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 (TBD)

🧩 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 (melted, wasted inventory). 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.” Next week we swap the missing probabilities back in and the criteria change accordingly.

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. 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.”

Click the criterion buttons below, Maximax, Maximin, Minimax regret (it rewrites the table as regret values), Hurwicz (then drag the optimism dial α), and Equal likelihood, and watch which row each one highlights. Start with Maximax vs Maximin and notice they disagree:

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. In the ice-cream table 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.

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. (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): \(\alpha\,(\text{best}) + (1-\alpha)\,(\text{worst})\). At \(\alpha = 0.5\) on the ice-cream table: Stock high \(= 0.5(2400) + 0.5(-600) = 900\); Stock medium \(= 0.5(1400) + 0.5(400) = 900\); Stock low \(= 0.5(700) + 0.5(700) = 700\). High and medium tie at 900, both ahead of low. Nudge the dial up to \(\alpha = 0.6\) and the tie breaks cleanly toward the optimist: Stock high \(= 0.6(2400) + 0.4(-600) = 1200\) vs. Stock medium \(= 0.6(1400) + 0.4(400) = 1000\) → Stock high. \(\alpha = 1\) is pure maximax, \(\alpha = 0\) is pure maximin. Now drag the dial slowly from 0 to 1 and watch which row the widget highlights. 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 the crossings are exact. 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 \(\alpha = 0.5\). 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.
• Equal likelihood (Laplace). With no reason to favor any state, treat them as equally likely and average: Stock high \(= (2400 - 600)/2 = 900\); Stock medium \(= (1400 + 400)/2 = 900\); Stock low \(= 700\). High and medium tie at 900, and notice 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.

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 (a later topic), 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 is a function of your optimism, not of 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 fresh numbers before the group practice. A community theater must announce its season opener before it learns whether subscription renewals come back strong or weak. Season profit (\$):

Work out maximax, maximin, and minimax regret yourself, then check:

• Maximax → Broadway revival. Row maxes are \$3,000 / \$6,000 / \$4,800; the revival’s best is biggest. (The high-variance “swing” option: the rights are expensive, so a weak year hurts.)
• Maximin → Family musical. Row mins are \$3,000 / \$1,000 / \$2,400; the musical’s worst (\$3,000) is the best of the worsts. (The flat risk-free row, same payoff in every state, because the school groups book it either way.)

• Minimax regret → New local play. The regret table is

  Season opener	Strong	Weak	Max regret
  Family musical	\$3,000	\$0	\$3,000
  Broadway revival	\$0	\$2,000	\$2,000
  New local play	\$1,200	\$600	\$1,200

  The local play’s worst-case regret (\$1,200) is the smallest, so the all-rounder hedges. The same three shapes turn up as in the ice-cream table, one swing, one flat risk-free row, one all-rounder, but they sit in different rows and each rule lands somewhere else. 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 your group’s names on the sheet and hand it to your TA before you leave. It counts toward participation.

• 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 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.