Questions on decision trees, expected monetary value and utility

Question 1

State what a square node and a circular node stand for on a decision tree, and where the costs are written.

Accepted 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.

Question 2

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.

Accepted 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.

Question 3

Working back through a decision tree, the three branches leaving one decision node have been valued at 40, 62 and 55. State the value written at the node, and what is done to the other two branches.

Accepted Answer

A decision node keeps the greatest value available under the criterion in use, so 62 is written at the node. B1. The branches worth 40 and 55 are rejected, and each is marked with a double stroke across it so that the tree shows which choices were turned down and why. B1 for striking the rejected branches. The world does not choose here, so there are no probabilities on these branches and no averaging: that is what separates a square node from a circle.

Question 4

A decision maker's utility of a sum of money x is √x. Find the utility of 49 and of 100, and state what the comparison says about this decision maker's attitude to risk.

Accepted Answer

√49 = 7 and √100 = 10. B1 for both. Money has been multiplied by rather more than two while utility has been multiplied by only 10/7, so the utility rises less than proportionally with money: each further pound is worth less to this decision maker than the one before, which is what risk averse means. B1 for the conclusion with the reason.

Question 5

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.

Accepted 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.

Question 6

State the worst outcome of the chosen option, and explain why a small firm might reject it despite its expected monetary value.

Accepted 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.

Question 7

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.

Accepted Answer

Utility measures how much each outcome is actually worth to the decision maker, rather than its cash value, and in this model 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.

Question 8

A project costing 50 pays 180 with probability p and 30 otherwise. The alternative is a certain payment of 70. Find the range of values of p for which the project has the higher expected monetary value.

Accepted Answer

The project's expected monetary value is 180p + 30(1 − p) − 50 = 150p − 20. It beats the certain 70 when 150p − 20 > 70, that is 150p > 90, so p > 0.6. M1 for the expected monetary value in terms of p, A1 for 150p − 20, M1 for comparing it with 70, A1 for p > 0.6. At p = 0.6 exactly, the two options tie at 70 and the decision maker is indifferent. A question of this shape is a sensitivity analysis: it says how far the estimate of p can be out before the decision itself changes.

Question 9

A firm must choose between project A, which pays 196 with probability 0.5 and 16 otherwise, and project B, which pays 121 with probability 0.5 and 81 otherwise. Compare the two on expected monetary value, and then again using a utility equal to the square root of the pay-off. Comment on the result.

Accepted Answer

Expected monetary values: A gives 0.5(196) + 0.5(16) = 106 and B gives 0.5(121) + 0.5(81) = 101, so on that criterion A is chosen.
Expected utilities: A gives 0.5(14) + 0.5(4) = 9 and B gives 0.5(11) + 0.5(9) = 10, so on utility B is chosen.
M1 for both expected monetary values, A1 for 106 and 101 with A preferred, M1 for both expected utilities, A1 for 9 and 10 with B preferred, B1 for the comment. The root utility rises less than proportionally with money, so it penalises A's wide spread of outcomes: B gives up 5 of average pay-off in exchange for far less risk, and a decision maker whose utility has that shape takes the trade.

Question 10

A firm may launch a product for 80, giving a 0.7 chance of revenue 200 and a 0.3 chance of revenue 30, or do nothing. All figures are in thousands. A student draws the first node of the tree as a circle and writes the probabilities 0.7 and 0.3 on the two branches leaving it. State the two faults in that drawing, find the expected monetary value of the launch and the decision, and explain why subtracting the 80 only at the very end still gives the right number here.

Accepted Answer

The first node is where the firm chooses between launching and doing nothing, so it is a decision node and must be drawn as a square. B1. Probabilities belong on the branches out of a chance node, where the world chooses; the two branches out of this node are the firm's own options and carry no probabilities at all. B1.
The chance node inside the launch branch is worth 0.7(200) + 0.3(30) = 140 + 9 = 149, and the launch costs 80, so its expected monetary value is 149 − 80 = 69. Doing nothing is worth 0, so the firm should launch. M1 for the chance node, A1 for 69 with the decision.
The 80 is met on every branch below that chance node, so averaging first and then subtracting gives the same answer as subtracting it on each branch: 0.7(120) + 0.3(−50) = 84 − 15 = 69. B1 for the cost being common to every branch. A cost met on only some branches has to be subtracted where it is met, because then it is not part of every average.

Question 11

A gamble pays 100 with probability 0.4 and nothing otherwise. The alternative is a certain 45. A decision maker's utility of a sum of money x is x². Compare the two options on expected monetary value and then on expected utility, and say what the shape of this utility says about the decision maker.

Accepted Answer

Expected monetary values: the gamble gives 0.4(100) + 0.6(0) = 40 against the certain 45, so on that criterion the certain 45 is chosen. M1 for both, A1 for 40 against 45 with the choice.
Expected utilities: the gamble gives 0.4(100²) + 0.6(0) = 4000, while the certain 45 has utility 45² = 2025, so on utility the gamble is chosen. M1 for both expected utilities, A1 for 4000 against 2025 with the choice.
This utility rises faster than proportionally with money: each further pound is worth more to this decision maker than the one before, which describes someone risk seeking, content to accept a smaller average return for the chance of a large one. B1.
A root utility would have gone the other way.

Question 12

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, 0.45 and 0.35. The licence figures are unaffected. Decide whether the survey is worth commissioning.

Accepted 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) + 0.45(90) + 0.35(20) = 40 + 40.5 + 7 = 87.5, less 60 gives 27.5, 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 27.5 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.

Question 13

An exploration company holds a licence it could sell now for 25. Instead it can drill a well for 40. With probability 0.3 the well strikes oil, giving revenue 200. Otherwise the well is dry, and the company must then choose: sell the site for 15, or deepen the well for a further 30, giving revenue 150 with probability 0.4 and nothing otherwise. All figures are in thousands. Draw the decision tree and advise the company.

Accepted Answer

The tree has a second decision node inside the dry branch, and working backwards means settling that later decision first.
Deepening is worth 0.4(150) + 0.6(0) = 60, less its cost of 30, giving 30; selling the site gives 15. So a dry well is followed by deepening, and the dry branch is worth 30.
The drilling chance node is then 0.3(200) + 0.7(30) = 60 + 21 = 81, less the 40 cost of the well, giving 41.
That beats selling the licence now for 25, so the company should drill, and deepen if the well is dry, with expected monetary value 41. B1 for a tree with the second decision node inside the dry branch, M1 for taking that decision first, A1 for 30 and the choice to deepen, M1 for the drilling chance node, A1 for 81 and 41, A1 for the advice.
The advice is a policy rather than a single act: what to do now, and what to do at the later node if it is reached, which is the same backwards logic as dynamic programming.

Question 14

A gamble pays 144 with probability 0.5 and 36 with probability 0.5. A decision maker's utility of a sum of money x is √x. Find the expected monetary value of the gamble and its expected utility, and hence the certain sum this decision maker values equally with the gamble. State what the difference between that certain sum and the expected monetary value measures.

Accepted Answer

The expected monetary value is 0.5(144) + 0.5(36) = 72 + 18 = 90. B1.
The two utilities are √144 = 12 and √36 = 6, so the expected utility is 0.5(12) + 0.5(6) = 9. M1 for weighting the utilities rather than the money, A1 for 9. Note that this is not the utility of 90, which is 9.49: averaging and taking the root do not commute.
The certain sum with the same utility satisfies √x = 9, so x = 81. M1 for solving √x = 9, A1 for 81. The decision maker is therefore indifferent between the gamble and a certain 81, and would take any certain sum above 81 in preference to the gamble.
The difference 90 − 81 = 9 is the amount of average pay-off this decision maker will give up in exchange for certainty. B1.
A decision maker with a root utility always values a gamble below its average, and the gap widens as the two outcomes are pulled further apart.

Question 15

A company can sell a patent now for 80, or develop it for 90. Development succeeds with probability p, giving revenue 260. If it fails, the company chooses between abandoning the project, which brings in nothing further, and a redesign costing 40 that gives revenue 150 with probability 0.5 and nothing otherwise. All figures are in thousands. Find the value of the later decision, and hence the values of p for which developing has the greater expected monetary value.

Accepted Answer

Work backwards, so the later decision is settled first. The redesign is worth 0.5(150) + 0.5(0) = 75, less its cost of 40, giving 35; abandoning is worth 0. M1 for the redesign's chance node, A1 for 35 and the choice to redesign.
A failed development is therefore worth 35, and the development branch is worth 260p + 35(1 − p) − 90 = 225p − 55. M1 for the branch in terms of p, A1 for 225p − 55.
Developing beats selling when 225p − 55 > 80, that is 225p > 135, so p > 0.6. M1 for the comparison with 80, A1 for p > 0.6.
At p = 0.6 exactly the two are both worth 80 and the criterion does not separate them. The redesign option matters to the answer: without it a failure would be worth 0 and the development branch 260p − 90, which would need p greater than 17/26, about 0.654, instead.