358. 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”.

358.1. Node types

• Decision nodes (squares ☐): branches are acts you choose
• Chance nodes (circles ◯): branches are states of nature, each with a probability
• Terminal nodes (triangles ▲): each leaf has a payoff (cost or revenue)

358.2. 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

358.3. 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.

358.4. Solving the example

Big plant: $𝐸\left[\text{payoff}\right]=0.6\cdot 80+0.4\cdot \left(-10\right)=48-4=44$ (millions).

Small plant: $𝐸\left[\text{payoff}\right]=0.6\cdot 40+0.4\cdot 20=24+8=32$ (millions).

Decision: build big. Expected value: $44$M.

358.5. 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.

358.6. Sensitivity analysis

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

• Tornado diagrams — rank inputs by impact on decision
• Spider plots — show decision threshold crossings
• Probability sensitivity — at what $𝑝$ does the recommended decision change?

358.7. Common applications

• Capital investment — go / no-go, big / small project
• Drug development — phase-by-phase, abandon if results poor
• Insurance / risk management — buy coverage vs self-insure
• Marketing campaigns — pilot test then full launch
• Litigation — settle vs go to trial

358.8. See also

• EMV — the standard criterion
• EVPI — value of resolving uncertainty
• EVSI — value of imperfect info
• Backward Induction — same algorithm