Decision Trees
Decision Trees · Decisions, Chance Events, and Expected Value
How do you compare the outcomes of choices when the situation is uncertain? Which choice is best isn’t settled by the choice alone. You also have to consider what could happen after you choose, how likely each outcome is, and what you gain or lose as a result. Launch a new product and you make a big profit if the market responds well but may take a loss if it responds poorly, and choosing not to launch has an opportunity cost too.
A decision tree lays out these choices, the uncertain events, and their outcomes in the shape of a tree so you can compare alternatives systematically. The decision tree in Ch. 07 of PART 3 · 13 Statistical learning & prediction is a model that learns prediction rules from data; the decision tree here is a tool a person draws by hand to analyze a choice.
What is a decision tree?
A decision tree breaks a decision into stages and shows, in order, which choices you can make, which events can follow, and what outcome each path leads to.
| Element | Meaning | Usual symbol |
|---|---|---|
| Decision node | A point where the decision maker chooses | Square □ |
| Chance node | A point where it’s uncertain which event will occur | Circle ○ |
| Outcome | The profit, cost, or utility at the end of each path | Endpoint and outcome value |
There’s an important distinction here. The decision maker controls the choices but can’t directly control chance events. A decision tree shows the two separately.
When do you need a decision tree?
If the alternatives are simple and the outcomes are certain, comparing them directly is enough. But a decision tree is useful in situations like these.
Deciding whether to launch and, if so, whether to do market research first
Judging whether to invest now or after checking market conditions
Deciding whether to continue further
When choices and uncertain outcomes are linked across several stages
It’s especially useful when today’s choice changes the options you’ll have later. If you do market research first, you can decide whether to launch after seeing the results (Section 05).
A worked example: should we launch a new product?
Suppose a company is considering launching a new product and has estimated net profit, including launch costs, as follows. These are assumptions for illustration, not forecasts for a real business.
| Item | Assumption |
|---|---|
| Probability the market responds well | 60% |
| Probability the market responds poorly | 40% |
| If the response is good | Net profit of $200,000 |
| If the response is poor | Net loss of $80,000 |
| If you don’t launch | No additional profit or loss ($0) |
To compare launching with not launching, you can’t look only at the biggest possible profit; you have to consider the possible outcomes together with how likely each one is.
Comparing alternatives by expected value
The expected value is what you get by multiplying each possible outcome by its probability and adding them all up.
pi: probability that outcome i occurs · xi: monetary value of that outcome · when all outcomes are included, the probabilities sum to 1
The expected net profit of not launching is $0, so if you go by expected value alone, launching is the better choice. But the expected value isn’t an amount you’re guaranteed to get. The actual outcome is either a $200,000 profit or an $80,000 loss, and an outcome of $88,000 never happens. The expected value is an average value for comparing uncertain outcomes on a single scale: the average you’d get if you made the same decision a very large number of times.
A decision tree is solved from right to left
In a decision tree, you start the calculation from the final outcomes and work back to the first choice. This is called backward induction.
Write the profit, cost, or utility of each path at its endpoint.
Multiply each possible outcome by its probability and add them up.
Pick the larger value if you’re maximizing expected profit, or the smaller if you’re minimizing cost, and prune the other branches.
Figure 3 is a two-stage problem: before launching, should you spend $10,000 on market research? We assumed the research comes back "positive" with probability 90% when the actual response is good and 20% when it’s poor. Working from the right end, the calculation goes like this.
The probability of a good response rises to 87.1%, so the expected value of launching is $163,871 → launch
The probability of a good response falls to 15.8%, so the expected value of launching is −$35,789 → hold ($0)
0.62 × $163,871 + 0.38 × 0 = $101,600, or $91,600 after the research cost
That’s $3,600 more than the $88,000 expected value of launching right away, so under these assumptions it pays to do the research first. The research is valuable because a "negative" result lets you avoid a launch that’s expected to lose money. The increase in expected value you get from the research ($13,600) is called the expected value of sample information (EVSI); if the research costs more than this, there’s no reason to do it. If you could know the outcome perfectly in advance, the gain would be $32,000, which is called the expected value of perfect information (EVPI). No research is worth more than this amount.
Updating the probabilities based on the research result is Bayes’ theorem (PART 3 · 11 Bayesian statistics).
What if the probability estimate is wrong?
Expected value depends heavily on the probabilities you put in. In Figure 4, the expected value of launching changes along a straight line with the probability of a good response p, and once p drops below about 28.6%, holding becomes the better choice. There’s plenty of room between the assumed 60% and the break-even point, so the conclusion holds even if the probability estimate is somewhat off. Finding the point where the conclusion flips like this is sensitivity analysis (Ch. 01), and if the break-even point is close to the assumed value, you should scrutinize the probability estimate more carefully.
Is the choice with the highest expected value always best?
Expected value is a useful criterion, but it doesn’t reflect how willing the decision maker is to take risks.
| Alternative | Outcome | Expected value |
|---|---|---|
| A | Get $80,000 for certain | $80,000 |
| B | $200,000 with 50% probability, $0 with 50% probability | $100,000 |
B has the higher expected value, but its outcome is also more variable. Someone who values stability might choose A, and someone willing to take risks for a bigger payoff might choose B. The difference isn’t a calculation error but a difference in how risk is valued, and a company or individual that can’t easily absorb a loss may choose based on a utility that accounts for risk (Ch. 03).
Cautions when using decision trees
Put in inaccurate probabilities or outcome values and the conclusion can be wrong even if the calculation is right.
Leave out unexpected costs or extra opportunities and the comparison of alternatives gets distorted.
Don’t mix measures, such as comparing one alternative by revenue and another by net profit.
Cover risk, fairness, and long-term strategic value with expected utility, sensitivity analysis, and multi-criteria methods.
In a decision tree analysis assuming a 60% probability of a good market response, the expected net profit of launching the new product was $88,000, higher than not launching ($0). Including pre-launch market research (cost $10,000) raised the expected net profit to $91,600, and the break-even probability above which launching is better was 28.6%.
Pick at choices, average chance by probability, and work back from the end.
- Structure: A decision tree lays out choices (□), uncertain events (○), and outcomes stage by stage.
- Expected value: The average value from multiplying outcomes by their probabilities and adding them up; it isn’t the amount you actually receive.
- Backward induction: Take the expected value at chance nodes and pick the best branch at decision nodes, working back to the first choice.
- Value of information: More research is worth something only if its result can change the decision, and its cost has to be less than the EVSI.
- Limits: The conclusion is sensitive to the probability estimates, and expected value alone doesn’t capture attitudes toward risk.
Up next: 03 Expected Utility and Risk Preference. We’ll see why the choice between alternatives with the same expected value can depend on your attitude toward risk.