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, pp. 587–596. Nine pages, and the middle three are software this course does not use. The chapter PDF is in Canvas Files.

Stop where “Sequential Decision Tree Analysis with Excel QM” begins on p. 596. Read the chapter for the method, and use the course’s tree builder instead of the chapter’s software.

Then the warm-up self-check, on expected value, the engine that drives every tree:

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

🧩 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 as a picture that a board can follow and a computer can solve?

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

Start with the county’s payoff table. Net benefit (\$M) for the three choices under the two seasons, with the epidemiologists’ odds of Severe 0.4 / Mild 0.6 along the top:

Expected value weights each payoff by its state’s probability and adds the products. The table opens on that rule; here is the arithmetic, in \$M:

Stockpile small has the highest expected value, \$52M. Unlike maximax (best possible payoff), maximin (best floor), and minimax regret (smallest worst-case regret), EV requires probabilities. The tree will make that weighting visible on its branches.

A decision tree draws the same problem left to right, in the order things happen, and it has exactly three kinds of parts:

• a square is a decision node: a fork you control (which stockpile to buy);
• a circle is a probability node, which is Taylor’s term; most people say chance node. It is a fork nature controls (the season), with a probability written on every branch leaving it;
• a branch is one alternative out of a square or one state out of a circle, and a branch that goes no further ends in a leaf carrying its payoff. On paper a leaf is simply the end of the line; the widgets below and the tree builder mark it with a small triangle.

Here is the stockpile table drawn that way. The square comes first because it is decided first, the circles next because the season is learned after the order is placed, and every leaf is a cell of the table:

One square, three branches, a circle on each, two branches out of every circle, six leaves. For a one-shot decision that is all a tree is: the payoff table redrawn as a flow, cell for cell.

Three drawing habits keep a tree honest. The practice sheet checks the first two, and the take-home’s first blocks check all three. Time runs left to right, so the first thing decided is the leftmost square and anything learned later sits to its right. Every branch out of a circle carries its probability, and the branches out of one circle sum to 1 (\(0.4 + 0.6\) here), because they cover every way that one uncertain event can resolve. And a cost sits on the branch where it is paid, in parentheses. Taylor writes the price of each investment on its branch and subtracts it once, at that branch; netting it out of every leaf below gives the same answer, but a cost shared by several leaves is then easy to count twice or not at all. A tree that breaks one of these folds back to the wrong number, and looks fine while it does.

Build the skill 2: fold it back, and read the answer off it

You solve a tree from the right, working backward toward the first square, with one rule for each shape:

• at a circle, take the probability-weighted average of its branches, their expected value, because you do not choose which branch happens;
• at a square, keep the best branch and prune the rest, because you do choose, and you will never take a worse branch when a better one is open.

Press Fold back above and watch the order. Each circle weights its severe-season payoff by 0.4 and its mild-season payoff by 0.6, then adds the two contributions. All values below are in \$ millions:

These are the three circles’ expected values: \$30M, \$52M and \$34M. Only then does the square keep its best branch, stockpile small, pruning the other two. If you would rather do each layer’s arithmetic yourself before the widget shows it, Step one layer advances one layer per press. Stockpile big holds the only 120 on the table and still folds to 30: the mild-season loss takes 18 of the severe branch’s 48 back. The tree agrees with the table, and for a one-shot decision it always will. The commonest mistake of the week is averaging at a square: that prices you as though you chose at random, and a board never does.

The surviving branch at every square, read left to right, is the decision strategy: the plan the tree recommends. For this one-square tree the strategy is one word, stockpile small. Wednesday makes it a sentence with an “if” in it.

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. The course’s own tree builder does the drawing and the arithmetic for you, in the browser with nothing to install. Build any tree, type payoffs, probabilities and costs onto its branches, and it folds back, marks the branches that survive, and names the strategy. Its Check tab hides the values and marks your own instead, and a named input such as \(p\) shows where the best first move changes, which is Wednesday’s flip probability.

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. This week the builder checks work you have done yourself: Monday’s group practice runs your hand fold through its Check tab, and the take-home has you rebuild your Excel tree in it and upload its picture with the workbook.

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.
• A tree hides nothing. Every assumption is a labelled branch: the probability you assumed, the payoff you estimated, the option you left out. A board can argue with a tree in a way it cannot argue with a cell formula, which is why the drawing is worth doing even when the arithmetic is easy.

Back to the puzzle

Stockpile small, worth \$52M, and the picture says so as plainly as the table did: three circles roll up to 30, 52 and 34, and the square keeps the middle branch. What the tree adds is not a different answer but room. The moment the county wants to decide something after the season shows its hand, the table has nowhere to put that second choice and the tree simply grows another square.

In-class group practice (wrap-up)

To close Session 1, turn a small payoff table into a tree and fold it back together by hand, then check your fold in the Check tab of the tree builder on one laptop. The closing challenge adds a second decision node, which you fold by hand and then build in the builder, and it is where Wednesday picks up.

Write every member’s name on the sheet. You have 15 minutes for the practice, then two minutes to photograph your group’s own attempt before the solutions are shown. One member uploads the photo to Canvas by the end of class; participation credit is for attempting the work.

The instructor uses the final eight minutes to demonstrate the handout’s method, check the main results with units, and explain a common mistake. This demonstration closes the session.

• Download: Week 5 group practice (Word)

Session 2: Wed, Sep 23 · Sequential decisions: decide, observe, decide again

🧩 Puzzle

The county does not have to commit everything blind. It could buy a seasonal forecast before choosing, or buy small now and keep the right to top up once the season shows its hand. What is a forecast worth, what is the right to decide later worth, and how do you draw either one?

Build the skill 1: a perfect forecast, as a decision tree

Start with the dream case: a perfect forecast that always tells the truth. Compare two plans on the very same stockpile problem: 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 rules never change: average the circles, best-branch the squares:

The two squares on the forecast branch choose first, because once the forecast is in, the county knows which season it is in; only then can the circle above them be averaged. Here is every number on the tree, from the right (\$ millions):

“Buy a perfect forecast” rolls back to \$102M against \$52M for “decide now,” a gap of exactly \$50M. That gap is the expected value of perfect information (EVPI), which you can check two other ways: EV with perfect information minus the best EV (\(102 - 52\)), and the smallest expected opportunity loss: regret is the best payoff in a state minus the payoff chosen. Stockpile small regrets \(120 - 70 = 50\) if Severe and \(90 - 40 = 50\) if Mild, so its expected regret is \(0.4(50) + 0.6(50) = 50\). Three doors, one \$50M, because they are one idea: the profit a blind chooser leaves on the table. It is the ceiling on what any forecast could be worth to a county that has to commit blind. A real forecast is sometimes wrong, so it is worth less; computing how much less needs the forecast’s track record, which is Week 6.

Notice the tree’s new shape. The first square is followed by a circle (what the forecast says) and then by squares again. A square sitting to the right of a circle is what “deciding after you know” looks like on paper, and it is the shape the rest of this session is about.

Build the skill 2: decide, observe, decide again

The perfect-forecast tree was “information, then a decision.” The county’s other plan, buy small now and keep the right to top up once the season shows, is that shape with no forecast in it. The first square keeps Monday’s three options; the season’s circle follows each one, Severe 0.4 and Mild 0.6; and on the buy-small arm a second square follows each season: top up or hold. So the tree runs square, circle, square: decide, watch what the world does, then make a second decision.

Every leaf Monday priced keeps its number: stockpile big now is 120 and −30, wait & see is −50 and 90, and holding is small’s 70 and 40. The supplier will also sell mid-season, at an emergency price. Topping up after a Severe read lands the county at 100, twenty short of the 120 it would have had by stockpiling big in the first place. Topping up after a Mild read lands it at 20, below the 40 it earns by holding, because it has paid the emergency price for supply the season never needed:

After buying small, each season offers a choice between two numbers rather than a single payoff, and a payoff table has no way to say so. That is the whole reason to draw the tree:

Fold it back right to left, and the rules apply at both square layers. The deep squares go first, because the buy-small circle cannot be averaged until you know what the county would do after each season. Press Fold back and the widget takes the same three layers, in this order (\$ millions):

The buy-small circle averages what the two deep squares chose, 100 and 40, never the four leaves behind them. Big and wait fold exactly as they did on Monday. Back at the first square, buy small now (\$64M) beats waiting (\$34M) and stockpiling big (\$30M).

The answer is not the words “buy small.” The surviving branch at every square, read left to right, is the decision strategy, a complete contingent plan: buy small now; if the season turns severe, top up; if it is mild, hold. A strategy names what to do now and what to do later for each outcome that could actually be reached, and a tree with a second square is answered by nothing less. It is worth \$64M, against the \$52M that Monday’s blind commitment was worth.

Build the skill 3: what the right to decide later is worth

Buying small now beats waiting by \$30M and stockpiling big by \$34M, but neither gap prices the top-up. Each one mixes two things: the right to choose later, and the difference between two supply plans. To price the choice on its own, change one thing only, whether the county may still choose, and keep everything else about buying small exactly as it is.

Suppose the county had promised the supplier up front that it would top up whichever season arrived. The Severe square still takes 100, because it would have topped up anyway, but the Mild square is stuck with 20 instead of the 40 it would have chosen, and the arm folds to \(0.4(100) + 0.6(20) = 40 + 12 = \$52\text{M}\). Promise the opposite, never to top up, and the Severe square is stuck with 70 instead of 100, and the arm folds to \(0.4(70) + 0.6(40) = 28 + 24 = \$52\text{M}\), which is Monday’s plain stockpile small. Either promise costs the county the same, so the value of keeping the second decision open reads the same from either side:

Here are the three ways to buy small side by side. The top arm keeps the second decision open; the other two have promised it away, so their branches run straight to the leaves they promised. Press Fold back and read the three circles, 64 against 52 and 52. The square keeps the open plan, and the 12 it wins by is the value of the choice:

The \$12M is what the county really buys by keeping the choice: not information for its own sake, but the right to change the plan once the season is known. Notice what it is not. It is not the \$34M between buying small with the option (64) and stockpiling big (30), because that gap also carries the difference between two supply plans. And it is not the \$50M a perfect forecast was worth, which buys the county its knowledge before it has to commit. \$12M buys something cheaper and more ordinary: the right to still be choosing once the season has shown its hand.

Beyond the book. Taylor folds sequential trees back and reads the strategy off them (pp. 593–595) but does not put a separate price on the later decision, or find the probability at which the first move flips. Both are standard decision-analysis extensions and both are asked for in this week’s take-home; the fold-back arithmetic underneath them is exactly Taylor’s.

Build the skill 4: how sure would you have to be, and what would knowing be worth

The 0.4 is the shakiest number on the tree, so vary it to see when the best plan changes. The tree is the county’s top-up tree from Build the skill 2, unchanged: buy small now, stockpile big now, or wait & see, with a second square after buying small. Once its deep squares have chosen, top up after a Severe season (100 beats 70) and hold after a Mild one (40 beats 20), each first move pays one number per season. These are the payoffs every calculation below uses, in \$ millions:

Call the probability of a severe season \(p\), so a mild one has \(1 - p\). The deep squares’ choices do not depend on \(p\) (100 beats 70 and 40 beats 20 whatever it is), so weighting each row by \(p\) and \(1 - p\) makes each first move a straight line in \(p\):

Buying small stays on top until one of the other two lines catches it. Waiting catches it where \(90 - 140p = 40 + 60p\), at \(p = 50/200 = 0.25\), where both are worth \$55M. Stockpiling big catches it where \(-30 + 150p = 40 + 60p\), at \(p = 70/90 = 7/9\), about 0.78, where both are worth about \$86.7M. Between 0.25 and 0.78, buy small and keep the top-up; below 0.25, wait; above 0.78, stockpile big. The epidemiologists’ 0.4 sits inside that band, 0.15 above its lower edge and about 0.38 below its upper one. So a board member who says “stockpile big, the season will be severe” is claiming the chance of a severe season is above seven in nine, and one who says “wait, it will be mild” is claiming it is below one in four. Those are the numbers each has to defend. The flip probability turns a disagreement about attitude into a disagreement about one estimate, which is a far easier argument to settle.

Once the deep squares have chosen, the first decision is a three-row payoff table again, with buy small’s row (“Small, then choose”) being the strategy’s 100 and 40, so you can compare its expected value as the odds change. The widget below opens on Move the odds with that table: the three first moves are lines in \(p\), they cross at 0.25 and 0.78, and the dot moves from wait to buy small to big as you drag across them. The dashed line above them is the oracle’s, and the bracket under it at the slider is the number the next paragraph works out:

What would knowing the season in advance be worth now? Draw the perfect-forecast shape from Build the skill 1 again, with today’s plan on its top arm: “buy small, then choose” is worth 64. “Know the season first” is a circle leading to two squares, and at each one you revisit the first decision under that outcome, because knowing changes what you would do, not just what you would earn. Knowing the season is severe, the county stockpiles big for 120 (buying small and topping up would get only 100); knowing it is mild, it waits for 90 (buying small and holding would get only 40). On a payoff table you averaged each column’s best cell; here you average what the two squares chose:

That is Build the skill 1’s 102 again, because a county that knows the season in advance never needs the top-up. What changes is the baseline. EVPI is measured against the best plan without information, and that is now the top-up plan at 64, not Monday’s blind 52:

The right to top up has already captured \$12M of the \$50M that knowing the season was worth to a county committing blind, so a forecast can add at most the other \$38M: \(50 = 12 + 38\). Press Fold back on the tree below: the four deep squares choose first (top up and hold on the top arm, big and wait on the bottom one), then the circles average to 64 and 102, and the first square keeps knowing first:

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. Information is worth paying for only if it could change what you do; once EVPI is small it cannot, so stop. Gathering more costs money, delay, and fatigue.
• 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 and which Week 6 measures.

Back to the puzzle

Both questions get a number, and they are different numbers. The right to top up later is worth \$12M: buying small while that second decision stays open is worth \$64M, and either promise that closes it, always top up or never, drops the plan back to Monday’s \$52M. A perfect forecast is worth \$50M to a county that must commit blind, but only \$38M to one that already holds the right to top up, because the top-up has bought \$12M of what knowing would; a realistic, imperfect forecast is worth less still, and Week 6 makes that precise with Bayesian revision and EVSI. Monday’s practice sheet already showed you the mechanism, since adding a second square to the trial-weekend branch lifted its value without changing a single probability or payoff. So the county buys small now, tops up if the season turns severe, holds if it does not, and declines any forecast priced above \$38M. The plan stands as long as the chance of a severe season stays between 0.25 and about 0.78. Decision analysis turned “how much is it worth to know more, or to keep our options open?” into dollar figures on one tree.

After-class check

A few questions spanning the whole week: reading and folding a tree, the strategy, the value of perfect information on a tree, the option value, and the flip probability. Retry as many times as you like; nothing is submitted.

Weekly take-home (graded: submit on Canvas)

The take-home is one Excel workbook and one picture of your tree, 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. Sketch it first, then in the Your model block list the squares and the circles in the order they occur, write the two fold-back rules in your own words, and name the one figure in the brief the tree never uses. 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 block that moves that probability. Next, build the same tree in the tree builder, with the pilot’s cost on its branch as a negative number, press Fold back, and check that its values agree with your workbook’s. File, then Download a picture (PNG), saves the picture you submit; a screenshot of the whole folded tree works too. Then you price the flexibility itself, find the uptake probability at which your advice reverses, work out what knowing uptake in advance would be worth, and state the strategy: what to do now, and after each read. Two short responses plus the required AI-use disclosure go on the Free responses sheet. Upload two files on Canvas: the completed .xlsx and the picture of your folded tree.