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Decision Theory Models Decision Tree _ Utility Theory

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									  Decision Theory Models
Decision Tree & Utility Theory
       Kusdhianto Setiawan
      Gadjah Mada University
                 Introduction
• What is a good decision?
  – Based on logic
  – Is rational model applied by all people in making
    logical decision?
  – What is rational model?
• Types of Decision-Making Environment
  – Under Certainty: tahu semua konsekuensinya
  – Under Risk: tahu probabilitas dari outcomes
  – Under Uncertainty: tidak tahu probabilitas dari
    outcomes
  Decision Making Under Risk
- Risky choice = Gamble that can yield
  various outcomes with different
  probabilities?

- Psychophysical Analysis relevant?

  - Daniel Bernoully (1978):
    - people are generally averse to risk and risk
      aversion decreases with increasing wealth.
    - People do not evaluate prospects by the
      expectation of their monetary outcomes, but rather
      by the expectation of subjective value of these
      outcome
- Von Neumann & Morgenstein (1947) –
  Concept of Rationality
  - Preference of rational decision making should
    follow:
     - Transitivity, if A is preferred to B and B is preferred to C,
       then A is preferred to C.
     - Substitution, if A is preferred to B, then an even chance
       to get A or C is preferred to an even chance to get B or
       C).
     - Dominance, if prospect A is at least as good as prospect
       B in every respect and better than B in at least one
       respect, then A should be preferred to B.
     - Invariance, preference order between prospects should
       not depend on the manner in which they are described.
     Expected Monetary Value
• EMV is the weighted sum of possible
  payoffs for each alternative (prospect)
• EMV (alternative i) =
  (payoff of first state of nature) x (probability of first state of nature)
  + (payoff of 2nd state of nature) x (probability of 2nd state of nature)
  + ……… + (payoff of last state of nature) x (probability of last state
  of nature).
• What does EMV means?
  – Nilai moneter (uang) yang akan kita terima secara rata-rata jika
    mengambil keputusan dalam kondisi tertentu (state of nature)
    berulang kali.
                EMV Continued
John Thompson Case
                              State of Nature
       Alternative                                EMV
                           Fav. Mkt    Unf. Mkt
 Construct a large plant   200.000     -180.000   10.000

 Construct a small plant   100.000      -20.000   40.000
      Do Nothing              0            0        0
      Probabilities          0.5          0.5
      Expected Value of Perfect
         Information (EVPI)
• EVPI merupakan harga dari perfect information,
  misal: jasa konsultan yang diharapkan akan
  memberikan informasi paling benar (harga
  tertinggi yang mungkin kita bayar).
• EVPI = expected value with perfect information
  (EVwPI)– maximum EMV
• EVwPI = (best outcome for the 1st SoN) x (P(1st
  SoN)) + …. + (best outcomes of last SoN) x
  (P(last SoN)).
                 Opportunity Loss
•    maximizing EMV = minimizing expected opportunity loss (EOL)




                                   State of Nature
          Alternative                                         EOL
                               Fav. Mkt       Unf. Mkt
                              200.000 –
    Construct a large plant                 0 – (-180.000)   90.000
                              200.000
                              200.000 –
    Construct a small plant                 0 – (-20.000)    60.000
                              100.000
         Do Nothing           200.000 – 0        0- 0        100.000
         Probabilities            0.5            0.5
        Sensitivity Analysis
• SA investigaes how our decision might
  change with different input data.
• EMV(large p) = 200.000P – 180.000(1-P)
               = 380.000P – 180.000
• EMV(small p) = 100.000P – 20.000(1-P)
               = 120.000P – 20.000
• EMV(do nothing) = 0P + 0(1-P) = 0
               Sensitivity Analysis
EMV Values




                                           EMV Do Nothing
                0.167       0.62
    -20.000
                           Probability of Favourable Market


    -180.000
Decision Making Under Uncertainty
•   Maximax (Optimistic Approach)
•   Maximin (Pessimistic Approach)
•   Equally Likely (Laplace)
•   Criterion of Realism (Hurwicz Criterion)
•   Minimax (based on opportunity loss)
      Marginal Analysis: Discrete
             Distribution
Example: Café’ du Donut
Buying price from the producer: $4/cartoon
Selling price to customer: $6/cartoon, then
Marginal Profit (MP) = 6 – 4 = $2/cartoon
Marginal Loss (ML) = $4, lets
P = probability that demand ≥ supply (or the
  probability of selling at least one additional unit)
1 – P = probability that demand will be less than
  supply.
• The Optimal Decision Rule
  P(MP) ≥ (1 - P)(ML)
  or P(MP) + P(ML) ≥ ML
  or P(MP + ML) ≥ ML
  or P ≥ ML/(MP+ML), meaning that:
  as long as the probability of selling one
  more unit (P) is greater than or equal to
  ML/(MP + ML), we would stock additional
  unit.
       Café’ Du Donut Case
                                                  P that Sales
                                   P that sales
                                                  will be at
                     Daily Sales   will be at
• P ≥ ML/(MP+ML) =                 this level
                                                  this level or
                                                  greater
  4/(4+2) = 4/6
                         4            0.05          1 ≥ 0.66
• P ≥ 0.66
                         5            0.15        0.95 ≥ 0.66


                         6            0.15         0.8 ≥ 0.66


                         7            0.20            0.65


                         8            0.25            0.45


                         9             0.1            0.2


                         10            0.1            0.1
     Marginal Analysis with Normal
              Distribution
•    Data Requirement
    –   The average or mean for the product, μ
    –   The standard deviation of sales, σ
    –   The Marginal Profit
    –   The Marginal Loss
•    Steps
    1. Determine P = ML/(MP+ML)
    2. Locate P on the Normal Distribution, and find Z for a
       given area under the curve, then find X*
       Z = (X* - m)/s
 Chicago Tribune Distributor Case
• Average Sales/day = 50 papers
• Standard Deviation = 10 papers
• Marginal Profit = 6 cents
• Marginal Loss = 4 cents
• Determine Stocking Policy!
Step 1
P = ML/(ML+MP)=4/(4+6)=0.4
Step 2
1 - 0.4 = 0.6 … look at the z table, and find for z
Z = 0.25 standard deviation from the mean
0.25 = (X*- 50)/10  X*=10(0.25) + 50 = 52.3 or 53 papers
Decision: The distributor should order 53 paper daily.

								
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