Decision Analysis

Should you launch the product? Drill the exploratory well? Settle the lawsuit or fight it? Each is a choice made now, whose payoff depends on things you can't control and don't yet know — the market, the geology, the jury. Gut feeling and slogans ("fortune favours the bold") are no help when millions ride on the answer. Decision analysis is the branch of operations research that gives such choices a spine of arithmetic: lay out the options and the uncertainties, attach numbers, and let the structure tell you what to do.

Two kinds of fork: things you choose, things that happen

Every decision problem is a tree with two kinds of branch point:

A decision node (drawn as a square) is a fork you control — you pick one branch: launch or don't, drill or don't.
A chance node (drawn as a circle) is a fork the world controls — the branches are outcomes with probabilities that add to one: good market or bad, oil or dry.
At the leaves sit the payoffs — the money (or utility) you end up with down each path.

The key summary of a chance node is its expected value: the probability-weighted average of what lies beyond it. Applied to money it has a special name — the expected monetary value, or EMV.

Folding back a tree

A firm can launch a new gadget or not. If it launches, the market turns out good with probability 0.6 (profit £200\text{k}) or bad with probability 0.4 (loss £50\text{k}). Not launching yields £0. To solve it we fold back — work right to left, replacing each chance node by its EMV and each decision node by its best branch:

At the chance node the EMV is 0.6 \times 200 + 0.4 \times (-50) = 120 - 20 = £100\text{k}. Back at the decision node we compare the two branches — launch is worth £100\text{k}, not launching is worth £0 — and pick the larger. Launch, with an expected value of £100\text{k}. Folding back always turns a sprawling tree into a single number and a recommended first move.

What is it worth to know the future?

Suppose a perfect oracle could tell you the market before you decided. You'd launch only when it says "good" (pocketing 200) and walk away when it says "bad" (pocketing 0). The expected value with perfect information is

\text{EV with PI} = 0.6 \times 200 + 0.4 \times 0 = £120\text{k}.

You already secure £100\text{k} without the oracle, so the extra the oracle buys you — the expected value of perfect information — is

\text{EVPI} = \text{EV with PI} - \text{best EMV} = 120 - 100 = £20\text{k}.

This is the ceiling on what any study, survey or test market is worth: no information, however good, can be worth more than perfect information. If a market-research firm quotes £30\text{k} for a forecast here, decline — it cannot pay for itself.

Because the best first move often depends on moves that come later. Real problems chain decisions and chances — test first, then decide whether to launch, then see the market — and intuition cannot juggle the branching probabilities. The tree makes the structure explicit and the fold-back is mechanical: average at circles, maximise at squares, right to left. It also lays bare which uncertainties actually move the decision, so you know where extra information (and money) is worth spending — exactly what EVPI quantifies.

Folding back on EMV treats £100\text{k} for sure as exactly equal to a coin that pays £200\text{k} or nothing — same average, so EMV is indifferent. But a founder betting the company might vastly prefer the sure thing, and an insurer will pay to shed variance it doesn't like. EMV is the right rule only for a risk-neutral decider. When variance genuinely hurts, replace money with utility — a curved function that bends down for large sums — and fold back on expected utility instead. The tree is the same; only the numbers at the leaves change.