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1 1 Slide © 2004 Thomson/South-Western Payoff Tables n The consequence resulting from a specific combination of a decision alternative and a state of nature is a payoff. n A table showing payoffs for all combinations of decision alternatives and states of nature is a payoff table. n Payoffs can be expressed in terms of profit, cost, time, distance or any other appropriate measure.
2 2 Slide © 2004 Thomson/South-Western Decision Trees n A decision tree is a chronological representation of the decision problem. n Each decision tree has two types of nodes; round nodes correspond to the states of nature while square nodes correspond to the decision alternatives. n The branches leaving each round node represent the different states of nature while the branches leaving each square node represent the different decision alternatives. n At the end of each limb of a tree are the payoffs attained from the series of branches making up that limb.
3 3 Slide © 2004 Thomson/South-Western Decision Making without Probabilities n Three commonly used criteria for decision making when probability information regarding the likelihood of the states of nature is unavailable are: the optimistic approach the optimistic approach the conservative approach the conservative approach the minimax regret approach. the minimax regret approach.
4 4 Slide © 2004 Thomson/South-Western Optimistic Approach n The optimistic approach would be used by an optimistic decision maker. n The decision with the largest possible payoff is chosen. n If the payoff table was in terms of costs, the decision with the lowest cost would be chosen.
5 5 Slide © 2004 Thomson/South-Western Conservative Approach n The conservative approach would be used by a conservative decision maker. n For each decision the minimum payoff is listed and then the decision corresponding to the maximum of these minimum payoffs is selected. (Hence, the minimum possible payoff is maximized.) n If the payoff was in terms of costs, the maximum costs would be determined for each decision and then the decision corresponding to the minimum of these maximum costs is selected. (Hence, the maximum possible cost is minimized.)
6 6 Slide © 2004 Thomson/South-Western Minimax Regret Approach n The minimax regret approach requires the construction of a regret table or an opportunity loss table. n This is done by calculating for each state of nature the difference between each payoff and the largest payoff for that state of nature. n Then, using this regret table, the maximum regret for each possible decision is listed. n The decision chosen is the one corresponding to the minimum of the maximum regrets.
7 7 Slide © 2004 Thomson/South-Western Expected Value Approach Calculate the expected value for each decision. The decision tree on the next slide can assist in this calculation. Here d 1, d 2, d 3 represent the decision alternatives of models A, B, C, and s 1, s 2, s 3 represent the states of nature of 80, 100, and 120.
8 8 Slide © 2004 Thomson/South-Western Decision Tree 126.96.36.199.188.8.131.52.2.4 d1d1d1d1 d2d2d2d2 d3d3d3d3 s1s1s1s1 s1s1s1s1 s1s1s1s1 s2s2s2s2 s3s3s3s3 s2s2s2s2 s2s2s2s2 s3s3s3s3 s3s3s3s3 Payoffs 10,000 15,000 14,000 8,000 18,000 12,000 6,000 16,000 21,000 22 33 44
9 9 Slide © 2004 Thomson/South-Western Expected Value for Each Decision Choose the model with largest EV, Model C. 33 d1d1d1d1 d2d2d2d2 d3d3d3d3 EMV =.4(10,000) +.2(15,000) +.4(14,000) = $12,600 = $12,600 EMV =.4(8,000) +.2(18,000) +.4(12,000) = $11,600 = $11,600 EMV =.4(6,000) +.2(16,000) +.4(21,000) = $14,000 = $14,000 Model A Model B Model C 22 11 44
10 Slide © 2004 Thomson/South-Western Expected Value of Perfect Information n Frequently information is available which can improve the probability estimates for the states of nature. n The expected value of perfect information (EVPI) is the increase in the expected profit that would result if one knew with certainty which state of nature would occur. n The EVPI provides an upper bound on the expected value of any sample or survey information.
11 Slide © 2004 Thomson/South-Western Expected Value of Perfect Information n EVPI Calculation Step 1: Step 1: Determine the optimal return corresponding to each state of nature. Determine the optimal return corresponding to each state of nature. Step 2: Step 2: Compute the expected value of these optimal returns. Compute the expected value of these optimal returns. Step 3: Step 3: Subtract the EV of the optimal decision from the amount determined in step (2). Subtract the EV of the optimal decision from the amount determined in step (2).
12 Slide © 2004 Thomson/South-Western Bayes’ Theorem and Posterior Probabilities n Knowledge of sample (survey) information can be used to revise the probability estimates for the states of nature. n Prior to obtaining this information, the probability estimates for the states of nature are called prior probabilities. n With knowledge of conditional probabilities for the outcomes or indicators of the sample or survey information, these prior probabilities can be revised by employing Bayes' Theorem. n The outcomes of this analysis are called posterior probabilities or branch probabilities for decision trees.
13 Slide © 2004 Thomson/South-Western Expected Value of Sample Information n The expected value of sample information (EVSI) is the additional expected profit possible through knowledge of the sample or survey information.
14 Slide © 2004 Thomson/South-Western Efficiency of Sample Information n Efficiency of sample information is the ratio of EVSI to EVPI. n As the EVPI provides an upper bound for the EVSI, efficiency is always a number between 0 and 1.
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