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

This week’s big question: suppose you know the probabilities of each outcome. How should those odds change your choice, what would it be worth to know the future for certain, and how far can you trust the odds themselves?

Before the week

Read: Taylor, Introduction to Management Science (13e), Chapter 12, pp. 582–587. Six pages, but two stretches inside them are not yours this week, so the real load is about four. The chapter PDF is in Canvas Files.

Stop where “Decision Trees” begins partway down p. 587. The Excel screenshot above that heading is the last of the software; everything after it is Week 5.

Two things to know before you open it:

• Read for the pattern, not the numbers. Taylor compares a real-estate investor’s apartment, office building, and warehouse using state probabilities. Our example compares a farmer’s crops. All three of this week’s tools are worked on that one investor table, so what carries over is the sequence of moves, not the figures.
• The reading covers Monday. Wednesday is past it. Each tool here is computed once, at one set of odds: the word “sensitivity” does not appear anywhere in the 61 pages of this chapter, and nothing in it asks how far the probabilities could shift before the answer changes. That question is Wednesday’s, and so is what happens to the value of information as they shift. The chapter also gives cost tables a single sentence, on p. 583; Wednesday spends a block on them, because your take-home is one.

Then the warm-up self-check, on weighted averages, the rule that state probabilities sum to 1, and a single expected value:

Session 1: Mon, Sep 14 · Expected value, expected opportunity loss & the value of perfect information

🧩 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. The regional forecast puts the season at Wet with probability 0.4 and Dry with probability 0.6. Net profit (in \$ thousands, so 50 means \$50,000) 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? And what would it be worth to know the weather before planting?

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, there are no defensible probabilities. Before computing, screen for dominance: remove a choice if another is at least as good in every state and better in at least one. 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 or pessimist criteria. You can average each row, weighting every payoff by how likely its state is. Taylor calls that row average the decision’s expected value; you will also see it written expected monetary value (EMV) when the payoffs are dollars, and the two names mean the same thing:

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

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. Three rows, three weighted averages:

• 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) \(= 0.4(20) + 0.6(35) = 8 + 21 = \mathbf{29}.\)

Soybeans wins at 31, meaning \$31,000 of expected profit. Corn’s 50 is the biggest number on the table and it still loses, because the season is more likely to be dry. No season ever pays 31, either: soybeans return 40 or 25, and 31 is what they average over many seasons with these odds. That 31 is the number every other tool on this page is measured against. Click Expected value in the widget below: a fourth column appears carrying 26, 31 and 29, one per crop, with soybeans marked. Check all three against the arithmetic above, not just the winner:

The widget prints the table’s own numbers, in \$ thousands. Leave its Reveal EVPI button alone for the moment; Build the skill 3 earns it.

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. Add each row’s two numbers and you can see why: corn \(26 + 15 = 41\), soybeans \(31 + 10 = 41\), wheat \(29 + 12 = 41\). Every row’s EV and EOL sum to the same 41, so ranking rows by smallest EOL is ranking them by largest EV, read upside down. (Watch the columns, too: 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.) Hold on to that constant 41 and the winning EOL of 10. Both are about to reappear.

The bars below say the same thing without the arithmetic. Each crop is one bar. Press Next once for its expected value, and again to stack its expected regret on the end: the longer the value, the shorter the regret, and every bar ends on the same line. The third press names that line, which is the next section’s job:

Build the skill 3: what perfect information would be worth (EVPI)

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:

There is the 41 from the regret table. It 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:

\(\text{EVPI} = \text{EV with PI} - \text{best EV without information}\).

The second term is the best row average you computed in Build the skill 1: soybeans’ 31. So \(\text{EVPI} = 41 - 31 = \mathbf{10}\).

That \$10,000 is what a perfect forecast is worth, and it is the ceiling on what any study, survey, or consultant can be worth, because no real information beats perfect information. Now click Reveal EVPI in the widget above: it highlights the best cell in each column and reports 41, 31, and the gap of 10.

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.

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

• A single play is not the long run. The expected value is the average over many repetitions, and a farmer plants once a year, so a 0.6-probability dry year can still ruin a single season even when EV says soybeans. Look at the floors: corn’s dry year pays 10 and soybeans’ pays 25. Had the forecast said Wet 0.7 instead, corn’s average would win, \(0.7(50) + 0.3(10) = 38\) against soybeans’ \(0.7(40) + 0.3(25) = 35.5\), and corn’s floor would still be 10. A farmer who cannot get through a 10 season plants soybeans anyway, giving up expected value for a floor. That instinct has a name, risk aversion, and it is exactly the gap between expected value and utility that Week 6 measures.
• 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.
• Nudge them before you bet the farm. Session 2 moves the wet probability up and down and finds the range over which soybeans stay the pick. If the answer survives a wide band of odds, the recommendation is robust; if a small change tips it, the choice hinges on a number you only guessed.

Back to the puzzle

Plant soybeans. Weighting each row by the odds put it on top at \$31,000 against corn’s \$26,000 and wheat’s \$29,000, and the regret table agreed from the other side: soybeans also carry the smallest expected regret, \$10,000. That same \$10,000 answers the second question. Knowing the weather in advance would lift the farmer’s expected profit from \$31,000 to \$41,000, so a perfect forecast is worth \$10,000 and nothing less than perfect can be worth more. Wednesday asks how much of this survives once you stop trusting the 0.4.

In-class group practice (wrap-up)

To close the session, work a fresh under-risk problem by hand in your group: expected value, the regret table with its EOL column, and the value of perfect information, all on one table.

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 4 group practice (Word)

Session 2: Wed, Sep 16 · How far to trust the odds, and what information is worth

🧩 Puzzle

Same farmer, same table. Three questions arrive before planting. The county extension agent asks how sure the farmer is about that 0.4: it came from thirty years of rainfall records, and the last decade has run wetter than the rest. A consultant offers a seasonal forecast for \$12,000, describing it as “highly accurate.” And the propane dealer needs the grain dryer’s fuel ordered this week, at the pre-season price and before any forecast could arrive: 3 loads or 5? Does Monday’s answer survive a different set of odds, is the forecast worth even a meeting, and how much propane should the farmer order?

Build the skill 1: nudge the odds and find the band where the answer holds

Call the wet probability \(p\), so the dry probability is \(1 - p\). Each crop’s expected value is then a straight line in \(p\): at \(p = 0\) it is the crop’s dry payoff, at \(p = 1\) its wet payoff, and in between it runs straight from one to the other. Written out:

• Corn: \(50p + 10(1-p) = 10 + 40p\), the steepest line, because corn’s wet and dry payoffs are furthest apart.
• Soybeans: \(40p + 25(1-p) = 25 + 15p\), a gentle rise.
• Wheat: \(20p + 35(1-p) = 35 - 15p\), the one line that falls as the season gets wetter.

Three straight lines on one pair of axes, so the pick can only change where two lines cross. Soybeans overtake wheat where \(25 + 15p = 35 - 15p\), at \(p = 1/3\) (both worth 30); corn overtakes soybeans where \(10 + 40p = 25 + 15p\), at \(p = 0.6\) (both worth 34). So: below one-third, plant wheat; between one-third and 0.6, plant soybeans; above 0.6, plant corn.

Now read the forecast against that band. The agent’s 0.4 sits inside it, so soybeans stand, but it sits closer to the wheat edge (about 0.07 away) than to the corn edge (0.2 away). A wetter decade can push the odds up a long way without changing the advice; a drier one flips it to wheat quickly. That asymmetry is the useful finding of a sensitivity check: not “soybeans, yes,” but “soybeans, unless the wet probability drops below a third.” A sensitivity check reports a band and a side to watch, never a bare yes.

The value of information moves with the odds too, and it has a line of its own. With a perfect forecast the farmer earns the best payoff in each state, so the oracle’s average is \(50p + 35(1-p) = 35 + 15p\): wheat’s dry 35 at \(p = 0\), corn’s wet 50 at \(p = 1\), and above every crop in between. EVPI at any \(p\) is the gap between that line and whichever crop is on top. Read it at a few wet probabilities:

Inside the soybean band EVPI is flat at 10, and the lines say why: soybeans’ \(25 + 15p\) climbs 15 per unit of \(p\), exactly as the oracle’s \(35 + 15p\) does, and a gap between parallel lines never changes. (Monday’s regret table says the same thing from the other side: soybeans regret 10 in both states, so their EOL is 10 whatever the odds.) Outside the band the ceiling falls, because the blind pick is now the crop that is right in the likelier state and the oracle has less left to fix. Each outer line meets the oracle’s at its own corner, wheat at \(p = 0\) (both 35) and corn at \(p = 1\) (both 50), so the gap closes as the odds lean. Quote EVPI with its odds: at 0.7 the ceiling is \$7,500, not \$10,000.

The widget below opens on Move the odds, so just drag the slider. It draws the three lines, marks both crossings (it rounds one-third to 0.33), and puts a dot on whichever crop is on top; the EV column re-computes at every \(p\). The dashed line above the three is the oracle’s, \(35 + 15p\), and the bracket under it at the slider is the EVPI. Drag through the band and the bracket stays at 10 while the dot climbs soybeans’ line; drag down to 0.33 and wheat has taken over; drag past 0.6 and corn does. At \(p = 0.7\) corn is worth 38, soybeans have risen from 31 to 35.5 and wheat has fallen from 29 to 24.5, and the bracket has shrunk to 7.5. The readout prices the crystal ball at every stop you make, exactly the rows of the table:

Build the skill 2: 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. The consultant wants \$12,000 for an imperfect seasonal forecast; decline it on sight, without asking a single question about accuracy. Even a flawless forecast is worth only \$10,000 to the planting decision, the only one it arrives in time to change, so no accuracy claim can lift an imperfect one above it. Had the price been \$3,000, the accuracy question would become the live one, and Week 6 is where it gets answered.
• Sometimes the answer is “don’t pay for data.” A small EVPI means information cannot change your decision enough to matter, and gathering it costs money and time (a planting window closes). Information is worth something only if it could change what you do; a forecast that confirms soybeans has told you nothing you would act on. EVPI tells you when to stop studying and start planting.

Build the skill 3: when the table is costs, everything flips

Every number in the farmer’s crop table is a profit, and your take-home is a cost table. You have already read one cost table, though: Monday’s regret table. Every cell in it is a loss (corn 0 and 25, soybeans 10 and 10, wheat 30 and 0), so it already reads lower is better: the column bests are the two zeros, and the winner is the smallest weighted average, soybeans at 10, not the largest. Read a cost table the same way. Taylor’s rule for it is one sentence: when the payoffs are costs, the best decision is the one with the lowest expected value, and everything else flips with it.

The propane order is the farmer’s own cost table. The grain dryer burns propane at harvest, and the farmer pre-buys it now at the dealer’s pre-season price of 10 a load (in \$ thousands, like the crop table): 3 loads or 5 loads. A wet season brings the grain in wet, and wet grain takes longer to dry, so it needs five loads; a dry season needs three. Every load short in a wet season is an emergency top-up at 16, and propane left over in a dry one is already paid for. So a cell is built the way your take-home’s cells will be, from what happens in that state: the loads you paid for, plus the top-ups the season forces. Three loads in a wet season, for instance, cost \(3(10) + 2(16) = 62\). The odds are Monday’s, Wet 0.4 and Dry 0.6:

Three loads win on expected cost, 42.8 against a flat 50, even though a wet season then costs 62 once the top-ups are paid. Regret is your cost minus the column’s lowest cost (the column bests are now 50 when it is wet and 30 when it is dry), so 3 loads regrets 12 in a wet season and 5 loads regrets 20 in a dry one. EV with PI weights each column’s lowest cost: \(0.4(50) + 0.6(30) = 38\). And EVPI is the best expected cost minus that: \(42.8 - 38 = \mathbf{4.8}\), which is once again the smallest EOL. It comes out positive, as a saving should; subtract the other way round and you have a sign error, not a discovery.

Two more habits flip with it. The dominance screen: on a cost table a row is dominated when it costs more in every column (neither order is: three loads cost more in a wet season, five in a dry one), and it goes before any probability is used. And what stays out of the table: a figure that is the same whichever row you pick cannot move the pick, so it never enters a cell, and the price of a forecast stays out too, because it is compared with EVPI rather than added to a cost. Same three tools, every “best” read as “lowest.” Write lower is better at the top of a cost table before you compute anything, and the profit-table habit of maximizing cannot creep in.

Try the three tools on the propane table below, which is set to read lower is better. Press Expected value and it keeps the lowest row, 3 loads at 42.8. Press Minimax regret and every cell becomes its cost minus the column’s lowest: 12 and 0 for 3 loads, 0 and 20 for 5 loads, the regrets inside the EOL column above. Press Reveal EVPI and it marks the cheapest cell in each column, reports 38 against 42.8, and gives the gap as 4.8, a saving. Move the odds draws the two cost lines with the oracle’s below them. Its slider is \(p = P(\text{Wet})\), the same \(p\) as Build the skill 1: three loads’ expected cost is \(62p + 30(1-p) = 30 + 32p\) and five loads’ a flat 50 whatever the weather, so the lines cross where \(30 + 32p = 50\), at \(p = 20/32 = 5/8\), which the widget rounds to 0.63. Drag up to 0.63, which is to say make a wet season a little likelier than five in eight, and five loads become the cheaper order:

In practice: every input is an estimate, and the model says so out loud

• The odds came from somewhere. Thirty years of records is a better source than a hunch, but a decade-long shift is exactly the case where the record and the future disagree. Say where the probability came from next to the recommendation, not instead of it.
• The payoffs are estimates too. Every cell rests on a price, a yield, and a cost assumption. A sensitivity check on the odds is the habit; the same check on a payoff cell is the same habit pointed elsewhere.
• A forecast’s accuracy is itself a number you need. “Highly accurate” is a claim, not a figure. Once a price sits below the EVPI ceiling, the value of the forecast depends on how often it is right, and that is Week 6’s whole job.

Back to the puzzle

The agent gets a band, not a yes: soybeans hold for any wet probability between one-third and 0.6, so a wetter decade can nudge the 0.4 upward a long way before corn takes over, while a slide below one-third flips the advice to wheat. The drier tail is the one to watch. The consultant gets no meeting: \$12,000 is above the \$10,000 that a perfect forecast would be worth to the planting decision, so no accuracy claim can rescue an imperfect one at that price. The dealer gets an order for three loads: an expected \$42,800 against \$50,000 for five, and three stay the cheaper order until the chance of a wet season passes 5/8, a little beyond the 0.6 where corn takes over, so a wetter decade would change the crop before it changed the order. Without paying the consultant, the farmer still plants soybeans (\$31,000) and pre-buys three loads of propane.

Where this heads next. Week 5 draws decisions like this one as decision trees, the only form that keeps working once decisions come in stages: decide, see what happens, decide again. Week 6 prices an imperfect forecast through its expected value of sample information (EVSI), which is always less than EVPI, and a forecast is worth buying only at a price below its EVSI.

After-class check

Questions spanning expected value, expected opportunity loss, the expected value of perfect information, how the pick moves with the odds, and cost tables, 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.