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Branches and Boxes
Quantitative Analysis · Chapter 5: Decision Analysis with ProbabilitiesSection 5.5 · lesson 24 of 5512 min · free with a free account
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In this lesson
How decision trees and payoff tables clarify complex business choices.
- Decide when a payoff table is sufficient and when a decision tree is needed
- Apply dominance analysis and report the breakeven probability
- Draw a tree in correct notation and solve it by rollback
- Compute EVPI and explain the ceiling it sets on information
Transcript
Everything said in the video, section by section (1,827 words). Read along, or search it.
Decision Analysis with Probabilities · Section 5.5
Branches and Boxes
Some decisions stand alone. Pick the supplier with the best price, and you are done. Others unfold in stages. Today's choice changes what tomorrow's options look like, and chance events reshape the playing field. Decision trees and payoff tables are the two complementary tools for those problems. They make a complicated choice visible, traceable, and defensible. In this section we build both of them. We solve them by rollback, and we read off the recommendation that emerges. You will learn when a table is enough, and when only a tree will do. You will use dominance analysis to shrink a problem before optimising it. And you will compute the expected value of perfect information, a hard ceiling on what any study can be worth.
Payoff Tables
Rows You Control, Columns You Do Not
Start with the simplest structured tool there is. The payoff table. It handles choice under uncertainty, and its power comes from one act of discipline. It forces you to separate what you decide from what merely happens to you. The rows are your alternatives. Build a large plant, build a small plant, do nothing. Those are yours to pick. The columns are states of nature. Strong demand, weak demand. Those you do not control at all. Every cell holds one number, the payoff if that alternative meets that state. That is the whole structure. There is nothing more to it. Payoff tables work best for single-stage decisions with a manageable number of options and outcomes. Perhaps five alternatives by five states. Push past that, and the cognitive load starts to outweigh the clarity you bought.
The EMV Criterion
Weight Every Payoff by Its Probability
Now we need a decision rule, and the usual one is expected monetary value. For each alternative, multiply every payoff by the probability of its state. Then sum across the row. That total is the alternative's E M V. Take the plant example, where strong and weak demand are equally likely. The large plant earns two hundred thousand dollars if demand is strong. It loses one hundred and eighty thousand if demand is weak. Half of two hundred, minus half of one hundred and eighty, gives ten thousand dollars. The small plant earns one hundred thousand, or loses twenty thousand, for an expected value of forty thousand. Doing nothing is worth zero, in either state. So under the E M V criterion, the small plant is the rational choice. But notice what bought that win. It gave up the largest upside to escape the largest downside.
Shrink, Then Test
Dominance and the Breakeven Probability
Two habits separate a good payoff-table analysis from a naive one. The first is dominance, and you apply it before you compute anything at all. An alternative is dominated if another is at least as good in every state, and strictly better in at least one. A dominated row can be deleted outright. It will never be the best choice, no matter what the probabilities turn out to be. That shrinks the table and exposes the genuinely competitive options. The second habit is sensitivity, because a recommendation can flip on a small probability change. Raise the chance of strong demand from a half to zero point six. The large plant's expected value climbs to forty-eight thousand dollars. Here is the trap. The small plant's value rises too, to fifty-two thousand, so it has not been overtaken. Set the two expressions equal and the real crossover sits at about zero point six one five. That crossover is the number to report, because it tells the decision-maker how confident the answer is.
Why Trees
When One Decision Leads to Another
Payoff tables collapse cleanly when the decision is a single shot. Many real decisions are not. They are sequential. A first choice produces information, that information changes the probabilities of later events, and a second choice follows. A two-dimensional table simply cannot hold that structure. Try it, and the table becomes unreadable long before it becomes wrong. A decision tree can hold it. The tree lays out every alternative path through the problem, from left to right. Time runs left to right, so decisions and chance events alternate naturally along each path. Each terminal point on the right carries a single number. That number is the payoff for that one particular sequence of choices and events.
Notation
Squares Decide, Circles Happen
The notation is strict, and being strict is exactly the point. A square is a decision node, and its branches are the alternatives the decision-maker chooses among. No probabilities are attached to them. The manager simply picks one branch. A circle is a chance node, and its branches are states of nature, each carrying a probability. Here is the rule that catches most drawing errors. The probabilities leaving any single circle must sum to exactly one. If they do not, a branch is missing, or an outcome has been counted twice. Terminal nodes are the endpoints on the right. Each holds the payoff for the unique path that reaches it, whether that is profit, cost, net present value or utility. And here is the structural trick. A decision node can follow a chance node. That is what lets a tree say, if the experiment is favourable, decide between A and B; if not, decide between C and D.
Rollback
Solve It Backwards
Once the tree is drawn, solving it is mechanical. You move right to left, from the terminal payoffs back toward the root. Each node gets the rule that fits its own type. At a chance node, compute the expected value by weighting each branch by its probability. That single number now replaces the chance node for the rest of the calculation. At a decision node, you choose. Take the highest expected value for a profit problem, the lowest for a cost problem. The winning branch's value becomes the node's value, and the losers are pruned - drawn, but marked as not chosen. Keep going until you reach the root, whose value is the optimal expected outcome. The surviving path from root to terminal is the optimal strategy. Why is working backwards legitimate? Because the best decision at any node depends only on what lies downstream. Not on how you arrived there. Solve the small sub-problems first, and you never enumerate every strategy in the tree.
Worked Example
A Pharma Clinical Trial Decision
Time for the full worked example. A pharmaceutical company has developed a candidate drug. It faces two linked decisions, and the first one changes what the second is worth. Decision one. Run clinical trials, or not? Trials cost five million dollars, but they produce real information about whether the drug actually works. There is a sixty per cent chance the trial result is favourable, forty per cent unfavourable. Decision two. Launch, and if so at what scale? The company can launch nationally, launch regionally, or not launch at all. With favourable evidence, success in the market is virtually assured. Without trials, it is a coin flip. So the root splits two ways, and a favourable trial opens a second decision with three launch options. An unfavourable trial forces a no-launch ending. You cannot launch a failed drug.
Terminal Payoffs
Every Path Ends in One Number
Every terminal is one complete story, so let us walk the endpoints. Start on the run-trials side, where the five million dollar fee is already spent. A favourable result then a national launch pays ninety-five million dollars, capturing the large market. A favourable result with a regional launch pays thirty-five million. Smaller market, smaller reward. If the trials succeed and the firm still walks away, it loses the trial cost alone. Five million. An unfavourable result gives the same minus five million, because the launch is off. Now the skip-trials side, where the firm is launching blind on a coin flip. A national launch pays a hundred million if the drug works, and loses sixty million if it fails. That loss covers a full recall and the reputational damage. A regional launch pays forty million on success, and loses twenty million on failure. Doing nothing pays zero. No launch, no cost, no revenue.
Rollback in Action
Right to Left, One Node at a Time
Now roll the tree back, right to left. Take the two chance nodes on the skip-trials branch first. A blind national launch. Half of a hundred million, minus half of sixty million, is twenty million dollars. A blind regional launch. Half of forty, minus half of twenty. Ten million dollars. Now the decision node sitting above them. Choose the best of twenty million, ten million, and zero. Skipping trials is worth twenty million, and the choice there is a national launch. Move across to the favourable-trial decision node. Choose the best of ninety-five million, thirty-five million, and minus five million. National again. Now the chance node just after the trials. Zero point six times ninety-five, plus zero point four times minus five. Fifty-seven million minus two million, which is fifty-five million dollars. At the root, compare fifty-five million against twenty million, and run the trials. If the result is favourable, launch nationally. If not, do not launch. The information the trial buys is worth far more than the five million it costs.
EVPI
The Ceiling on What Information Can Be Worth
The pharma case proved that imperfect information was worth buying, because a trial shifts probabilities without settling them. So here is the natural follow-up question. What would perfect information be worth? Picture a crystal ball that tells you, before you spend anything, exactly which state will occur. The expected value of perfect information is that ceiling, the upper bound on what any such effort could be worth. No real-world test, study or survey can beat it. Work it out here, using the fifty-fifty success odds from the blind launch. If you knew the drug would work, you would launch nationally with no trials, for a hundred million. If you knew it would fail, you would do nothing, for zero. So the expected value with perfect information is fifty million dollars. Subtract the twenty million you could earn with no information at all. E V P I is thirty million, measured against that no-information baseline. One honest wrinkle before you move on. The trial appears to add thirty-five million, above that ceiling. It resolves once you notice something. Favourable versus unfavourable is not the same split as the drug working or failing.
Cautions
What the Tree Does Not Tell You
Four cautions, and none of them is optional. First, the probabilities are estimates, and the recommendation is only as good as the numbers you assigned. Always test sensitivity to changes of plus or minus ten per cent in each probability. Second, the payoffs may be uncertain too, since terminal values are usually point estimates of whole distributions. Replace them with expected values, or lay a simulation over the top if the spread matters. Third, risk attitude. Expected value treats a fifty million dollar loss and an equal gain as symmetric. For a risk-averse decision-maker, expected utility can give a different answer entirely. Fourth, tree size discipline. Trees grow fast. Prune dominated and clearly inferior branches early. Aim for ten to thirty terminal nodes, not hundreds. To close. Tables for single-stage decisions, trees for multi-stage ones. Squares decide, circles happen, and every circle's branches sum to one. Solve by rollback, averaging at the chance nodes and maximising at the decision nodes.
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