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**Decision Theory Models Decision Tree & Utility Theory**

Kusdhianto Setiawan Gadjah Mada University

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**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

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**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

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**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.

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**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.

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**EMV Continued John Thompson Case Alternative State of Nature EMV**

Fav. Mkt Unf. Mkt Construct a large plant 10.000 Construct a small plant 40.000 Do Nothing Probabilities 0.5

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**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)).

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Opportunity Loss maximizing EMV = minimizing expected opportunity loss (EOL) Alternative State of Nature EOL Fav. Mkt Unf. Mkt Construct a large plant – 0 – ( ) 90.000 Construct a small plant – 0 – ( ) 60.000 Do Nothing – 0 0- 0 Probabilities 0.5

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Sensitivity Analysis SA investigaes how our decision might change with different input data. EMV(large p) = P – (1-P) = P – EMV(small p) = P – (1-P) = P – EMV(do nothing) = 0P + 0(1-P) = 0

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**Sensitivity Analysis EMV Values EMV Large Plant EMV Small Plant**

EMV Do Nothing 0.167 0.62 Probability of Favourable Market

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**Decision Making Under Uncertainty**

Maximax (Optimistic Approach) Maximin (Pessimistic Approach) Equally Likely (Laplace) Criterion of Realism (Hurwicz Criterion) Minimax (based on opportunity loss)

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**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.

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**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.

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**Café’ Du Donut Case P ≥ ML/(MP+ML) = 4/(4+2) = 4/6 P ≥ 0.66**

Daily Sales P that sales will be at this level P that Sales will be at this level or greater 4 0.05 1 ≥ 0.66 5 0.15 0.95 ≥ 0.66 6 0.8 ≥ 0.66 7 0.20 0.65 8 0.25 0.45 9 0.1 0.2 10 P ≥ ML/(MP+ML) = 4/(4+2) = 4/6 P ≥ 0.66

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**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 Determine P = ML/(MP+ML) Locate P on the Normal Distribution, and find Z for a given area under the curve, then find X* Z = (X* - m)/s

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**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 = 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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