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Decision Analysis1 DSC 3120 Generalized Modeling Techniques with Applications Part III. Decision Analysis

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Decision Analysis2 u A Rational and Systematic Approach to Decision Making u Decision Making: choose the “best” from several available alternative courses of action u Key Element is Uncertainty of the outcome We, as decision maker, control the decision Outcome of the decision is uncertain to and uncontrolled by decision maker (controlled by nature)

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Decision Analysis3 Example of Decision Analysis You have $10,000 for investing in one of the three options: Stock, Mutual Fund, and CD. What is the best choice? Question: Do you know the choices? Do you know the best choice? What is the uncertainty? How do you make your choice?

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Decision Analysis4 Components of Decision Problem u Alternative Actions -- Decisions There are several alternatives from which we want to choose the best u States of Nature -- Outcomes There are several possible outcomes but which one will occur is uncertain to us u Payoffs Numerical (monetary) value representing the consequence of a particular alternative action we choose and a state of nature that occurs later on

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Decision Analysis5 Payoff Table

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Decision Analysis6 An Example State of Nature Alternative

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Decision Analysis7 Three Classes of Decision Models u Decision Making Under Certainty Only one state of nature (or we know with 100% sure what will happen) u Decision Making Under Uncertainty (ignorance) Several possible states of nature, but we have no idea about the likelihood of each possible state u Decision Making Under Risk Several possible states of nature, and we have an estimate of the probability for each state

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Decision Analysis8 Decision Making Under Uncertainty u LaPlace (Assume Equal Likely States of Nature) Select alternative with best average payoff u Maximax (Assume The Best State of Nature) Select alternative that will maximize the maximum payoff (expect the best outcome-- optimistic) u Maximin (Assume The Worst State of Nature) Select alternative that will maximize the minimum payoff (expect the worst situation-- pessimistic) u Minimax Regret (Don’t Want to Regret Too Much) Select alternative that will minimize the maximum regret

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Decision Analysis9 Payoff Table Example: Newsboy Problem

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Decision Analysis10 LaPlace Criterion Example: Newsboy Problem

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Decision Analysis11 Maximax Criterion Example: Newsboy Problem

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Decision Analysis12 Maximin Criterion Example: Newsboy Problem

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Decision Analysis13 Minimax Regret Criterion: Step 1 Example: Newsboy Problem

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Decision Analysis14 Minimax Regret: Step 2 (Regret or Opportunity Loss Table) Example: Newsboy Problem

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Decision Analysis15 Decision Making Under Risk In this situation, we have more information about the uncertainty--probability

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Decision Analysis16 Decision Making Under Risk u Maximize Expected Return (ER) ER i = (p j r ij ) = p 1 r i1 + p 2 r i2 +…+ p m r im Where ER i = Expected return if choosing the i th alternative ( A i ), (i = 1, 2, …, n) p j = The probability of state j ( S j ) r ij = The payoff if we choose alternative A i and S j state of nature occurs

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Decision Analysis17 Expected Return & Variance Example: Newsboy Problem

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Decision Analysis18 Decision Making Under Risk u High return is good, but on the other hand, low risk is also important u Variance -- a measure of the risk Variance i = p j (r ij - ER i ) 2 Where p j = The probability of state j (S j ) r ij = The payoff if choose A i and S j occurs ER i = Expected return for alternative A i

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Decision Analysis19 Expected Value of Perfect Information u EVPI measures the maximum worth (value) of the “Perfect Information” that we should pay for in order to improve our decisions EVPI = ER w/ perfect info. - ER w/o perfect info. ER w/ perfect info. = p j max(r ij ) ER w/o perfect info. = max(ER i ) = max( p j r ij )

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Decision Analysis20 Calculate EVPI Example: Newsboy Problem ER w/ PI ER w/o PI EVPI

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Decision Analysis21 Expected Opportunity Loss (EOL) u We can also use EOL to choose the best alternative u Minimizing EOL = Maximizing ER both criteria yield the same best alternative EOL i = p j OL ij where p j = The probability of state j (S j ) OL ij = The opportunity loss if choose A i and S j occurs s min(EOL i ) = EVPI

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Decision Analysis22 Expected Opportunity Loss Example: Newsboy Problem EVPI

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Decision Analysis23 Decision Making with Utilities u Problem with Monetary Payoffs People do not always just look at the highest expected monetary return to make decisions; they often evaluate the risk Example: A company wants to decide to develop a new product or not

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Decision Analysis24 Decision Making with Utilities u Utility -- combines monetary return with people’s attitude toward risk u Utility Function -- a mathematical function that transforms monetary values into utility values Three general types of utility functions 0 MV Utility 0 MV Utility 0 MV Utility (1) Risk-Averse(2) Risk-Neutral(3) Risk-Seeking

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Decision Analysis25 Risk-Averse Utility Function Utility u Properties of Risk-averse Utility Function non-decreasing: more money is always better concave: utility increase for unit ($100, e.g.) increase of money is decreasing (extra money is less attractive) Dollars

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Decision Analysis26 How to Create Utility Function u Method I. Equivalent Lottery Start with two endpoints A (the worst possible payoff) and B (the best possible payoff) and assign U(A) = 0 and U(B) = 1 Then to find the utility for a possible payoff z between A and B, select the probability p (=U(z)) such that you are indifferent between the following two alternatives –receive a payoff of z for sure –receive a payoff of B with probability p or a payoff of A with probability 1 - p

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Decision Analysis27 How to Create Utility Function u Method II. Exponential Utility Function where x is the monetary value, r>0 is an adjustable parameter called risk tolerance First, the value of r can be estimated such that we are indifferent between the following choices a payoff of zero a payoff of r dollars or a loss of r/2 dollars with chance Then the utility for a particular monetary value x can be found using the above assumed exponential utility function

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Decision Analysis28 Expected Utility Example: Newsboy Problem

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