No rule escapes the doctrinal paradox

The anchor’s first rung sets out the doctrinal paradox. Three judges must decide whether a contract was valid, whether it was breached, and — as the law requires — whether damages are owed, which is exactly the conjunction of the first two. Each judge is internally consistent; the majority verdicts on the three questions are not.

The natural response is to look for a better rule. Majority voting is one rule among many, most aggregation problems admit several, and the paradox looks like the sort of thing a cleverer procedure would sidestep. It is the response every reader has, and it is the right response — the question is only whether the search succeeds.

No procedure sidesteps it, and the reason is not subtle enough to need a subtle proof: on a small agenda the rules can simply be listed.

What kind of rule is being considered #

The theorem’s conditions are restrictions on what a rule may look like, and each is doing work.

Universal domain. The rule takes an answer from every possible profile of consistent individual judgements. It is not allowed to say “this combination cannot arise” — because it can, and a court cannot refuse to sit.

Anonymity. Swapping two judges’ positions does not change the verdict. That excludes dictatorships and weighted votes, and it is what makes a rule a function of the counts rather than of who voted how.

Systematicity. The verdict on a proposition depends only on the pattern of votes on that proposition, and the same function is used for every proposition. That excludes rules that treat the conclusion differently from the premises.

And collective rationality. The output is a complete and consistent set of judgements: a view on every proposition and its negation, with no contradiction. That is the condition the individuals are already required to satisfy, asked of the group — which is why it is called collective rationality and why demanding it seems so modest.

An anonymous systematic rule on this agenda is therefore nothing more than a function from the number of yes-votes to a verdict. With three judges there are four possible counts — nought, one, two or three — so there are sixteen such rules, and every one can be examined.

That reduction is the whole reason an exhaustive search is possible, and it is worth appreciating how much the two conditions buy. Without anonymity a rule would be a function of who voted how, which for three judges on three propositions is a function on a space of five hundred and twelve points; without systematicity each proposition could have its own rule. Together they collapse the space to sixteen. The conditions that make the theorem hard to satisfy are the same ones that make it easy to check.

The two conditions, and why they conflict #

Two of the four conditions bite, and they bite in opposite directions.

Completeness forces the rule to say something about $p$ and about $\neg p$, and never both. A rule deciding $p$ from the count $k$ decides $\neg p$ from the count $3 - k$, so completeness requires

$$\text{rule}(k) = 1 - \text{rule}(3 - k)$$

for every $k$. That is a strong condition: it fixes half the rule’s values from the other half, and it rules out both constant rules and the unanimity rule at a stroke.

Consistency forces the verdict on the conjunction to agree with the verdicts on the two premises, at every profile.

The hero figure tabulates both conditions for all sixteen rules. Some satisfy the first; some satisfy the second; none satisfies both.

Which rules survive each condition #

Reading the table by column is more informative than reading the verdict, so it is worth doing.

The rules that are consistent and not complete are those that fail to take a view somewhere. The two constants — always yes, always no — are consistent trivially and ignore the judges entirely. Unanimity is consistent and incomplete: where two judges accept $p$ and one accepts $\neg p$, it accepts neither.

Unanimity is the interesting survivor, because it is a real rule that people use. It is consistent, it responds to the votes, and it is anonymous and systematic. What it is not is decisive: it frequently returns no verdict at all, which for a court is not an option.

It is worth noticing that this is exactly the escape a criminal jury takes. A unanimous jury is coherent — it never convicts on premises it has not all accepted — and it hangs, often, which is the incompleteness showing up as an outcome the system has a procedure for. The impossibility is not avoided there; it is absorbed by having somewhere to put a non-verdict, and a body without that option cannot use the escape.

And the rules that are complete include majority, which is the paradox’s own case, and three others that are stranger — the rule that says yes on a minority and no on a majority, for instance, which is complete by the same arithmetic and is not something anybody would propose — and every one of them contradicts itself on some profile.

The strange ones are worth not dismissing. The completeness condition is a formal one, and it admits rules nobody wants; the search covers them because excluding them would require a further condition, and adding conditions to reach an impossibility weakens the result. An impossibility is stronger the fewer conditions it uses, so the right move is to let the absurd rules in and observe that they fail too.

So the impossibility has a precise shape: a rule may be decisive or it may be coherent, and choosing decisiveness is what a court has to do.

The three conditions in that figure are worth naming separately from the theorem, because a court has to satisfy all of them and they are not mathematical requirements.

The conclusion is what the case turns on. Damages are owed or they are not, and the court has to say which; “no verdict” is not available.

The reasoning is public. A court states its findings on the premises as well as its conclusion, so both are on the record and an inconsistency between them is visible.

And the judges are equals. No judge’s view counts twice, which is anonymity, and it is a constitutional requirement rather than a modelling convenience.

So a court is committed to universal domain, anonymity, and collective rationality before the mathematics starts, and the theorem says those three cannot be had together with systematicity. Which of the four to give up is the subject of the rung above.

How often it happens #

The impossibility says some profile causes trouble, which would be a small matter if such profiles were rare. They are not.

The figure counts them: over all ways three judges can hold consistent positions on two premises, a substantial fraction produce a clash between the two procedures. The proportion depends on the agenda and on how the judges’ positions are distributed, and for the plausible distributions it is not small.

That is what separates this from a technicality. The Condorcet cycle in voting is also possible for every profile-independent rule, and is comparatively rare under many models of how people vote; the doctrinal paradox is not rare, because the propositions are logically related and the relation is exactly what creates the clash.

The general principle is that the more connected an agenda is — the more the propositions imply one another — the more often aggregation fails. An agenda of unrelated propositions has no paradox at all, because there is nothing for the verdicts to contradict.

The general theorem #

The three-judge two-premise case is a complete instance and it is not the theorem. List and Pettit’s result, from 2002, is that on any agenda satisfying a mild connectedness condition, no rule is universal, anonymous, systematic and collectively rational.

The connectedness condition is worth stating because it identifies exactly what makes an agenda dangerous: the agenda must contain a minimal inconsistent subset of at least three propositions. That is, some three propositions cannot all hold together, though any two of them can.

Two premises and their conjunction have that shape: $p$, $q$ and $\neg(p \wedge q)$ cannot all hold, and any two of them can — which is what “minimal” means and is exactly the configuration that lets three judges each reject a different one. The paradox is a property of the agenda rather than of any rule, and an agenda without such a subset has no paradox.

There is a converse worth having, because it says the connectedness condition is exactly right rather than merely sufficient. If an agenda has no minimal inconsistent subset of size three or more — if every inconsistency involves only two propositions, which means only a proposition and its negation — then majority voting is perfectly well behaved and there is no paradox. So the theorem’s hypothesis is a characterisation, and an agenda is dangerous precisely when three of its propositions conflict without any two doing so.

Later work weakens the conditions considerably. Dropping systematicity and keeping only independence — the verdict on a proposition depends only on votes on that proposition, but different propositions may use different rules — leaves the impossibility standing, with dictatorship as the only escape. That is Dietrich and List’s version and it is the exact analogue of Arrow’s theorem for judgements rather than preferences.

It is worth being explicit that “how often” depends on a model of how judges’ views are distributed, and the figure’s count assumes every consistent combination is equally likely — the impartial culture assumption, which nobody believes and everybody uses for a first estimate.

Real courts are not impartial cultures: judges’ positions correlate, because they are reasoning about the same evidence. Correlation cuts both ways. Judges who agree entirely produce no paradox; judges who split along one axis and agree on the other produce it constantly. So the practical frequency is a fact about the deliberating body rather than about the arithmetic, and the arithmetic supplies only the bound that it cannot be zero.

What it is not #

Two readings are tempting and wrong.

It is not that majorities are unreliable. Every individual majority verdict here is perfectly reliable — it is what most judges think about that question. What fails is the combination, and no single verdict is at fault.

It is not that the judges are being irrational. Every one of them holds a consistent set of views, which the figure requires. The inconsistency belongs to the collective and is produced by the aggregation, which is a genuinely different thing from anybody being wrong.

And it is not Arrow’s theorem in disguise, though it is closely related. Arrow’s is about aggregating preference orderings; this is about aggregating sets of judgements. Each can be embedded in the other, and the embeddings are not trivial; the two were proved fifty years apart and the connection was noticed later.

The relationship is worth stating precisely because it is often stated loosely. An ordering is a set of judgements about pairwise comparisons, subject to consistency conditions — transitivity chiefly — so a preference aggregation problem is a judgement aggregation problem on a particular agenda. In that reading Arrow’s theorem is a special case, and the judgement framework is the more general one.

Where it was noticed #

The paradox has two independent discoveries, forty years apart, and the second is the one that started a subject.

Lewis Kornhauser and Lawrence Sager described it in 1986, in a law review, as a problem for appellate courts — they called it the doctrinal paradox, and their concern was practical: which of the two procedures should a court adopt, and does the choice change outcomes? It changes outcomes.

Christian List and Philip Pettit generalised it in 2002 into a theorem about aggregation, at which point it stopped being a curiosity about courts and became the founding result of judgement aggregation — a field which now has the same shape as social choice theory, with a menu of conditions, a menu of impossibilities, and a catalogue of which combinations survive.

And Vilfredo Pareto had the essential observation in 1906, in a footnote about a committee reasoning from premises, which nobody followed up.

The pattern is the one that recurs whenever an impossibility is found: the phenomenon is noticed as an oddity in a particular setting, restated as a theorem about a class, and then recognised as an instance of something more general still. Kornhauser and Sager’s contribution was to see that a court’s procedure is a choice; List and Pettit’s was to see that no choice works; and the field since has been mapping exactly which conditions can be dropped.

What the pictures cannot show #

The table’s rows are rules and a reader cannot tell which is which. Each row is a string of yes and no verdicts against vote counts, and identifying majority among them takes a moment’s arithmetic. The figure is a census rather than an exhibit, and its content is the two columns at the right.

Sixteen rules is the whole space and only for this agenda. The figure’s exhaustiveness is real and its scope is one agenda with one shape. The general theorem quantifies over agendas, and the search establishes an instance of it.

Anonymity and systematicity are assumed, not tested. The rules examined are the anonymous systematic ones because those are the ones describable by a vote count. A rule that reads the names, or treats the conclusion differently, is outside the table entirely — and dropping those conditions is exactly how the escapes below are found.

The two searches are at three and five judges and the theorem is about every number. Both find the same verdict, which is evidence and not a proof; what makes it a theorem is the combinatorial argument about minimal inconsistent sets, which is where the parity of the count does not appear at all.

And the profiles are not drawn. Each row’s consistency verdict summarises a search over sixty-four profiles for the three-judge case, and only one of those profiles appears in a figure. What a rule does on the other sixty-three is a computation reported as a word.

Where the ladder goes next #

The next rung takes the impossibility as given and asks what a body actually does: decide the premises or decide the conclusion, which are the two live options and disagree exactly where this rung’s search says they must.

Named here as debts. The general theorem’s proof, which is a combinatorial argument about minimal inconsistent sets and is stated above rather than given. And the escape routes — supermajority rules, restricted domains, sequential procedures — each of which drops one condition and is a subject of its own.

Sideways, the preference version of the same obstruction is Arrow’s, the exhaustion over a small space of rules is the method the voting essays use throughout, the logical structure of an agenda is what quantifier order is about in a different setting, and the paradox itself is the anchor’s first rung.

Sideways: four conditions and no rule is the same shape of argument about preferences rather than propositions, and a share of the votes is not a share of the power is what happens when the aggregation is of influence instead.

What is worth carrying away #

When a paradox appears under one rule, the question is whether it is a fact about the rule or about the problem, and enumerating the rules settles it.

Majority voting produces the doctrinal paradox, and the instinct is to blame majority voting. Listing all sixteen anonymous systematic rules shows that every one of them fails, so the paradox belongs to the agenda: propositions that imply one another cannot be voted on independently and stay coherent.

The habit worth taking is to enumerate the alternatives when the space is small enough. Sixteen rules is a table; the general theorem is a page of combinatorics; and the table settles the question completely for the case anybody would ask about first.