these weights..

U(M) = 0.26*1 + 0.74*0 = 0.26U(V) = 0.26*(6/7) + 0.74*0.5 = 0.593U(T) = 0.26*(3/7) + 0.74*1 = 0.851U(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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MCDM - Swing Weights

Use hypothetical combinations to determine weights

Base option = worst on all attributesOther options - “swings” one of the

attributes from worst to bestDetermine your rank preference, find

weights

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Add 1 attribute to car (cost)

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

Swing weight table:Benchmark 25mpg, $50k, 3 Comf

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Stochastic Dominance “Defined”

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 #1CRP below for 2 strategies shows

“Accept $2 Billion” is dominated by the other.

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Stochastic Dominance (again) 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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First-Order Stochastic Dominance (FOSD) 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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FOSD

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

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FOSD Example

Option A Option B

Profit ($M) Prob.

0 ≤ x < 5 0.25 ≤ x < 10 0.310 ≤ x < 15

0.4

15 ≤ x < 20

0.1

Profit ($M) Prob.

0 ≤ x < 5 05 ≤ x < 10 0.110 ≤ x < 15

0.5

15 ≤ x < 20

0.3

20 ≤ x < 25

0.1

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First-Order Stochastic Dominance

00.20.40.60.8

1

0 5 10 15 20 25Profit ($millions)

Cumulative Probability

AB

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Second-Order Stochastic Dominance (SOSD) 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)..

€

U(x)dFA (x)0

x

∫ ≥ U(x)dFB (x)0

x

∫

€

[FB (x) − FA (x)]dx0

x

∫ ≥ 0,∀x

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SOSD Example

Option A Option B

Profit ($M) Prob.

0 ≤ x < 5 0.15 ≤ x < 10 0.310 ≤ x < 15

0.4

15 ≤ x < 20

0.2

Profit ($M) Prob.

0 ≤ x < 5 0.35 ≤ x < 10 0.310 ≤ x < 15

0.2

15 ≤ x < 20

0.1

20 ≤ x < 25

0.1

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Second-Order Stochastic Dominance

00.20.40.60.8

1

0 5 10 15 20 25Profit ($millions)

Cumulative Probability

AB

Area 2

Area 1

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SOSD

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SD and MCDM

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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Reading pdf/cdf graphs

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