Week 3: Decision Making Under Uncertainty (criteria)

This week’s big question: you have to commit to a plan before you learn how things turn out, and you have no probabilities to lean on. With nothing but a table of payoffs, how do you make a defensible choice?

Note: Monday, Sep 7 is Labor Day. No class. We meet Wednesday only this week, so the load is intentionally light.

Before the week

Read: Taylor, Introduction to Management Science (13e), Chapter 12, pp. 574–582. Chapter 12 runs 61 pages and carries Weeks 3 through 6, so take only this slice now. Nine pages, and the last two sections are software:

Stop where “Decision Making with Probabilities” begins on p. 582. That is Week 4, and it is the opposite situation: there you get odds, and the criteria change completely.

Taylor’s real-estate example runs every criterion on one table, which is exactly what Wednesday does with a different one. Read for the pattern, not the numbers: each criterion is a different way of squeezing a row down to one number, and the whole lesson is that they do not agree.

Then the warm-up self-check, on reading a payoff table:

Session 1: Wed, Sep 9 · Choosing when you don’t know the odds

🧩 Puzzle

The ice-cream vendor. Before the season starts, a vendor must prep stock: high, medium, or low, and then live with whatever summer arrives: Hot or Cool. A hot summer rewards a big stock; a cool one punishes it (unsold stock cleared at a loss). The catch: the vendor has no probabilities: no weather model, no history they trust. With only the payoffs below, how much should they stock?

Stock level	Hot summer	Cool summer
Stock high	\$2,400	−\$600
Stock medium	\$1,400	\$400
Stock low	\$700	\$700

Build the skill 1: the payoff table and the optimist/pessimist criteria

Where are we? Decision problems split by how much you know about the future:

• Certainty. You know which state will happen, so you just pick the best payoff. (Boring; not our week.)
• Risk. You don’t know the state, but you have probabilities for it. That is what makes expected value possible, and we get to it in Week 4.
• Uncertainty. You don’t know the state and you have no probabilities. That is this week.

Everything below is the toolkit for decision-making under uncertainty: how to choose well when you cannot put a number on “how likely.”

A payoff table lays out every decision alternative (a row you control) against every state of nature (a column you don’t), with the resulting payoff in each cell. The rows are the stock levels, the columns are the summers, and the vendor picks a row before nature picks a column.

Where a cell comes from. The table is season profit, and every cell is built from the vendor’s own prices. Stock low, medium, and high are 350, 700, and 1,200 units, bought at \$4 each; a unit sells for \$6; whatever is left when the season ends clears at \$2. A hot summer buys everything up to 1,200 units, a cool one buys 450. Sales are the smaller of stock and demand, the rest is salvage, and the whole stock is paid for whatever the summer does:

Take Stock medium in a cool summer: 450 units sell at \$6 for \$2,700, the 250 unsold clear for \$500, and all 700 units cost \$2,800, so the cell is \$2,700 + \$500 − \$2,800 = \$400. The rest of the table comes out of the same line: in a hot summer every unit sells at a \$2 margin (350 units make \$700, 1,200 make \$2,400), and Stock high in a cool summer is \$2,700 + \$1,500 − \$4,800 = −\$600. This week’s take-home hands you the prices, not the table, so the cells are yours to build.

First, look for a dominated row. Before applying any criterion, check whether one decision is beaten in every state by another, a row that loses under Hot and under Cool. That row is dominated and you can delete it; nothing will ever pick it. Said the other way round, a row that beats another row in every state dominates it, and a row that beats every other row in every state is simply the answer.

What it looks like when one fires. Suppose the vendor could also stock medium through a pricier distributor: the same tubs, \$200 more in cost whatever the summer does, so \$1,200 under Hot and \$200 under Cool. Set that beside plain Stock medium’s \$1,400 and \$400. It loses under Hot and under Cool, so Stock medium dominates it and it leaves the table before any criterion runs. That took two comparisons and no attitude toward risk at all, which is what makes it the free move.

Among the three rows the vendor actually has, though, no row is dominated: Stock high wins under Hot, Stock low wins under Cool, and Stock medium is best in neither but worst in neither. That is exactly why the criteria below can disagree. There is no free lunch to throw away first. The take-home adds a row like the pricier-distributor one and expects you to strike it out before any criterion runs.

When you have no probabilities, you can’t average the columns, so you fall back on criteria that summarize each row by a single number that reflects your attitude toward risk:

• Maximax (optimist). Look only at each row’s best outcome, then pick the row with the biggest best. “Assume the good case and go for the upside.”
• Maximin (pessimist). Look only at each row’s worst outcome, then pick the row with the biggest worst. “Protect the floor, guarantee the best worst case.”

On the ice-cream table, the two corners of each row are:

Maximax keeps the Row best column and takes its biggest, so it stocks high, chasing the \$2,400. Maximin keeps the Row worst column and takes its biggest, so it stocks low, guarding the \$700 floor. They disagree on the first try, and neither is wrong: each encodes an attitude toward risk, not an error.

Press Maximax and then Maximin in the widget below and watch which row each one highlights. Its last column carries every row’s score under the rule you pressed: maximax fills it with the Row best column, \$2,400 / \$1,400 / \$700, and maximin with the Row worst column, −\$600 / \$400 / \$700, the same three rows ranked in opposite orders. The other three buttons are the rules Build the skill 2 teaches next; come back to them there:

Build the skill 2: regret, and a couple of in-betweens

Optimist and pessimist ignore how much you’d kick yourself for guessing wrong. Regret fixes that, and it goes by several names in the textbook: minimax regret, the Savage criterion (after its inventor), or opportunity loss. They are the same idea. For each column (state), find the best payoff anyone could have gotten in that state; a cell’s regret is that column-best minus the cell’s own payoff, the opportunity you missed by not having chosen the winner.

Work one cell out loud first. In a Hot summer the best anyone got is \$2,400 (Stock high). So Stock low’s Hot-summer regret is \$2,400 − \$700 = \$1,700: that is how much you left on the table by playing it safe when the heat actually came. Do that for every cell and you get the full regret table:

Minimax regret then summarizes each row by its largest regret (the right-hand column above) and picks the row whose worst-case regret is smallest: \$1,000 for Stock medium beats \$1,300 and \$1,700. So minimax regret lands on Stock medium: a third row, different from both maximax and maximin. Notice that every column carries a zero: the row that won that state regrets nothing. (Click Minimax regret in the widget. It swaps the payoffs for the regret values and marks each row’s worst regret, landing on the same Stock medium.)

Watch the direction. Maximax and maximin scan across a row (best/worst of the decision you’d own). Regret is computed down a column first (column-best minus each cell), and only then do you take the max across the row. Mixing up “max over the row” and “best over the column” is the single most common regret-table mistake.

Two more criteria sit between optimist and pessimist, and Taylor expects you to actually compute them, not just name them:

• Hurwicz (realism). Blend each row’s best and worst with a coefficient \(\alpha\) (your optimism dial): \(\alpha\,(\text{best}) + (1-\alpha)\,(\text{worst})\). \(\alpha\) is how much weight the good case gets, and what is left goes to the bad case, so the two weights always spend exactly 1. \(\alpha = 1\) is pure maximax, \(\alpha = 0\) is pure maximin, and \(\alpha\) is your own number, not data. On the ice-cream table:
  • At \(\alpha = 0.6\). Stock high \(= 0.6(2400) + 0.4(-600) = 1200\); Stock medium \(= 0.6(1400) + 0.4(400) = 1000\); Stock low \(= 0.6(700) + 0.4(700) = 700\) → Stock high. Press Hurwicz in the widget above: its dial opens at this 0.6 and names Stock high at \$1,200.
  • From \(\alpha = 0\) to \(1\). Drag the dial slowly and watch which row the widget highlights; the chart under the dial draws the same three lines and marks the crossings, so you do not have to hunt for them. It does not simply slide between the maximin and the maximax answers: it passes through all three rows. Each row’s Hurwicz value is a straight line in \(\alpha\), so a crossing is one equation. Stock low is flat at \(700\); Stock medium is \(400 + 1000\alpha\); Stock high is \(-600 + 3000\alpha\). Medium overtakes low where \(400 + 1000\alpha = 700\), at \(\alpha = 0.3\), and high overtakes medium where \(-600 + 3000\alpha = 400 + 1000\alpha\), at \(\alpha = 0.5\). Right at that crossing the two rows tie: \(0.5(2400) + 0.5(-600) = 900\) and \(0.5(1400) + 0.5(400) = 900\), both ahead of low’s \(700\), and the widget names the tie instead of hiding it. So below 0.3 stock low, between 0.3 and 0.5 stock medium, above 0.5 stock high. The in-between criterion really can pick the in-between row, and you can now name exactly which attitudes it takes to get there.
  • What the crossings decide. Each crossing is a threshold on you, not a curiosity of the chart. The 0.5 says you have to weight the hot summer more than the cool one before stocking high beats stocking medium; the 0.3 says a vendor only three-tenths optimistic already leaves the safe row. So state your \(\alpha\) before you look at the payoffs, then read off which side of each crossing you are on. Argue about the side, not the second decimal: \(\alpha = 0.6\) and \(\alpha = 0.9\) give the same order, \(0.45\) and \(0.55\) do not. And a row that is never on top at any \(\alpha\) is out as far as Hurwicz is concerned, the same free move as dominance, which is worth checking before you sweep the dial at all.

• Equal likelihood (Laplace). With no reason to favor any state, weight each summer 0.5 and read the row average (with three states, each would weigh a third):

  Stock level	Hot × 0.5	Cool × 0.5	Row average
  Stock high	\$2,400	−\$600	(2,400 − 600)/2 = \$900
  Stock medium	\$1,400	\$400	(1,400 + 400)/2 = \$900
  Stock low	\$700	\$700	(700 + 700)/2 = \$700

  High and medium tie at 900, ahead of low (press Equal likelihood in the widget and it names the tie rather than picking one), and this “neutral” rule still disagrees with maximin, which picked low. The criterion, not the data, drives the choice. Equal likelihood is really just expected value with made-up equal probabilities: next week we replace those 0.5s with real ones and call it expected value. Refusing to pick an attitude is itself an attitude; even the neutral rule is a choice.

In practice: the criterion reveals your risk attitude, not the truth

• “No probabilities” is rarer than it looks. Genuinely zero information is unusual; you usually have some prior: a hunch, a base rate, last year’s weather. Pure no-probability criteria are the honest fallback when that prior is too weak to trust, not the everyday default.
• The criterion encodes you. Maximax, maximin, minimax regret, Hurwicz, and Laplace can each pick a different decision from the same table. That spread isn’t a bug. The criterion you choose is a statement about how much downside you can stomach. A cash-strapped vendor leans pessimist; a vendor chasing a breakout season leans optimist.
• People don’t even obey their own criterion. Almost everyone buys insurance, even though a pure payoff-maximizer “should” decline a bet whose expected cost exceeds its expected payout. Why? Because a dollar lost when you are broke hurts more than a dollar gained when you are comfortable. Money is not the same as satisfaction. That gap is the idea of utility (Week 6 puts a curve on it), and it is the deepest reason these criteria describe an attitude, not a truth.
• The payoffs are estimates too. Every number in the table is a forecast of profit under assumptions about price, spoilage, and demand. A criterion applied to shaky payoffs is still shaky, so sanity-check the cells before you trust the recommendation.

Back to the puzzle

The criteria genuinely disagree, and that’s the lesson:

• Maximax → Stock high. Its best case is the biggest of any row (\$2,400), so the optimist swings for the hot summer.
• Maximin → Stock low. Its worst case (\$700) is the best of the three worst cases, so the pessimist locks in the safest floor. Note Stock low is the same in both states. A flat, risk-free row: it neither wins big nor loses, so it always appeals to maximin.
• Minimax regret → Stock medium. Its largest possible regret is \$1,000 (vs. \$1,300 for high and \$1,700 for low): the smallest worst-case regret, so it hedges in the middle.
• Equal likelihood → a tie. Stock high and Stock medium both average \$900, ahead of Stock low’s \$700. Averaging the columns as though they were equally likely is already a preview of next week.
• Hurwicz → whichever row your dial names. It is the one criterion here that can land on any of the three: Stock low below \(\alpha = 0.3\), Stock medium between 0.3 and 0.5, Stock high above 0.5 (and a tie at each crossing). That is the week’s point stated as plainly as it gets: the answer depends on your optimism, not on the table.

Three reasonable people, three different stock levels, from one table, in the uncertainty environment (no probabilities). The math didn’t pick for them; their attitude toward risk did. So the defensible answer isn’t a number alone. It’s a sentence: “We’re in uncertainty, I’m protecting against a bad season, so by maximin I stock low.” Name the environment, name the criterion your risk attitude implies, then state the decision in words. That is the whole point of the exercise.

Try it yourself: a second table (answers below)

Get one clean rep on a changed table before the group practice. A caterer offers the vendor a fixed-price contract for the season instead of stocking the stand: a flat \$1,000, whatever the summer does. Screen first: \$1,000 beats Stock low’s \$700 under Hot and under Cool, so the contract dominates Stock low, which leaves the table before any criterion runs. What is left, as season profit (\$):

None of these three dominates another. Work out maximax, maximin, and minimax regret yourself, then check:

• Maximax → Stock high. Row bests are \$1,000 / \$2,400 / \$1,400; the optimist still swings for the hot summer.
• Maximin → Caterer’s contract. Row worsts are \$1,000 / −\$600 / \$400; the contract’s \$1,000 is the best of the worsts. (The flat risk-free row again, with a higher floor than Stock low ever had.)

• Minimax regret → Stock medium. The Hot column’s best is still Stock high’s \$2,400, but the Cool column’s best is now the contract’s \$1,000, so the regret table is

  Option	Hot	Cool	Max regret
  Caterer’s contract	\$1,400	\$0	\$1,400
  Stock high	\$0	\$1,600	\$1,600
  Stock medium	\$1,000	\$600	\$1,000

  Stock medium’s worst-case regret (\$1,000) is still the smallest, but look at the Cool column: the contract raised its best from Stock low’s \$700 to \$1,000, so Stock high’s and Stock medium’s cool-summer regrets both rose by \$300, and Stock high’s worst regret went from \$1,300 to \$1,600. Regret is measured against each column’s best, so an option that tops a column moves every other row’s regret in it, whether or not the rule then picks that option. And the flat row now sits on top, so answering by position from the ice-cream table gets all three wrong: read the table, not the layout.

In-class group practice (wrap-up)

To close the session, work a fresh under-uncertainty problem by hand in your group: build the regret table from scratch, and reason about three rules that each point at a different choice.

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 3 group practice (Word)

After-class check

Questions spanning payoff tables, the decision environments, dominance, and all five criteria: maximax, maximin, minimax regret (the Savage criterion), Hurwicz, and equal likelihood, several of them applied to a small table. Retry freely; nothing is submitted.

Weekly take-home (graded: submit on Canvas)

Everything for the take-home lives in one Excel workbook, and this week no payoff table is given to you. The model sheet opens with a case brief for a fall-festival pumpkin booth: a grower’s price quoted by the case, a booth price, what the compost partner pays, and two figures that belong to neither. In the Your model block you name the acts and the states of nature and write out how a single payoff cell is built, then you build the table, screen it for a dominated option, and work all five of this week’s criteria, including Hurwicz and equal likelihood. They do not agree, which is the point.

The last cell asks for something the brief does not hand you. The committee gives you one optimism weight, but someone on it will argue for a different one, so you find the tipping point: the weight at which your Hurwicz answer would change, and what it changes to. Then you check whether the middle order is ever the Hurwicz pick on its own at any weight at all. The answer is not the same as it was for the ice-cream table above, and working out why is the assignment. One short recommendation 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.