The nearest consistent verdict

Three judges each answer two questions and a third that follows from them: is the first premise true, is the second, and is their conjunction true? Each judge answers consistently. A majority says yes to the first premise, a majority says yes to the second, and a majority says no to the conclusion — a combination no judge holds, and one nobody could hold.

The two standard repairs each privilege one part of the agenda. Deciding the premises takes the majority on each premise and derives the conclusion, overruling the majority on the conclusion. Deciding the conclusion takes the majority there and leaves the premises to look after themselves. A third repair treats all three questions alike. Among the verdicts that are consistent, pick the one that disagrees with the judges least.

On the classic profile the answer is a tie. Accepting everything disagrees with the second and third judges twice each; accepting only the first premise disagrees with the first and third judges twice each; accepting only the second, with the first and second. Each costs four disagreements. Rejecting everything costs five. Three verdicts are equally close and the rule cannot say which.

Distance, and the rule it defines #

A judgement set on this agenda is three answers, and the natural distance between two sets is the number of questions on which they differ. The distance-based rule — studied under that name by Gabriella Pigozzi in 2006, and in older form as a way of merging inconsistent databases — returns the consistent set whose total distance from all the judges is smallest.

Three properties follow immediately. The rule is consistent by construction, since it only ever returns a consistent set. It is anonymous, since the total does not care which judge is which. And it is neutral between the propositions: nothing in the count distinguishes a premise from the conclusion.

What it gives up, among the conditions of the impossibility theorem that lists the four ways out, is the requirement that the verdict on each proposition depend only on the votes on that proposition. Here the verdict on the conclusion can depend on how the votes on the premises are distributed, which is exactly what allows it to be consistent. It also gives up something the theorem takes for granted, which is that a rule returns one answer.

The total has a simpler description than it appears. Each question contributes its minority — the judges who disagree with the majority there — plus, if the chosen set overrules the majority on that question, the size of the majority’s margin. So the rule is: when the majority’s answers are inconsistent, overrule it on the questions where it is narrowest. When the majority is consistent the rule returns it unchanged, since overruling nothing is cheapest.

The same thing on a cube #

The three answers are three coordinates, each 0 or 1, and the eight possible answer patterns are the corners of a cube.

Four corners are consistent: yes to both premises and the conclusion, yes to one premise only, and no to everything. The other four are not, because their third coordinate disagrees with the first two. The judges sit on consistent corners, and the majority, computed coordinate by coordinate, is the median of their corners — the corner that takes each coordinate from the larger group.

A median of consistent points need not be consistent, and here it is not: $110$ says yes, yes, no. The distance-based rule moves it back to the nearest consistent corner. On this profile three consistent corners are adjacent to $110$, one step along each of the three edges out of it, and the cube’s symmetry makes the tie visible. The first two are exchanged by swapping the premises; the third is a different kind of repair, changing the conclusion rather than a premise, and it lies at the same distance only because all three margins are equal.

This geometric picture is the right one for the general theory. The majority is always the median; the paradox is that the consistent corners are not closed under taking medians; and a repair is a rule for projecting back.

The same cube, with the same distance, is the setting of error-correcting codes, and the resemblance is exact. There, a code is a chosen set of corners, a transmitted word arrives with some coordinates flipped, and decoding means moving it to the nearest codeword. Here the consistent judgement sets are the code, the majority’s pattern is the received word, and the distance-based rule is nearest-codeword decoding. The four consistent corners of this cube form a very poor code — the inconsistent corner $110$ is one step from three of them at once, so a single “error” there cannot be corrected unambiguously — and that poverty is the three-way tie. A good code spreads its codewords so that every corner has a unique nearest one; an agenda’s consistent sets are not chosen for that property and rarely have it.

Five judges, and a tie every time #

A tie on one profile might be an accident of three judges. It is not.

With five judges the majority on each question is three, four or five, and its margin over the minority is one, three or five. For the majority to contradict itself, at least three judges must accept each premise while at most two accept both — which forces the two premise majorities to overlap in at most two judges. Work through the possibilities and the margins that can arise on a troubled profile are few: either all three questions are carried by one vote, or one premise is carried by three while the other and the conclusion are carried by one, or both premises by one and the conclusion’s rejection by three.

In every one of those cases, two of the three possible repairs cost the same. When all margins are one, all three repairs tie. When the conclusion’s rejection is overwhelming, overruling it is out of the question and the two premises tie with each other. And when one premise is carried by three, the other premise and the conclusion tie. On every troubled profile of five judges the equally weighted rule returns a tie. It is consistent and anonymous and it is never decisive exactly where a decision was needed.

Weighting the conclusion #

The tie can be broken only by counting some disagreements as worse than others, and the natural choice is to weight the conclusion — the question the court was actually convened to decide — differently from the reasons given for it.

With the conclusion weighted by a half, overruling the majority on the conclusion is cheapest whenever its margin is small, and the rule accepts the conclusion as the premises imply: it becomes the premise-based procedure on 120 of the 150 profiles. On the remaining 30, where four of five judges reject the conclusion, even the discounted cost of overruling them is too high, and the rule instead drops a premise — but cannot say which, since both premises are carried by one vote.

With the conclusion weighted double, the rule never overrules the majority on the conclusion; it keeps the rejection and drops a premise, uniquely on 60 profiles and with a tie between the two premises on the other 90. It has become the conclusion-based procedure, with the premise to abandon chosen by margin where the margins differ.

The two ends of the scale correspond to a real argument in law. Lewis Kornhauser and Lawrence Sager, who named the doctrinal paradox in the 1980s, framed it as a choice between issue-by-issue voting and outcome voting on multi-judge courts, and noted that appellate courts in the United States use both, often without saying which. A weight below one on the conclusion is issue-by-issue voting with a tie-break; a weight above one is outcome voting with a rule for choosing the reasons. The distance count does not decide between them. It makes the choice explicit, as a single number a court would have to announce.

So the distance-based rule is not a third procedure standing apart from the other two. It is a family, parameterised by how much the conclusion matters relative to its reasons, with the premise-based procedure at one end and the conclusion-based procedure at the other. The equally weighted member, which seemed the principled one, is the point where the family’s two ends balance, and balancing is exactly where the ties are.

A judge who knows the rule #

A procedure that looks at the pattern of votes across questions invites a judge to vote strategically across them. Under the premise-based procedure the incentive is familiar. A judge who accepts the first premise, rejects the second and wants the conclusion rejected is safe; but a judge who accepts both premises only mildly, and cares most about the conclusion, can sometimes change the court’s conclusion by misreporting one premise, since the conclusion is computed from the premise majorities and the judge’s own conclusion-vote is ignored.

The distance-based rule does not remove that incentive; it spreads it out. Because the verdict on each question depends on the margins on every question, a judge can shift the outcome on the conclusion by shifting a margin on a premise, and whether doing so pays depends on the whole profile rather than on one question. Franz Dietrich and Christian List proved in 2007 that, for judges who prefer verdicts closer to their own, a rule is immune to this kind of manipulation exactly when it decides each proposition independently and responds to extra support in the right direction. The distance-based rule is designed precisely not to decide propositions independently — that is how it achieves consistency — so on some profile some judge gains by misreporting.

This is the same trade that the four conditions on voting rules and the impossibility for judgements impose, seen from the side of incentives. Independence is what makes a rule hard to game, and independence is what the paradox forbids a consistent rule to have.

A court that cannot choose #

What a tie means for an institution is worth stating plainly. A court that adopts the equally weighted rule has not avoided the doctrinal paradox; it has converted it into a declaration that three verdicts are equally good. Something outside the rule must then decide — a presiding judge’s casting vote, a default for the defendant, a coin. Each of those is a rule that privileges something, and the privilege has merely been moved out of the distance count and into the tie-break.

That is a general lesson about escapes from impossibility theorems. The theorem proves that no rule has all of a list of properties. A rule that seems to have all of them, apart from one that nobody listed, usually turns out to have given up the unlisted one in a costly way. Here the unlisted property is resoluteness — returning a single verdict — and the distance-based rule gives it up on precisely the profiles the theorem is about.

The same lesson appears in the treatment of the strategic side of voting, where rules escape manipulation by being random or by being indecisive, and in the core of a cooperative game, where the stable divisions can be many or none. Collective choice keeps offering the same bargain: a clean criterion, satisfied by a set of outcomes rather than by one.

The same rule for rankings #

Distance-based aggregation is older in the theory of preferences than in the theory of judgements, where it is Kemeny’s rule, proposed by John Kemeny in 1959.

A ranking of three candidates is a judgement set on the agenda of three pairwise questions: is $a$ preferred to $b$, is $b$ preferred to $c$, is $a$ preferred to $c$. Six of the eight answer patterns are rankings; the other two are cycles. Condorcet’s paradox is the majority landing on a cycle — the same median problem as the court’s, on a different cube. Kemeny’s rule returns the ranking that contradicts the fewest pairwise preferences, and on the pure cycle it ties three ways, each tie cutting the cycle at a different link.

Kemeny’s rule is well understood. H. Peyton Young and Arthur Levenglick characterised it in 1978 as the unique rule with a certain list of consistency properties, and Young later argued it was what Condorcet himself had been reaching for. It also has a cost the three-judge pictures hide: John Bartholdi, Craig Tovey and Michael Trick proved in 1989 that computing a Kemeny ranking is NP-hard, so for many candidates no efficient method is known. The judgement version inherits the same difficulty. Finding the nearest consistent verdict means searching the consistent sets, and on a large agenda there are too many to search.

What the tables cannot show #

Every table here is exhaustive over one small agenda — two premises and their conjunction — and over three or five judges. The claim that every troubled profile of five judges ties at equal weights is a search result on 1,024 profiles, backed by the margin argument; for seven judges or more the margins can differ in more ways and ties are no longer forced, so the result is a fact about small courts rather than a theorem about all of them.

The distance used throughout is the plain count of disagreements. Other distances give other rules — weighting some judges, or counting a disagreement on the conclusion as two — and the sweep over the conclusion’s weight explores exactly one direction in that space. Which distance is right is not a mathematical question, and nothing drawn here settles it.

And the tables treat every profile as equally likely. A real court’s judges are correlated, and whether the troubled profiles arise often, or which margins they tend to have, depends on the court. The counts describe the space of possible disagreements, not the frequency of any of them.

The question it leaves: which agendas need a repair at all #

Every rule on this page, and every escape in the essay before it, is a response to one agenda: two premises and a conclusion that depends on both. Some agendas never produce the paradox. A body voting on two unconnected questions can take the majority on each and never contradict itself, and so can a body voting on a chain of thresholds — whether a sentence should exceed one year, two years, three.

What distinguishes them is a property of the agenda’s logic rather than of any rule: whether its smallest inconsistent sets of judgements have three members or only two. That condition, the median property, decides exactly when majority voting is safe, and it is drawn in the essay on agendas that cannot contradict themselves. The distance-based rule is the repair; the median property says when no repair is needed.

The cube already hints at the answer. On the chain of thresholds, the consistent corners are the four patterns $000$, $100$, $110$ and $111$, and the median of any three of them is again one of them — take any three judges’ thresholds and the middle one is a threshold too. Closure under medians is the whole condition, and it holds for some agendas and fails for others for a reason that can be read off their smallest contradictions.

A verdict nobody voted for #

The distance-based rule answers the paradox with the most even-handed repair available, and the pictures show its price. The majority is a median, the median of consistent positions need not be consistent, and the nearest consistent verdict is frequently not one verdict but several at the same distance.

Choosing among them requires saying which disagreements matter more. Weight the conclusion lightly and the rule becomes the premise-based procedure; weight it heavily and it becomes the conclusion-based one; weight it equally and it declines to choose. The two procedures a court was already choosing between turn out to be the two ends of a single scale, and the even-handed middle of that scale is exactly where the doctrinal paradox, instead of disappearing, turns into a tie.

That is not a defeat for the distance idea. It is a precise statement of what the paradox was about. The majority’s contradiction was never a failure of counting; it was an unanswered question about priority — whether a court’s reasons or its decision should prevail when they part — and a rule that treats every question alike cannot answer a question about priority. The distance count gives the question a number to be set, and every consistent procedure is some setting of it.