--- name: Decision Tree status: draft description: DECISION family. A branching map of a sequential choice under uncertainty — decision nodes you control, chance nodes nature decides, payoffs at the leaves — solved by folding back to the optimal path. sources: - title: "Raiffa, Howard (1968), Decision Analysis: Introductory Lectures on Choices under Uncertainty" url: https://openlibrary.org/works/OL2620279W --- # Decision Tree ## Why it matters A decision tree lays out a choice that unfolds over time — your decisions, the uncertain events that follow each one, and the payoff at the end of every path — as a diagram that branches left to right. Square nodes are decisions *you* control; circular nodes are chance events *nature* controls, with probabilities on their branches; the leaves carry the values you end up with. Its whole job is to stop you from deciding off the most-likely outcome or the most-feared one, and instead weigh the *entire* spread of futures, each in proportion to how likely it is — and to make that weighing something you can actually compute. For example: you can lease equipment for a fixed $50k a year, buy it now and bet on uncertain resale value, or wait six months to see where the market goes and then choose. In prose the argument circles — "but what if prices crash," "but what if we wait and miss the window." Drawn as a tree, the wait branch sprouts a chance node (market turns bear, base, or bull), each outcome carries a number, and folding those numbers back shows the wait costs you six months of carrying charges for information that, here, isn't worth what it costs. The tree turns an argument that went in circles into a path you can defend. - **What it shows.** A sequence of decisions and the uncertain events between them, with probabilities and payoffs, laid out so the optimal path — and what it depends on — becomes computable. - **When to reach for it.** A decision with real sequence (choices now versus choices after you learn more), genuine uncertainty you can put rough probabilities on, and payoffs you can put numbers on. - **How to read it.** Start at the root on the left and move right through time; at each square you choose, at each circle nature rolls the dice, and each leaf is where you land — then fold the values back from the leaves to find the best opening move. - **What you'd miss without it.** The branches you'd never have priced — the low-probability outcome that dominates the math, and the worth of *waiting to learn more* before you commit, which the tree makes a number you can compare against its cost. - **Where it misleads.** It draws the future as a clean tree with crisp probabilities, so the precision of the rolled-back number can feel like certainty when the inputs were guesses; and a decision with real feedback over time doesn't collapse into a tree at all. ## How to read it Picture the diagram drawn left to right, with time flowing the same way. At the far left is the **root** — the first thing that happens. Each fork is a **node**, and the node's *shape* tells you who is in control. A **square** is a **decision node**: a point where *you* choose, and the branches leaving it are the options on the table. A **circle** is a **chance node**: a point where *nature* decides — an uncertain event — and each branch leaving it carries a **probability**, with the probabilities on any one circle summing to one. At the far right, the branches end in **leaves**, and each leaf carries a **payoff**: the value (dollars, or a utility number) you receive if the path that reaches it actually happens. A path from root to leaf is one complete story of how the decision could play out. You solve the tree by working *backwards* — the move decision analysts call **folding back** (or rolling back). Start at the leaves and move right-to-left. At each **chance node**, you can't choose, so you *average*: the node's value is its **expected value**, the probability-weighted sum of its branches (a 70%-chance $100 branch and a 30%-chance –$40 branch fold to 0.7×100 + 0.3×(−40) = $58). At each **decision node**, you *can* choose, so you take the best: the node's value is the value of its single highest branch, and you mark that branch as the choice. Keep folding until you reach the root. What you're left with is the **value of the whole decision** and — by reading the marked branches forward again — the **optimal path** through it. That backward pass is what the tree is *for*. It mechanizes a calculation prose can't hold in its head: it weighs every outcome by its likelihood, so a fat payoff hiding behind a long-shot probability gets counted at its true weight and no more. And because a *wait-and-learn* option shows up as its own branch — decide now, versus first run a chance node that reveals something, then decide — the tree makes the **value of information** computable too: the gain from learning before you commit, set against what the learning costs. A sequential decision under uncertainty, normally argued in circles, becomes a picture you fold back to an answer. ## When to use it The decision tree belongs to the **DECISION family** of diagrams — the ones that structure a choice — and within that family it is the *sequential-choice-under-uncertainty* artifact: the tool for a decision that unfolds over time, where chance intervenes between your moves and the payoffs can be quantified. That places it between two relatives, and knowing the boundaries is how you pick the right one: - An **Influence Diagram** captures the *same* decision in a more compact form — decisions, uncertainties, and value drawn as a small graph of nodes and arrows rather than an exhaustive tree. It is the right choice when the tree would explode into thousands of duplicated branches, or for talking through structure with a stakeholder; the cost is that it *hides* the branch-by-branch detail the tree shows. Reach for the tree when you want every path and payoff visible and computable; reach for the influence diagram when you want the structure at a glance. - A **plain pro/con list** handles the other extreme — a single go/no-go choice with no sequence and no probabilities to weigh. It's faster and lighter, and for a one-shot decision it's often enough; the tree would be overkill. Reach for a decision tree when the choice has real sequence, the uncertainties take rough probabilities, the payoffs take numbers, and the cost of getting it wrong justifies the work. Skip it when the choice is single-shot with no estimable odds (a pro/con or comparison is enough), when the structure alone is the point (use an influence diagram), or when the decision is genuinely dynamic with feedback that won't flatten into a clean tree (that wants a simulation, not a tree). ## How Ora builds it Ora produces a decision tree from a **semantic spec** — a structured description of the **decision nodes** (your choices and their options), the **chance nodes** (the uncertain events, each branch tagged with a **probability**), and the **payoffs** at the leaves. Folding back is part of the spec, not an afterthought: each node is annotated with its **fold-back value** (expected value at chance nodes, best-branch value at decision nodes) and each decision node with the branch it selected, so the optimal path is carried in the data. That spec is rendered to a diagram laid out **left to right** by time, with the three node types drawn as distinct shapes — and because shape alone isn't accessible, the shapes are reinforced with text labels and the diagram ships with an outline view and alt-text describing the decisions, the chance branches with their probabilities, the leaf payoffs, and the rolled-back optimal strategy. The diagram is the visual face of Ora's **Decision Under Uncertainty** mode: when you ask "I face this sequential choice — lay it out and tell me the best path," that mode builds the structure, assigns the probabilities and payoffs, runs the fold-back, and this artifact is how it shows its work — the path it recommends *and* what that recommendation hinges on. The technique is the foundation of formal **decision analysis**, developed in the 1960s by **Howard Raiffa** at Harvard and Ronald Howard at Stanford, and set out in Raiffa's *Decision Analysis: Introductory Lectures on Choices under Uncertainty* (1968). The diagrammatic form was popularized for managers by John F. Magee's pair of *Harvard Business Review* articles, "Decision Trees for Decision Making" and "How to Use Decision Trees in Capital Investment" (both 1964). It has since become standard equipment in capital-investment analysis, clinical decision-making, oil-and-gas exploration, and legal strategy. ## Related - **Influence Diagram** — the compact member of the DECISION family: the same decision drawn as a small graph, trading the tree's branch-by-branch detail for structure at a glance. - **Tornado Chart** — the companion that tests how sensitive the tree's recommendation is to its inputs, ranking which uncertain probability or payoff the optimal path depends on most. - **Quadrant Matrix** — a lighter DECISION-family tool for sorting options along two axes when there's no sequence or probability to fold back. - **Decision Under Uncertainty** (mode) — the analytical operation this diagram renders: framing the choices, assigning probabilities and payoffs, and folding back to the optimal path. ## Sources - [Raiffa, Howard (1968), Decision Analysis: Introductory Lectures on Choices under Uncertainty](https://openlibrary.org/works/OL2620279W)