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Decision analysis questions
Draw the choices and the chances as a tree, work the averages back to the first decision, and take the best. Then ask whether the average was ever the right thing to compare.
6 original questions · 25 marks · the decision analysis notes · Decision Mathematics 2
Every question here is written for this library rather than taken from a past paper. Write your answer out before opening the worked one: the answers award marks point by point, and the marks are easier to see when you have something of your own to compare against.
State what a square node and a circular node stand for on a decision tree, and where the costs are written.
Worked answer
A square is a decision node, where you choose. A circle is a chance node, where the world chooses, with a probability on each branch. Costs are written on the branch where they are incurred and subtracted there, rather than gathered up at the end, so that every option is compared net of what it costs to take. B1 for the decision node, B1 for the chance node, B1 for the placing of costs.A chance node has a 0.4 branch paying 250 and a 0.6 branch paying 60, and reaching it costs 45. Find its expected monetary value.
Worked answer
0.4(250) + 0.6(60) = 100 + 36 = 136. Subtracting the cost of 45 leaves an expected monetary value of 91. M1 for 0.4(250) + 0.6(60), A1 for 91. Write the 136 in the circle and the 91 beyond the cost, so both numbers are on the tree.A firm can develop a product in-house for 60, giving a 0.5 chance of revenue 200, 0.3 of 90 and 0.2 of 20; or licence it for 25, giving 0.7 of 110 and 0.3 of 40; or do nothing. All figures are in thousands. Find the expected monetary value of each option and state the decision.
Worked answer
In-house: 0.5(200) + 0.3(90) + 0.2(20) = 100 + 27 + 4 = 131, less 60 gives 71.
Licence: 0.7(110) + 0.3(40) = 77 + 12 = 89, less 25 gives 64.
Do nothing: 0.
The highest is 71, so develop in-house. M1 for an expected value at a chance node, A1 for 71, A1 for 64, A1 for choosing in-house, B1 B1 for the double strokes and the value in the square. Strike through the other two branches at the decision node and write 71 in the square. The two double strokes and the value in the square are worth a mark each, and a bare statement of the decision loses them.State the worst outcome of the chosen option, and explain why a small firm might reject it despite its expected monetary value.
Worked answer
With probability 0.2 the revenue is only 20 against a cost of 60, a loss of 40 thousand. Expected monetary value is an average over many repetitions, and this decision is made once. A firm that could not absorb a loss of 40 would rightly take the licence, whose worst outcome is 40 − 25 = 15, a gain. B1 for the loss of 40, B1 for expected monetary value being a long-run average, B1 for the firm being unable to absorb the loss. The calculation is correct and still not the whole answer.Explain what utility is, and describe how a utility of the square root of the pay-off changes the comparison of a gamble paying 200 or nothing with equal chance against a certain 80.
Worked answer
Utility measures how much each outcome is actually worth to the decision maker, rather than its cash value, and for most people it rises less than proportionally with money.
The gamble's expected monetary value is 0.5(200) = 100, above 80. Its expected utility is 0.5 times the square root of 200, about 7.07, while the certain 80 has utility the square root of 80, about 8.94. On utility the certain 80 wins, and the two criteria disagree. B1 for what utility is, B1 for the expected monetary value of 100, M1 for the two expected utilities, A1 for the conclusion.A firm can develop a product in-house for 60, giving a 0.5 chance of revenue 200, 0.3 of 90 and 0.2 of 20; or licence it for 25, giving 0.7 of 110 and 0.3 of 40; or do nothing. All figures are in thousands, and without further information the best option is in-house, with expected monetary value 71. Before deciding, the firm can commission a survey costing 10. It reports favourably with probability 0.6, after which the in-house revenue probabilities become 0.7, 0.2 and 0.1; otherwise they become 0.2, 0.4 and 0.4. The licence figures are unaffected. Decide whether the survey is worth commissioning.
Worked answer
After a favourable report, in-house is worth 0.7(200) + 0.2(90) + 0.1(20) = 140 + 18 + 2 = 160, less 60 gives 100, beating the licence's 64, so choose in-house.
After an unfavourable report, in-house is worth 0.2(200) + 0.4(90) + 0.4(20) = 40 + 36 + 8 = 84, less 60 gives 24, so choose the licence at 64 instead.
The survey's chance node is therefore 0.6(100) + 0.4(64) = 60 + 25.6 = 85.6, less its cost of 10 gives 75.6.
That beats the 71 available without a survey, so commission it. M1 for the favourable branch, A1 for 100 and the in-house choice, M1 for the unfavourable branch, A1 for 24 and the licence choice, M1 for the survey's chance node, A1 for 75.6, A1 for commissioning the survey. The survey is worth 75.6 − 71 = 4.6 thousand, and the whole of that gain comes from the unfavourable branch, where it stops the firm developing in-house.
The same practice on paper: the printable workbook for this topic, questions and a worked answer book.
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