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Mutli-Attribute Decision MakingEliciting Weights

Scott MatthewsCourses: 12-706 / 19-702

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Admin Issues

HW 4 due todayNo Friday class this week

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Multi-objective Methods

Multiobjective programming Mult. criteria decision making (MCDM)Is both an analytical philosophy and a

set of specific analytical techniques Deals explicitly with multi-criteria DM Provides mechanism incorporating values Promotes inclusive DM processes Encourages interdisciplinary approaches

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2:1 Tradeoff Example

Find an existing point (any) and consider a hypothetical point you would trade for. You would be indifferent in this trade

E.g., V(30,9) -> H(31,7) H would get Uf = 6/10 and Uc = 4/7 Since we’re indifferent, U(V) must = U(H) kC(6/7) + kF(5/10) = kC(4/7) + kF(6/10) kC (2/7) = kF(1/10) <=> kF = kC (20/7) But kF + kC =1 <=> kC (20/7) + kC = 1 kC (27/7) = 1 ; kC = 7/27 = 0.26 (so kf=0.74)

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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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Marginal Rate of Substitution

For our example == 1/2Which is what we said it should be(1 unit per 2 units)

M ij =ik i

+x −i−x

jk i+x −

i−x

0.26 / 7

0.74 /10

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Eliciting Weights for MCDM

2:1 tradeoff (“pricing out”) is example about eliciting weights (i.e., 2:1 )

Method was direct, and was based on easy quantitative 0-1 scale

What are other options to help us?

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Ratios

Helpful when attributes are not quantitative Car example: color (how much more do we like

red?) First ask sets of pairwise comparison questions Then set up quant scores Then put on 0-1 scale This is what MCDM software does (series of

pairwise comparisons)

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MCDM - Swing Weights

Use hypothetical extreme combinations to determine weights

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

attributes from worst to bestDetermine rank preference, find

weights

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Choosing a Car

Car Fuel Eff (mpg) Comfort IndexMercedes 25 10Chevrolet 28 3Toyota 35 6Volvo 30 9Which dominated, non-dominated?

Dominated can be removed from decision BUT we’ll need to maintain their values for

ranking

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Swing Weights Table

Combinations of varying all worst attribute values with each best attribute

How would we rank / rate options below? Combo Ran

kRate

Weight

Base 25 F, 3C 3 0

Fuel 35F, 6C

Comfort 25F, 10C

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Example

Worst and best get 0, 100 ratings by default

If we assessed “Fuel” option highest, and suggested that “Comfort” option would give us a 20 (compared to 100) rating..

Combo Rank

Rate

Weight

Benchmark

25 F, 3C 3 0 0

Fuel 35F, 3C 1 100 100/120

Comfort 25F, 10C

2 20 20/120

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Outcome of Swing Weights

Each row is a “worst case” utility and best case utility

E.g., U(“Fuel” option)= kf*Uf(35) + kc*Uc(6) U(Fuel)= kf*1 + 0 = kf

Same for U(comfort) option => kc

We assessed swing weights as utilities Utility of swinging each attribute from worst

to best gives us our (elicited) weights

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So how to assess?

Proportional scoring ~ risk neutralRatios - good for qualitative attributes

First do qualitative comparisons (eg colors) Then derive a 0-1 scale

Incorporate risk attitudes (not neutral) We have used mostly linear utility Risky has lower utility

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MCDM with Decision Trees

Incorporate uncertainties as event nodes with branches across possibilities See “summer job” example in Chapter

4.

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Still need special (external) scales. And need to value/normalize them Give 100 to best, 0 to worst, find scale

for everything between (job fun) Get both criteria on 0-100 scales!

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Next Step: Weights

Need weights between 2 criteria Don’t forget they are based on whole

scale e.g., you value “improving salary on

scale 0-100 at 3x what you value fun going from 0-100”. Not just “salary vs. fun”

12-706 and 73-359 21Proportional Scoring for Salary, Subjective Rankings for Fun

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Notes

While forest job dominates in-town, recall it has caveats: These estimates, these tradeoffs, these

weights, etc. Might not be a general result.

Make sure you look at tutorial at end of Chapter 4 on how to simplify with plugins

Read Chap 15 Eugene library example!

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How to solve MCDM problems

All methods (AHP, SMART, ..) return some sort of weighting factor set Use these weighting factors in

conjunction with data values (mpg, price, ..) to make value functions

In multilevel/hierarchical trees, deal with each set of weights at each level of tree

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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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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 0.2

5 ≤ x < 10 0.3

10 ≤ x < 15

0.4

15 ≤ x < 20

0.1

Profit ($M) Prob.

0 ≤ x < 5 0

5 ≤ x < 10 0.1

10 ≤ 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 25

Profit ($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 0.1

5 ≤ x < 10 0.3

10 ≤ x < 15

0.4

15 ≤ x < 20

0.2

Profit ($M) Prob.

0 ≤ x < 5 0.3

5 ≤ x < 10 0.3

10 ≤ 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 25

Profit ($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)