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Maximin is the pessimist's rule
Score each act by its worst payoff and take the act whose worst is best. It guards against the worst state and cares about nothing else.
Lesson · Decision and Rationality · Ignorance
Maximin, Maximax and Minimax Regret
When dominance is silent and no probabilities are available, a rule must turn each row of the table into one score. Maximin scores the worst case, maximax the best, and minimax regret the largest shortfall from what the state allowed. They can disagree on one table, and the disagreement is the lesson.
maximin: best of the row minimums
maximax: best of the row maximums
regret = column best − your payoff
minimax regret: smallest worst regret
regret ignores a gain every act shares
Three rules, three scores per row
Core ideas
What this lesson establishes
A rule for ignorance reduces each act to a single number computed from its row of the table, and then picks the best number. What differs is which number.
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Maximax is the optimist's rule
Score each act by its best payoff and take the act whose best is highest. It chases the best state and ignores the risk.
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Regret is measured against the state
In each state the best act pays something. The regret of an act in that state is the best payoff minus what it paid. Minimax regret takes the act whose largest regret is smallest.
The material
Three scores from one table
The umbrella table again, scored three ways, and then a table where two rules part.
“The Decision Matrix and Dominance” left the umbrella undecided. Take pays 2 in rain and 1 when dry; leave pays minus 3 and 3. Suppose you have no idea how likely rain is. The three rules here each give an answer, and they do not all give the same one.
Definition · Maximin
Score each act by its minimum payoff across the states. Choose the act with the largest such minimum. On the umbrella table take scores 1 and leave scores minus 3, so maximin takes the umbrella.
Definition · Maximax
Score each act by its maximum payoff. Choose the act with the largest maximum. Take scores 2 and leave scores 3, so maximax leaves it at home.
Definition · Minimax regret
Build the regret table first: in each column, subtract every payoff from the column's best. Score each act by its largest regret and choose the smallest score.
regret rain dry
take 0 2
leave 5 0
In rain the best payoff is 2, from take, so take regrets nothing and leave regrets 2 − (−3) = 5. When it is dry the best is 3, from leave, so leave regrets nothing and take regrets 3 − 1 = 2. The largest regrets are 2 for take and 5 for leave, so minimax regret also takes the umbrella.
On this table, then, two rules say take and one says leave. The disagree preset is a table where maximin and minimax regret part. A safe act pays 3 in a weak market and 3 in a strong one; a bold act pays 2 and 10. Maximin takes the safe act, whose worst is 3 against 2. But the safe act regrets 7 in the strong market, where the bold one would have paid 10, while the bold act regrets at most 1. Minimax regret takes the bold act.
Minimax regret is maximin applied to a different table, the regret table, and that table is relative: each column is measured from its own best. So anything every act shares in a state drops out of the regret table and cannot move the answer. It does move maximin. The shifted preset adds a bonus of 10 to the rain column for both acts, as if rain brought an allowance whatever you carried. The regret table is unchanged and still picks take, but take's worst is now 1 and leave's worst is 3, so maximin switches to leave.
None of the three is the right rule. Each encodes an attitude: maximin says the worst state is the one to plan for, maximax that the best state is, and minimax regret that the thing to avoid is having chosen worse than the state allowed. The lab computes what each says exactly; which attitude to hold is left to you.
Interactive
A decision table under every rule
the same payoffs, eight ways of choosing, each computed exactly
Method
Applying the three rules
Do the three scores in the order below and compare the verdicts.
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Write each row's minimum and maximum
Take the smallest and largest entry of each row. These two columns give maximin and maximax at once.
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Build the regret table
For each state, find the best payoff in its column, then subtract each act's payoff from it. Every entry is zero or more, and each column has a zero.
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Score the regret rows by their largest entry
Each act's score is its worst regret. The smallest worst regret wins.
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Compare and look for a disagreement
If the three rules pick the same act, the choice does not depend on attitude. If not, say which state's worst case is driving each.
Three rules, one table
Score 0 / 0Choose the answer you can justify.
Where maximin and minimax regret part
Two acts in two markets. Safe pays 3 and 3; bold pays 2 and 10.
maximin safe min 3 bold min 2 picks safe
maximax safe max 3 bold max 10 picks bold
regret safe 0, 7 bold 1, 0
worst regret safe 7 bold 1 picks bold
maximin and minimax regret pick different acts
Safe guards against the poor market and pays for it with a regret of 7 in the good one. Bold risks a loss of 1 against safe in the weak market and wins 7 in the strong. A decision maker who dreads having acted unwisely will take bold; one who dreads the worst outcome will take safe.
Common mistakes
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Treating minimax regret as maximin on a different table, and stopping thereIt is maximin applied to the regret table, and the correction is in the word different. Regret is measured from each state's own best, so it is blind to a gain every act shares in a state, while maximin reads the raw payoffs. Add 10 to the rain column of the umbrella and minimax regret still takes the umbrella; maximin switches to leave.
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Reading a rule's verdict as the verdictEach rule answers a question about the table: what is the worst, what is the best, what is the biggest shortfall. The same table can answer differently, as disagree does. A choice defended by a rule is defended by the attitude the rule carries.
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Using these rules when the probabilities are in fact knownMaximin ignores a state however unlikely it is. If you know rain has a one-in-a-thousand chance, planning only for it throws away what you know, and the next lesson's rule uses it.
Check your understanding
✓
Finish when you can score one table three ways and name the disagreement.
Given a decision table, you should be able to compute the maximin and maximax scores, build the regret table, apply minimax regret, and say which rules agree and which part, with the state that drives the difference.
Note: The rule menu carries one rule for ignorance this lesson does not teach. Laplace weights every state equally and takes the best average, which makes it expected utility with the probabilities left at one over the number of states; it has its own place in Justice and Collective Choice, where the same menu is used for a different question. The rules after it need probabilities, and the next lesson is where they begin.