Decision Trees 2026: Expected Value Before Opinion

Squares you control and circles you do not, in a decision tree

A decision node is a square. You pick the branch, so you take the best one and the tree keeps that value. A chance node is a circle. The world picks, so you weight every branch by its probability and add them up. Nothing else happens in a decision tree.

Almost every wrong answer comes from putting a square where a circle belongs. Someone treats a competitor's response as a choice they control, takes the branch they prefer, and produces a number that is too high in a way no amount of checking will reveal. The arithmetic is clean. The node type is wrong.

Four habits prevent it, and they take about a minute.

1. Name the owner of every node in the margin. Us, the regulator, the competitor, the plant.
2. Read the tree out loud and say who decides at each fork. Any owner other than the client makes it a circle.
3. Check that every circle's probabilities sum to one. A circle that sums to 0.9 has a branch missing.
4. Write the units on the payoffs. Net present value and annual profit cannot sit in the same column.

The order also matters. Time runs left to right and value rolls right to left. You cannot value the first decision until you have valued everything downstream of it, which is why a tree gets built forwards and solved backwards.

The catalyst decision tree, rolled back

A specialty chemicals business has a new catalyst that works in the lab. Commercializing it means a 48 million dollar plant. The engineering team puts a 55 percent chance on the catalyst holding its yield at industrial scale.

If it holds, the plant is worth 90 million dollars of net present value. If it does not, the plant still gets built and runs on the old chemistry, and it is worth negative 30 million. So the tree has one square with two branches, and one of those branches leads to a circle.

Roll back the commit-now branch first. The chance node is 0.55 times 90 plus 0.45 times negative 30. That is 49.5 minus 13.5, so 36.0 million dollars.

Now the second branch. The team can run a 6 million dollar pilot for nine months that settles the yield question before any plant money is committed. Assume the pilot is conclusive, which is the strongest case for testing and therefore the right place to start.

With a conclusive pilot, the 45 percent outcome no longer costs anything, because you simply do not build. The 55 percent outcome still pays 90 million, nine months later. At a 10 percent discount rate, nine months of delay multiplies it by 1.10 to the power of negative 0.75, which is 0.9310, so 90 million becomes 83.79 million.

Expected value of the test-first branch is 0.55 times 83.79, which is 46.08, minus the 6 million pilot, so 40.08 million dollars. Call it 40.1 million.

What the information is actually worth in expected value

Test first beats commit now by 40.08 minus 36.0, which is 4.08 million dollars. That figure is the answer to the question the board asked, and it decomposes into exactly two pieces.

Read the top two rows together and the ceiling appears. The information is worth 13.50 million of avoided loss and it costs 3.41 million in delay, so the most a rational board would pay for it is 13.50 minus 3.41, which is 10.09 million dollars. At 6 million the pilot clears comfortably. At 11 million it does not, even though it is the same pilot answering the same question.

That ceiling is the output a decision tree produces and a discussion does not. It also tells you what to negotiate: a pilot that takes four months instead of nine cuts the delay cost, which raises the ceiling and makes a more expensive pilot affordable.

One more threshold is worth having in your pocket. Commit now and test first are equal in value when the probability of success is low enough that the avoided loss shrinks to the pilot cost plus delay. Solve it and the two branches meet near 66 percent. Above that confidence level the pilot is not worth nine months, and below it the pilot pays. Engineering said 55 percent, so the pilot pays, and you now know how wrong engineering would have to be for that to change.

The bar usually sits outside the decision tree

A decision tree returns expected value in dollars. Clients rarely underwrite in expected value alone, and this is where candidates lose cases that they computed correctly.

Oliver Wyman publishes the cleanest illustration of it. In its oil and gas case, a gas operator considers distributing drinking water. Revenue is "Rs. 750,000,000, assuming 300 operational days per year" and total annual costs are "Rs. 575,000,000 or 57.5 crore", so the business earns 17.5 crore a year before tax. The published recommendation still says no: entering "does not align with the client's investment criteria for a five-year payback period", checked 30 September 2026 at oliverwyman.com.

The business made money and the answer was no. The bar was a payback period, and no amount of expected value arithmetic reaches it. So the opening move in a tree-shaped case is a question: what does the client underwrite on, expected value, payback, downside floor, or a capital limit that simply cannot be exceeded.

A downside floor is the one that most often overrides a tree. Our commit-now branch has a 45 percent chance of losing 30 million. If the business cannot absorb a 30 million dollar loss without breaching a covenant, the expected value of 36 million is irrelevant and the pilot is mandatory at almost any price. Bain writes that it is "not looking for a single right answer" and watches how candidates "build on ideas as the conversation evolves", checked 29 September 2026 at bain.com. A constraint appearing halfway through is exactly that kind of evolution.

Reading a tree out in the right order

The tree is arithmetic, and the recommendation is a sentence. Say the decision, then the number, then the condition that would reverse it.

Our own grader reports a closing recommendation on four lines, and no consulting firm publishes a marking scheme of its own. The weights below are ours, read from the rubric code, each out of one hundred.

Rework it with the numbers moved

Change the downside and watch the ceiling move. At a downside of 60 million instead of 30, the commit-now branch becomes 0.55 times 90 plus 0.45 times negative 60, which is 49.5 minus 27, so 22.5 million. The loss avoided becomes 0.45 times 60, which is 27 million, and the ceiling on pilot spending rises to 27 minus 3.41, or 23.59 million. Doubling the downside more than doubles what the test is worth.

Then change the delay. A four-month pilot discounts 90 million by 1.10 to the power of negative one third, which is 0.9687, so roughly 87.19 million. Delay cost falls to 0.55 times 2.81, which is about 1.55 million, and the ceiling rises from 10.09 to 11.95 million. Removing five of the nine months is worth about 1.9 million dollars, which tells the engineering director what acceleration is worth buying.

Charles River Associates tells candidates to "show your math", checked 30 September 2026 at crai.com. On a rollback that means writing each node value on the page as you get it, so the corrections land on one node instead of on the whole tree.

Time the rollback in the case math drill, which scores Answer, Working, Units and Interpretation separately, and check a payback bar in the breakeven calculator. Both are free and need no account. Neither will supply your probabilities, which is the part the tree cannot compute for you. For the version of this tool that decomposes a quantity rather than a choice, see driver trees, and for the structure that decides which branches exist at all, see issue trees.