MutliAttribute Decision Making Eliciting Weights

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Mutli-Attribute Decision MakingEliciting Weights

• Scott Matthews
• Courses 12-706 / 19-702

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Admin Issues

• HW 4 due today
• No 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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21 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) -gt H(31,7)
• H would get Uf 6/10 and Uc 4/7
• Since were indifferent, U(V) must U(H)
• kC(6/7) kF(5/10) kC(4/7) kF(6/10)
• kC (2/7) kF(1/10) ltgt kF kC (20/7)
• But kF kC 1 ltgt kC (20/7) kC 1
• kC (27/7) 1 kC 7/27 0.26 (so kf0.74)

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With these weights..

• U(M) 0.261 0.740 0.26
• U(V) 0.26(6/7) 0.740.5 0.593
• U(T) 0.26(3/7) 0.741 0.851
• U(H) 0.26(4/7) 0.740.6 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
• 1/2
• Which is what we said it should be
• (1 unit per 2 units)

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Eliciting Weights for MCDM

• 21 tradeoff (pricing out) is example about
  eliciting weights (i.e., 21 )
• 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 attributes
• Other options - swing one of the attributes
  from worst to best
• Determine rank preference, find weights

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Choosing a Car

• Car Fuel Eff (mpg) Comfort
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  Index
• Mercedes 25 10
• Chevrolet 28 3
• Toyota 35 6
• Volvo 30 9
• Which dominated, non-dominated?
• Dominated can be removed from decision
• BUT well 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?

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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..

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Outcome of Swing Weights

• Each row is a worst case utility and best case
  utility
• E.g., U(Fuel option) kfUf(35) kcUc(6)
• U(Fuel) kf1 0 kf
• Same for U(comfort) option gt 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 neutral
• Ratios - 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
• Dont 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

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Proportional 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 gt z A) Pr(Profit gt 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 DominanceExample 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

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Second-Order Stochastic Dominance (SOSD)

• How to compare 2 lotteries based on risk
• Given lotteries/distributions w/ same mean
• So were 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 Example

• Option A

• Option B

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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)