Week 5: Decision Trees & Sequential Decisions

This week’s big question: real decisions come in stages. Decide, see what happens, decide again. How do you lay the whole sequence out and find the best first move? A decision tree does exactly that, and “folding it back” finds the answer.

Before the week

Read: Taylor, Introduction to Management Science (13e): Chapter 12, “Decision Analysis,” pp. 587–596 (focus: decision trees and sequential decisions). The book solves its trees with add-ins we do not use; read it for the method and use the tools listed below instead.
Warm-up self-check, expected value, the engine that drives every tree:

Session 1: Mon, Sep 21 · Building and folding back a tree (TBD)

🧩 Puzzle

The outbreak stockpile. A county health office must decide how much protective supply to buy before flu season. Stockpile big pays off hugely in a severe season but wastes money in a mild one; stockpile small is steadier; wait & see gambles on a mild season. The catch: nobody knows yet whether the season will be severe or mild. Which choice is best, and how do you lay the whole decision out so a computer (or a graph) can solve it?

Build the skill 1: from a payoff table to a tree

Quick recap from Weeks 3–4: a decision problem is a payoff table (one row per decision, one column per state of nature), and with state probabilities you rank decisions by expected value. Press Expected value below to recall how stockpile-small wins at \$52M (all five no-probability criteria (maximax, maximin, minimax regret, Hurwicz, and equal-likelihood) are Week 3’s topic, recapped briefly here):

Maximax (optimistic). Assume the best state happens; pick the biggest best-case payoff.
Maximin (pessimistic). Assume the worst; pick the best worst-case.
Minimax regret: minimize how much you’d kick yourself afterward.
Hurwicz (coefficient of optimism): a tuned compromise between the pure optimist and the pure pessimist. Pick a dial $\alpha$ between 0 and 1 and score each row as $\alpha\,(\text{best}) + (1-\alpha)\,(\text{worst})$, taking the best and worst payoffs in that row; $\alpha = 1$ is maximax, $\alpha = 0$ is maximin, and anything between blends the two. At a middling $\alpha = 0.4$ the stockpile rows score big $0.4(120) + 0.6(-30) = \$30\text{M}$, small $0.4(70) + 0.6(40) = \$52\text{M}$, wait $0.4(90) + 0.6(-50) = \$6\text{M}$. Stockpile small still wins.
Equal-likelihood (Laplace): if you genuinely cannot rank the states, treat them as equally likely and just average the row. Here that is big $\tfrac{1}{2}(120-30) = \$45\text{M}$, small $\tfrac{1}{2}(70+40) = \$55\text{M}$, wait $\tfrac{1}{2}(-50+90) = \$20\text{M}$, stockpile small again.

Notice they disagree: the optimist stockpiles big, the cautious planner stockpiles small. With no probabilities, “best” genuinely depends on the decision-maker’s stance. (Four of the five no-probability criteria here land on stockpile small (only maximax picks big) but that is not guaranteed; on a different table they can each point at a different row.)

Build the skill 2: add probabilities: expected value

If you can estimate the probability of each state (here, severe 0.4, mild 0.6), you can compute each decision’s expected value (its probability-weighted average payoff) and pick the largest. Press Expected value in the table above: stockpile-small wins with an EV of \$52M. The same logic drawn as a decision tree shows all three options side by side, fold the expected values back from the right:

The tree folds back to stockpile small (\$52M), beating big (\$30M) and wait (\$34M), matching the table.

A square is a decision (you choose); a circle is a probability node, often called a chance node (probabilities choose). Squares take the best branch; circles take the expected value.

Drawing trees with software

Trees get unwieldy fast once a second decision stage appears, and by Week 6 you will be folding back trees with branch probabilities you revised yourself. Two tools do the drawing and the arithmetic for you, and either is fine for this course:

SilverDecisions runs in the browser with nothing to install. Build the tree, type payoffs and probabilities onto the branches, and it folds back and marks the optimal path.
SimpleDecisionTree is an Excel add-in, useful if you would rather keep everything in one workbook. The add-in file is posted in Canvas.

Do the first fold-back by hand anyway. The point of this week is that you can read a tree and know why a branch was pruned, and software that does it invisibly leaves you unable to say why. Week 6 folds a revised tree on top of this one, so the hand method has to be yours before then.

In practice: where do the numbers come from?

A decision model is only as good as its payoffs and probabilities, and in the real world both are estimated:

Probabilities are often subjective. “0.4 chance of a severe season” usually comes from history, expert judgment, or a forecast, not a known truth. Always ask whose estimate it is and how it was made.
Payoffs hide assumptions. A “\$120M benefit” bundles costs, prices, and outcomes that may shift. Run the numbers again with different but plausible values (a quick sensitivity check) before trusting a recommendation.
Missing or biased data. If your history excludes the worst past seasons (they were “unusual” and dropped), your probabilities will understate risk. Garbage in, garbage out.
The criterion is a judgment, not a fact. The same payoff table can be read through maximax, maximin, Hurwicz, equal-likelihood, or expected value, and the recommended choice can legitimately differ from one to the next. The method does not remove the judgment about how much risk you will tolerate, it forces you to state it out loud as a criterion, so others can see exactly which assumption drove the answer.

Back to the puzzle

There is no single “right” choice until you state your stance: an optimist stockpiles big, a worst-case planner stockpiles small, and an expected-value decision-maker (severe 0.4 / mild 0.6) also lands on stockpile small (\$52M). The method didn’t remove the judgment. It made the judgment explicit and defensible, which is the real goal.

In-class group practice (wrap-up)

To close Session 1, turn a small payoff table into a tree and fold it back together, then the closing challenge adds a second decision node, which is where Wednesday picks up. Write your group’s names on the sheet and hand it to your TA before you leave. It counts toward participation.

Download: Week 5 group practice (Word)

Session 2: Wed, Sep 23 · The value of information & sequential decisions (TBD)

🧩 Puzzle

Before committing, the county could commission a seasonal forecast that hints whether the season will be severe or mild. A good forecast isn’t free, and it isn’t perfect. How much is information worth, and when is it worth buying?

Build the skill 1: expected value of perfect information (EVPI)

Start with the dream case: a perfect forecast that always tells the truth. With it, you’d pick the best decision in each state. The gap between that and your best choice without information is the expected value of perfect information (EVPI). The absolute most any forecast could be worth. Press Reveal EVPI in the table:

EVPI here is \$50M, so a forecast costing more than \$50M is never worth it, no matter how good.

Three doors to the same \$50M. That \$50M shows up no matter how you look for it. (1) The table gap: EV-with-perfect-information \$102M minus the best ordinary EV \$52M. (2) The tree gap you are about to see below. (3) The smallest expected opportunity loss (EOL), the probability-weighted regret of each decision. Building the EOL column, the regrets fold to big \$72M, stockpile small \$50M, wait \$68M (for small: $0.4(120-70) + 0.6(90-40)$ uses regret 50 in both states, so EOL $= \$50\text{M}$). The minimum EOL is \$50M for stockpile small. Exactly the EVPI. That identity is not a coincidence: the EOL of the best decision is the EVPI, because the regret you cannot avoid is precisely the value a perfect forecast would hand back to you.

Build the skill 2: a perfect forecast, as a decision tree

EVPI is easiest to see as a sequential decision tree on the very same stockpile problem. Compare two plans: decide now (commit to the EV-best choice, stockpile small) versus buy a perfect forecast and then pick the best response to whichever season it reveals. The fold-back rule never changes: expected-value the circles, best-branch the squares:

“Buy a perfect forecast” rolls back to \$102M versus \$52M for “decide now”, a gap of exactly \$50M, the same EVPI the table gave. That’s the ceiling on what any forecast could be worth. A real forecast is imperfect (it’s sometimes wrong), so it is worth less than the EVPI; computing exactly how much less requires revising the probabilities with the forecast’s track record, a topic for Week 6.

Build the skill 3: a genuinely sequential tree: decide, observe, decide again

The tree above is “information, then a decision.” The skill this chapter is named for goes one layer deeper: decide, watch what the world does, then make a second decision. Here is a small original example with a real second decision node: square, circle, then square again:

The campus food truck. An operator can Pilot small now (a cart with a short lease) or Lease a full kitchen up front. Demand for the semester turns out High or Low, a coin-flip each (p = 0.5). If they pilot and demand is High, a brand-new decision opens up: Expand into a second cart, or Stay small. Net profit (\$000): Lease → High \$90, Low −\$20; Pilot then Low → \$20; Pilot then High → Expand \$80 or Stay \$50.

Fold it back right to left, and the rule applies at both square layers. Start at the deepest decision: if you piloted and saw High demand, you choose the better branch, Expand (\$80) over Stay (\$50), so that node is worth \$80. Now the chance node above it: pilot averages $0.5(80) + 0.5(20) = \$50\text{k}$. The lease chance node averages $0.5(90) + 0.5(-20) = \$35\text{k}$. Back at the first square, Pilot small now (\$50k) beats Lease full kitchen (\$35k). The pilot wins not because it is timid but because it keeps the second decision open. You only commit to expansion after you have seen demand, which is exactly what “sequential” buys you. (One sanity check at every circle: the branch probabilities leaving a chance node must sum to 1. Here $0.5 + 0.5 = 1$, because they cover every way that one uncertain event can resolve.)

The surviving choice at every square is the decision strategy: a complete contingent plan saying what to do now and what to do later for each signal or state that could actually be reached.

In practice: information isn’t free or perfect

Imperfect signals. Real tests and forecasts are wrong sometimes. A model that ignores false positives/ negatives overstates the value of information.
Sampling bias. A survey of people who answer the phone, or sensors only in wealthy neighborhoods, gives a biased read of the “state of nature.” Who’s missing from the data often matters more than who’s in it.
Do not over-test. Gathering information has a cost (money, delay, fatigue). Once EVPI is small, stop. More data will not change the decision.
Expected value is the long-run rule, not always the right one. EV picks the choice that wins on average over many repetitions. But a county that runs the flu decision once a year, and could be wiped out by a single unprepared severe season, may rationally reject the EV-best option to avoid that one catastrophe. That is risk aversion, and it is the same instinct that makes people buy insurance whose expected value is negative. Repeated, survivable decisions lean on EV; one-shot, ruinous ones lean on utility, which weights a disaster far more heavily than its raw probability suggests.

Back to the puzzle

A forecast is worth buying only if its price is below the value it adds. A perfect forecast here is worth \$50M (the EVPI). The ceiling; a realistic, imperfect forecast is worth less. Either way, decision analysis turns “how much is it worth to know more?” into a dollar figure. Week 6 makes the imperfect case precise, Bayesian revision and EVSI.

After-class check

A few questions spanning the whole week: criteria, expected value, trees, and the value of information. Retry as many times as you like; nothing is submitted.

Weekly take-home (graded: submit on Canvas)

Everything for the take-home goes in one Excel workbook, built around a two-stage decision: a hospital system can pilot a telehealth service and decide again once uptake is known, or roll it out system-wide now. No tree is drawn for you. In the Your model block you list the squares and the circle in the order they occur and write the two fold-back rules in your own words. Then you turn a peer system’s launch record into a probability, net the rollout payoffs, and fold the tree back using live formulas that reference your probability cell. Typed-in numbers will not survive the last two blocks. Finally you price the flexibility itself, find the uptake probability at which your advice reverses, and work out what knowing uptake in advance would be worth. Two short responses plus the required AI-use disclosure go on the Free responses sheet. Upload the completed .xlsx on Canvas. That one file is your entire submission.