1 Civil Systems Planning Benefit/Cost Analysis Scott Matthews Courses: / / Lecture 16

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

and Recall: Choosing a Car Example  CarFuel Eff (mpg) Comfort  Index  Mercedes25 10  Chevrolet283  Toyota356  Volvo309

and “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)

and With these weights..  U(M) = 0.26* *0 = 0.26  U(V) = 0.26*(6/7) *0.5 =  U(T) = 0.26*(3/7) *1 =  U(H) = 0.26*(4/7) *0.6 =  Note H isnt really an option - just “checking” that we get same U as for Volvo (as expected)

and 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

and 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

and 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 F A (z) ≤ F B (z) for all z

and Stochastic Dominance: Example #1  CRP below for 2 strategies shows “Accept $2 Billion” is dominated by the other.

and 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

and 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: F A (x) is cdf of A, F B (x) is cdf of B.  F B (x) ≥ F A (x) for all x, with one strict inequality  or.. for any non-decr. U(x), ∫U(x)dF A (x) ≥ ∫U(x)dF B (x)  Expected value of A is higher than B

and FOSD Source:

and FOSD Example  Option A  Option B Profit ($M)Prob. 0 ≤ x < ≤ x < ≤ x < ≤ x < Profit ($M)Prob. 0 ≤ x < 50 5 ≤ x < ≤ x < ≤ x < ≤ x < 250.1

and

and 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)..

and SOSD Example  Option A  Option B Profit ($M)Prob. 0 ≤ x < ≤ x < ≤ x < ≤ x < Profit ($M)Prob. 0 ≤ x < ≤ x < ≤ x < ≤ x < ≤ x < 250.1

and Area 2 Area 1

and SOSD

and 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)

and Reading pdf/cdf graphs  What information can we see from just looking at a randomly selected pdf or cdf?