355. Decision Trees

A decision tree is the standard graphical model for decision analysis. Branches represent decisions or random outcomes; leaves carry payoffs; the tree’s root is “now”.

355.1. Node types

355.2. Example: build big plant or small?

                    [build big]
   ☐──────────────────●(0.6 high demand) → $80M
   |                  ●(0.4 low demand)  → $-10M
   |
   |   [build small]
   └──────────────────●(0.6 high demand) → $40M
                      ●(0.4 low demand)  → $20M

355.3. Folding-back (rollback) algorithm

The algorithm to solve a decision tree: work from leaves backward to root.

At each chance node: compute expected payoff
    E[payoff] = Σ p_i × payoff_i over branches
At each decision node: pick the branch with the best expected payoff
    Tag the node with that branch's value (and which act to pick)
At the root: read off the optimal first decision and its expected value

This is just backward induction applied to a decision tree.

355.4. Solving the example

Big plant: (millions).

Small plant: (millions).

Decision: build big. Expected value: M.

355.5. Multi-stage trees

Decisions can be sequenced: act, observe state, act again. Each stage adds a layer of (decision → chance) before reaching a payoff. Fold back as before — each chance / decision node still uses local expected value / argmax.

355.6. Sensitivity analysis

After solving, vary inputs (probabilities, payoffs) and re-solve. Identify which inputs matter most. Standard tools:

355.7. Common applications

355.8. See also