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flowchart TD
A(["Option A"])
B(["Option B"])
C(["Option C"])
A -->|"beats, 2 of 3<br/>(voters 1 and 3)"| B
B -->|"beats, 2 of 3<br/>(voters 1 and 2)"| C
C -->|"beats, 2 of 3<br/>(voters 2 and 3)"| A
10 Making Group Decisions: Voting, Social Choice, and Ensembles
Every multi-agent system eventually holds an election, and hardly any knows it. Sample a model five times and keep the most common answer: an election with a franchise of five. Ask three reviewers whether the patch should merge; set a judge over a panel of candidates; break a tie with a third agent — elections, every one. Whoever assembles these arrangements is practising social choice — the theory of how many voices become one decision — the way M. Jourdain spoke prose: fluently, daily, with no idea the discipline exists. This chapter is where the bill for that innocence is presented.
The theory, for once, does not merely predate the agents by decades; it predates the telegraph. Borda and Condorcet laid out most of the subject quarrelling before the 1780s Académie Royale — Condorcet proving en route the jury theorem this book leans on hardest — and Arrow’s 1951 axiomatisation showed the paradoxes were no accident. The agents arrived, as ever, last — to an apparatus waiting since before the storming of the Bastille.
One assumption does quiet work through the first half and needs naming: the voters vote sincerely. The reader of Chapter 9 should already smell trouble — a ballot is a message, a message among interested parties is a move, and an agent that cares about the outcome will be tempted to vote for what works rather than what it thinks. How badly that undermines the enterprise is a theorem too, and it arrives on schedule.
10.1 One Choice from Many: The Aggregation Problem
Strip any group decision to its skeleton: voters, alternatives, and each voter’s expression of where it stands — a favourite, a ranking, a score, a yes or no. The collected expressions are a profile, and the object at the centre of social choice is the aggregation rule: the function that turns a profile into a collective verdict (Brandt et al., 2016). The thing to internalise: the rule is a designed artefact. Everything beyond the inputs — what counts as winning, how ties break, who weighs what — somebody chose. In a multi-agent system that somebody is the engineer, and the rule is not even metaphorically a function: it is a function, sitting in the codebase, usually unexamined and quite often unnamed.
Name a few, then. Self-consistency (Section 3.10) samples a model several times and returns the most common answer: plurality rule — most votes wins — over an electorate of samples. And a judge model set over a panel of candidate outputs is a rule so distinctive that it has a technical name in the theory, which Section 10.6 will produce with due ceremony. None of these is wrong; what this chapter exists to end is choosing among them without noticing a choice was made — Table 10.1 decodes the elections hiding in the common patterns.
| Agent-system construct | The classical rule it is | The personality it inherits |
|---|---|---|
| Self-consistency: sample n times, keep the most common answer | Plurality over an electorate of samples | Blind to all but each voter’s top choice; vote-splitting unless ballots are clustered by meaning |
| Ensemble that averages its members’ scores | Score voting (a linear opinion pool); the Borda count only when the scores encode ranks | Consensual: rewards the broadly acceptable answer |
| Ensemble that takes the median | The median rule | Stubborn towards outliers |
| Merge gate: proceed only if all three reviewers approve | Unanimity rule | Cautious, bought at the price of paralysis |
| Two-of-three reviewer sign-off | Majority rule | May’s safe harbour: uniquely fair on a binary question |
| Tie broken by consulting a third agent | A three-voter election | Majority rule’s, inherited — including the correlation risk when the third shares the tied pair’s model |
| Judge model ranking a panel of candidates | Dictatorship (Section 10.6) | Transitive and cycle-free — but the judge’s biases become the group’s policy |
| Pairwise elimination of candidates by a judge | Sequential pairwise vote (a tournament) | Agenda-sensitive: when the judge’s verdicts cycle, the running order picks the winner |
Before any rule can be judged, one question must be put to the vote itself: is there a right answer? Sometimes not: dividing a budget between speed and safety, the votes express preferences, and no measurement could reveal the choice mistaken; the aggregation answers to fairness. Sometimes there is. When three reviewers vote on whether the patch contains a bug, it either does or does not; the votes express judgements — estimates of a fact — disagreement signals error, and the aggregation answers to a standard fairness knows nothing of: accuracy. Preference aggregation is politics; judgement aggregation is measurement. The ballots look identical, and the difference decides which mathematics applies — Section 10.3’s impossibility theorems bite hardest on preferences, Section 10.4’s hopeful statistics belong to judgements — so misdiagnosing a vote’s species means consulting the wrong theory.
Diagnosis takes practice; real votes are frequently hybrids: “which of these five patches should ship?” mixes fact (which are correct) with taste (which style to prefer). The working test is counterfactual: if an oracle revealed the truth, would the disagreement dissolve? Run the test on each vote a system takes and most sit near the judgement end — agent systems mostly aggregate estimates of facts — which is encouraging. Not wholly, though; the classical demonstration of why deserves a place in every practitioner’s head.
Suppose a patch merges exactly when it is correct and adequately tested — a conclusion resting on two premises. Three reviewers assess both: one finds it correct but poorly tested; one, well tested but incorrect; one, sound on both counts. Premise by premise: a majority says correct, a majority says well tested, both pass, so the policy says merge. Conclusion by conclusion: only one reviewer supports merging, two-to-one against. The same three honest ballots yield opposite verdicts depending on which question was put to the vote — and nothing in logic or fairness singles out either count. This is the discursive dilemma, and List and Pettit proved it no curiosity: under mild conditions, no procedure can guarantee consistent collective judgements on logically connected propositions while treating voters and propositions even-handedly (2002). The moral is immediate: an orchestrator that polls its reviewers premise-wise and one that polls them verdict-wise are different constitutions that will, on honest inputs, sometimes ship different code — a substantive choice currently made, in most systems, by whoever wrote the prompt.
10.2 Counting Heads: Voting Rules and Their Surprises
In a memoir printed in the Académie’s volume for 1781, the Chevalier de Borda attacked a method so obvious nobody had thought it needed defending: one vote each, most votes wins (1781). His demonstration — that the method can crown a candidate whom a majority would defeat in any head-to-head contest — set the pattern every later discovery would follow: take a rule that looks like plain common sense, construct a profile of perfectly reasonable votes, and watch it do something indefensible. The lesson of two and a half centuries of such constructions: every rule has a personality that follows it into any system adopting it.
Plurality — each voter names one favourite, the most-named wins — is the rule everyone reinvents first, and its vice is the one Borda exposed: it hears only first choices. Its signature failure is vote-splitting: two similar alternatives divide their supporters, and a third that most voters rank last wins with a plurality neither rival could match — the spoiler effect, where a candidate who cannot win changes who does.
The failure travels straight into Section 3.7’s machinery: self-consistency is plurality over an electorate of samples, and plurality’s blindness comes with it. Sample a model ten times on a hard problem and the correct answer may arrive as “7”, “seven”, and “the answer is 7”, while a tempting wrong answer arrives, each time, in identical form; counted naively, the wrong answer wins on a technicality. So self-consistency implementations must cluster answers by meaning before counting — semantic equivalence as electoral reform, and the difference between a working rule and a spoiler-prone one. An election is only as sound as its notion of which ballots name the same candidate.
Borda’s remedy was to make the ballot say more. In the Borda count, each voter submits a full ranking; with m alternatives, a first place earns m-1 points, down to zero; the highest total wins. The rule listens to the whole of every ranking, and its personality is consensual, favouring the broadly acceptable over the passionately supported and widely loathed — often what a team scoring candidate outputs wants. Its vices are two. The count is exquisitely menu-sensitive: add or remove an also-ran and the points shift beneath the contenders, so the choice between A and B can be reversed by an irrelevant C’s arrival. And the same sensitivity is a lever: a voter who buries a rival — ranking it last from tactics, not conviction — moves real points, and Borda’s airy reply, that his method was intended only for honest men, has been quoted back at him ever since. Among self-interested agents that reply is a bug report filed in advance; the theorem that makes it general arrives in the next section.
Condorcet, Borda’s colleague and rival, proposed a sterner standard (1785): put every pair to a head-to-head majority vote, and if some alternative beats every rival, it — the Condorcet winner — ought to win. It has a claim to be what majority rule means with more than two options, and it sharpens Borda’s complaint into a criterion: plurality stands condemned because it can elect a candidate who is not the Condorcet winner — and so, embarrassingly, can Borda’s own count, which is roughly where the two men’s civility ended.
Then Condorcet turned the method on itself and found the trapdoor. Three voters rank three options: A \succ B \succ C; B \succ C \succ A; C \succ A \succ B. Count the pairs: A beats B two to one; B beats C two to one; C beats A two to one. The majorities form a cycle (Figure 10.1) — the group prefers A to B to C to A — and there is no Condorcet winner. Nothing is wrong with the voters, each holding a perfectly consistent ranking; it is the collective preference that fails to be an ordering. The reader of Chapter 9 has seen the shape — individually impeccable inputs, indefensible aggregate — and Section 10.1’s discursive dilemma is its judgement-flavoured sibling. It is the field’s founding humiliation; Arrow will explain why it cannot be tidied away.
The cycle’s sting is what it hands to whoever controls the order of business: put the options to sequential pairwise votes, winner meets the next. Start with A against B and the outcome is C; start with B against C and it is A; start with A against C and it is B. Same voters, same preferences, three different decisions — chosen, in effect, by whoever set the agenda. The modern instance: an orchestrator whittling candidates through pairwise comparisons by a judge model is running exactly this tournament, and whenever the judge’s pairwise verdicts cycle — the biases catalogued in Section 10.6 make intransitive verdicts an observed fact rather than a theoretical nicety — the bracket, not the judgement, is picking the winner.
Assemble the specimens (Table 10.2) and the lesson states itself: plurality, Borda, and pairwise majority, each a reasonable reading of “let the group decide”, can return three different winners from the same profile of honest votes. The rule is not a neutral pipe from preferences to outcomes; it is a co-author of the result, and the only question is whether its contribution was chosen or inherited from whichever example the framework documentation used. At which point an optimist asks: why not design a better rule, with the virtues and without the vices? The question was asked with full seriousness in 1951, and the answer — no, provably — is next.
| Rule | Ballot it asks for | How the winner is chosen | Personality | Signature pathology |
|---|---|---|---|---|
| Plurality | One favourite named | The most-named alternative wins | The rule everyone reinvents first — so obvious it seems to need no defending | Vote-splitting / the spoiler effect; blind to all but first choices |
| Borda count | A full ranking | Points by rank (m-1 down to 0); highest total wins | Consensual — rewards broad acceptability over passionate support | Menu-sensitivity (an irrelevant also-ran can flip A versus B); rewards burying a rival |
| Pairwise majority (Condorcet) | A full ranking, read as head-to-head contests | The alternative that beats every rival (the Condorcet winner) | A sterner standard — arguably what majority rule means with more than two options | May name no winner at all: majorities can cycle (Condorcet’s paradox); agenda-sensitive |
10.3 Arrow’s Theorem: The Price of Consistency
Arrow’s contribution was to stop auditioning rules and change the question. The subject had proceeded specimen by specimen — propose a rule, find its embarrassment, propose another — until in 1951 Kenneth Arrow, in a doctoral thesis of unassuming size and immodest consequence, inverted it (1951): write down the properties any acceptable rule must surely have, each so mild that refusing it seems perverse, then ask what rules satisfy them all. The method is the axiomatic turn the modern subject is built on; its first application produced the most famous negative result in the social sciences.
The conditions deserve stating, because each sounds like nothing at all. A rule — taking every voter’s full ranking of three or more alternatives, returning a collective ranking — should have universal domain: it accepts any profile. It should respect unanimity: if every voter ranks A above B, the group does too. It should satisfy independence of irrelevant alternatives: the verdict between A and B should depend only on how the voters compare A and B, not on where anyone ranks some third option C — Borda’s menu-sensitivity promoted to a requirement. And non-dictatorship: no voter whose personal ranking simply is the collective ranking, whatever everyone else submits. Four conditions, each a platitude.
Arrow’s theorem is that no rule satisfies all four. Not none discovered so far: none, provably, in the space of all possible rules — any procedure meeting the first three is a dictatorship, and any that avoids dictatorship violates one of the others somewhere. The pathologies of Section 10.2 were therefore never defects of craftsmanship awaiting a cleverer design: Condorcet’s cycle was the first sighting of a theorem, Borda’s menu-sensitivity is where his rule pays Arrow’s tax, and the optimist’s question receives its final answer. There is no such rule; the search that had run since the Académie was not unfinished but impossible.1
What should an engineer do with an impossibility theorem? Not despair, not dismiss — the productive reading is the one this book gave Section 9.2’s PPAD result: a map of what cannot be bought, so that trade-offs are chosen rather than stumbled into. Every aggregation rule in every multi-agent system violates at least one Arrow condition; the design question is which one, and on purpose. Surrendering independence yields the Borda family — serviceable, consensual, menu-sensitive: a fair trade with a fixed menu and honest voters. Surrendering universal domain is subtler and often the engineer’s best move: the theorem needs the awkward profiles to bite, and when preferences share structure — every voter ranking the options along one dimension, cheapest-to-dearest or safest-to-boldest — the cycles vanish and the median voter’s preference beats all others, a result owed to Black (1948), the standing reward for asking agents to express preferences on a scale. Domain restriction is impossibility-avoidance by design: shape what can be voted on and the premises quietly fail to apply.
One escape remains, the clean one: Arrow’s theorem requires three or more alternatives, and with exactly two something better than possibility holds. May proved that for two alternatives simple majority is the only rule satisfying anonymity (voters interchangeable), neutrality (alternatives interchangeable), and positive responsiveness (a gained vote turns a tie into a win, never hurts) (1952) — a uniqueness theorem where its elders are impossibilities. This is the safe harbour, and why the binary gate is the aggregation this book keeps steering towards: merge or don’t, ship or hold, pass or fail are votes May underwrites, while choose among these five candidates is a vote Arrow haunts. Where a decision can honestly be decomposed into two-option questions, the theory’s blessing is unqualified — with the caveat already served: chain binary votes through connected propositions and the discursive dilemma waits, so the decomposition must be chosen with the same care as the rule.
The second impossibility is the one Borda waved away. Gibbard and Satterthwaite independently proved that every rule satisfying two conditions — each of the three or more alternatives can actually win, and no voter dictates — is manipulable: at some profiles a voter does strictly better misreporting its preferences than voting sincerely (1973; 1975). Not the Borda count specifically: every rule. Strategic voting is structural to aggregation among agents with stakes, and the reader of Chapter 9 will recognise the register: a ballot is a message, a message among interested parties is a move (Section 9.5), and sincere voting is, among self-interested agents, exactly the cheap talk that chapter taught us to price. An agent that grades candidate patches soon learns that inflating its favourite’s score and burying the rival moves the outcome; nothing but indifference keeps a voting agent honest, and no rule can change this. What can change is the cost of the manoeuvre — manipulation can be made computationally hard, the founding trick of computational social choice and part of Section 10.7’s business.
The ledger so far is grim: cycling majorities, co-author rules, an impossibility where the perfect rule should be, a guarantee that whatever survives can be gamed. But every entry belongs to the preference species of Section 10.1. When the voters are instead estimating a truth, the question is no longer fairness to rankings but accuracy about the world, probability replaces paradox, and the very man whose cycle began the trouble proved the subject’s most optimistic theorem. Condorcet’s jury is where the good news lives.
10.4 The Wise Crowd: Condorcet’s Jury Theorem
Put the chapter’s two conditions together — two options, so May’s harbour holds; a fact of the matter, so accuracy is the standard — and you arrive where everything works better than anyone has a right to expect. A jury votes on a question with a right answer: bug or no bug. Suppose each juror is competent — correct with probability p greater than one half, however modestly — and that the jurors err independently. Then — the jury theorem — the majority verdict is more likely correct than any single juror’s, its accuracy rises with every pair of jurors added, and it approaches certainty as the jury grows (1785). It is the most optimistic theorem in the book, published four years before the Revolution by the man whose paradox opened the chapter — and the mathematical licence behind every ensemble the reader has ever run.
The magic is only the law of large numbers wearing a wig: noise, being independent, cancels as votes accumulate, while the signal — the pull of the actual answer — points the same way every time and adds up. The numbers are computable exactly: with n jurors, n odd, each correct, independently, with probability p, the majority verdict is right precisely when at least (n+1)/2 jurors are, an event of probability
P_n \;=\; \sum_{k=(n+1)/2}^{n} \binom{n}{k} p^{k} (1-p)^{n-k},
a sum a few lines of Python will evaluate. Three jurors, each right 60% of the time, deliver a majority verdict right about 65% of the time — a modest gain; make it a hundred and one such jurors and the majority is right around ninety-eight times in a hundred. Mediocrity, aggregated, becomes excellence, provided only that the mediocrities are different mediocrities: each juror wrong in its own way, right in the same one.
The promise is cheaply kept: with comb counting, the sum P_n transcribes directly as
from math import comb
def p_majority(n: int, p: float) -> float:
return sum(comb(n, k) * p**k * (1 - p) ** (n - k)
for k in range((n + 1) // 2, n + 1))
p_majority(3, 0.6) # -> 0.648
p_majority(101, 0.6) # -> 0.979...its two trailing calls replaying the paragraph’s arithmetic.
Hold the celebration until the dark twin is on the record. Run the arithmetic with p below one half and every conclusion reverses sign: the majority is now more likely wrong than any single juror, and as the jury grows it converges, implacably, on the wrong answer. Aggregation is an amplifier, not a purifier: it takes whatever tendency the jurors share and turns the volume up; a committee of the systematically mistaken is not wiser than its members, merely more sure. The corollary deserves italics it will not get: adding more voters helps only if the voters are better than chance on this question. Scaling out a jury of agents that misunderstand the task buys not robustness but a unanimous, confident, and thoroughly aggregated mistake — at ten times the token cost of asking one of them.
The theorem’s empirical face was supplied, reluctantly, by Francis Galton, who at a 1906 Plymouth livestock fair collected the 787 paid entries in a guess-the-weight-of-the-ox competition, expecting to document the vox populi’s foolishness; the middlemost estimate came to 1,207 pounds against a true dressed weight of 1,198 — within 0.8% — and he published it in Nature as Vox Populi (1907). His version generalises the jury from verdicts to estimates — the median of many guesses — and the moral, branded later as the wisdom of crowds, survives the change of arithmetic: a crowd of the individually unremarkable, if its errors are independent, is collectively formidable. The italicised clause is the entire theory; everything else is bookkeeping.
Read as engineering, the theorem is a specification with two clauses. Competence is the tractable one: measurable, per voter, per domain — the track record reputation (Section 9.4) accumulates and provenance (Chapter 6) records — required only to clear one half. Independence is the treacherous one: correlation creeps in by every door. Jurors who share a training, a briefing, or a bias share errors; jurors who see one another’s votes before casting their own stop being jurors and become a cascade, each ballot polluted by those before it — which is why deliberation sits uneasily beside the theorem: letting the jury talk pools its information and correlates its errors in the same breath — a trade taken up in Chapter 11. The correlated-jury theory’s headline is mercifully simple (Ladha, 1992): correlation does not abolish the theorem; it taxes it — correlated jurors count for less than their number, until in the limit a thousand jurors are one juror, photocopied.
A thousand jurors that are one juror photocopied: the reader has met such a jury, and probably ran one this week. Samples from a single model share its training, biases, and blind spots wholesale; the independence clause is precisely the one the modern machinery flunks, and how much of the theorem’s promise survives is what the next section weighs.
10.5 Juries of Correlated Minds: Ensembles and Self-Consistency
Machine learning has run Condorcet’s theorem as an engineering programme since before anyone thought to call it that. An ensemble — several models trained and made to vote — outperforms its members under exactly two conditions, and Dietterich’s classic analysis names them: the members must be accurate, better than chance, and diverse, wrong in different places (2000). Competence and independence, relabelled for the laboratory. The celebrated methods are, read through this chapter, devices for manufacturing the second condition: bagging retrains each member on a different resampling; random forests hide different features from each tree, decorrelating by enforced ignorance. The discipline never cited the Marquis, but it spent two decades learning to build what he specified: juries whose members err apart.
Self-consistency — Section 3.7’s device, plurality over an electorate of samples — draws every juror from a single model (2022): one training run, one set of weights, one prompt, the only wedge of independence the sampling temperature: real but thin, and what it decorrelates explains both why the method works and where it stops. Temperature randomises the slips — the dropped carry, the wrong turn taken once — scattering flawed paths across many different wrong answers while sound paths, pulled by the same truth, converge on the same right one; the majority harvests the convergence, as the theorem promises. What temperature cannot touch is the model’s convictions: a systematic misconception, learned in training and shared by every sample, arrives in juror after juror wearing the same confident face. Against slips, self-consistency is Condorcet’s machine in good order; against convictions, the photocopied juror of Section 10.4 — a thousand ballots, one mind.
The shared mind’s signature is visible in the accuracy curve: plot accuracy against samples and it climbs, then flattens — the height of the plateau set not by how many jurors you can afford but by the correlated component of their error, the part no counting cancels. This is Ladha’s tax collected in tokens (1992): past the knee, every further sample purchases a verdict already delivered. The engineering is blunt: measure the curve; stop at the knee; and when the plateau sits below the accuracy you need, hear what the flatness says: the residual error is systematic, and the remedy is not a larger jury of the same mind but a jury of different minds.
Different minds must be procured, and the options form a hierarchy of expense. Cheapest is prompt diversity: rephrase, reorder, vary the worked examples — decorrelating the errors the framing induced, which (as Section 9.6’s framing-sensitivity attests) are not few. Next, method diversity: one agent solving by running code, another by reasoning in prose — the arithmetic slip and the mistranslated requirement are different species of error, and a vote across them cancels both. Dearest and strongest is model diversity: members from different laboratories, architectures, and training recipes — the closest thing on offer to Condorcet’s independent jurors, though the frontier laboratories drink from heavily overlapping corpora, so even this independence is partial. The organising principle is the one bagging already knew: diversify at the level where your errors correlate — by phrasing, vary the phrasing; by method, the method; by world-view, the model — and accept that the last correlation can be thinned but not abolished.
One refinement remains. Jurors are rarely equal, and counting them equally wastes what is known about them: the optimal rule weights each vote by the log-odds that its caster is right — for juror i of competence p_i, errors independent as before,
w_i \;\propto\; \log \frac{p_i}{1 - p_i},
so that a juror at 90% deserves several times the ballot of a juror at 60%, and one at 50% is rightly weightless (Nitzan & Paroush, 1982). The result licenses the weighted ensembles the frameworks offer, with one warning: the weight must come from the measured track record — the reputation of Section 9.4, the provenance-backed standing of Chapter 6 — never from the juror’s announced confidence, which in a language model is calibrated to persuade rather than predict (Section 3.13). Weighting by self-assurance does not sharpen a jury; it hands the gavel to whichever member bluffs most fluently.
Everything in this section has aggregated verdicts. But much of what a team needs aggregated is a grading — which of five drafts is best — where fact and taste blend, and practice has drifted from panels toward a single arbiter set over the candidates to rank them: the arbiter that now sits in half the evaluation pipelines in production. The theory has a name for a lone voice whose ranking simply is the collective ranking; Section 10.1 promised due ceremony, and the ceremony is next.
10.6 The Model as Judge: Dictatorship by Design
The arrangement just described — one model set over the candidates, its ranking final — has a precise name, and the introduction can at last be performed. A decision procedure in which one participant’s ranking simply is the collective ranking is what Arrow’s fourth condition exists to forbid: a dictatorship. The word trails connotations the design does not deserve — nobody seized power — but the technical reading repays the discomfort, for dictatorship is the one arrangement Arrow’s theorem permits: satisfy the other three conditions and a dictator is what you must have. Read constructively, that says a dictatorship delivers exactly what aggregation cannot: a single mind’s ranking is automatically transitive, cycles are impossible, menu-sensitivity never arises, and no agenda-setter can play the order of comparisons against it. The judge model is not a lazy shortcut around social choice but one of the two coherent responses to Arrow — buy consistency at the price of one voter’s everything — and this section prices the purchase.
One precision: Arrow’s dictator is strictly a voter whose ranking the rule copies while everyone else’s is ignored, so where the candidates’ authors submit no rankings there is no profile and nothing to forbid — the election not rigged but declined, a move Section 10.7 will endorse; the name earns its keep where there was an electorate to overrule, and either way one mind’s ranking is transitive, menu-proof, and agenda-proof.
The empirical case for paying the price was made by Zheng and colleagues, whose study gave the pattern its field name — LLM-as-a-judge — and its licence (2023). Set a strong model to judge chat responses and its verdicts agree with human preferences over 80% of the time — about as often as the humans agree with each other; by that unnerving standard the judge is as good a proxy for the crowd as another member. Add the operational virtues — one call where a jury costs n, verdicts with reasons attached — and the design’s conquest of practice explains itself. For grading at pipeline volume there is usually no electorate worth convening; the choice is not judge-versus-jury but judge-versus-nothing.
But a dictator’s quirks are the constitution, and the same study that licensed the judge catalogued its tastes. Position bias: judges favour the candidate shown first — swap the order and a comfortable share of verdicts swaps with it. Verbosity bias: longer answers are graded better, at fixed correctness. And self-preference: a judge rates work higher when it is its own — traced by Panickssery and colleagues to self-recognition, the models able to distinguish their own generations and inclined to favour what they recognise (2024): nepotism implemented in the weights. Set these beside the jury and the structural point emerges: a jury’s idiosyncrasies are noise, scattered and cancelled by the count; a dictator’s idiosyncrasies are policy, applied uniformly to every case before the court. And policy, once stable, is a target. Candidates tuned against a judge — by selection, prompt iteration, or reinforcement — learn the judge’s tastes rather than the task’s truth: Section 10.3’s manipulation made easy, one voter to study, its leanings stable, documented, and in several cases published.
The remedies are constitutional law — constraints on the dictator’s discretion, each targeting a named taste. Position bias dissolves under symmetrisation: judge each pair in both orders and accept only agreements — a systematic lean converted into noise, a two-member jury from one judge’s two sittings. Verbosity is dampened by length norms and by rubrics that separate what a holistic score smears together — and a rubric is the discursive dilemma deployed as a tool: premise-wise judgement (is it correct? is it tested? is it clear?) constrains verdict-wise discretion, replacing taste with something closer to written law, at the price already disclosed: the decomposition becomes the substantive choice. Self-preference is met by separation of powers: the judge must never be the author, nor share the author’s weights, of anything it grades. Beneath all sits the audit: a judge is a jury of one, so the system’s accuracy simply is the judge’s competence p on this distribution — measurable against ground truth rather than assumed from fluency. An unaudited judge is not an evaluation pipeline; it is a preference pipeline with delusions of objectivity.
Dilute the dictatorship, then: convene several judges and let them vote. Panels help — but notice what convening one does. A panel needs an aggregation rule, so Arrow returns at the meta-level; its members, drawn from the same few laboratories, are correlated jurors, so Section 10.5‘s ceiling returns too; and a panel with interested members reopens strategic voting one floor up. The aggregation problem recurs at every layer; the panel is wise exactly insofar as its judges are competent and diverse — the same two clauses, billed again. The going refinement, letting the candidates or their advocates argue before the panel votes, trades the jurors’ last independence for a pooling of their information — a bargain Chapter 11 will weigh in full: whether letting the agents argue makes the verdict better or merely better-defended (2024).
The inventory is complete — two species, a shelf of rules with documented personalities, the impossibilities that bound them, the jury that redeems them, and the two modern architectures, correlated jury and constitutional dictator. What remains is the question all of it was for: how should an engineer actually choose?
10.7 Choosing How to Choose
An engineer before the inventory needs a procedure, and the inventory supplies one — three questions that turn “how should we aggregate this?” from a vibe into a diagnosis. First, which species? A fact of the matter makes the vote a jury, with Section 10.4’s good statistics available on their two conditions; without one, the vote is a constitution-writing exercise, and Arrow’s country must be entered with the map out. Second, how many options? Two, and May’s harbour holds — majority rule, uniqueness theorem, no small print; three or more, and every rule pays one of Arrow’s taxes, the question being which tax the application can best afford. Third, who has stakes? Disinterested voters can be counted at face value; interested ones must be assumed, per Gibbard–Satterthwaite, to vote for what works — a rule that survives honest voters and collapses under strategic ones was tested against the wrong electorate.
Run the team’s own votes through the checklist (Table 10.3) and the answers fall out fast. Ship or hold is the strong case — binary, factual, put to testers with no stake — and majority rule over independent checks carries both May’s uniqueness and Condorcet’s convergence: the vote to lean on hardest, the reason this book keeps steering collective decisions toward gates. Divide the budget fails the third question so decisively — pure preference, all parties interested — that it should scarcely be a vote at all; what it should be is Chapter 12’s business. The two hybrid votes between the poles take structure: binary gates where the logic allows, premise-wise rubrics for the rest, a constitution for the judge (Section 10.6) — the decomposition itself, as ever, the substantive choice.
| The vote | Species | Options | Who holds the stakes | Prescription |
|---|---|---|---|---|
| Ship or hold | Judgement (a fact) | Two | Disinterested testers | Majority rule over independent checks — carries both May’s uniqueness and Condorcet’s convergence; the vote to lean on hardest. |
| Choose among five candidate patches | Hybrid (fact and taste) | Many | — | Spend structure: decompose into binary gates where the logic allows, then rubric the remainder premise-wise — the decomposition is itself the substantive choice. |
| Grade at scale | Grading (fact and taste) | Many | — | A judge, which is a constitution — symmetrise, separate powers, audit against ground truth (Section 10.6). |
| Divide the budget | Preference (a taste) | Many | All parties interested | Scarcely a vote at all — give intensity a price and make it a market (Chapter 12). |
Where stakes cannot be removed, manipulation is managed rather than mourned, and the tools come in three grades. The bluntest is architectural: keep the voters disinterested — evaluators who do not author, jurors who gain nothing by the verdict — Section 10.6’s separation of powers generalised into a staffing principle. The subtlest is computational, the founding trick of computational social choice: Bartholdi, Tovey, and Trick observed that Gibbard–Satterthwaite only guarantees a profitable manipulation exists, not that it can be found, and exhibited a rule easy to count but NP-hard to manipulate (1989) — complexity conscripted as a shield, the theorem outflanked rather than overturned. The honest asterisk: worst-case hardness is a picket fence, not a wall — typical instances are often easy to manipulate — so the shield thickens rather than protects. The most durable tool is the oldest: visible voting histories, manipulation costly in reputation (Section 9.4), so gaming the count does not pay twice.
The deepest freedom is the one the checklist saves for last: not choosing a better rule but declining to hold the election. Voting compresses everything a voter knows into a mark on a ballot, and sometimes the compression is the mistake. When disagreement stems from information — each agent has seen different evidence — pool the evidence rather than counting conclusions: let them argue, trading the jury’s independence for something richer (Chapter 11). When disagreement stems from interest — the budget, the scarce slot — a ballot cannot express how much anyone cares, only which way; intensity wants a price, the count wants to be a market, and that is Chapter 12. And when one agent demonstrably knows best, the theory counsels deference: Section 10.5’s log-odds weights, pushed to their limit by a juror far more competent than the rest, concentrate the decision on the expert — a dictatorship again, but earned, held on a track record measured against the world (Section 9.5), and revocable the day the record turns. The dictator the chapter began by forbidding returns at its end with a licence: legitimacy, for aggregation as for everything else in Part IV, is a matter of what the arrangement can show for itself.
Everything so far has put agents in the voting booth; the newest deployment reverses the casting, making the rule a model and the electorate human. Tessler and colleagues built a language-model mediator for small-group deliberation — it drafts a group statement from written opinions, collects critiques, and redrafts — and in experiments with thousands of participants the machine’s statements were preferred to a human mediator’s and left groups measurably less divided (2024). Read with this chapter’s instruments, the mediator is Section 10.6’s dictator in a gentler office — it holds the pen, and its drafting choices are the constitution — buying Section 10.4’s forbidden trade made carefully: deliberation that pools information without seating the loudest voice at the head of the table. An aggregation rule that can read is a genuinely new instrument, and the scrutiny it owes is this chapter’s oldest content: who chose the rule, what it rewards, and an audit of the pen-holder’s tastes, by someone the pen-holder does not grade.
So the chapter’s thesis can be paid in full. Every multi-agent system holds elections; the constitution is code; and the difference between a system whose collective judgements can be trusted and one whose cannot is rarely the brilliance of the voters — it is whether anyone chose the rule, checked the species, measured the competence, guarded the independence, audited the judge. The engineer who has read this far practises social choice the way M. Jourdain spoke prose, but now knowingly, which changes what the practice is worth. One assumption has governed even this chapter’s strategic corners: the agents expressed positions and the rule did the rest — nobody made an offer or changed anyone’s mind. The moment the voters begin to talk — to bargain over the budget rather than ballot on it, to argue the patch’s merits rather than grade them — aggregation gives way to negotiation, and the question stops being how to count positions and becomes how positions move. That is Chapter 11, where the agents finally argue — and where we ask the evidence, rather than the enthusiasm, whether arguing makes them any cleverer.
10.8 Summary
- Every multi-agent system holds elections, acknowledged or not. Self-consistency sampling, ensemble averaging, reviewer sign-offs, tie-breaking, and judge models are all voting rules, and a rule chosen by accident carries the theory’s consequences regardless.
- Preferences and judgements are different cargo. Aggregating what agents want is a fairness problem with no right answer to track; aggregating what agents believe is an estimation problem with one. The dismaying theorems mostly govern the first, the heartening ones the second.
- The pathologies are not accidents. Plurality splits votes, Borda rewards burying, majorities happily cycle — and whoever controls the agenda then controls the outcome. Arrow proved no rule for three or more options satisfies a short list of innocuous requirements at once; May marks the safe harbour (two options, majority rule); Gibbard–Satterthwaite adds that every non-trivial rule is manipulable — among agents with stakes, the ballot is a move, as Chapter 9 taught.
- Condorcet’s jury theorem is the good news, and its conditions are an engineering checklist. Voters better than chance, erring independently, converge on truth as the electorate grows; if either condition fails, the arithmetic amplifies error instead.
- Self-consistency is a jury and a judge model is a dictator. Samples from one model are jurors who attended the same school — the growth curve flattens as fast as their errors correlate, so ensemble diversity is independence engineering by another name; and the single arbiter holds the office Arrow’s fourth condition forbids: cheap, consistent, decisive, its biases enthroned as the constitution.
- Choosing how to choose is a design act, not a default. There is no neutral aggregation rule; manipulation can be priced out but not abolished; and some collective decisions should not be votes at all — they should be arguments (Chapter 11), markets (Chapter 12), or deference to demonstrated competence.
10.9 Exercises
Exercise 1. The orchestrator puts three candidate refactorings, A, B, and C, to a panel of nineteen evaluation agents, each of which returns a full ranking, and the profile comes back: six rank A \succ C \succ B; four rank B \succ A \succ C; three rank B \succ C \succ A; four rank C \succ A \succ B; and two rank C \succ B \succ A. (a) Determine the plurality winner. (b) Determine the Borda winner, showing all three scores and checking that they sum to 3 \times 19 = 57 — say why they must. (c) Tally all three pairwise contests; identify the Condorcet winner, if one exists, and the candidate that loses every contest it enters. (d) The three rules return three different winners: explain each verdict from the rule’s personality in Section 10.2 — what each rule read in the ballots and what it ignored — and state the relation between the plurality winner and the pairwise tallies that makes this profile Borda’s 1781 complaint in its strongest form. (e) Suppose the orchestrator instead eliminated candidates by sequential pairwise comparison under some running order: show that here the agenda cannot matter, and state the general fact about sequential elimination that this profile instantiates and the next exercise’s does not.
Exercise 2. An orchestrator whittles four candidate patches, W, X, Y, and Z, through sequential pairwise elimination: the first two candidates on the agenda are compared by a judge model, the winner meets the third, that winner meets the fourth, and the last survivor ships. Careful probing shows the judge’s pairwise verdicts are stable: it prefers X to Y, Y to Z, Z to X, and each of X, Y, Z to W. (a) Show that no ranking of the four candidates is consistent with these verdicts, and that the profile has no Condorcet winner but does have a Condorcet loser. (b) There are 4! = 24 agendas; determine, by exhaustive enumeration in code or by hand with a symmetry argument, how many agendas crown each candidate. (c) The orchestrator privately favours Z: exhibit an agenda under which Z ships, trace the eliminations, and state the general recipe for crowning any chosen member of the cycle. (d) Prove that no agenda ships W. (e) A colleague proposes drawing the agenda uniformly at random, “to be neutral”: say exactly what lottery this implements, and draw the operational lesson for any pipeline in which a judge’s pairwise verdicts might cycle — which the biases catalogued in Section 10.6 make an observed fact, not a theoretical nicety.
Exercise 3. The team’s policy is that a patch merges exactly when it is correct and adequately tested. Three reviewer agents each assess both premises, and every premise-judgement is independently right with probability 4/5, independence holding across reviewers and across premises. Two constitutions are on the table: premise-wise — majority vote on each premise separately, merge iff both premises pass — and conclusion-wise — each reviewer votes to merge iff it judged both premises true, and the majority of merge-votes decides. (a) For a merge-worthy patch (both premises in fact true), compute each constitution’s probability of reaching the right verdict, in exact fractions. (b) Repeat for a patch that is in fact correct but inadequately tested. (c) Let \pi be the proportion of merge-worthy patches in the gate’s traffic, the remainder all being of the correct-but-poorly-tested kind: find the threshold value of \pi at which the two constitutions’ overall accuracies cross, and say which constitution the team should adopt, and by how many points, if roughly 35% of its candidate patches are merge-worthy. (d) Explain the mechanism behind the flip — consider two reviewers whose single slips land on different premises — and reconcile the whole exercise with List and Pettit’s impossibility (Section 10.1): what exactly does the theorem forbid, and what does it leave the engineer free to measure and choose?
Exercise 4. The ship-or-hold gate is to be manned by a jury of tester agents, each independently correct with probability p = 0.7, each costing 400 tokens per verdict; a wrong collective verdict costs an expected 40{,}000 tokens of rework. Write P_n for the majority accuracy of Section 10.4. This is Section 3.7’s self-consistency drill — the same P_n(0.7) and smallest-odd-n-to-a-threshold — recast as an economic optimum: not how many samples reach an accuracy, but how many jurors are worth their tokens. (a) Compute P_3 and P_5 exactly. (b) Find the smallest odd n for which P_n \ge 0.99. (c) The budget office proposes instead minimising the expected total cost \mathrm{cost}(n) = 400\,n + 40{,}000\,(1 - P_n): find the minimising odd n, and compare its cost with the lone-tester gate and with the jury of (b). (d) Show that the optimum in (c) obeys a marginal rule — enlarge the jury by two so long as the accuracy gain, priced in avoided rework, exceeds the 800 tokens the pair costs — exhibit the two marginal gains that bracket the optimum, and say in one sentence why “how many jurors?” is an economic question and not a purely statistical one.
Exercise 5. The gate’s questions are not all alike: a fraction h = 1/4 of them are hard, and each tester is right with probability p_{\mathrm{easy}} = 0.9 on the easy ones but only p_{\mathrm{hard}} = 0.4 on the hard ones. (a) Compute the testers’ marginal competence \bar{p} over the mixed stream, confirm it clears one half comfortably, and compute the accuracy that a naive reading of the jury theorem — P_n(\bar{p}), writing P_n(p) for the majority accuracy at competence p — promises at n = 21. (b) Justify the true accuracy A_n = (1-h)\,P_n(p_{\mathrm{easy}}) + h\,P_n(p_{\mathrm{hard}}) and compute it at n = 1, 3, 5, 7, 21. (c) Find \lim_{n \to \infty} A_n, the odd n at which A_n peaks, and the first odd n at which the jury is worse than a single tester. (d) Explain the shape: which half of Section 10.4 governs each slice of the stream, what should an engineer measure to detect the situation, what will the votes on the hard slice look like as n grows — and restate the italicised clause of the chapter that the comfortable marginal figure \bar{p} fails to honour.
Exercise 6. The companion repository’s foundations/algorithms/jury.py carries the chapter’s three tools: p_majority, the Nitzan–Paroush weights, and effective_jury_size, the design-effect heuristic n_{\mathrm{eff}} = n / (1 + (n-1)\rho). (a) For samples from one model with pairwise error correlation \rho = 1/4, compute n_{\mathrm{eff}} at n = 25 and its limit as n \to \infty; the n required to reach n_{\mathrm{eff}} = 3.5 and n_{\mathrm{eff}} = 3.9; and the \rho at which twenty-five samples would be worth ten independent jurors — say what the middle pair of answers prices. (b) Now give the correlation a structure: each juror, independently with probability c, copies one shared draw, and otherwise draws privately; shared and private draws are each correct with probability p. Show that the correlation between two jurors’ correctness indicators is c^2, then write a seeded simulation extending the module’s vocabulary and confirm the estimate empirically at c = 0.5 and c = 0.25. (c) Simulate majority accuracy at p = 0.7 for both values of c at n = 5, 25, 101, 1001; derive analytically the limiting accuracy as n \to \infty, including the threshold c^* at which its character changes; and reconcile the early peak and slow decline of the c = 0.5 column with Exercise 5. (d) Confront the heuristic with the structure: what plateau does n_{\mathrm{eff}} \to 1/\rho suggest at \rho = 1/4, what does the simulation deliver, which of the chapter’s claims about correlated juries survive the confrontation intact — and how do the copy events and the private draws map onto the slips and convictions of Section 10.5?
Further exercises for this chapter continue in the web edition’s exercise bank.
The proof is shorter than the fame suggests: Geanakoplos’s pivotal-voter argument runs to about a page (Geanakoplos, 2005) and shows where each platitude earns its keep.↩︎