Concept

Aggregation

Combining several individuals' positions into one collective position. Whether it can be done coherently depends on the logical structure of what is being aggregated, and the impossibility theorems say when it cannot.

Named by 5 essays across one field — each of them below, with the objects they name alongside it.

3 consistent judges, and a majority that is not. A table of judges against three questions, every judge's row internally consistent, with the majority answer to each question underneath forming a combination no judge holds.

The court that contradicts itself

Three judges each answer three questions, and each answers them consistently. Take the majority on each question separately and the answers no longer hang together — the body as a whole endorses a combination no member of it holds, and no rearrangement of the procedure removes the problem.

applied · Judgement aggregation

16 rules, and none that survives. A table of every systematic anonymous aggregation rule for 3 judges: one row per rule, showing the verdict it gives at each count of yes-votes, whether it decides every proposition, and whether it is consistent. No row has both.

No rule escapes the doctrinal paradox

A court whose members each hold a consistent position can reach an inconsistent verdict by majority. One such case is easy to build, which invites the hope that a better rule would avoid it — and every rule that responds to the votes at all fails somewhere.

applied · Judgement aggregation

Where the two procedures part company. A table of 3 judges' verdicts on two premises and the conclusion each is committed to, with the two majorities at the foot disagreeing about the conclusion.

Deciding the premises or the conclusion

A body that cannot be both decisive and coherent has to choose which. The two live options are to vote on the reasons and let the verdict follow, or to vote on the verdict and let the reasons look after themselves — and they reach opposite answers on exactly the profiles the impossibility identifies.

applied · Judgement aggregation

How good each judge must be, as the premises multiply. k = 1: 0.5000; k = 2: 0.7071; k = 3: 0.7937; k = 4: 0.8409; k = 5: 0.8706; k = 6: 0.8909; k = 7: 0.9057; k = 8: 0.9170; k = 9: 0.9259; k = 10: 0.9330.

Every premise makes the verdict vote stricter

A court that must find several things true before it finds liability can vote on each of them or on the verdict. With two elements, judges right seven times in ten are enough for either. With five, the verdict vote needs judges right eighty-seven times in a hundred, or a larger court makes it worse — while the premise vote still needs only better than even. And when any one ground is enough instead of all, the two procedures swap their failures.

applied · Judgement aggregation

A party that loses every issue wins the vote. Groups A: 110, B: 101, C: 011, D: 000, E: 000; X wins 3–2; issues won by X: 0 of 3.

A party that loses every issue can win

Two parties take opposite sides on three issues, and every voter backs the party they agree with on more of them. The party that wins the election can be on the losing side of all three issues — in a fifth of random electorates it loses most of them, and in about one in a hundred it loses every one. A majority of voters can even find themselves outvoted on most of the questions decided. Only when issues are settled by three-to-one majorities on average is that ruled out.

applied · Voting rules

Named alongside it

The objects these essays reach for when they reach for this one.

Consistency Judgement aggregation Majority Voting rule Exhaustive search Impossibility Logic Majority rule Proposition Axiom Condorcet jury theorem Condorcet paradox

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