Utility Tree (Decision Tree) for Expected Value
A utility tree is a graphical representation that maps out possible outcomes of a decision, their probabilities, and associated utilities (e.g., monetary gains) to calculate expected values, enabling comparison of different choices.
THE IDEA
Utility Tree (Decision Tree)
A utility tree (also called a decision tree) helps evaluate decisions with probabilistic outcomes by calculating expected values. The tree branches represent possible scenarios, each with a probability and an outcome value. The expected value of a decision is the sum of (probability × outcome) for all branches.
Example from Exercise 41: Jelly Beans III
- Jar contents: 90 jelly beans (45 red, 36 yellow, 9 green).
- Bets:
- Yellow-Yellow-Green (Y-Y-G): pays $100
- Red-Yellow-Green (R-Y-G): pays $90
- Yellow-Yellow-Yellow (Y-Y-Y): pays $80
- Red-Red-Green (R-R-G): pays $70
- Process: Pick three beans consecutively with replacement, win if you match the exact sequence.
- Probabilities:
- Y-Y-G: (36/90) × (36/90) × (9/90) = 0.4 × 0.4 × 0.1 = 0.016
- R-Y-G: (45/90) × (36/90) × (9/90) = 0.5 × 0.4 × 0.1 = 0.02
- Y-Y-Y: (36/90)^3 = 0.4^3 = 0.064
- R-R-G: (45/90) × (45/90) × (9/90) = 0.5 × 0.5 × 0.1 = 0.025
- Expected monetary payout: If losing pays nothing and there is no entry cost, expected value = probability × payoff.
- Y-Y-G: 0.016 × $100 = $1.60
- R-Y-G: 0.02 × $90 = $1.80
- Y-Y-Y: 0.064 × $80 = $5.12
- R-R-G: 0.025 × $70 = $1.75
- Best bet by expected payout: Y-Y-Y. The highest win probability alone would not establish that result; the payoff also matters.
Note: The four probabilities do not sum to 1 because they are not collectively exhaustive—many other sequences (e.g., Y-R-G) are possible but ignored in this analysis.
Include the value of losing outcomes and any costs when comparing net returns. For non-monetary outcomes, the assigned utilities must meaningfully represent the decision-maker’s preferences; arbitrary scores do not automatically have that interpretation.
When to use it
Evaluating different investment portfolios with varying risks and returns.
Choosing between marketing strategies that have different probabilities of success.
Deciding whether to pursue a new product launch based on estimated outcomes.
Comparing betting or gambling options by expected monetary value.
FROM THEORY TO EVERYDAY LIFE
In practice
Jelly Bean Bets
A jar with 90 jelly beans (45 red, 36 yellow, 9 green) offers four bets: Y-Y-G ($100), R-Y-G ($90), Y-Y-Y ($80), R-R-G ($70). Using a utility tree, the expected values are $1.60, $1.80, $5.12, and $1.75 respectively. With no entry cost and no payout for other sequences, Y-Y-Y has the highest expected payout. Different costs or losing outcomes could change the comparison.
A MOMENT TO REFLECT
Put it to work.
Why don't the probabilities of the four scenarios in the jelly bean example sum to 1?
How would the decision change if the payoffs were non-monetary (e.g., happiness on a scale of 0–100)?
In what real-life situations might a utility tree help you compare risky options?
CONTINUE THE CONVERSATION
Source texts
- The Thinkers Toolkit by Morgan D Jones