How to use the decision tree expected value calculator
- Write the tree as an indented outline, two spaces per level. Start a line with "decision:" or "chance:", or write an outcome as "name = payoff".
- Give each branch of a chance node a percent at the start of its line. The percents under one chance node must add up to 100.
- Put an amount in square brackets at the end of a line for a cost or gain on that branch, such as [-200000].
- Read the expected value of the best plan, the best choice at each decision and the table that shows how the answer changes as one probability changes.
Formula
leaf value = its bracketed amount + its payoff
chance node value = bracketed amount + sum of (percent / 100 x value of each branch)
decision node value = bracketed amount + the largest value among its choices
The tree is rolled back from the payoffs on the right to the top. At a decision the page keeps the best choice and discards the rest, and the first choice wins a tie.
Launch now, test first or do nothing
Launching nationally now costs $200,000 and brings $600,000 with a 55 percent chance of strong demand and $100,000 with a 45 percent chance of weak demand. Its expected value is -200,000 + 0.55 x 600,000 + 0.45 x 100,000 = $175,000.
Testing one region first costs $40,000. If the test is positive (50 percent), you can launch for $200,000 with 80 percent strong demand, worth $300,000, or stop and get 0. A negative test gives 0. The value is -40,000 + 0.5 x 300,000 = $110,000. Doing nothing is worth $0, so launching now is best. Varying the chance of strong demand, the best choice changes from testing to launching at 42 percent.
How to write the outline
Each line is one node. A node that makes a choice starts with "decision:" and its children are the options. A node where chance decides starts with "chance:" and its children each start with a percent. A line like "Strong demand = 600000" is a payoff at the end of a path.
Costs go in square brackets on the line of the choice that causes them. A line such as "chance: Launch nationally [-200000]" charges $200,000 whenever that branch is taken. Lines starting with # are notes and are ignored.
What expected value can and cannot tell you
Expected value is the average result if you could make the same choice many times. For a one-time decision it ignores how painful a loss would be, so look at the worst outcome of each choice as well. The percents are your estimates, which is why the sensitivity table matters: if the best choice flips for a small change in one estimate, the decision is fragile.
For a single set of outcomes with no choices, use the Expected Value Calculator. To compare options on several weighted criteria when payoffs are not money, use the Decision Matrix Calculator. To check a probability before you use it, see the Probability Calculator.
Assumptions and limits
- Amounts are shown with a dollar sign, but you can use any unit of value. All amounts must use the same unit.
- Percents under each chance node are exact and must total 100. The page does not rescale them.
- The first choice wins when two choices have the same value.
- The sensitivity table varies the first branch of one chance node and keeps the others in proportion to each other.
- Up to 200 lines are accepted, and nothing typed here leaves your browser.
Frequently asked questions
How is the sensitivity table calculated?
The page sets the first branch of the chosen chance node to 0, 10, 20 and so on up to 100 percent, shares the remaining percent among the other branches in their original proportions, and rolls the whole tree back each time.
Why does a line fail with a message about percents?
Every branch of a chance node needs a percent, and they must add up to 100. Branches of a decision take no percent. The message names the node, so you can find the line.
Can a decision come after a chance?
Yes. A chance branch can lead to another decision, as in the example where a positive test leads to a choice about launching. Indent it under the branch.
What does the amount in brackets do?
It is added to the value whenever the branch is taken, before the value of what follows. Use a negative amount for a cost and a positive one for a gain that does not depend on later events.