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Slides 8a: Introduction

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Decision Analysis A set of alternative actions We may chose whichever we please A set of possible states of nature Only one will be correct, but we dont know in advance A set of outcomes and a value for each Each is a combination of an alternative action and a state of nature Value can be monetary or otherwise

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Decision Analysis Certainty Decision Maker knows with certainty what the state of nature will be - only one possible state of nature Ignorance Decision Maker knows all possible states of nature, but does not know probability of occurrence Risk Decision Maker knows all possible states of nature, and can assign probability of occurrence for each state

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Decision Making Under Certainty

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Decision Making Under Ignorance – Payoff Table Kelly Construction Payoff Table (Prob. 8-17)

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Decision Making Under Ignorance Maximax Select the strategy with the highest possible return Maximin Select the strategy with the smallest possible loss LaPlace-Bayes All states of nature are equally likely to occur. Select alternative with best average payoff

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Maximax: The Optimistic Point of View Select the best of the best strategy Evaluates each decision by the maximum possible return associated with that decision (Note: if cost data is used, the minimum return is best) The decision that yields the maximum of these maximum returns (maximax) is then selected For risk takers Doesnt consider the down side risk Ignores the possible losses from the selected alternative

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Maximax Example Kelly Construction

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Maximin: The Pessimistic Point of View Select the best of the worst strategy Evaluates each decision by the minimum possible return associated with the decision The decision that yields the maximum value of the minimum returns (maximin) is selected For risk averse decision makers A protect strategy Worst case scenario the focus

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Maximin Kelly Construction

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Decision Making Under Risk Expected Return (ER)* Select the alternative with the highest (long term) expected return A weighted average of the possible returns for each alternative, with the probabilities used as weights * Also referred to as Expected Value (EV) or Expected Monetary Value (EMV) **Note that this amount will not be obtained in the short term, or if the decision is a one-time event!

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Expected Return

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Expected Value of Perfect Information EVPI measures how much better you could do on this decision if you could always know when each state of nature would occur, where: EVUPI = Expected Value Under Perfect Information (also called EVwPI, the EV with perfect information, or EV C, the EV under certainty) EVUII = Expected Value of the best action with imperfect information (also called EV Best ) EVPI = EVUPI – EVUII EVPI tells you how much you are willing to pay for perfect information (or is the upper limit for what you would pay for additional imperfect information!)

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Expected Value of Perfect Information

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Using Excel to Calculate EVPI: Formulas View Kelly Construction

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A newsvendor can buy the Wall Street Journal newspapers for 40 cents each and sell them for 75 cents. However, he must buy the papers before he knows how many he can actually sell. If he buys more papers than he can sell, he disposes of the excess at no additional cost. If he does not buy enough papers, he loses potential sales now and possibly in the future. Suppose that the loss of future sales is captured by a loss of goodwill cost of 50 cents per unsatisfied customer. The Newsvendor Model

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The demand distribution is as follows: P 0 = Prob{demand = 0} = 0.1 P 1 = Prob{demand = 1} = 0.3 P 2 = Prob{demand = 2} = 0.4 P 3 = Prob{demand = 3} = 0.2 Each of these four values represent the states of nature. The number of papers ordered is the decision. The returns or payoffs are as follows:

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State of Nature (Demand) Decision Payoff = 75(# papers sold) – 40(# papers ordered) – 50(unmet demand) Where 75¢ = selling price 40¢ = cost of buying a paper 50¢ = cost of loss of goodwill

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Now, the ER is calculated for each decision: State of Nature (Demand) Decision ER Prob ER 1 = -40(0.1) + 35(0.3) – 15(0.4) – 65(0.2) = -12.5ER 2 = -80(0.1) – 5(0.3) + 70(0.4) + 20(0.2) = 22.5ER 3 = -120(0.1) – 45(0.3) + 30(0.4) – 105(0.2) = 7.5ER 0 = 0(0.1) – 50(0.3) – 100(0.4) – 150(0.2) = -85 Of these four ERs, choose the maximum, and order 2 papers

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ER(new) = 0(0.1) + 35(0.3) + 70(0.4) + 105(0.2) State of Nature Decision Prob = 59.5 ER(current) = 22.5 EVPI = 59.5 – 22.5 = 37.0 cents

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maxmax The decision that yields the maximum of these maximum returns (maximax) is then selected. maximum This method evaluates each decision by the maximum possible return associated with that decision. Maximax Criterion: Maximax Criterion: The Maximax criterion is an optimistic decision making criterion.

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max min Then, the decision that yields the maximum value of the minimum returns (maximin) is selected. Maximin Criterion: Maximin Criterion: The Maximin criterion is an extremely conservative, or pessimistic, approach to making decisions. minimum Maximin evaluates each decision by the minimum possible return associated with the decision.

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So, using the 3 criteria, we made the following decisions regarding the newsvendor data: CriteriaDecision CriteriaDecision Maximin Cash FlowOrder 1 paper Expected ReturnOrder 2 papers Maximax Cash FlowOrder 3 papers

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risk-averse Most people are risk-averse, which means they would feel that the loss of a certain amount of money would be more painful than the gain of the same amount of money. Utility functions Utility functions in decision analysis measure the attractiveness of money. Utility can be thought of as a measure of satisfaction. THE RATIONALE FOR UTILITY

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Utility Dollars Typical risk-averse utility function: Go from $400 to $500 results in A gain in utility of 0.06

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To illustrate, first suppose you have $100 and someone gives you an additional $100. Note that your utility increases by U(200) – U(100) = – = Now suppose you start with $400 and someone gives you an additional $100. Now your utility increases by U(500) – U(400) = – = This illustrates that an additional $100 is less attractive if you have $400 on hand than it is if you start with $100.

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Utilities and Decisions under Risk Summary: Utility Utility is a way to incorporate risk aversion into the expected return calculation. Calculating a utility function is out of the scope of this course, but it can be calculated by a series of lottery questions (e.g., Would you prefer one million dollars or a 50% chance of earning five million?).

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