Sit in on a leadership team arguing about a big call and you will mostly hear numbers: the forecast, the cost, the upside. Underneath, the real disagreement is usually about structure, namely what the options actually are, what cannot be known until later, and which commitments land before which answers arrive. Structure is something you can draw. This piece is about drawing it, and about what the drawing tells you that the numbers alone never will.
The map before the maths
Strip any contested decision to its bones and three kinds of thing remain: the moments you control (you can launch now, or run a pilot first), the moments you don’t (demand will be strong or weak, and no meeting can vote on which), and the order they arrive in (you must commit before demand reveals itself, which is the whole problem). Most cross-table disagreement is one of those three wearing a number as a disguise.
So draw it before you price it. Put a small square wherever you choose, a small circle wherever the world chooses, and read time left to right. Branches out of squares are options; branches out of circles carry probabilities; every path ends at a payoff. The drawing has a name, a decision tree, and the name is the least interesting thing about it. The value is that a structure on paper can be disputed line by line, where a structure in someone’s head cannot.
Here is the example this article runs on, the same one that sits in Branch as its defaults. A product is ready. Launch now, and the result turns first on demand, strong in about 4 in 10 futures and weak in about 6 in 10, then on how well the rollout is executed inside each. Pilot first, and you spend time and money to scale only what works: demand decides again, and then it is how well the pilot converts. Everything is in thousands, net of all costs, and every probability and payoff below is an input you are allowed to dispute. That is rather the point of writing them down.
Folding back
Pricing the tree takes one rule, applied from the right-hand edge backwards. At every circle you don’t get to pick the branch, so take the probability-weighted average of what lies beyond it. At every square you do get to pick, so keep the best branch and cross out the rest. Repeat until you reach the first square. The procedure is called rolling back (or folding back) the tree, and that is the entire algorithm: no solver, no simulation, a pencil does it.
Walk it on the example. On the launch side, strong demand with a smooth rollout pays +400 and a rough one +120; about 7 in 10 rollouts go smoothly, so the strong-demand circle is worth +316. Weak demand offers only ways to lose, −80 or −250, and averages −131. Weight the two by the 4-in-10 and 6-in-10 demand odds and launching now is worth +47.8. The pilot side folds the same way: +220 if demand turns out strong, −20 if weak, +76.0 overall. At the square the comparison is one number against another, and the pilot branch keeps it.
launch · weak demand: .70 × (−80) + .30 × (−250) = −131
launch now: .40 × 316 + .60 × (−131) = +47.8
pilot · strong demand: .80 × 260 + .20 × 60 = +220
pilot · weak demand: .50 × 20 + .50 × (−60) = −20
pilot first: .40 × 220 + .60 × (−20) = +76.0
+76.0 beats +47.8, so the folded tree says pilot first, by an expected +28.2.
The average and the lottery
The first series on this site spent three articles on one thesis, that an average is not a decision, and it applies with full force to the two numbers we just produced. Nothing pays +47.8 and nothing pays +76.0. Those are averages over futures, and no future is average.
What each strategy actually hands you is a lottery, and the tree gives you the whole of it. Launch now ends somewhere negative in about 6 of 10 futures, and at worst that is −250: a full launch into demand that was never there. Pilot first ends negative in about 3 of 10 futures and never worse than −60, because the pilot is the most you can lose. Launching keeps the bigger prize, +400 against +260 (the price of arriving later with pilot money spent), and pays for it with the deeper, likelier hole. That trade, not 47.8 against 76.0, is the decision. A number without its spread is a decision waiting to go wrong.
Notice what it took to produce those spreads: nothing. No simulation, no ten thousand draws. When the outcomes are few and discrete, multiplying the probabilities along each path gives the whole distribution exactly; the tree is its own answer. Simulation earns its keep when the chain of uncertainties is long and continuous, a boundary the endnote comes back to.
What would you pay to know?
Here is the tree’s quiet gift. Suppose, purely hypothetically, you could learn demand before committing: a perfectly reliable signal, strong or weak, delivered in advance. You would no longer need a strategy; you would need a lookup. Signal says strong: launch, that branch is worth +316. Signal says weak: pilot, −20, the smallest available regret. About 4 in 10 signals would say strong and about 6 in 10 weak, so deciding with the answer in hand is worth +114.4 on average, against the +76.0 of deciding without it.
The difference, +38.4, is what the answer itself is worth: the most you should ever pay to know demand before you commit. In plain words, it is the price of being allowed to switch strategies after the truth arrives instead of before. The literature calls it the expected value of perfect information, EVPI, and it converts a vague itch (“shouldn’t we do more research first?”) into arithmetic. No market study costs nothing. One that costs less than the value of what it tells you is a bargain; one that costs more is theatre. In this example a study quoted at 50 has no case to make, and one quoted at 10 rather does. This is the consulting question hiding inside every “should we do more analysis?”.
value of the answer: 114.4 − 76.0 = +38.4
Two honest footnotes before anyone books the fieldwork. Real studies are not perfect signals, so a real study is worth less than the 38.4 ceiling, and pricing an imperfect one is a longer calculation for another day. And the ceiling is only as good as the tree it fell out of: move the probabilities and it moves with them. Even so, a ceiling is useful on contact. Any analysis costing more than a perfect answer cannot be worth buying, however reassuring it would feel to commission it.
The bridge
Each strategy’s lottery yields the three-number summary the rest of this site runs on: a 10th percentile, a median and a 90th. Launch now: −250 / −80 / +400. Pilot first: −60 / +20 / +260. Take either triple to Elicit as low / typical / high and you can compare the strategies as full distributions rather than as two averages.
One caveat travels with them: a four-outcome lottery is coarse by construction, and its percentiles are steps, not a curve. The 10th percentile of launch now is −250 only because nearly 2 in 10 futures sit exactly there. Treat the triples as a faithful summary of a deliberately blunt object, not as the smooth tail behaviour of the world.
What this draft holds still. Probabilities enter the tree as single numbers (.40 strong, .70 smooth), and the series on judgement under uncertainty is about how hard those single numbers are to get honestly. Demand and execution are treated as independent, which is a modelling choice, not a fact about the world. And payoffs sit at the leaves as point values: when the chain of uncertainties is long and continuous rather than four discrete endings, folding back stops being the right tool and simulation starts, the move described in From sensitivity to simulation. Draft note: copy and figure to be reconciled against Branch before publish.