MGMT20005 Chap.2 Decision Trees and Expected Value
Decision Trees and Expected Value
Week 2 adds probabilities and time to the decision model. Squares represent decision nodes, circles represent chance nodes, and terminal points carry complete path payoffs. The tree is read chronologically from left to right but solved from right to left. At a chance node, outcomes are mutually exclusive and collectively exhaustive, so their probabilities sum to one and their payoffs are averaged by probability.
At a decision node, management selects the branch with the best expected value for the stated objective. A risk profile then restores information hidden by the average: it lists the possible realised payoffs and their probabilities. Expected value supports comparison, but it is not a guaranteed outcome and need not override a manager's explicit risk preference.
What this chapter covers
- 01
Decision nodes, chance nodes, branches and terminal payoffs
- 02
Mutually exclusive and collectively exhaustive chance outcomes
- 03
Expected value as a probability-weighted average
- 04
Right-to-left rollback through a decision tree
- 05
Conditional path probabilities in staged decisions
- 06
Risk profiles and the limits of expected value
Worked example · free
Compare a beta release with a national launch
- 1The cautious-market probability is 1 minus 0.45, which equals 0.55. The two chance outcomes sum to one.
- 2EV(beta) = 0.45 times 38 plus 0.55 times 24 = 17.10 plus 13.20 = 30.30.
- 2EV(national) = 0.45 times 105 plus 0.55 times minus 27 = 47.25 minus 14.85 = 32.40.
- 1Select National under expected value because 32.40 exceeds 30.30 by 2.10, while acknowledging the possible loss of 27.
Key terms
- Decision node
- A point in a tree where the decision maker controls which branch is chosen.
- Chance node
- A point where an uncertain event selects among probability-labelled outcome branches.
- Terminal payoff
- The complete consequence at the end of a path after all local costs and benefits are included.
- Expected value
- A probability-weighted average of possible payoffs used to compare strategies under risk.
- Rollback
- The right-to-left process of averaging at chance nodes and optimising at decision nodes.
- Risk profile
- The probability distribution of payoffs generated by a selected strategy.
Decision Trees and Expected Value FAQ
Why is a decision tree solved from right to left?
A decision made today depends on the values of later branches. Starting at terminal payoffs lets you calculate each final chance-node value, carry that value back to an earlier decision, retain the best branch, and continue until the root has one optimal contingent strategy. Left-to-right calculation would reach decisions before their consequences were valued.
When do probabilities multiply along a path?
Multiply when successive conditional events must all occur for one terminal path. For example, the probability of passing a test and then seeing strong demand is the pass probability times the conditional strong-demand probability. Add probabilities only for mutually exclusive paths when combining them or checking that a complete outcome set sums to one.
Does the largest expected value always settle the decision?
It settles the recommendation under an expected-value criterion. A manager can still prefer a lower-EV strategy because of downside exposure, liquidity, survival constraints or another risk preference. If so, the answer should report both expected values and the risk feature that justifies departing from the EV ranking rather than altering the arithmetic.
What should be shown on a drawn decision tree?
Use distinct shapes for decision and chance nodes, label every branch, place probabilities on chance branches and include every local cost or benefit in the terminal payoff. Show the expected value beside each chance node and mark the retained branch at each decision node so the final strategy can be audited.
Exam move
Practise trees in two passes. On the first pass, ignore arithmetic and audit the structure: decision versus chance nodes, branch order, conditional labels and terminal consequences. On the second pass, solve from the far right. Multiply conditional probabilities along complete paths where needed, calculate probability-weighted values at chance nodes, and select the best branch at decision nodes.
Check that sibling chance probabilities sum to one and that the terminal path probabilities under the retained strategy also sum to one. Close with a recommendation that quotes the expected-value comparison and names a material downside. This prevents an expected value from being reported as though it were the payoff management will actually receive. Build one tree again from a blank page after checking the solution.
Label the node types, probabilities, conditional decisions and terminal values from memory, then compare structure before checking arithmetic. If your value differs, trace one branch at a time from the terminal outcome to the root. This separates a probability error from a rollback error and makes revision faster.
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