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BA 452 Lesson C.3 Statistical Decision Making ReadingsReadings Chapter 13 Decision Analysis

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BA 452 Lesson C.3 Statistical Decision Making OverviewOverview

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Overview Decision Formulation lists alternatives, uncertain states of nature (hot, cold, …), and resulting consequences. Decision Formulation is especially important when a decision is unprecedented. Decision Making without Probabilities assigned to the states of nature is possible for optimists (who assume the best happens) and for pessimists (who assume the worst happens). Expected Value is the average consequence of a sequence of decisions, according to the central limit theorem. Hence, people facing repeated decisions should maximize expected value. Backward Induction finds your optimal sequence decisions by making your last decision first. — So, before your first cigarette, think about your last. Decision Tree Formulation pictures a decision with nodes and branches that lists alternatives, uncertain states of nature, and consequences. Decision Trees are useful for a sequence of decisions.

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BA 452 Lesson C.3 Statistical Decision Making Decision Formulation

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BA 452 Lesson C.3 Statistical Decision Making Overview Decision Formulation lists alternatives, uncertain states of nature (hot, cold, …), and resulting consequences. Decision Formulation is especially important when a decision is unprecedented (outside your experience). Decision Formulation

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BA 452 Lesson C.3 Statistical Decision Making n Decision theory and decision analysis help people (including business people) make better decisions. They identify the best decision to take. They identify the best decision to take. They assume an ideal decision maker: They assume an ideal decision maker: Fully informed about possible decisions and their consequences.Fully informed about possible decisions and their consequences. Able to compute with perfect accuracy.Able to compute with perfect accuracy. Fully rational.Fully rational. n Decisions can be difficult in two different ways: The need to use game theory to predict how other people will respond to your decisions. The need to use game theory to predict how other people will respond to your decisions. The consequence of decisions, good and bad, are stochastic. The consequence of decisions, good and bad, are stochastic. That is, consequences depend on decisions of nature.That is, consequences depend on decisions of nature. Decision Formulation

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BA 452 Lesson C.3 Statistical Decision Making n A decision problem is characterized by decision alternatives, states of nature (decisions of nature), and resulting payoffs. n The decision alternatives are the different possible actions or strategies the decision maker can employ. n The states of nature refer to possible future events (rain or sun) not under the control of the decision maker. States of nature should be defined so that they are mutually exclusive (one or the other) and collectively exhaustive (one will happen). States of nature should be defined so that they are mutually exclusive (one or the other) and collectively exhaustive (one will happen). There will be either rain or sun, but not both.There will be either rain or sun, but not both. Decision Formulation

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BA 452 Lesson C.3 Statistical Decision Making n The consequence resulting from a specific combination of a decision alternative and a state of nature is a payoff. n Payoffs can be expressed in terms of profit, cost, time, or distance. n For a single (one-shot) decision, a payoff table shows payoffs for all combinations of decision alternatives and states of nature. n For a sequence of decisions, a game tree shows payoffs for all combinations of decision alternatives and states of nature. Decision Formulation

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BA 452 Lesson C.3 Statistical Decision Making Decision Making without Probabilities

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10 BA 452 Lesson C.3 Statistical Decision Making Overview Decision Making without Probabilities assigned to the states of nature is possible for optimists (who assume the best happens) and for pessimists (who assume the worst happens). Decision Making without Probabilities

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11 BA 452 Lesson C.3 Statistical Decision Making n Two commonly used criteria for decision making do not require probability information regarding the likelihood of the states of nature: the optimistic approach. the optimistic approach. the conservative (or pessimistic) approach. the conservative (or pessimistic) approach. Decision Making without Probabilities

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12 BA 452 Lesson C.3 Statistical Decision Making n The optimistic approach would be used by an optimistic decision maker. n The decision with the largest possible payoff is chosen. For example, play the lottery whenever possible. For example, play the lottery whenever possible. Other examples? Other examples? n If the payoff table were in terms of costs, the decision with the lowest cost would be chosen. For example, buy the cheapest car (and hope it gets you to work). For example, buy the cheapest car (and hope it gets you to work). Other examples? Other examples? Decision Making without Probabilities

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13 BA 452 Lesson C.3 Statistical Decision Making n The conservative approach would be used by a conservative (or pessimistic) 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.) For example, drive a Hummer at low speeds on surface streets because you might die if you drive a Honda at high speeds on the freeway. For example, drive a Hummer at low speeds on surface streets because you might die if you drive a Honda at high speeds on the freeway. n If the payoff were 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.) For example, buy a television with proven low maintenance cost. For example, buy a television with proven low maintenance cost. Decision Making without Probabilities

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14 BA 452 Lesson C.3 Statistical Decision Making Question: Consider the following problem with three decision alternatives and three states of nature with the following payoff table representing profits: Decision Making without Probabilities States of Nature Decisions s1s1s1s1 s2s2s2s2 s3s3s3s3 d1d1d1d144-2 d2d2d2d203 d3d3d3d315-3

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15 BA 452 Lesson C.3 Statistical Decision Making An optimistic decision maker would use the maximax approach. Choose the decision that has the largest single value in the payoff table. Maximum Maximum Decision Payoff Decision Payoff d 1 4 d 1 4 d 2 3 d 2 3 d 3 5 d 3 5 Maximaxpayoff Maximaxd ecision States of Nature s1s1s1s1 s2s2s2s2 s3s3s3s3 d1d1d1d144-2 d2d2d2d203 d3d3d3d315-3 Decision Making without Probabilities

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16 BA 452 Lesson C.3 Statistical Decision Making A conservative decision maker would use the maximin approach. List the minimum payoff for each decision. Choose the decision with the maximum of these minimum payoffs. Minimum Minimum Decision Payoff Decision Payoff d 1 -2 d 1 -2 d 2 -1 d 2 -1 d 3 -3 d 3 -3 Maximindecision Maximinpayoff States of Nature s1s1s1s1 s2s2s2s2 s3s3s3s3 d1d1d1d144-2 d2d2d2d203 d3d3d3d315-3 Decision Making without Probabilities

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17 BA 452 Lesson C.3 Statistical Decision Making Here is a practical application: n Pittsburgh Development Corporation (PDC) bought land for a new condominium complex. There decision is the size of the complex: d 1 = a small complex with 30 condominiums d 1 = a small complex with 30 condominiums d 2 = a medium complex with 60 condominiums d 2 = a medium complex with 60 condominiums d 3 = a large complex with 90 condominiums d 3 = a large complex with 90 condominiums n The future demand for condominiums is uncertain. It depends on the result of an election. There are two possibilities for the election, and so for demand: s 1 = strong demand for the condominiums s 1 = strong demand for the condominiums s 2 = weak demand for the condominiums s 2 = weak demand for the condominiums Decision Making without Probabilities

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18 BA 452 Lesson C.3 Statistical Decision Making The minimum information needed to complete the formulation of the problem is the payoff table for profit. Find maximax and maximin solutions. States of Nature States of Nature s 1 s 2 s 1 s 2 d d Decisions d Decisions d d d Decision Making without Probabilities

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19 BA 452 Lesson C.3 Statistical Decision Making Expected Value

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20 BA 452 Lesson C.3 Statistical Decision Making Overview Expected Value is the average consequence of a sequence of decisions, according to the central limit theorem. Hence, people facing repeated decisions should maximize expected value. Expected Value

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21 BA 452 Lesson C.3 Statistical Decision Making n Most decision makers are neither optimists nor pessimists (at least not in the extreme form just discussed). n Decision making then requires probability information regarding the likelihood of the states of nature. Expected Value

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22 BA 452 Lesson C.3 Statistical Decision Making n Expected value approach Once probabilistic information regarding the states of nature is assessed, one may use the expected value (EV) approach. Once probabilistic information regarding the states of nature is assessed, one may use the expected value (EV) approach. Here the expected return for each decision is calculated by summing the products of the payoff under each state of nature and the probability of the respective state of nature occurring. Here the expected return for each decision is calculated by summing the products of the payoff under each state of nature and the probability of the respective state of nature occurring. The decision yielding the best expected return is chosen. The decision yielding the best expected return is chosen. Expected Value

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23 BA 452 Lesson C.3 Statistical Decision Making Question: Burger King is considering opening a new restaurant on Main Street. It has three different models, each with a different seating capacity. Burger Prince estimates that the average number of customers per hour will be 80, 100, or 20. Here is the payoff table for the three models: Average Number of Customers Per Hour Average Number of Customers Per Hour s 1 = 80 s 2 = 100 s 3 = 120 s 1 = 80 s 2 = 100 s 3 = 120 Model A $10,000 $15,000 $14,000 Model A $10,000 $15,000 $14,000 Model B $ 8,000 $18,000 $12,000 Model B $ 8,000 $18,000 $12,000 Model C $ 6,000 $16,000 $21,000 Model C $ 6,000 $16,000 $21,000 Expected Value

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24 BA 452 Lesson C.3 Statistical Decision Making Choose the model with largest expected value if the probabilities of states s 1, s 2, and s 3 are.4,.2, and.4 Expected Value

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25 BA 452 Lesson C.3 Statistical Decision Making Answer: Compute expected values when the probabilities of states s 1, s 2, and s 3 are.4,.2, and.4: Average Number of Customers Per Hour Average Number of Customers Per Hour s 1 = 80 s 2 = 100 s 3 = 120 s 1 = 80 s 2 = 100 s 3 = 120 Model A $10,000 $15,000 $14,000 Model A $10,000 $15,000 $14,000 Model B $ 8,000 $18,000 $12,000 Model B $ 8,000 $18,000 $12,000 Model C $ 6,000 $16,000 $21,000 Model C $ 6,000 $16,000 $21,000 EV(A) =.4(10,000) +.2(15,000) +.4(14,000) = $12,600 EV(B) =.4(8,000) +.2(18,000) +.4(12,000) = $11,600 EV(C) =.4(6,000) +.2(16,000) +.4(21,000) = $14,000 Choose the model with largest EV, Model C. Expected Value

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26 BA 452 Lesson C.3 Statistical Decision Making Backward Induction

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27 BA 452 Lesson C.3 Statistical Decision Making Overview Backward Induction finds your optimal sequence decisions by making your last decision first. — So, before your first cigarette, think about your last. Backward Induction

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28 BA 452 Lesson C.3 Statistical Decision Making Use backward induction (and draw a decision tree, if needed) to help make the following sequential decisions, where later decisions depend on earlier decisions: 1.Should Dell Computer invest $5 million in research to create a faster external hard drive for computers? The probability of a successful project is 0.5.The probability of a successful project is 0.5. If the project were successful, it requires a new $20 million production facility to make the new products.If the project were successful, it requires a new $20 million production facility to make the new products. If the new products were made, demand and revenues are uncertain:If the new products were made, demand and revenues are uncertain: With probability 0.5, demand is high and revenue is $59 million.With probability 0.5, demand is high and revenue is $59 million. With probability 0.3, demand is med. and revenue is $45 million.With probability 0.3, demand is med. and revenue is $45 million. With probability 0.2, demand is low and revenue is $35 million.With probability 0.2, demand is low and revenue is $35 million. 2.If the project were successful, should Dell Computer sell its rights in the project for $25 million? Backward Induction

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29 BA 452 Lesson C.3 Statistical Decision Making Step 1: Replace uncertain payoffs with expected value: 1.Should Dell Computer invest $5 million in research to create a faster external hard drive for computers? The probability of a successful project is 0.5.The probability of a successful project is 0.5. If the project were successful, it requires a new $20 million production facility to make the new products.If the project were successful, it requires a new $20 million production facility to make the new products. If the new products were made, expected revenueIf the new products were made, expected revenue = 0.5 x $59 million x $45 million x $35 million = $50 million, so expected profits are $25 million. With probability 0.5, demand is high and revenue is $59 million.With probability 0.5, demand is high and revenue is $59 million. With probability 0.3, demand is med. and revenue is $45 million.With probability 0.3, demand is med. and revenue is $45 million. With probability 0.2, demand is low and revenue is $35 million.With probability 0.2, demand is low and revenue is $35 million. 2.If the project were successful, should Dell Computer sell its rights in the project for $25 million? Backward Induction

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30 BA 452 Lesson C.3 Statistical Decision Making Step 2: Make the second decision: 1.Should Dell Computer invest $5 million in research to create a faster external hard drive for computers? The probability of a successful project is 0.5.The probability of a successful project is 0.5. If the project were successful, it requires a new $20 million production facility to make the new products.If the project were successful, it requires a new $20 million production facility to make the new products. If the new products were made, expected revenueIf the new products were made, expected revenue = 0.5 x $59 million x $45 million x $35 million = $50 million, so expected profits are $25 million. With probability 0.5, demand is high and revenue is $59 million.With probability 0.5, demand is high and revenue is $59 million. With probability 0.3, demand is med. and revenue is $45 million.With probability 0.3, demand is med. and revenue is $45 million. With probability 0.2, demand is low and revenue is $35 million.With probability 0.2, demand is low and revenue is $35 million. 2.If the project were successful, should Dell Computer sell its rights in the project for $25 million? Do not sell rights in the project (and earn profit $(25-5) = $20 million, since keeping the rights is worth $25 million profit. Backward Induction

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31 BA 452 Lesson C.3 Statistical Decision Making Step 3: Replace uncertain payoffs with expected value: 1.Should Dell Computer invest $5 million in research to create a faster external hard drive for computers? Expected profit = 0.5 x $25 million (successful) x (-$5 million) = $10 million, which is positive and so worth the investment. The probability of a successful project is 0.5.The probability of a successful project is 0.5. If the project were successful, it requires a new $20 million production facility to make the new products.If the project were successful, it requires a new $20 million production facility to make the new products. If the new products were made, expected revenueIf the new products were made, expected revenue = 0.5 x $59 million x $45 million x $35 million = $50 million, so expected profits are $25 million. With probability 0.5, demand is high and revenue is $59 million.With probability 0.5, demand is high and revenue is $59 million. With probability 0.3, demand is med. and revenue is $45 million.With probability 0.3, demand is med. and revenue is $45 million. With probability 0.2, demand is low and revenue is $35 million.With probability 0.2, demand is low and revenue is $35 million. 2.If the project were successful, should Dell Computer sell its rights in the project for $25 million? Do not sell rights in the project (and earn profit $(25-5) = $20 million, since keeping the rights is worth $25 million profit. Backward Induction

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32 BA 452 Lesson C.3 Statistical Decision Making Decision Tree The backward induction steps leading to the conclusion to invest the $5 million in research can be seen by drawing a decision tree, where nodes are either your decisions (moves) or nature’s decisions (moves). Draw and solve a decision tree for the original decision problem: 1.Should Dell Computer invest $5 million in research to create a faster external hard drive for computers? The probability of a successful project is 0.5.The probability of a successful project is 0.5. If the project were successful, it requires a new $20 million production facility to make the new products.If the project were successful, it requires a new $20 million production facility to make the new products. If the new products were made, demand and revenues are uncertain:If the new products were made, demand and revenues are uncertain: With probability 0.5, demand is high and revenue is $59 million.With probability 0.5, demand is high and revenue is $59 million. With probability 0.3, demand is med. and revenue is $45 million.With probability 0.3, demand is med. and revenue is $45 million. With probability 0.2, demand is low and revenue is $35 million.With probability 0.2, demand is low and revenue is $35 million. 2.If the project were successful, should Dell Computer sell its rights in the project for $25 million? Backward Induction

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33 BA 452 Lesson C.3 Statistical Decision Making Decision Tree

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34 BA 452 Lesson C.3 Statistical Decision Making Overview Decision Tree Formulation pictures a decision with nodes and branches connecting nodes that lists alternatives, uncertain states of nature (hot, cold, …), and resulting consequences. Decision Trees are especially useful for a sequence of decisions. Decision Tree

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35 BA 452 Lesson C.3 Statistical Decision Making Decision Tree Example: Consider the following decision tree, where square nodes are decision nodes and round nodes are chance nodes. 1.Each chance node has branches marked with their probabilities. For example, branch C occurs with probability At the end of each branch is its payoff. For example, branch S ends in payoff 300, and branch D ends in payoff Use backward induction and expected-payoff maximization to determine the optimal initial choice of A or B. Explain. Decision Tree

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36 BA 452 Lesson C.3 Statistical Decision Making Review Questions You should try to answer some of the following questions before the next class. You will not turn in your answers, but students may request to discuss their answers to begin the next class. Your upcoming Final Exam will contain some similar questions, so you should eventually consider every review question before taking your exams.

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37 BA 452 Lesson C.3 Statistical Decision Making Review 1: Backward Induction

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38 BA 452 Lesson C.3 Statistical Decision Making Question: Make the following decisions: 1.Dante development Corporation is considering bidding on a contract for a new office building complex. The cost of preparing a bid is $200,000.The cost of preparing a bid is $200,000. If you bid, you will win the contract with probability.8.If you bid, you will win the contract with probability.8. If you win the contract, you pay $2,000,000 to be a partner in the project.If you win the contract, you pay $2,000,000 to be a partner in the project. 2.If you win the contract, you consider selling your share in the project for $2,100,000. If you do not sell, there is uncertain revenue.If you do not sell, there is uncertain revenue. 70% chance of revenue $3,000,000.70% chance of revenue $3,000, % chance of revenue $2,500,000.20% chance of revenue $2,500, % chance of revenue $0.10% chance of revenue $0. Review 1: Backward Induction

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39 BA 452 Lesson C.3 Statistical Decision Making Answer: Step 1: Replace uncertain payoffs with expected value: 1.Dante development Corporation is considering bidding on a contract for a few office building complex. The cost of preparing a bid is $200,000.The cost of preparing a bid is $200,000. If you bid, you will win the contract with probability.8.If you bid, you will win the contract with probability.8. If you win the contract, you pay $2,000,000 to be a partner in the project.If you win the contract, you pay $2,000,000 to be a partner in the project. 2.If you win the contract, you consider selling your share in the project for $2,100,000. If you do not sell, there is uncertain revenue.If you do not sell, there is uncertain revenue. 70% chance of revenue $3,000,000.70% chance of revenue $3,000, % chance of revenue $2,500,000.20% chance of revenue $2,500, % chance of revenue $0.10% chance of revenue $0. If you do not sell, expect revenue =.7 x $3M +.2 x $2.5M +.1 x $0 = $2,600,000.If you do not sell, expect revenue =.7 x $3M +.2 x $2.5M +.1 x $0 = $2,600,000. Review 1: Backward Induction

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40 BA 452 Lesson C.3 Statistical Decision Making Step 2: Make the second decision: 1.Dante development Corporation is considering bidding on a contract for a few office building complex. The cost of preparing a bid is $200,000.The cost of preparing a bid is $200,000. If you bid, you will win the contract with probability.8.If you bid, you will win the contract with probability.8. If you win the contract, you pay $2,000,000 to be a partner in the project.If you win the contract, you pay $2,000,000 to be a partner in the project. 2.If you win the contract, do not sell your share. Your expected revenue is $2,600,000. Review 1: Backward Induction

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41 BA 452 Lesson C.3 Statistical Decision Making Step 3: Replace uncertain payoffs with expected value: 1.Dante development Corporation is considering bidding on a contract for a few office building complex. The cost of preparing a bid is $200,000.The cost of preparing a bid is $200,000. If you bid, you will win the contract with probability.8.If you bid, you will win the contract with probability.8. If you win the contract, you pay $2,000,000 to be a partner in the project, expect revenue $2,600,000, and so expect profit $400,000 = $2.6M - $0.2M - $2M.If you win the contract, you pay $2,000,000 to be a partner in the project, expect revenue $2,600,000, and so expect profit $400,000 = $2.6M - $0.2M - $2M. If you loose the contract, expect profit = -$200,000If you loose the contract, expect profit = -$200,000 If you bid, expect profit =.8 x $400, x (-$200,000) = $280,000.If you bid, expect profit =.8 x $400, x (-$200,000) = $280,000. Review 1: Backward Induction

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42 BA 452 Lesson C.3 Statistical Decision Making Make your decisions: 1.Dante development Corporation is considering bidding on a contract for a few office building complex. If you bid, expect profit =.8 x $400, x (-$200,000) = $280,000.If you bid, expect profit =.8 x $400, x (-$200,000) = $280,000. Since expected profit is positive if you bid, and zero if you do not bid, then bid.Since expected profit is positive if you bid, and zero if you do not bid, then bid. 2. If you win the contract, do not sell your share. Review 1: Backward Induction

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43 BA 452 Lesson C.3 Statistical Decision Making Decision Tree The backward induction steps leading to the conclusion to bid on the contract and not sell your share if you win the bid can be seen by drawing a decision tree, where nodes are either your decisions (moves) or nature’s decisions (moves). Draw and solve a decision tree for the original decision problem: 1.Dante development Corporation is considering bidding on a contract for a new office building complex. The cost of preparing a bid is $200,000.The cost of preparing a bid is $200,000. If you bid, you will win the contract with probability.8.If you bid, you will win the contract with probability.8. If you win the contract, you pay $2,000,000 to be a partner in the project.If you win the contract, you pay $2,000,000 to be a partner in the project. 2.If you win the contract, you consider selling your share in the project for $2,100,000. If you do not sell, there is uncertain revenue.If you do not sell, there is uncertain revenue. 70% chance of revenue $3,000,000.70% chance of revenue $3,000, % chance of revenue $2,500,000.20% chance of revenue $2,500, % chance of revenue $0.10% chance of revenue $0. Review 1: Backward Induction

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44 BA 452 Lesson C.3 Statistical Decision Making BA 452 Quantitative Analysis End of Lesson C.3

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