1
Sort, then compare from the bottom
List each distribution from the worst-off up. Compare the first entries, and only if they are equal compare the second, and so on. The first difference decides. This is leximin, short for lexical maximin.
Lesson · Justice and Collective Choice · Distributive justice
The Difference Principle and Leximin
The difference principle ranks distributions by their worst-off member, then by the next worst when the first are tied, and so on. An unequal distribution that lifts the bottom beats an equal one that does not. The wrong model is that the principle demands equality.
compare the worst-off first
tied there? compare the next worst
raising the floor beats equal shares
the principle allows inequality that pays
Leximin, step by step
Core ideas
What this lesson establishes
The previous lesson ended on a tie at the bottom. Three ideas turn the maximin rule into a ranking.
2
Inequality is permitted, on a condition
The principle does not prefer unequal distributions. It permits an inequality if the worst-off are better off with it than with any alternative, and forbids it otherwise.
3
The justification is an argument about incentives
The usual defence of an unequal distribution is that the prospect of higher rewards draws out effort and talent, and that the extra product reaches the bottom. That is a premise about how people behave, not an output of the rule.
The material
Leximin on two distributions
A rule that compares sorted lists, a case where it favours inequality, and a case where it does not.
The second part of Rawls's second principle of justice, the difference principle, says that social and economic inequalities are to be arranged so that they are to the greatest benefit of the least advantaged. As a rule for comparing two distributions of welfare it comes to this: prefer the one in which the worst-off person is better off. If their positions are the same, the person next up decides, and so on.
Definition · Leximin
Leximin ranks two distributions by sorting each from lowest to highest and comparing them entry by entry from the bottom. The first position at which they differ decides, and the distribution with the larger entry there is ranked higher.
Take the distributions (5, 5, 5) and (6, 9, 20). Sorted, they are as they stand. The first entries are 5 and 6, and 6 is larger, so the second distribution is ranked higher. The 9 and the 20 are never looked at. The same verdict would stand if the top entry were 20 000, or if it were 7.
That is where the principle parts company with equality. A pure egalitarian prefers (5, 5, 5) to (6, 9, 20) because it is level, and levelling is what they are after. The difference principle prefers the second, because everyone is no worse off in it, and the worst-off are better. Preferring the first would mean denying the worst-off a unit of welfare so that others would not have more than they do. “Priority, Equality and Levelling Down” takes that cost as its subject.
When the bottoms tie, leximin looks upward. Compare (5, 8, 8) with (5, 6, 20). The first entries are both 5. The second entries are 8 and 6, so the first distribution wins, even though the total of the second is far larger. Maximin alone would have called the pair a tie; leximin finishes the job.
The incentive argument
Why would anyone choose the unequal distribution? The standard answer runs: if talented people are paid more, they will do work that produces more for everyone, and some of that increase reaches the worst-off. In the numbers, it is the step from 5 to 6 at the bottom that is bought by the 9 and the 20 above.
This is an argument, and it has a premise the lab cannot check. If the high rewards do not in fact raise the floor, the unequal distribution has nothing to recommend it, and leximin ranks it below an equal one with the same bottom. The lab ranks the two vectors you type; whether the first caused the second is what the philosophy is about.
Example · When equality is the better answer
Suppose the unequal distribution is (5, 8, 20) and the equal one is (5, 5, 5). The floors tie at 5, and the next entries are 8 and 5, so leximin ranks the unequal one higher. Now suppose the unequal distribution is (4, 8, 20). The floor is lower, and leximin ranks the level one higher, however large the top entry.
Three limits. Rawls applied the principle to the holdings of primary goods, such as income, rights and opportunities, and not to a single scale of welfare; he took the worst-off to be a representative group and not a named person; and he described the lexical form used here, which looks upward when the floors tie, as a refinement his argument did not need, since he expected the position of the worst-off to settle the cases that matter. The lab compares vectors of numbers, which is the cleanest form of the rule and not the whole of his view.
Interactive
Two distributions, six rules
the same people ranked by total, average, priority, the worst off, and sufficiency
Method
Ranking two distributions by leximin
Four moves. The only way to go wrong is to stop looking after the first entry when it ties.
1
Sort each distribution from the lowest entry up
Order matters in the comparison and the identity of the person does not. The lab sorts for you and shows the order.
2
Compare the first entries
If one is larger, that distribution is ranked higher and the rest is irrelevant, however much larger the later entries are in the other.
3
If they tie, compare the second entries, then the third
Move up one position at a time and stop at the first difference. If every entry matches, the distributions are ranked equal.
4
Ask what produced the numbers
If an inequality is to be defended, say what it does for the bottom entry. The justification is a claim about incentives or output, and it is a premise of the argument and not a result of the ranking.
Reading a leximin verdict
Score 0 / 0Choose the answer you can justify.
Leximin on two pairs
First the case where the floor is raised, then the case where the floors tie.
A = 5, 5, 5 B = 6, 9, 20 first entries 5 < 6
B ranks above A, whatever the 9 and 20 are
A = 5, 8, 8 B = 5, 6, 20 first entries 5 = 5
second entries 8 > 6, so A ranks above B
In the second pair B has the larger total, 31 against 21, and a total rule would choose it. Leximin does not add, so the larger total cannot buy a lower second entry.
Common mistakes
1
Thinking the difference principle demands equalityIt permits any inequality that leaves the worst-off better off than they would otherwise be. (6, 9, 20) beats (5, 5, 5) because 6 is more than 5, and the 9 and 20 count neither for nor against. Equality is the answer only when no inequality raises the floor.
2
Stopping at a tie in the first entryEqual floors do not make equal distributions. Compare (5, 8, 8) and (5, 6, 20): maximin finds a tie, and leximin goes on to the second entries and finds that 8 beats 6.
3
Reading the ranking as a defence of inequalityThe rule ranks the vectors it is given. That the unequal one has the higher floor is a premise somebody has to establish. If the high rewards did nothing for the bottom, the same rule would rank them below a level distribution with the same floor.
Check your understanding
✓
Finish when you can rank two distributions by leximin and say what an inequality has to do to be allowed.
Given two lists of welfare levels, sort them, find the first position at which they differ and name the winner; then state, for an unequal winner, the claim about incentives that the verdict depends on.
Note: Try a distribution whose lowest entry is zero and the others enormous: leximin ranks it below any distribution whose lowest entry is positive. That is the rule at its most uncompromising, and it is why critics ask whether a lexical priority is too strong.