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## PowerPoint Slideshow about ' Civil Systems Planning Benefit/Cost Analysis' - linore

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Recall: Choosing a Car Example

- Car Fuel Eff (mpg) Comfort
- Index
- Mercedes 25 10
- Chevrolet 28 3
- Toyota 35 6
- Volvo 30 9

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“Pricing out”

- Book uses $ / unit tradeoff
- Our example has no $ - but same idea
- “Pricing out” simply means knowing your willingness to make tradeoffs
- 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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With these weights..

- 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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MCDM - Swing Weights

- 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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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 #1

- CRP 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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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)..

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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?

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