Decision analysis

Why it matters

Some decisions are made once, under real uncertainty: build a big plant or a small one, launch now or run a pilot first, bid on a contract or not. There is no long-run average to optimize, and nobody knows what demand will be. Decision analysis gives a disciplined way to lay out the choices, the uncertain events and the payoffs, and to compare options fairly. A decision tree is also one of the best tools you have for explaining a recommendation to managers, because they can see every assumption.

Building blocks of a decision tree

• Decision nodesPoints in a tree, drawn as squares, where you choose the best branch. (drawn as squares): you choose one branch.
• Chance nodesPoints in a tree, drawn as circles, where nature picks a branch with probabilities that sum to 1. (drawn as circles): nature chooses a branch, with probabilities that sum to 1.
• End points: the payoff of that path, usually profit or cost.

The tree is drawn left to right in the order things happen, and solved right to left (rolling backSolving a tree from the end points to the first decision, averaging at chance nodes and taking the best at decision nodes.):

• At a chance node, take the expected value: the sum of probability times value.
• At a decision node, take the best branch and note which one it is.

The expected value of an option is often called its EMVExpected monetary value: the probability-weighted average payoff of an option. (expected monetary value).

Worked example: plant size

A company must decide how to serve a new market. Demand will be strong with probability 0.6 or weak with probability 0.4. Profits are in $k.

• Build large: 200 if demand is strong, -60 if weak.
• Build small: if demand is strong, the company can then expand (profit 140) or stay small (profit 90). If demand is weak, profit is 20.
• Wait: profit 0.

[decide] -- large -- (demand) -- 0.6 strong --> 200
         |                   \-- 0.4 weak   --> -60
         |-- small -- (demand) -- 0.6 strong --> [decide] -- expand --> 140
         |                   |                            \-- stay   -->  90
         |                   \-- 0.4 weak   --> 20
         \-- wait --> 0

Roll back from the right.

small, strong demand: best of expand (140) and stay (90) = 140
EMV(small) = 0.6 * 140 + 0.4 * 20 = 84 + 8 = 92
EMV(large) = 0.6 * 200 + 0.4 * (-60) = 120 - 24 = 96
EMV(wait)  = 0

The EMV-best choice is build large (96), with small close behind (92). The plan is a full strategy: "build small, and expand if demand is strong" is how the small option is valued.

Sensitivity: how sure must we be?

Let p be the probability of strong demand.

EMV(large) = 200p - 60(1 - p) = 260p - 60
EMV(small) = 140p + 20(1 - p) = 120p + 20

They are equal when 140p = 80, so p = 0.571. Our estimate of 0.6 is only just above the break-even point. If the true probability is 0.5, small wins (80 versus 70). That is worth telling the decision maker: "the choice hinges on whether you believe strong demand is more than about 57% likely."

The value of information

Suppose a market study could tell us the demand in advance. With perfect information we would pick the best option for each outcome:

strong: max(200, 140, 0) = 200
weak:   max(-60, 20, 0)  = 20
EV with perfect information = 0.6 * 200 + 0.4 * 20 = 128
EVPI = 128 - 96 = 32

The expected value of perfect information (EVPIExpected value of perfect information: the most any study of the uncertainty could be worth.) is 32 ($k): never pay more than that for any study, because no real study is perfect. A real, imperfect study is valued with Bayes' rule (the expected value of sample information), using what you learned about conditional probabilities.

Beyond expected value

EMV assumes the decision maker is risk neutralJudging options only by their expected value, ignoring how bad the worst outcomes are.. The large plant loses 60 with probability 0.4; a small firm might not survive that. Two ways to handle risk:

• Show the risk profile: the distribution of outcomes for each option, not just the average.
• Use a utility function that values losses more heavily than gains, and roll back expected utility instead of expected money.

Probabilities come from data, models (like the simulations in the last lesson) or structured expert judgement. Write down where each one came from.

Common mistakes

• Rolling back in the wrong direction. Always solve from the end points back to the first decision.
• Probabilities that do not sum to 1 at a chance node.
• Double counting costs, for example subtracting the plant cost at the end point and again on the branch.
• Ignoring later decisions. The option to expand is what makes the small plant attractive.
• Reporting only the EMV. Show the break-even probabilities and the worst case too.

Quick recap

• Decision trees use decision nodes (choose the best) and chance nodes (take expected values).
• Roll back from right to left to find the best strategy and its EMV.
• Sensitivity analysis finds the break-even probabilities.
• EVPI caps what information is worth; risk attitude may justify a lower-EMV option.

Try it

In the Do tab, write a recursive function that rolls back a decision tree and picks the best option.