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Acts, states, payoffs
An act is something you can choose. A state is a way the world might be that you do not control. A payoff is how things go for you if you choose that act and the world is in that state. The table has one cell for each pair.
Lesson · Decision and Rationality · Ignorance
The Decision Matrix and Dominance
A decision problem is a table of acts against states with a payoff in every cell. Dominance is the one rule that needs nothing but the table: it removes an act that some other act matches or beats in every state. When no act dominates, the table is silent, and the rest of the course is about what to add.
rows: acts columns: states
cell: the payoff of an act in a state
a dominates b: at least as good in every
state, and better in at least one
dominated acts are dropped, no odds needed
A table, and one rule that needs nothing else
Core ideas
What this lesson establishes
Every rule in the next lessons reads the same object, so it is worth being exact about what the object holds.
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Dominance compares acts column by column
Act a dominates act b when a is at least as good in every state and better in at least one. The comparison is state by state, never one act's best cell against another's.
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No odds are needed
Dominance uses no probabilities and no attitude to risk, which is why it is so hard to resist when it applies and so seldom decides anything.
The material
Reading a decision table
The table first, then the rule, then the way it falls silent.
Whether to take an umbrella is a decision with two acts and two states. If it rains, you are glad to have taken it; if it stays dry, you carried it for nothing. Put numbers on how each of the four cases goes for you and you have a decision table.
act rain dry
take 2 1
leave -3 3
The first row reads: if you take the umbrella, you score 2 in rain and 1 when it is dry. The numbers are yours to choose, and they stand for how good each case is; the lab will not tell you whether a soaking is worth minus 3, only what follows if it is.
Definition · Dominance
Act a dominates act b when a pays at least as much as b in every state, and strictly more in at least one. A dominated act can be removed from the table without loss. If one act dominates all the others, it is the choice.
The umbrella table has no dominance. Take beats leave in rain, 2 to minus 3, and leave beats take when it is dry, 3 to 1. Neither act is at least as good in both columns, so the lab reports that no act dominates and picks none.
Example · Adding a worse act
Add a third act, take a coat, paying 1 in rain and 0 when dry. Take-the-umbrella pays 2 and 1 in the same states, more in each, so it dominates the coat and the coat can be struck out. The table is unchanged in what it recommends, and the dominated act drops out. In the dominated preset the status line under the table names the acts that survive, take and leave, and the coat is not among them; the choice tile still reads none, because neither survivor dominates the other.
The lab's definition is the weak one, and the weak preset shows why that matters. Take pays 2 and 1; leave pays 2 and 0. They tie in rain, take wins when it is dry, so take dominates leave even though it is not better everywhere. A strict version, better in every state, would leave this table undecided.
The best possible outcome is not the test. Leaving the umbrella has the highest cell in the table, a 3, but that cell sits in only one column and the same act has the lowest cell, minus 3, in the other. Dominance asks about each column in turn.
There is one condition on using dominance, and it is easy to break. The states must not depend on the act. If you reason that you will pass the exam whether or not you study and then skip the revision, you have treated passing as a state when it is partly an outcome of the act. “Newcomb's Problem” returns to exactly this worry with a table that does seem to dominate.
Interactive
A decision table under every rule
the same payoffs, eight ways of choosing, each computed exactly
Method
Applying dominance to a table
Work pair by pair, and write down which state decided each comparison.
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Fix the states first
Check that no state depends on what you choose. If one does, the table is mis-built and dominance may mislead.
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Compare two acts column by column
Write the two rows one over the other and compare each state. a dominates b only if no column favours b and at least one favours a.
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Strike dominated acts and repeat
Remove any act that another dominates, then compare what is left. Dominance is transitive, so the order of striking does not matter.
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Ask whether one act is left
If one remains, it is the choice. If several remain, dominance has done all it can and another principle has to choose.
Dominated or not
Score 0 / 0Choose the answer you can justify.
The umbrella, then a coat
Take, leave and coat in the states rain and dry, payoffs as in the lab. Compare the acts in pairs.
take vs leave rain: 2 > -3 dry: 1 < 3 no dominance
take vs coat rain: 2 > 1 dry: 1 > 0 take dominates
leave vs coat rain: -3 < 1 dry: 3 > 0 no dominance
left after striking the coat: take, leave
no single act dominates all the others, so no choice
The coat went, and the decision did not get any easier. That is the usual shape of dominance reasoning: it removes the plainly bad acts and leaves the real question where it was.
Common mistakes
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Choosing the act with the best possible outcomeLeaving the umbrella has the highest cell in the table, a 3, and it is not the best act: it has the lowest cell too, minus 3. One cell does not rank an act. Dominance compares two acts state by state, and the umbrella table shows the best cell and the best act coming apart.
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Dropping an act that merely does well in the likely stateAn act that is worse in some state is not dominated by an act that is better in the likely one. Dominance does not weigh states, so it cannot say that one column matters more; that is a job for the probabilities introduced later in the course.
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Treating a tie as a reason to keep both actsIf one act ties the other in every state but one and wins that one, it dominates. The weak preset has take and leave tied in rain, and take wins in the dry state, so take is the choice. Equal payoffs in a column are no reason to retain the weaker act.
Check your understanding
✓
Finish when you can lay out a table and say what dominance does to it.
Given a decision described in words, you should be able to write it as acts against states, name any dominated act with the state-by-state comparison that shows it, and say when dominance leaves the choice undecided.
Note: The lab ships with the dominance rule and with no probabilities, which is the situation of this lesson. The rule menu has the others, which the next two lessons use; a rule that needs odds will say so rather than invent them.