1
Priority weights by level
A prioritarian counts a gain more when the person gaining has less. The weight depends on how well off the person is in absolute terms, not on how others are doing.
Lesson · Ethics and the Arithmetic of Welfare · Consequentialism
Priority, Equality and Levelling Down
A priority rule gives a gain more weight the lower the recipient starts. An equality rule looks at the spread. Compute a priority score and the Gini coefficient for two distributions, and use the levelling-down case to show that the two can disagree.
priority: a gain counts more the lower down
g(x) = x up to the knee, then half
Gini 0: all equal; larger: more spread
levelling down: more equal, better for no one
Priority against equality
Core ideas
What this lesson establishes
The sum treats a unit of welfare the same wherever it falls. Two families of views object, for different reasons, and the lab can tell them apart.
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Equality looks at the spread
An egalitarian cares about inequality as such, which the Gini coefficient measures: 0 for a perfectly equal distribution, larger as the welfare spreads.
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Levelling down separates them
Bringing the better off down to the level of the worse off lowers the Gini and benefits no one. Priority sees nothing gained; a rule that ranks by inequality alone sees an improvement.
The material
Weights, spreads and the levelling-down case
A concave weighting, a measure of inequality, and the distribution where they come apart.
Definition · Prioritarian score
Choose a knee. A person’s welfare x counts as x up to the knee and counts for only half of what lies above it, so g(x) = x for x ≤ knee and g(x) = knee + (x − knee)/2 above. The score of a distribution is the sum of g over its people.
This is the simplest weighting that has the right shape: a unit of welfare counts for less the more a person already has. A smoother curve would shift the numbers and keep the structure, and the knee is a parameter to move, not a fact.
Take the priority preset, with the knee at 8, A = 6, 14 and B = 9, 9. Under A, g(6) = 6 and g(14) = 8 + 3 = 11, a score of 17. Under B, each g(9) = 17/2, a score of 17. The lab prints a tie, although the totals are 20 and 18. The six counts in full; the fourteen is discounted.
Definition · Gini coefficient
The Gini coefficient is the sum of |xᵢ − xⱼ| over every ordered pair of people, divided by 2·n² times the mean. It is 0 when everyone has the same, and grows as the welfare spreads.
For 1, 30, 2 the unordered gaps are 29, 1 and 28, so the ordered sum is 116, the divisor is 2·9·11 = 198, and the Gini is 58/99. For 10, 10, 10 it is 0.
Before the levelling-down case, a simpler one shows what the Gini does not measure. In the first preset, A = 10, 10, 10 and B = 5, 5, 5 both have Gini 0, so a rule that ranks by inequality alone has no preference between them. Priority prefers A, 27 to 15, because every person is better off. The Gini reads the spread and is blind to the level.
The levelling-down case is the one where the better off are brought down to the worst off and no one gains. In the levelled preset, A = 10, 2 has Gini 1/3 and B = 2, 2 has Gini 0. Ranked by inequality alone, B wins. Priority scores A at 11 and B at 4, and prefers A. The person at 2 is no better off under B, and the person at 10 is worse off.
That is the levelling-down objection. The egalitarian has replies: that equality is one good among several, so a levelled outcome can be worse all things considered while still better in the one respect. The lab ranks by one thing at a time and cannot say how the respects should be combined. What it shows is that priority and equality are different views, even though both favour the worse off in the ordinary case.
Interactive
Two distributions, six rules
the same people ranked by total, average, priority, the worst off, and sufficiency
Method
Applying a priority weight and an inequality measure
Five steps. The last compares the two verdicts.
1
Choose the knee
State where the weighting turns. It is a parameter of the view and should be named before anything is computed.
2
Transform each welfare number
Replace x by g(x): unchanged up to the knee, half the excess above it.
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Add the transformed numbers
That sum is the score. Compare the two scores; equal scores tie.
4
Compute the Gini of each distribution
Sum the gaps over every ordered pair, divide by 2·n² times the mean. A zero means perfect equality.
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Compare the verdicts
Does the lower Gini go with the higher priority score? Where it does not, the two views disagree, and the case is a levelling-down case.
Priority or equality
Score 0 / 0Choose the answer you can justify.
Priority and Gini on two distributions
Knee at 8. Weight each welfare number, add, then compute the Gini.
knee 8: g(x) = x up to 8, else 8 + (x - 8)/2
A = 6, 14 g = 6, 11 score 17 total 20
B = 9, 9 g = 17/2 each score 17 total 18
priority says a tie, 17 against 17
Gini of A = 1/5 Gini of B = 0
The two distributions are equal in priority terms and very different in spread. A rule that ranks by Gini alone would prefer B, and one that ranks by total would prefer A; priority is the position in between.
Common mistakes
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Treating prioritarianism as egalitarianismPriority gives weight to a person’s level, not to the gap between people. When everyone is brought down by half the Gini has no preference and priority prefers A; when only the better off are brought down the Gini prefers B and priority still prefers A. The views agree often and are not the same.
2
Reading a Gini of 0 as a good outcomeThe Gini measures the spread and says nothing about the level. Everyone at 5 has the same Gini as everyone at 10.
3
Taking the knee as a factThe knee and the half slope are parameters of this lab’s weighting. Move the knee and the tie in the priority preset moves with it; the structure of the view remains.
Check your understanding
✓
Finish when you can apply a weighting and a Gini and say where they differ.
Given two distributions, you should be able to compute the prioritarian score and the Gini coefficient of each, rank them under both, and say whether the case is one of levelling down.
Note: Several variants of both views exist, including sufficientarianism, which counts only whether people fall below a threshold. The lab supports that rule too, and Justice and Collective Choice returns to it.