Put a payoff table beside a prior and the whole of a one-shot decision under uncertainty is a tree two levels deep. Folding it back is the recursion again, with two kinds of node instead of one. What the fold makes possible is a question the table alone cannot answer: what would it be worth to know the state before deciding, and what is this particular noisy signal worth? The first is a ceiling that no signal can exceed. The second can be exactly nothing, for a signal that looks perfectly respectable.
Core ideas
Three ideas. The first is the fold, the second is the ceiling, and the third is the one that makes the ceiling worth having.
Work from the leaves back. A chance node takes the probability-weighted average of what its branches fold to; a decision node takes the largest, and remembers which branch that was. Nothing else changes: values are written only once everything below them is settled, and the tree is filled from the ends. The branch a decision node remembers is the plan, and the number it holds is not.
Knowing the state before choosing lets you pick the best act in each state separately, so its value is Σ P(state) × max over acts. That has the maximum inside the average, which is the swap “The Stochastic Recursion” warned about — and here the swap is the point rather than a mistake, because it is describing a different and better-informed decision-maker. The difference between the two orders is EVPI, and no signal can be worth more.
EVSI is computed by finding, for each signal, how likely that signal is and what the posterior over states becomes; then taking the best act under each posterior; then averaging those over the signals. A tree with no likelihood in it has no EVSI at all, and the lab reports it as absent rather than producing a number from the branches — a shape can always be given a number, and that is precisely the danger.
The material
The two node rules, the two prices, and the signal that costs nothing to ignore.
Given acts a, states s, a payoff π(a, s) and a prior p(s), the prior value is max over a of Σₛ p(s) π(a, s) and the perfect-information value is Σₛ p(s) max over a of π(a, s). Their difference is EVPI.
A signal is a likelihood P(g | s). For each signal g, P(g) = Σₛ p(s) P(g | s) and the posterior is p(s) P(g | s) / P(g). The value with the signal is Σ_g P(g) max over a of Σₛ posterior × π(a, s), and EVSI is that minus the prior value.
The lab builds the tree from the payoff table rather than asking you to type one, because typing a tree is harder than typing the thing a tree is drawn from and the tree is not the model. With no signal it is one decision node over a chance node per act. With a signal it is a chance node over the signals, and inside each branch of it the same decision again, now over the posterior.
Twenty and nothing. Perfect information here is worth 20, which is a great deal against a decision currently worth 10, and this signal captures none of it. The reason is visible in the likelihood: the two rows are identical, so the signal is as likely in one state as in the other, the posterior comes out equal to the prior, and no decision changes. A signal has to discriminate to be worth anything, and looking informative is not the same thing.
The lab also enumerates strategies. A strategy here is a choice of act for each signal, and the best it finds is worth 10, which is what the fold says. That enumeration prices each strategy straight from the joint distribution with no tree, no posterior and no fold anywhere in it, so it is an independent witness rather than a restatement.
Payoffs: building returns 100 if things turn out good and loses 20 if they do not; waiting returns nothing either way. The prior is 3/10 good. So building is worth 16 on the prior, waiting 0, and the best act is to build. Knowing the state first is worth 30, so EVPI is 14.
The survey is right four times in five when things are good and three times in four when they are not. A favourable survey then has probability 83/200 and leaves a posterior of 48/83 good, under which building is worth 4100/83; an unfavourable one has probability 117/200 and a posterior of 4/39 good, under which waiting at 0 beats building. The whole thing is worth 41/2, so EVSI is 9/2 — which is 9/28 of EVPI, and no more than EVPI, as it must be.
Notice what the survey buys. Without it you build whatever happens; with it you build after a favourable reading and wait after an unfavourable one. A signal is worth something exactly when it changes what you do, and 9/2 is the price of that change. If both readings had left building as the best act, EVSI would have been zero however sharply the posteriors moved — which is a strange and useful fact, and it is why information has to be priced against a decision and never in the abstract.
A payoff table with a cautious act paying 20 whatever happens, a middle act paying 0, 40 and 45, and a bold one paying -30, 20 and 80, against a prior of 1/4, 1/2, 1/4. On the prior the three are worth 20, 125/4 and 45/2, so the middle act wins. Perfect information is worth 45 and EVPI is 55/4.
With the signal the whole thing is worth 257/8, so EVSI is 7/8 — a genuine but small fraction of a large EVPI. The signal is informative and mostly fails to change the act: only the weaker of the two readings moves the decision, and it moves it to the cautious act rather than the bold one.
The payoffs are known exactly and are in one currency. The prior is known exactly, which is the assumption a reader should be least comfortable with: EVSI depends on it, and a prior that was really a guess produces a price for information that is also a guess, stated to the last fraction. The likelihood is known exactly too, which is to say somebody has measured how often this signal is right in each state — and if they have not, the number on the page is the value of an imagined instrument.
And the objective is still the expectation. A payoff of -40 with probability one half sits in the average as comfortably as anything else, which is fine if the decision is one of many and ruinous if it is not. Nothing in the fold can raise that question; the fold will happily return a precise expected value for a gamble nobody should take.
Interactive
EVPI bounds what any signal can be worth; EVSI says what this one is worth
Method
Five steps, and the order matters: two of them are cheap and rule out the third entirely.
One weighted average per act. This is what you would do knowing nothing more, and every price below is measured against it — so getting it wrong shifts both EVPI and EVSI by the same amount and neither looks odd.
For each state take the best act in that state, then average over the prior. It is the same numbers in the other order and it is never smaller. The difference is EVPI, and it is the ceiling.
No signal can be worth more than knowing the state outright. If the asking price is above EVPI the answer is no, and it took two weighted averages to find out. This is the cheapest useful thing on the page.
P(g) = Σₛ p(s) P(g|s), then the posterior by Bayes. Check that the posteriors are distributions before going on; the lab checks that each state's column of the likelihood sums to 1, because a column is a distribution over what you might see.
That is EVSI. Then ask the question the arithmetic cannot: does the best act actually differ between signals? If it does not, EVSI is zero no matter how sharply the posteriors moved, and the signal is not worth buying at any price.
The same procedure on three payoff tables. Only the last column differs in kind: one signal is worth a reasonable share of the ceiling, one a small share, and one nothing at all.
The middle column is the one that repays a second look. EVPI is 55/4, which is large next to a decision worth 125/4, and the signal captures 7/8 of it — under a tenth. It is a perfectly informative signal in the ordinary sense: the posteriors move a long way. It is nearly worthless because the act it recommends is almost always the one you would have chosen anyway.
For a rehearsal, take the build-or-wait instance and make the survey perfect — a likelihood of 1 0 in the first row and 0 1 in the second. EVSI should rise to exactly EVPI, 14, and not a fraction more. Then make it perfectly wrong, 0 1 and 1 0, and predict the answer before running it: a signal that is always backwards is just as informative as one that is always right.
The harder rehearsal: hold the likelihood fixed and move the prior. There is a prior at which building and waiting are equally good, and EVSI is largest somewhere near it. Find it, and then say in words why information is worth most when the decision is closest — and worth nothing when the decision is not in doubt, however uncertain the state is.
You should be able to apply the two node rules from the leaves back, compute the prior value and the perfect-information value and know which has the maximum on the inside, use EVPI as a ceiling before touching Bayes, compute posteriors and EVSI when the ceiling does not settle it, and explain why a signal that never changes the act is worth nothing however sharp it looks.
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