UPI Study › Courses › Quantitative Analysis › Free lessons › Lesson 5.3 Free with a free account
Choosing in the Dark
Quantitative Analysis · Chapter 5: Decision Analysis with ProbabilitiesSection 5.3 · lesson 22 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: Decision Analysis with Probabilities →← Data That Informs ChoicesNext lesson: Decision Analysis with Probabilities →
In this lesson
Three frameworks for rational decisions when no probabilities exist.
- Build payoff tables that stay auditable as the grid grows
- Prune dominated alternatives before any criterion is applied
- Construct the opportunity-loss matrix column by column
- Read Hurwicz crossovers to find where a recommendation flips
Transcript
Everything said in the video, section by section (1,816 words). Read along, or search it.
QAN-201 - Quantitative Analysis · Section 5.3
Choosing in the Dark
Chapter two point three gave you five ways to choose when probabilities are unknown. Maximax, Maximin, Minimax Regret, Hurwicz, and Laplace. This chapter does not add a sixth criterion, it makes those same five usable on genuinely messy problems. More alternatives, more states of nature, real money on the line. Three additions carry the weight. Building the payoff table carefully, once the grid stops being small. Building the regret table as a formal matrix operation. And pruning dominated alternatives before you compute anything at all. By the end, you will run a five location decision from raw table to written memo.
What Changes
The Same Criteria, at Realistic Scale
Start with what does not change. The five criteria are exactly the criteria you already know. Maximax still picks the best of the best cases, and Maximin still maximises the floor. Nothing about the arithmetic of a criterion is different at scale. What changes is everything that happens before a criterion runs. The rigour with which you build the payoff structure, and the pre-processing that keeps it manageable. Three additions matter most here. First, careful construction of the payoff table as the dimensions grow. Second, formal construction of the opportunity loss table, one column at a time. Third, dominance analysis, which deletes alternatives no criterion could ever choose. Together these turn a textbook exercise into something a decision maker can actually deploy.
The Data Structure
Payoff Tables at Realistic Scale
A payoff table is a data structure, so treat it like one. Each row is a single alternative, and each column is a single state of nature. Each cell records the payoff for that pairing: profit, cost, net present value, or utility. With m alternatives and n states, the table holds m times n cells. That product is the source of every complication that follows. Three habits keep a larger table workable. First, define alternatives and states cleanly. Alternatives must be mutually exclusive and collectively exhaustive, and states discrete and non overlapping. Every realisation the world might produce should map to exactly one column. Second, use a single payoff metric throughout, because mixing dollars, utility points and qualitative ratings breaks every criterion. Third, write down the assumption behind each column, because a one line note per scenario is what lets someone audit the table later.
Step Zero
Dominance: Pruning Before You Compute
Before you apply any criterion, scan the table for dominance. Alternative i dominates alternative j if, in every single state of nature, i does at least as well. And it must do strictly better in at least one of those states. Take the toy table: A returns sixty, ninety and one hundred twenty across three states. B returns forty, seventy and one hundred, so B loses to A everywhere and can simply be deleted. Why is that safe? Because no rational decision maker would ever pick it. Optimist, pessimist, or anything in between, there is always something at least as good. Pruning dominated rows shrinks the table, simplifies every later calculation, and makes the analysis easier to explain. But notice what dominance does not tell you: A is not automatically the right answer. C beats A in state one, so C survives and might still win on some criterion. Dominance only filters out the alternatives that no criterion could ever rescue.
Opportunity Loss
The Regret Table as a Matrix Operation
The regret table, also called the opportunity loss table, has exactly the same shape as the payoff matrix. And it is built one column at a time, never one row at a time. For a given state of nature, find the largest payoff in that column. Then subtract every cell in the column from that maximum. The cell matching the column maximum gets a regret of zero: you chose perfectly. Every other cell gets a positive number, and that number is what you left on the table. Minimax Regret then runs the other way, along the rows. For each alternative take its worst regret, and pick the alternative whose worst regret is smallest. Two properties are worth carrying with you. Every column of a regret matrix contains at least one zero, by construction. And a dominated alternative never has lower regret than its dominator in any column, which is a second reason pruning is safe.
When Alpha Decides
Hurwicz Sensitivity
Hurwicz is the criterion with a dial on it. The score is alpha times the best payoff, plus one minus alpha times the worst. Alpha is the coefficient of optimism, running from zero to one. Here is the useful structural fact: Hurwicz is linear in alpha, so every alternative plots as a straight line. At alpha equals zero the line sits at the worst case payoff, at alpha equals one, the best. With several alternatives in play, plot all the lines together. The picture reveals more than a winner. It reveals the structure of the decision. Which options are pure pessimist choices, and which only become attractive at high alpha. Crossover points mark the alpha values at which the recommendation changes. If one alternative wins across a wide band of alpha, risk calibration barely matters. If a small nudge in alpha flips the winner, you have found the real question.
Worked Example
Cascade Logistics: Five Sites, Four Futures
Now the full worked example: Cascade Logistics is a mid sized third party logistics firm serving the American Midwest. It must commit to one new regional distribution hub, and five candidates are on the table. Atlanta, a large city with expensive real estate. Memphis, an established logistics hub at middling cost. Phoenix, a growth market and the riskier bet. Dallas, high cost but with broad reach. And Kansas City, central and cheap. Future demand depends on which of four macroeconomic scenarios unfolds over the next decade. A recession, with freight volumes falling twenty per cent. Slow growth, with freight flat. Steady expansion, with freight rising five per cent a year. Or a boom, an e-commerce surge of twelve per cent a year. Cascade's analysts agree that no probability can be reliably assigned to those scenarios. That is what puts this problem under uncertainty rather than under risk.
Step 1
Build the 5 × 4 Payoff Table
Step one is the payoff table itself, five alternatives by four states, twenty cells to fill. The metric is projected ten year net present value, in thousands of dollars. One metric, one horizon, applied uniformly. That discipline is the whole point. Atlanta runs one twenty, two twenty, three ten and three eighty across the four scenarios. Memphis is flatter, from one eighty up to three twenty. Phoenix is the outlier: it loses forty thousand dollars in a recession and earns four hundred sixty thousand in a boom. Dallas runs ninety, two hundred, two sixty and three hundred. Kansas City barely moves, from two ten to two ninety. Read the shape before you read the arithmetic. Phoenix has the highest ceiling and the only negative floor, Kansas City the highest floor and the lowest ceiling. That contrast, ceiling against floor, is really what this decision is about.
Step 2
Dominance Eliminates Dallas
Step two comes before any criterion: check for dominance. Put Dallas beside Atlanta, column by column. One twenty beats ninety, two twenty beats two hundred, three ten beats two sixty, three eighty beats three hundred. Atlanta is strictly better in every single column. So Dallas is dominated by Atlanta, and it leaves the analysis immediately. As it happens Memphis also dominates Dallas, but only one dominator is needed to delete a row. The conclusion is worth stating plainly: no criterion, optimist, pessimist, or any mixture, would ever recommend Dallas over Atlanta. Now check the survivors against each other. Kansas City beats Memphis in the recession column, two ten against one eighty. But it loses under steady expansion, two seventy against two eighty, so neither dominates the other. Similar checks clear the rest, and we go forward with four alternatives.
Step 3
Build the Regret Table
Step three, build the regret table on the surviving four, with column maxima recomputed without Dallas. In a recession, the best available result is two ten, from Kansas City. Under slow growth it is two fifty, again Kansas City. Under steady expansion, three twenty from Phoenix, and in a boom, four sixty, Phoenix again. Now subtract every cell from the top of its own column. Atlanta's regrets come out at ninety, thirty, ten and eighty. Memphis gives thirty, ten, forty and one forty; Phoenix gives two fifty, one hundred, zero and zero. Kansas City gives zero, zero, fifty and one seventy. Now take the worst regret in each row: ninety for Atlanta, one forty for Memphis, two fifty for Phoenix, one seventy for Kansas City. The smallest is ninety, so Minimax Regret picks Atlanta. Look closely at Phoenix, because it scores a perfect zero in two of the four columns, and still finishes last on this criterion. One catastrophic column, the recession, decides its fate.
Step 4
All Five Criteria on the Surviving Four
Step four, run the remaining criteria on the same four alternatives. Maximax compares best cases: three eighty, three twenty, four sixty, two ninety, and the largest is four sixty, so Phoenix. Maximin compares worst cases: one twenty, one eighty, minus forty, two ten, and the highest floor is two ten, so Kansas City. Minimax Regret we already have: ninety is the smallest worst regret, so Atlanta. Hurwicz at an alpha of one half is more interesting. For Atlanta, half of three eighty plus half of one twenty gives two fifty. Memphis and Kansas City also give two fifty, while Phoenix, dragged down by its negative floor, gives only two ten. So three alternatives tie, and the criterion tells us nothing. Laplace assumes each state is equally likely and averages the row. Atlanta averages two hundred fifty seven point five, the highest of the four. Memphis and Kansas City both average two hundred fifty five, Phoenix two hundred twenty two point five. Atlanta appears in three of the five recommendations, and the disagreement between criteria is a map of the risk question.
Steps 5 and 6
Sensitivity, Then the Memo
Because Hurwicz returned a three way tie, sensitivity analysis is not optional here. Sweep alpha from zero to one, and watch which line is on top. Three zones appear. Below an alpha of one half, Kansas City wins. It has the highest floor of any survivor, two ten, and the pessimist rewards floors. Between one half and roughly two thirds Atlanta takes over, a floor of one twenty with a ceiling of three eighty. Above two thirds Phoenix wins, the pure upside play, but that means accepting a possible loss of forty thousand dollars in a recession. Watch Memphis, because it is competitive everywhere and yet never strictly wins at any alpha. At the pessimist end it loses to Kansas City, in the middle to Atlanta, at the top to Phoenix. Now write the memo. Atlanta wins under Minimax Regret and under Laplace. It ties under Hurwicz at one half and wins across the whole moderate range. And given Cascade's stated risk posture, moderate rather than extreme on either side, Atlanta is the most defensible choice. There is a general lesson here. When several criteria agree, the recommendation is robust. When they disagree, the disagreement itself names the question the decision maker must answer.
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 Choosing in the Dark
What is dominance in decision analysis, and why check for it first?
How do you find the point where a Hurwicz recommendation changes?
How should a large payoff table be built so it stays reliable?
What should you conclude when decision criteria under uncertainty disagree?
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 →