Week 4: Decision Making Under Risk (EV, EOL, EVPI)

This week’s big question: last week you had to choose with no odds to lean on. Now suppose you do know the probabilities of each outcome. How should those odds change your choice, and what would it be worth to know the future for certain?

Before the week

• Read: Taylor, Introduction to Management Science (13e): Chapter 12, “Decision Analysis,” pp. 582–587 (focus: decision making with probabilities: expected value, expected opportunity loss, and EVPI).
• Warm-up self-check, weighted averages and computing a single expected value:

Session 1: Mon, Sep 14 · Expected value & expected opportunity loss (TBD)

🧩 Puzzle

The farmer. Before the season starts, a farmer must commit to one crop: Corn, Soybeans, or Wheat, and then live with whatever weather arrives. Unlike last week, the farmer does have odds: the regional forecast puts the season at Wet with probability 0.4 and Dry with probability 0.6. Net profit (in \$ thousands) depends on both choices:

Crop	Wet (0.4)	Dry (0.6)
Corn	50	10
Soybeans	40	25
Wheat	20	35

Corn loves a wet year and suffers in a dry one; wheat is the reverse; soybeans are steady. With these odds in hand, which crop should the farmer plant?

Build the skill 1: average each row with the probabilities (expected value)

Two ground rules the table quietly relies on. The states of nature (Wet, Dry) must be mutually exclusive and exhaustive: exactly one happens, and their probabilities sum to 1 (here \(0.4 + 0.6 = 1\)). Known state probabilities put this problem in decision-making under risk; under uncertainty, Week 3 had the same payoff-table structure but no defensible probabilities. And before you compute anything, run the Week 3 dominance screen. The farmer’s three crops are not dominated (corn wins the wet column, wheat wins the dry column, soybeans wins neither outright) which is precisely why the odds have to do the deciding.

When you know the probability of each state of nature, you no longer have to fall back on optimist/pessimist criteria. You can average each row, weighting every payoff by how likely its state is. When the payoffs are dollars, the textbook calls that row average its expected monetary value (EMV), the decision-specific form of expected value:

\(\text{EV(decision)} = \sum_{\text{states}} P(\text{state}) \times \text{payoff}\).

It’s the long-run average profit you’d earn if you faced this exact decision many times. The expected value criterion simply picks the decision with the highest EV. Click Expected value below and watch each row’s weighted average appear. The bar that wins is the one with the best long-run payoff:

(The widget labels the unit “\$”, but every payoff here is in thousands of dollars, so an EV of 31 means \$31,000.) Once you’ve seen the EVs, click the widget’s “Reveal EVPI” button: it averages the best-per-state payoffs (\(0.4 \times 50 + 0.6 \times 35 = 41\)) and reports \(\text{EVPI} = 41 - 31 = 10\). The exact ceiling we’ll derive by hand in Session 2. Worth a peek now so the idea isn’t a surprise on Wednesday.

Build the skill 2: the same answer through regret (expected opportunity loss)

There’s a second lens that always agrees with EV but tells you how much you’d kick yourself. For each state, find the best payoff anyone could have earned in that column; a cell’s opportunity loss (regret) is that best minus the cell’s own payoff. Averaging a row’s regrets by the same probabilities gives its expected opportunity loss:

\(\text{EOL(decision)} = \sum_{\text{states}} P(\text{state}) \times \big(\text{best in column} - \text{payoff}\big)\).

The best-in-column payoffs are Wet → 50 and Dry → 35, so subtract each cell from its column’s best to build the full regret table, then weight each row by the same odds:

The smallest expected regret is Soybeans at 10. The very same crop that maximized EV. That’s not a coincidence: minimizing EOL always selects the same decision that maximizes EV. (Watch the columns: corn has zero regret when it’s wet but the worst regret when it’s dry; the EV winner is whichever row keeps its weighted regret smallest.) Seeing all three EOLs side by side proves the agreement instead of just asserting it, and the winning EOL, 10, is the number that will reappear in Session 2 as the value of perfect information.

In practice: EV is a long-run average, and the odds are estimates

• EV assumes the “long run.” The expected value is the average over many repetitions. A farmer plants once a year, so a 0.6-probability dry year can still ruin a single season even when EV says soybeans. When the decision is one-shot and a bad outcome is catastrophic, weigh risk, not just the average. Concrete case: a small bakery can pay \$400 for a one-day event-cancellation rider, against a ~1% chance a storm wipes out a \$9,000 booth investment. By the average, you should skip it. The expected loss avoided is only \(0.01 \times 9{,}000 = \$90\), far below the \$400 premium. Yet an owner who cannot absorb a one-shot \$9,000 hit rationally buys anyway: they are willing to give up a little expected value to avoid a rare disaster. That instinct has a name, risk-averse vs. risk-seeking behavior, and it’s exactly the gap between expected value and utility that Week 6 picks up.
• Where do the probabilities come from? “Wet 0.4, Dry 0.6” is itself a forecast, from climate records, a weather service, or a hunch. Garbage odds make a confident-looking EV that’s still wrong. Always ask who estimated them and how.
• Run a sensitivity check. Before betting the farm, nudge the probabilities and see whether the best crop flips. If soybeans win across a wide band of “wet” probabilities, the recommendation is robust; if a small change tips it to corn, the choice hinges on a number you only guessed.

Back to the puzzle

Averaging each row with the odds (in \$ thousands):

• EV(Corn) \(= 0.4(50) + 0.6(10) = 20 + 6 = \mathbf{26}.\)
• EV(Soybeans) \(= 0.4(40) + 0.6(25) = 16 + 15 = \mathbf{31}.\)
• EV(Wheat) \(= 0.4(20) + 0.6(35) = 8 + 21 = \mathbf{29}.\)

The highest expected value is Soybeans at \$31,000, so the EV criterion plants soybeans. Notice the odds did the work: corn’s huge wet-year payoff isn’t enough when the season is more likely to be dry. (And EOL agrees. Soybeans also have the smallest expected regret, \$10,000.)

In-class group practice (wrap-up)

To close the session, work a fresh under-risk problem by hand in your group. Write your group’s names on the sheet and hand it to your TA before you leave. It counts toward participation.

• Download: Week 4 group practice (Word)

Session 2: Wed, Sep 16 · Expected value of perfect information (EVPI) (TBD)

🧩 Puzzle

Same farmer, same odds. A consultant offers a long-range forecast that would tell the farmer, before planting. Exactly whether the season will be Wet or Dry. With that certainty, the farmer would always plant the best crop for the actual weather. How much is that perfect forecast worth? It can’t be more than the profit you’re already leaving on the table, but how do you put a dollar figure on it?

Build the skill 1: what perfect information would be worth

Imagine a perfect oracle. In a Wet year (probability 0.4) the farmer would plant the wet-year winner, Corn, for 50; in a Dry year (probability 0.6) they’d plant the dry-year winner, Wheat, for 35. Averaging those best-per-state payoffs by the odds gives the expected value with perfect information:

This is not a decision you can actually make today (you can’t plant corn and wheat). It’s the average profit you’d earn if you could always react correctly. The expected value of perfect information is how much that beats your best blind choice:

\(\text{EVPI} = \text{EV\,with\,PI} - \text{best EV without information} = 41 - 31 = \mathbf{10}\).

So a perfect forecast is worth \$10,000, and that’s the ceiling on what any study, survey, or consultant is worth, because no real information is better than perfect.

Two routes, one number. There are two ways to reach the \$10,000, and they always land on the same value:

• Route A (information): \(\text{EV with PI} - \text{best EV} = 41 - 31 = 10.\)
• Route B (regret): the smallest EOL from the regret table (soybeans’ \$10,000) is the answer.

That identity, \(\text{EVPI} = \text{EOL of the best decision}\), holds for every decision problem. The intuition is short: the leftover regret you can’t dodge with your best blind choice is exactly the profit a perfect forecast would recover for you. So if you’ve already built the EOL table, you’ve already computed EVPI, no second pass needed.

Build the skill 2: folding back a decision tree

A decision tree draws the same problem left to right. Three pieces of canonical vocabulary, and you have the whole language:

• a square decision node = a choice you control (which crop to plant);
• a round probability node (often called a chance node) = an outcome nature controls (the weather, with the given probabilities);
• a branch = one alternative (out of a square) or one state (out of a circle), ending in a leaf payoff.

You solve it by folding back from the right: at each circle, average the branches by their probabilities; at each square, keep the best branch (and prune the rest). Watch it roll up:

The three chance nodes fold back to 26, 31, and 29 (exactly the EVs from Session 1) and the decision node keeps the best, Soybeans = 31. For this one-shot problem a tree is just the payoff table redrawn as a flow.

Why bother, then? Because trees do the one thing a flat table cannot: model a decide → observe → decide again sequence, where a second decision depends on an outcome you can’t see yet when you make the first. That structure won’t fit in a single column of a payoff table, which is exactly why decision trees exist. Week 5 is built around it: we’ll fold back genuinely sequential trees there, keeping a later decision open until the world reveals which branch you’re on. For now the takeaway is just that the tree above and the payoff table are the same object, and the tree is the version that keeps working once the decisions come in stages.

In practice: EVPI is a budget, not a guarantee

• EVPI is an upper bound on data. Real forecasts are never perfect, so the value of an actual study is less than EVPI. If a consultant charges \$12,000 for an imperfect seasonal forecast, decline it on sight. Even a flawless one is worth only \$10,000 here.
• The bound rides on the same shaky odds. EVPI was computed from “Wet 0.4, Dry 0.6.” Change those probabilities and the value of information moves with them. If the odds are 50–50, the answer is different, so a sensitivity check matters here too.
• Sometimes the answer is “don’t pay for data.” A small EVPI means information can’t change your decision enough to matter; you’re better off acting now. EVPI tells you when to stop studying and start planting.

Back to the puzzle

Perfect information would lift the farmer’s expected profit from \$31,000 (blind, best EV) to \$41,000 (always right), a gain of EVPI = \$10,000. Read it as a procurement budget for information: \$10,000 is the most the farmer should ever hand any consultant, lab, or survey, because nothing beats a perfect forecast. Since real forecasts are imperfect, the sensible offer is well under \$10,000. Without paying anyone, the tree still folds back to Soybeans (\$31,000) as the best blind choice.

Where this heads next. Real studies are never perfect, so the value of an actual forecast, the expected value of sample information (EVSI). Is always less than EVPI. EVPI is the ceiling; later in the course we’ll ask what imperfect information is worth and how to update odds once a (fallible) signal arrives.

After-class check

Questions spanning expected value, expected opportunity loss, the expected value of perfect information, and reading a folded-back decision tree, including a numeric EVPI. Retry freely; nothing is submitted.

Weekly take-home (graded: submit on Canvas)

Everything for the take-home goes in one Excel workbook, and the payoff table is yours to build. The model sheet opens with a case brief: a university IT department buying exam-season loaner laptops, priced by the pallet, with a credit for every machine that goes unloaned and an expensive emergency rental for every request it cannot fill. Everything is a cost, so lower is better. In the Your model block you name the acts and states and write out how one cost cell combines those three pieces; then you build the table, screen it for a dominated option, and rank the options on expected cost and on expected opportunity loss before putting a ceiling on what a demand survey could 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.