Decision Theory Part 2
This homework is harder and longer, but it will finish up Chapter 4. For the final, you will have to be able to do a probability tree for Bayes’ rule as you in problem 1, and you will have to be able to evaluate (or ‘prune’) a large decision tree that will be given to you as in problem 2, and you will have to be able to draw from scratch a small decision tree as in the start of problem 3. You will NOT have to do an entire tree from scratch, including Bayes’ rule for the final. So if your time is tight for this assignment, concentrate on the first 3 problems.
Bayes’ Rule Review
- Olive Construction Company is determining whether it should submit a bid for a new shopping center. In the past, Olive’s main competitor, Base Construction Company, has submitted bids 70% of the time. If Base company does not bid on a job, the probability that Olive Construction Company will get the job is 50%. If Base Construction bids on the job, the probability that Olive will get the job is 25%.
- What are the prior probabilities (use notation as in class)?
- What are the given conditional probabilities (use notation as in class)?
- What is the resulting probability tree?
- Use your tree to calculate the following: If Olive gets the job, what is the probability that Base Construction did not bid?
- What is the probability that if Olive gets the job, Base Construction did bid?
- What is the probability that Olive gets the job?
- Pruning the large decision tree
Part of the challenge of running a retail business is forecasting what demand for an item will be in the future, so that it can be ordered and produced now for that future demand. Stella runs a small local chain of appliance stores in an area that has been devastated by fires. For the last several years stainless steel has been the most popular finish for kitchen appliances. However, it appears that dark colors (e.g., black stainless steel) are now becoming more fashionable. Stella needs to order appliances now in order to satisfy the demand coming up in the next few months for contractors rebuilding homes devastated by the fire, as well as her usual demographic of couples wanting to remodel their kitchen. She decides to have a mix of stainless steel and black stainless steel, with other colors special ordered since they are asked for so rarely. She can either go with the current trend and order a majority of stainless steel products (D1), or she can order a majority of black stainless steel appliances, hoping that the new trend for sleeker, more modern colors takes hold. She figures out the profit she can make from each decision, depending on whether the demand is higher for the traditional stainless steel (S1), or higher for the newer, black stainless steel finishes (S2)
The payoff table Stella came up with is below:
| S1 | S2 | |
| D1 | 300 | 100 |
| D2 | 50 | 250 |
Stella decided that there was a 70% chance that stainless steel would still remain more popular than its darker counterparts.
However, after some consideration, Stella realized that stainless steel has been the most popular option since she has owned her stores. While many of those who were burned out were older and would likely replace their appliances with a more traditional look, she is wondering if those who want to sell, might be selling to younger couples who might want a more modern, sleek look. She decides to consider hiring a marketing research firm to help her make her decision, since this was likely to be a big year for her. After due consideration and struggling with her long-unused QBA skills with Bayes’ Rule, she came up with the following large decision tree:
Figure out Stella’s optimal decision strategy etc. by calculating the following (5 pts each subpart. Make sure to show your work and give explanations for decisions to get full credit. OK to show work on tree- just indicate that’s where the work is below.):
- EMV at node 10.
- EMV at node 11
- What decision do you make at node 5? Explain your answer or no credit.
- EMV at node 6
- EMV at node 7
- What decision do you make at node 3? Explain your answer or no credit.
- EMV at node 8
- EMV at node 9
- What decision do you make at node 4? Explain your answer or no credit.
- What is the EV at node 2?
- What decision do you make at node 1? Explain or no credit.
- What is the EV without SI?
- What is the EV with SI?
- What is the EV of SI?
- Without taking the cost of the extended forecast into account, what is Stella’s decision strategy?
- Supposing the cost of a marketing forecast is $5,000, would Stella still make the same decision at node 1 as she had earlier? Explain your answer or no credit.
- What is Stella’s optimal decision strategy taking the cost of the marketing research into account?
- What is the EV WITHOUT PI?
- What is the EV WITH PI?
- What is the EV OF PI?
- What is the efficiency of the sample information in Stella’s case?
- (Decision theory redux – the whole thing put together, including Bayes)
A machine shop owner is attempting to decide whether ot purchase a new drill press, a lathe, or a grinder. The return from each will be determined by whether the company succeeds in getting a government military contract. The profit or loss from each purchase and probabilities associated with each contract outcome are shown in the following payoff table.
Note: Probability of getting a contract is 0.40
| Purchase | Contract | No Contract |
| Drill press | $40,000 | -$8,000 |
| Lathe | $20,000 | $4,000 |
| Grinder | $12,000 | $10,000 |
- Which are the decisions? Why?
- Which are the states of nature? Why?
- On a separate sheet of paper, draw the decision tree for this problem, paying attention to the proper nodes. You will be drawing an augmented tree later, so this tree should take up 1/3 the page or less.
- Calculate the EV for each decision on the right hand side of the tree. You can do this at the ends of the tree branches if you left enough room.
- What is your best decision? Why?
- Calculate the EV OF PI. (this means you have to also calculate EV with PI and EV without PI) Show your work.
The above parts should reflect what I did in class for the small decision tree on Monday.
The machine shop owner is considering hiring a consultant with ties to the military to ascertain whether or not the shop will get the contract. The consultant is a former military officer who uses personal contacts in the military to find out such information. After talking to other shop owners who have hired this consultant, the owner has found he has a probability of .7 of getting a favorable report from the consultant, given that the contract is awarded to the shop, and a probability of .8 of getting an unfavorable report, given that the contract is not awarded.
- There are conditional probabilities given in the above description. If we denote favorable by F and unfavorable by U, and A for contracted awarded and N for contract not awarded, which of the following gives the proper notation?
P(A|F) or P(F|A)?
P(N|U) or P(U|N)?
What are the values for the conditional probabilities you were given ?(give that with P(X|Y) = 0.xx, not just the number 0.xx)
- Use Bayes Rule to calculate P(F), P(U), P(A|F), P(A|U), P(N|F) and P(N|U).Use a probability tree for your work. Show ALL your work below.
- Use your probabilities from g to draw an augmented tree. The shop owner has to decide whether to hire the consultant or not. The original tree was without a consultant. The new part of the tree will reflect the decisions he has to make WITH a consultant. This should parallel the trees we have done in class. Draw this on the same sheet of paper as the original tree for this problem.
- “Prune” the tree by calculating the relevant EVs and making the relevant decisions. Show these on your tree. If you need extra space for work, show it below, but label which node you are working on.
- Determine the optimal decision strategy.
- What is the EV OF SI for this problem? (note: you have to calculate EV with SI and EV wo SI, but you should be able to get those from your big tree. )
- What is the efficiency of sample information for this problem?
