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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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