UPI Study › Courses › Quantitative Analysis › Free lessons › Lesson 2.3 Free with a free account
Decision-Making Under Uncertainty
Quantitative Analysis · Chapter 2: Decision-Making Models and RiskSection 2.3 · lesson 7 of 5512 min · free with a free account
Free preview
Create a free account to keep watching
It takes a minute and costs nothing. Your free account unlocks all of Chapter 1 and 3 more lessons of Quantitative Analysis.
Lesson complete
Earn the 3 college credits for Quantitative Analysis
Enrolling adds a quiz after each lesson, the graded work and the official transcript. Recommended for credit by NCCRS.
Next lesson: Business Risk Analysis and the Goals of Risk Management →← Decision-Making Under Certainty and RiskNext lesson: Business Risk Analysis and the Goals of Risk Management →
In this lesson
Strategies when outcomes are known but probabilities are not.
- Separate decisions under risk from decisions under uncertainty
- Apply Maximax, Maximin, Minimax Regret, Hurwicz and Laplace
- Build a regret table from the column maxima
- Read conflicting recommendations as information, not noise
Transcript
Everything said in the video, section by section (1,757 words). Read along, or search it.
QAN-201 - Quantitative Analysis · Section 2.3
Decision-Making Under Uncertainty
Welcome to Chapter two point three, Decision-Making Under Uncertainty. In the previous section, every state of nature carried a probability. You computed an expected value, and the arithmetic did the deciding for you. This section takes that comfort away. Here you can list what might happen, but you cannot say how likely any of it is. There is no expected value to compute. So the question itself changes. Instead of asking what is most likely, you ask what kind of decision-maker you are. We will meet five classical criteria, and then watch them disagree with each other. That disagreement turns out to be the most useful thing they produce.
Risk vs Uncertainty
When You Cannot Assign Probabilities
Start with the distinction that organises the whole topic. A decision under risk has known probabilities. A decision under uncertainty does not. That is the only difference, and it changes the entire toolkit. You still have a set of alternatives to choose between. You still have a list of the states of nature that could occur. And you still have a payoff for every alternative and state pair. The only thing missing is the probability attached to each state. Think of a brand-new product in a market nobody has studied. Or a geopolitical event with no precedent, or a technology that may simply not work. In each case you can name the outcomes, but you cannot weight them honestly.
The Two Extremes
Maximax and Maximin
The first two criteria are mirror images of one another. Maximax is the rule of the unbridled optimist. For each alternative, find its best possible payoff, then choose the highest of those. The question it asks is blunt. If luck is on your side, which choice rewards you most? Maximin turns the telescope around, finding each alternative's worst possible payoff, then again picking the highest. In three words, maximise the floor. Maximax suits a venture fund that needs one breakout success to cover dozens of failures. It is dangerous whenever the worst case could put the firm out of business. Maximin suits capital-heavy projects, and choices made for people who cannot absorb a loss. But notice what both rules quietly do. Each throws away half the payoff table. So under Maximin a safe but mediocre option always beats a promising one, even when the risk is tiny.
Opportunity Loss
The Minimax Regret Criterion
Maximax and Maximin both look only at absolute payoffs, but Minimax Regret asks a different question altogether. How much will you regret this choice afterwards, once you see what nature actually did? Regret also goes by another name. Opportunity loss. It is the gap between what you got, and the best you could have got in that state. Building the regret table is a three-step recipe. Step one. Take each column, one state of nature, and find the largest payoff in it. That is the best you could have done, had you known the future in advance. Step two. Subtract every cell in that column from the column maximum. A regret of zero means that alternative was the right call under that state. Step three. Read across each row, find its largest regret, and pick the smallest of those. In temperament this criterion usually lands between the two extremes we just met.
Weighted Optimism
The Hurwicz Criterion
Most real decision-makers are neither pure optimists nor pure pessimists. The Hurwicz criterion gives that middle ground an actual number. It uses one tunable parameter, the coefficient of optimism, written as alpha. Alpha runs from zero to one. The Hurwicz value is alpha times the best payoff, plus one minus alpha times the worst. Set alpha to one and the second term disappears. You are back to Maximax. Set alpha to zero and the first term disappears. You are back to Maximin. Everything in between is a weighted blend of the two extremes. A common starting value is zero point five, but the right number depends on the person deciding. Here is the part that matters most in practice. Because the answer moves as alpha moves, Hurwicz is the natural home of sensitivity analysis.
Equal Likelihood
The Laplace Criterion
The last criterion takes the most pragmatic view of all. Laplace is sometimes called the principle of insufficient reason. If you have no basis to favour one state over another, treat them as equally likely. The Laplace value is then simply the average payoff across the row. One over n, times the sum of the payoffs, where n is the number of states. Notice what has quietly happened here. The problem has turned back into a decision under risk, and you pick the highest average. The appeal is balance. Every outcome counts, and no single scenario dominates the verdict. The weakness is the assumption itself, because equal likelihood is an assumption, not a fact. If you have any reason to think the states differ in probability, Laplace can mislead you.
Side By Side
Comparing the Five Criteria
Put the five criteria side by side and a clear spectrum appears. Maximax is the optimist, suited to low-stakes bets where the upside is the prize. Its blind spot is catastrophic loss. Maximin is the pessimist, built for survival-critical, one-shot, irreversible decisions. Its blind spot is attractive upside. Minimax Regret is hindsight-averse, and fits settings where regret gets judged after the fact. It is sensitive to any extreme outcome sitting inside a column. Hurwicz is the tunable one, for when a risk posture can honestly be written as a number. Laplace is neutral, and suits genuine ignorance about the probabilities. There is no algorithmic way to declare one of these the right one. And when they disagree, treat that as a feature rather than a bug.
Worked Example
Brightleaf Analytics Picks a Market Entry
Time to run all five criteria over a single problem. Brightleaf Analytics is an early-stage software startup choosing a market-entry strategy. Strategy A is an aggressive launch, B is a balanced rollout, C is a cautious pilot. Each performs differently depending on which of four demand scenarios actually unfolds. Those scenarios run from a cold market, through slow and steady, up to a boom. The market is new, so no probabilities can be reliably assigned. That is what makes this uncertainty. Every figure is first-year profit, measured in thousands of dollars. The aggressive launch runs from minus eighty in a cold market, to three hundred and fifty in a boom. The balanced rollout runs from thirty to two hundred and twenty, and the cautious pilot is flattest, sixty to one hundred and forty. Maximax reads the row bests. The largest is three hundred and fifty, so it picks the aggressive launch. Maximin reads the row worsts. Minus eighty, thirty, and sixty. The largest of those is sixty, so Maximin picks the cautious pilot. Two criteria, two opposite answers, and we have barely begun.
Step 3
Building the Regret Table
Now build the regret table, using the three steps from earlier. First, the column maxima. Under a cold market the best anyone achieves is sixty, from the cautious pilot. Under slow demand it is ninety, from the balanced rollout. Under steady demand it is one hundred and eighty, from the aggressive launch. And in a boom it is three hundred and fifty, again from the aggressive launch. Now subtract every cell from the top of its own column. The aggressive launch regrets one hundred and forty in a cold market, its single worst entry. Yet it regrets nothing at all under steady demand, or in a boom. The cautious pilot looks safe until the boom arrives, where it regrets two hundred and ten. The row maxima come out at one hundred and forty, one hundred and thirty, and two hundred and ten. The smallest is one hundred and thirty, so Minimax Regret picks the balanced rollout.
Steps 4 and 5
Hurwicz and Laplace on the Same Table
Step four applies Hurwicz with alpha at zero point five, weighting each strategy's best and worst payoffs equally. For the aggressive launch, half of three hundred and fifty, plus half of minus eighty, gives one hundred and thirty-five. For the balanced rollout, half of two hundred and twenty plus half of thirty gives one hundred and twenty-five. For the cautious pilot the blend comes out at exactly one hundred. The highest is one hundred and thirty-five, so Hurwicz picks the aggressive launch. But look hard at the margin. Ten points, on values well above a hundred. Drop alpha to zero point four and the ranking flips back toward the balanced rollout. So the recommendation turns over somewhere between those two settings of alpha. That is not a mandate. It is a warning that the answer hinges on risk appetite. Step five applies Laplace, giving each of the four scenarios a probability of one quarter. Average the aggressive launch row and you get one hundred and twenty-two point five. Average the balanced rollout row and you get exactly the same number, so the two tie. Preferring the smaller downside as a tie-breaker sends Laplace to the balanced rollout.
Step 6
Five Criteria, Three Different Answers
Step six. Line the five verdicts up next to one another. Maximax says A, the aggressive launch, on a best case of three hundred and fifty. Maximin says C, the cautious pilot, on a floor of sixty. Minimax Regret says B, with a maximum regret of one hundred and thirty. Hurwicz at zero point five says A again, valued at one hundred and thirty-five. Laplace ties A and B at one hundred and twenty-two point five. The five criteria split three ways, and no strategy dominates the others. That is not a failure of the method. That is the method working. The split tells Brightleaf's leadership exactly where the real question sits. Almost everything hinges on one thing, the firm's tolerance for that eighty thousand dollar loss in a cold market. Not on better forecasting, and not on cleverer arithmetic.
Key Takeaways
The Criteria Clarify the Trade-Off; They Do Not Decide It
So how should Brightleaf actually decide? If losing eighty thousand dollars would not threaten survival, the aggressive launch is defensible, and its upside genuinely dominates. If a loss that size would be existential, the balanced rollout is the better balance. It never loses money, and it minimises regret. The cautious pilot is right only when survival is the dominant concern. Notice that the criteria did not decide anything by themselves. They clarified the trade-off, so that leadership could make an informed call. To recap. Uncertainty means the states are known, but the probabilities are not. Maximax picks the best best-case. Maximin picks the best worst-case. Minimax Regret minimises the largest opportunity loss, built from the column maxima. Hurwicz blends best and worst through alpha, and testing alpha shows how robust the recommendation really is. Laplace assumes equal probabilities and takes the highest average, which is only as good as that assumption. When criteria disagree, the disagreement itself shows where the risk-attitude call really sits.
About Quantitative Analysis at UPI Study
Quantitative Analysis (Math 250) is a self-paced online course of 55 video lessons in 13 chapters, from decision analysis and simulation to inventory and queuing models, project scheduling, linear programming and transportation models. It carries a NCCRS credit recommendation for 3 semester credits.
Every lesson transcript is free on pages like this one, and the videos are free with a free UPI Study account. Enrolling adds a practice quiz after each lesson, the graded work and final exam that earn the credit, lifetime access, and an official transcript on completion. UPI Study has helped over 130,000 students with college credits. ACE and NCCRS have over 2,100 cooperating colleges that will consider accepting credits; the final decision is with the universities.
Enroll now → Course details Download the syllabus All Math courses Every course with PRO →
Compare: Statistics vs Quantitative Analysis
Why take it here
Why students take Quantitative Analysis with UPI Study
Where the credit goes
Universities that work with NCCRS
Quantitative Analysis carries a NCCRS credit recommendation. These universities work with NCCRS; some are UPI Study's signed partners. ACE and NCCRS have over 2,100 cooperating colleges that will consider accepting credits; the final decision is with the universities.
Thomas Edison State UniversitySigned partner
University of the PeopleSigned partner
Charter Oak State CollegeSigned partner
Strayer UniversitySigned partner
Columbia Southern UniversitySigned partner
Indiana TechSigned partner
Penn StateWorks with NCCRS
Southern New Hampshire UniversityWorks with NCCRS
SUNYWorks with NCCRS
+many more cooperating universitiesFind my college →Find my college → Not listed? Ask Athena in the corner of this page, or book a 15-minute call.
FAQ
Questions students ask us
About Decision-Making Under Uncertainty
What is the difference between the maximax and maximin criteria?
How do you build a regret table for the minimax regret criterion?
What is the Hurwicz criterion and the coefficient of optimism?
What is the Laplace criterion in decision-making under uncertainty?
About the Quantitative Analysis course
How many college credits is Quantitative Analysis?
How much does Quantitative Analysis cost and what is included?
Which universities accept UPI Study credits?
How long does Quantitative Analysis take?
How are the exams done?
How do my credits reach my college?
Ready for the full course? Earn 3 credits in Quantitative Analysis.
55 video lessons, a quiz after each one, and the graded work that earns the credit. Recommended for credit by NCCRS.
3 credits · NCCRSEnroll →