Maths › Decision Mathematics 2 › Decision analysis
Decision analysis
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.
Builds on Dynamic programming and Conditional probability.
IN THIS TOPIC
- Draw a decision tree and work expected monetary values back to the first decision.
- Explain the limits of expected monetary value and where utility helps.
COMMON MISCONCEPTION
The best course of action is always the one with the highest expected monetary value.
Trees and expected values
A decision node, drawn as a square, is a point where you choose. A chance node, drawn as a circle, is a point where the world chooses, with probabilities on the branches. Pay-offs go at the ends, and any cost incurred along a branch is subtracted at the point it is met.
Working back, a chance node takes the expected monetary value of its branches, which is the probability-weighted average. A decision node takes the best value available, and the branches it rejects are struck through. Repeat back to the first decision and you have both the answer and the reason for it.
WORKED EXAMPLE
Choosing a launch
A product can be launched nationally, costing 100, with a 0.6 chance of revenue 300 and 0.4 of 50; regionally, costing 40, with 0.6 of 150 and 0.4 of 30; or not at all. All figures in thousands. Which is best on expected monetary value?
National: 0.6(300) + 0.4(50) − 100 = 180 + 20 − 100 = 100.
Regional: 0.6(150) + 0.4(30) − 40 = 90 + 12 − 40 = 62.
Doing nothing: 0.
The national launch has the highest expected monetary value, so it is chosen on that criterion.
Where the average is the wrong measure
Expected monetary value is an average over many repetitions, and a one-off decision is not many repetitions. A firm that would be ruined by the bad outcome may quite reasonably prefer a smaller certain gain. The criterion produces a number, and the decision still has to be made by somebody.
Utility replaces money with a measure of how much each outcome is actually worth to the decision maker, and for most people that rises less than proportionally with money. Maximising expected utility can then favour the safer option even where the expected monetary value is lower, and that is the appropriate answer when the stakes are large compared with the resources available.
GUIDED PRACTICE
Reading the risk
In the launch example, a competitor offers to buy the product outright for 80 thousand. Compare the options and comment.
Show the working
The certain sale is worth 80, against the national launch's expected 100.
On expected monetary value the launch wins by 20.
But the launch has a 0.4 chance of returning only 50 against a cost of 100, a loss of 50.
A firm that could not absorb that loss should take the 80. The expected monetary value calculation is correct and still not the whole answer.
ASSESSMENT FOCUS
- Use squares for decision nodes and circles for chance nodes, and label every probability.
- Subtract costs along the branch where they are incurred, not at the end.
- Write the expected monetary value at each chance node and strike through rejected branches.
CHECK YOURSELF
A chance node has a 0.3 branch paying 200 and a 0.7 branch paying 50, and reaching it costs 60. Find the expected monetary value.
Show a hint
Average the branches, then subtract the cost.
Show the answer
0.3(200) + 0.7(50) = 60 + 35 = 95, and subtracting the cost of 60 leaves an expected monetary value of 35.
Chance nodes take the probability-weighted average and decision nodes take the best available, worked back to the first decision.
Expected monetary value is an average over repetitions, so for a one-off decision with large stakes utility may favour the safer option.
WORKBOOK
Printable practice for this topic: original exam-style questions with room to work, and a fully worked answer book. Free to use; please do not redistribute or sell.
Or read them with their worked answers on the decision analysis questions page.
CHECK YOUR PROGRESS
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- Draw a decision tree and work expected monetary values back to the first decision.
- Explain the limits of expected monetary value and where utility helps.
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