What the agenda leaves standing

The doctrinal paradox showed a court of three whose majorities accept two premises and reject the conclusion that follows from them. The impossibility theorem showed that no rule treating judges alike and questions alike can avoid it. And the agenda’s own logic decided where the trouble lives: majority is safe exactly when every set of answers that cannot all be true has only two members.

That leaves a question the theorems answer only in outline. Drop the demands that judges and questions be treated alike, and keep the ones that seem hardest to give up: each question is settled by the votes on that question and nothing else, a unanimous court is followed, a judge switching to yes never turns the court’s yes into a no, and the verdicts hang together on every possible profile. Which rules are left?

The answer is not one list but four, because it depends on the agenda. This essay finds the four lists by trying every rule, and then finds what in the agenda’s logic decides which list it gets.

Every rule, tried #

For three judges, a rule for one question is a function from the eight possible patterns of votes to a verdict. Requiring that it follow a unanimous court fixes two of the eight entries, and requiring that it respond in the right direction cuts the remaining possibilities to eighteen functions. A rule for an agenda of three questions is a choice of one function per question, so there are $18^3 = 5{,}832$ candidate rules; for two questions, $18^2 = 324$.

Each candidate was run on every profile — every way of giving each judge a consistent set of answers — and kept only if its verdict was consistent every time. The search was done twice, once plainly and once pruning candidates that an earlier question already rules out, and the two counts agree.

The four rows could hardly be more different. On two unconnected questions every rule survives, since no answer to one question constrains the other. On the chain — “is $x$ at least one?” and “is $x$ at least two?” — a hundred and twenty-nine of three hundred and twenty-four survive, majority on both among them. On the conjunction, seven of five thousand eight hundred and thirty-two. On the ranking, where a court decides whether A beats B, whether B beats C and whether A beats C, and must not produce a cycle, three.

The three survivors on the ranking are the three dictators: the rules that copy one judge’s answers on every question. That is Arrow’s theorem for three voters and three options, found by search rather than by proof, and with the extra monotonicity condition it is the whole of what survives. The seven on the conjunction are the three dictators and four more.

Seven rules, one shape #

The four extra survivors on the conjunction are the rules in which a fixed set of two or three judges decides, each question being accepted exactly when every judge in the set accepts it. With all three judges in the set it is unanimity: accept only what everyone accepts. With two it is a partnership in which either partner can block. With one it is a dictator again. A set of judges with this power is called an oligarchy, and every non-empty set of the three judges gives one: seven rules.

The search found something stronger than that these seven work. It found that nothing else does, and in particular that the same set must decide all three questions. A rule that let judges 1 and 2 decide the first premise and judges 2 and 3 the second fails somewhere, and so does any mixture of an oligarchy on one question with some other function on another.

The divided court shows why oligarchy is consistent. A set of judges that must all agree accepts a premise only when every member does, so when it accepts both premises, every member accepts both, and every member therefore accepts the conclusion; the set accepts the conclusion too. When it accepts the conclusion, every member accepts it, so every member accepts each premise. The oligarchy’s verdict is the answer that all of its members share, and a shared answer is consistent because each member’s answer is.

It is also clear from the table what the consistency costs. On this court, the majority accepts both premises and rejects the conclusion. The dictatorship of judge 1 accepts everything; the dictatorships of judges 2 and 3 each reject a premise; and every larger oligarchy rejects the conclusion and at least one premise — judges 2 and 3 together reject both — overruling a majority on a premise to do it. The rules that treat more than one judge as authoritative achieve consistency by giving each member a veto.

Four judges, the same lists #

Three judges is a small court, and a pattern that holds for three may be an accident of three. The same search with four judges tries far more rules: there are a hundred and sixty-six monotone unanimous functions of four votes, and more than four and a half million candidate rules on a three-question agenda.

The lists have the shape the three-judge search suggested. On the ranking, four survivors, one dictator per judge. On the conjunction, fifteen, one oligarchy per non-empty set of judges: $2^4 - 1$. For $n$ judges the conjunction keeps $2^n - 1$ rules and the ranking keeps $n$, and the counts for three and four are what those formulas give.

The search also answers a worry about the monotonicity condition. Perhaps it was doing the work, and without it more rules would survive. For three judges the search was repeated without it, over all sixty-four unanimous functions per question — more than a quarter of a million rules on the three-question agendas — and the survivors on the conjunction and the ranking were exactly the same seven and three. On these agendas, a rule that is consistent and unanimous is monotone whether or not that was asked of it.

Why the chain is generous #

The chain’s hundred and twenty-nine survivors are worth a closer look, because they are the opposite case: not a short list of forced institutions, but a large family with a one-line description.

On the chain, a judge who accepts “at least two” also accepts “at least one”, so the judges who accept the stricter question are always among those who accept the weaker. The only way a court can contradict itself is to accept the stricter question and reject the weaker. A pair of rules avoids that exactly when every pattern of votes that carries the stricter question would also carry the weaker one — when the rule for “at least two” is never more willing than the rule for “at least one”. That is the whole condition, and the survivors are all the pairs of rules ordered that way. Among eighteen functions of three votes, a hundred and twenty-nine such ordered pairs exist, and the search counts exactly those.

Majority on both questions is one of them. So is majority on the weaker question and unanimity on the stricter; so is one judge deciding the weaker question alone while a set containing that judge decides the stricter one. What these have in common is not a shape but an inequality, and an inequality between two rules is something a body can arrange by choosing its rules, whereas the conjunction and the ranking leave nothing to arrange.

The unconnected pair is more generous still, because it has no condition at all. Any rule for the first question can sit beside any rule for the second, since no answer to one constrains the other; all three hundred and twenty-four candidates survive. The four agendas therefore run from no constraint, through a single inequality, to a forced shape, and finally to a forced person. The agendas are no larger from one row to the next — two or three questions each — so what changes along the table is how tightly the questions are tied together.

Which answers force which #

The four agendas are all small, and in three of them some answers constrain others. What separates the ranking, which forces a dictator, from the conjunction, which allows oligarchies, has to be something about how those constraints are arranged.

The tool that exposes it is conditional entailment. Take one of the smallest sets of answers that cannot all be true — accepting both premises and rejecting the conclusion, say. Hold all but two of its members fixed. Then accepting one of the remaining two forces rejecting the other: with the second premise accepted, accepting the first premise forces accepting the conclusion. Draw an arrow for every such forcing.

On the conjunction, the arrows fall into two groups. The accepting answers reach one another: accepting the conclusion forces accepting each premise, and each premise, with the other held, forces the conclusion. The rejecting answers reach one another too. And there are arrows from the accepting group to the rejecting one: given that the conclusion is rejected, accepting one premise forces rejecting the other. But there is no arrow back. Nothing on this agenda ever forces a yes from a no.

On the ranking, every answer reaches every other. Klaus Nehring and Clemens Puppe named this property total blocking: an agenda is totally blocked when conditional entailment connects every answer to every other, in both directions. Elad Dokow and Ron Holzman proved in 2010 that on a totally blocked agenda — with one further technical condition that excludes agendas behaving like arithmetic modulo two — every independent, unanimous, consistent rule is a dictatorship. That is the ranking’s row of the table, and it does not need monotonicity.

The conjunction is not totally blocked, and the missing direction is exactly the room the oligarchies use. An oligarchy leans towards no: any member can turn a yes into a no, and nothing can turn a no into a yes. On an agenda where rejections never force acceptances, that bias is safe. On the ranking it is not, because a rejection of A over B is the acceptance of B over A, and every rejection on that agenda forces something.

What a veto costs #

An oligarchy is consistent and, unless its set is one judge, no member dictates. It is still a strange institution, and its cost can be counted rather than argued about.

Measured by agreement with the majority on each question, a pair of judges does best of the seven: it overrules the majority on a fifth of all questions, against nearly a quarter for a dictator and nearly a third for unanimity. That is not a good score, but it is the best available among consistent rules of this kind.

Measured by the conclusion, the picture changes. The majority finds for the conclusion in about a sixth of profiles. A single judge finds for it in a quarter, which is more often than the majority does, because a dictator who accepts both premises is followed wherever the others stand. A pair finds for it in one profile in sixteen, and unanimity in one in sixty-four — only when every judge accepts both premises. The larger the set, the more rarely the court ever says yes.

For a court deciding liability, a bias toward no is the familiar default for the defendant, and some bodies would accept it. For a committee deciding whether to act, the same bias is a committee that never acts. The search does not choose between those; it shows that on this agenda the choice among consistent, independent rules is a choice of who holds a veto.

Complete quota rules #

The rules most bodies actually use are quota rules: accept a question when at least a fixed number of judges accept it. Majority is one; unanimity is another. A quota rule treats judges alike, so on the conjunction it can only be an oligarchy if the set is everyone. The grid below tries every quota rule for five judges on the chain and on the conjunction.

On the chain, the safe pairs are those whose quota for “at least two” is no lower than the quota for “at least one”. That is the whole condition. Anyone who accepts the stricter question also accepts the weaker, so if enough judges accept the stricter to meet a quota, at least as many accept the weaker, and the weaker quota is met whenever it is no higher.

On the conjunction, of a hundred and twenty-five triples of quotas only one is safe: five of five on every question. This seems to contradict the result for supermajorities, which found that a quota above two-thirds keeps the conjunction consistent. It does not. That result concerns rules that may leave a question undecided — accepting a premise when enough judges accept it, rejecting it when enough reject it, and saying nothing in between. The rules here must answer every question, and a rule forced to say either yes or no loses consistency the moment its quota drops below unanimity. What the supermajority buys, on this agenda, is consistency paid for in silence; what the complete rule must pay is a veto.

What the search cannot show #

The search is exhaustive over its rules and its profiles, and the counts are exact. But it covers four small agendas and courts of three and four judges. The formulas $2^n - 1$ and $n$ are what the general theorems give, and the search confirms them at two sizes; it does not prove them for every size.

Nor is total blocking computed here for large agendas. On four agendas the graph is small enough to draw; on an agenda of dozens of interlinked propositions, finding every smallest inconsistent set is itself a hard search, and deciding whether the graph connects everything is harder still. The classification tells a body what its agenda forces only if the body can work out which kind of agenda it has.

And the conditions were chosen, not derived. Independence — deciding each question from its own votes — is the one the whole subject is about, and the reason to want it is that it makes a rule hard to manipulate: a judge cannot change the verdict on one question by misreporting another. Unanimity and monotonicity are weaker still. A body willing to give up independence has the premise-based and conclusion-based procedures and the nearest consistent verdict, none of which appears in this search because none of them is independent.

Still open: agendas between the extremes #

The ranking is totally blocked and forces a dictator; the conjunction is blocked in one direction and admits oligarchies; the chain is safe for majority. Real agendas sit in between and mix all three kinds of structure: some questions linked by chains, some by conjunctions, some by cycles. What survives on a given mixed agenda follows in principle from its entailment graph, and characterisations exist for many classes of agenda, but reading the answer off a large agenda remains a computation rather than an inspection.

There is also the matter of how the survivors should be compared. The dictator, the pair and unanimity are all consistent; they differ in how often they overrule the majority and in which direction they lean, and the figures above count both. Which of those counts should matter more is not a mathematical question. It depends on what a wrong yes costs against a wrong no, which is the question a court’s standard of proof already answers and a committee’s quorum rules already answer — differently, and for reasons the aggregation theory does not supply.

The logic chooses the institution #

Four agendas, one set of conditions, and four different lists of survivors: everything on the unconnected pair, the respectful rules on the chain, the oligarchies on the conjunction and the dictators on the ranking. None of the difference came from the rules. It came from which answers force which, and in particular from whether that forcing ever runs back from a rejection to an acceptance.

That is a useful way to read the whole of this subject’s impossibility. The theorem does not say that consistent collective judgement is impossible. It says that on some agendas it requires a dictator, on others a veto, and on others nothing at all — and that which of those a body faces is decided by the logical shape of the questions it has agreed to answer before any judge has voted.