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- Overview

- Decision Formulation

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 theory and decision analysis help people (including business people) make better decisions.
- They identify the best decision to take.
- They assume an ideal decision maker:
- Fully informed about possible decisions and their consequences.
- Able to compute with perfect accuracy.
- Fully rational.
- 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 consequence of decisions, good and bad, are stochastic.
- That is, consequences depend on decisions of nature.

- A decision problem is characterized by decision alternatives, states of nature (decisions of nature), and resulting payoffs.
- The decision alternatives are the different possible actions or strategies the decision maker can employ.
- The states of naturerefer 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).
- There will be either rain or sun, but not both.

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

Decision Making without Probabilities

- Decision Making without Probabilities

Decision Making without Probabilities

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

- Two commonly used criteria for decision making do not require probability information regarding the likelihood of the states of nature:
- the optimisticapproach.
- the conservative(or pessimistic) approach.

Decision Making without Probabilities

- The optimistic approach would be used by an optimistic decision maker.
- The decision with the largest possible payoff is chosen.
- For example, play the lottery whenever possible.
- Other examples?
- 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).
- Other examples?

Decision Making without Probabilities

- The conservative approach would be used by a conservative (or pessimistic) decision maker.
- 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.
- 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.

Decision Making without Probabilities

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

An optimistic decision maker would use

the maximaxapproach. Choose the decision that has the largest single value in the payoff table.

Maximum

DecisionPayoff

d1 4

d2 3

d3 5

Maximaxdecision

Maximax

payoff

Decision Making without Probabilities

A conservative decision maker would use

the maximinapproach. List the minimum payoff for each decision. Choose the decision with the maximum of these minimum payoffs.

Minimum

DecisionPayoff

d1 -2

d2 -1

d3-3

Maximin

decision

Maximin

payoff

Decision Making without Probabilities

- Here is a practical application:
- Pittsburgh Development Corporation (PDC) bought land for a new condominium complex. There decision is the size of the complex:
- d1 = a small complex with 30 condominiums
- d2= a medium complex with 60 condominiums
- d3= a large complex with 90 condominiums
- 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:
- s1 = strong demand for the condominiums
- s2= weak demand for the condominiums

Decision Making without Probabilities

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

s1s2

d1 8 7

Decisionsd2 14 5

d3 20 -9

- Expected Value

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.

- Most decision makers are neither optimists nor pessimists (at least not in the extreme form just discussed).
- Decision making then requires probability information regarding the likelihood of the states of nature.

- Expected value 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.
- The decision yielding the best expected return is chosen.

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

s1 = 80 s2 = 100 s3 = 120

Model A $10,000 $15,000 $14,000

Model B $ 8,000 $18,000 $12,000

Model C $ 6,000 $16,000 $21,000

Choose the model with largest expected value if the probabilities of states s1, s2, and s3 are .4, .2, and .4

Answer: Compute expected values when the probabilities of states s1, s2, and s3 are .4, .2, and .4:

Average Number of Customers Per Hour

s1 = 80 s2 = 100 s3 = 120

Model A $10,000 $15,000 $14,000

Model B $ 8,000 $18,000 $12,000

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.

Backward Induction

- Overview
- Backward Induction finds your optimal sequence decisions by making your last decision first. — So, before your first cigarette, think about your last.

- Use backward induction (and draw a decision tree, if needed) to help make the following sequential decisions, where later decisions depend on earlier decisions:
- 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.
- 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:
- 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 0.2, demand is low and revenue is $35 million.
- If the project were successful,should Dell Computer sell its rights in the project for $25 million?

- Step 1: Replace uncertain payoffs with expected value:
- 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.
- 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 revenue
- = 0.5 x $59 million + 0.3 x $45 million + 0.2 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 0.3, demand is med. and revenue is $45 million.
- With probability 0.2, demand is low and revenue is $35 million.
- If the project were successful,should Dell Computer sell its rights in the project for $25 million?

- Step 2: Make the second decision:
- 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.
- 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 revenue
- = 0.5 x $59 million + 0.3 x $45 million + 0.2 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 0.3, demand is med. and revenue is $45 million.
- With probability 0.2, demand is low and revenue is $35 million.
- 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.

- Step 3: Replace uncertain payoffs with expected value:
- Expected profit = 0.5 x $25 million (successful) + 0.5 x (-$5 million) = $10 million, which is positive and so worth the investment.
- The probability of a successful project is 0.5.
- If the new products were made, expected revenue
- = 0.5 x $59 million + 0.3 x $45 million + 0.2 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 0.3, demand is med. and revenue is $45 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.

- 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:
- The probability of a successful project is 0.5.
- 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 0.3, demand is med. and revenue is $45 million.
- With probability 0.2, demand is low and revenue is $35 million.
- If the project were successful,should Dell Computer sell its rights in the project for $25 million?

Decision Tree

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 Example: Consider the following decision tree, where square nodes are decision nodes and round nodes are chance nodes.
- Each chance node has branches marked with their probabilities. For example, branch C occurs with probability .25.
- At the end of each branch is its payoff. For example, branch S ends in payoff 300, and branch D ends in payoff 20.
- Use backward induction and expected-payoff maximization to determine the optimal initial choice of A or B. Explain.

- 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.

- Review 1: Backward Induction

- Question: Make the following decisions:
- Dante development Corporation is considering bidding on a contract for a new office building complex.
- The cost of preparing a bid is $200,000.
- 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 consider selling your share in the project for $2,100,000.
- If you do not sell, there is uncertain revenue.
- 70% chance of revenue $3,000,000.
- 20% chance of revenue $2,500,000.
- 10% chance of revenue $0.

- Answer:
- Step 1: Replace uncertain payoffs with expected value:
- Dante development Corporation is considering bidding on a contract for a few office building complex.
- The cost of preparing a bid is $200,000.
- 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 consider selling your share in the project for $2,100,000.
- If you do not sell, there is uncertain revenue.
- 70% chance of revenue $3,000,000.
- 20% chance of revenue $2,500,000.
- 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.

- Step 2: Make the second decision:
- Dante development Corporation is considering bidding on a contract for a few office building complex.
- The cost of preparing a bid is $200,000.
- 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,do not sell your share. Your expected revenue is $2,600,000.

- Step 3: Replace uncertain payoffs with expected value:
- Dante development Corporation is considering bidding on a contract for a few office building complex.
- The cost of preparing a bid is $200,000.
- 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 loose the contract, expect profit = -$200,000
- If you bid, expect profit = .8 x $400,000 + .2 x (-$200,000) = $280,000.

- Make your decisions:
- If you bid, expect profit = .8 x $400,000 + .2 x (-$200,000) = $280,000.
- 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.

- 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:
- Dante development Corporation is considering bidding on a contract for a new office building complex.
- The cost of preparing a bid is $200,000.
- 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 consider selling your share in the project for $2,100,000.
- If you do not sell, there is uncertain revenue.
- 70% chance of revenue $3,000,000.
- 20% chance of revenue $2,500,000.
- 10% chance of revenue $0.

BA 452 Quantitative Analysis

End of Lesson C.3
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