Cardinality of Social Groups

In 1268 the Republic of Venice settled on a way to choose its doge that still looks like a puzzle invented to torment students. The Great Council drew thirty of its members by lot. The thirty were cut to nine by lot. The nine named forty. The forty were cut to twelve by lot, and the twelve named twenty-five. The twenty-five were cut to nine, the nine named forty-five, the forty-five were cut to eleven, and the eleven named forty-one. Those forty-one finally elected the doge, who needed at least twenty-five of their votes.

The procedure was meant to defeat the great families, who could fix any single vote but couldn't steer ten rounds of alternating chance and choice. What stands out now is that every one of those numbers was picked on purpose. The Venetians treated the size of a body as part of what it was. Nine people pick differently than forty-five. A body of eleven can be leaned on in ways a body of forty-one cannot.

Mathematicians call the number of members in a set its cardinality. This essay is about the cardinality of human groups, and its claim is that the number is never a neutral detail. Somewhere between two and three, between five and fifteen, between a hundred and a few thousand, a group stops being a bigger version of what it was and becomes a different kind of thing, held together by different forces. Growing a Movement as a Design Problem used Robin Dunbar's nested rings of 5, 15, 50, and 150 to ask how a movement should size its cells. This essay starts lower and climbs higher, from the pair to the polity, and asks at each step what changes in the minds of the members and in the way the group has to be run.

Arithmetic Determinism -- number matters

Some of what changes is plain counting. Among five people there are ten possible pairs. Among fifteen there are 105. Among fifty there are 1,225, and among 150 there are more than eleven thousand. Members grow by addition and relationships grow by multiplication, so a group always outruns any one member's ability to keep track of it sooner than its headcount suggests.

In 1933 a Lithuanian-born management consultant named V. A. Graicunas worked this out for the boss's side of the desk. A manager doesn't only deal with each subordinate one at a time, but also with subordinates in combination, and has to reckon with how they deal with each other. Graicunas counted all three kinds of relationship and got a formula that grows explosively. A manager with five direct reports has 100 relationships to track. With six, 222. With twelve, more than 24,000. His paper became the mathematical footing for the old management rule that nobody should supervise more than five or six people whose work interlocks.

The arithmetic explains why size matters. It doesn't explain what each size feels like from the inside, or what kind of order becomes possible or necessary at each one.

Other ways of sorting groups

Size isn't the only way to classify human groups, and the older classifications are worth setting beside it. In 1909 the American sociologist Charles Horton Cooley named the primary group: the family, the children's play group, the neighborhood of elders, circles bound by face-to-face association and by a sense of "we" that comes before any purpose. Later sociologists set the secondary group against it: the firm, the school, the club, the professional body. These are larger, organized around a task, and their members deal with each other through roles more than as whole persons.

Anthropologists sort groups instead by what makes someone a member. Some groups are made of kin: the household, the extended family, the lineage, the clan. Some are made of place: the camp, the village, the ward, the town, the city. Some are made of identity: the ethnic group, the nation, the caste, the congregation. And some are made of choice: the guild, the association, the party, the firm.

None of these schemes is about number, but each one tracks it closely. Primary groups are nearly always small, because face-to-face association only works at small sizes. Kinship reaches as far as anyone can trace descent, and past that point it has to invent something. Place runs from the camp to the metropolis. Identity groups tend to be the largest of all, because a category can hold any number of people. Choice is the exception. Voluntary groups come in every size, which is part of why they matter so much. Read against the count, the older classifications show which kinds of bond can hold at which sizes. To see how, we have to go up the count one step at a time.

One and two

The German sociologist Georg Simmel was one of the first to take this question seriously. His 1908 Soziologie has a chapter called "The Quantitative Determination of the Group," and it begins with the pair.

A dyad, Simmel noticed, is the only group that depends entirely on each of its members. If either one leaves, the group no longer exists. There is no majority to outvote anyone and no third party to appeal to. Every decision is unanimous or it isn't a decision. This is what gives the pair its intensity, in friendship, marriage, partnership, and rivalry alike, and also its fragility. Each member knows the whole thing rests on the two of them, and that the other can end it.

In organizational terms the dyad is the contract. Two parties agree, each can walk away before agreeing, and the agreement binds only them. It is the form of social order closest to pure consent, because the exit and the dissolution are the same act. Much of what this collection of essays means by voluntary association is an attempt to keep larger arrangements as close as possible to that condition.

Three

Add one person and everything changes. Simmel thought the move from two to three was the biggest jump in the whole sequence, bigger than any that came after.

With three, a group can survive the loss of a member. More importantly, it acquires a structure that stands above any individual in it. Simmel described three roles the third person can play. The mediator stands between the other two and helps them settle what they couldn't settle alone. The tertius gaudens, the "third who rejoices," profits from the other two's quarrel, as a buyer profits when two sellers compete. And whoever plays divide et impera sets the other two against each other deliberately in order to rule both.

All three roles depend on the same new fact: with three, there can be a majority. Two can agree against one. That is the first appearance of power over a member who hasn't consented, and it arrives at the smallest possible size.

Solomon Asch's conformity experiments in the early 1950s found the psychological side of this. Asch had a subject judge the lengths of lines alongside confederates who gave a wrong answer in unison. One confederate barely moved anyone. Two had some effect. Three produced nearly the full effect, and adding more confederates after that changed surprisingly little. The pressure of a group is already fully present at three.

Asch's less famous finding matters more for liberty. When he planted a single confederate who gave the right answer, conformity collapsed, to roughly a quarter of its previous level. One ally was enough. A dissenter facing a unanimous group is under enormous pressure, and a dissenter with one companion is a different person. The freedom to disagree depends in practice on the freedom to find one other person who disagrees too. Every regime that has tried to suppress dissent has understood this, which is why the right to assemble, even in twos and threes, is usually the first right to go.

Four to six: the table

The next threshold shows up in ordinary conversation. Dunbar and two colleagues watched freely forming conversations in the mid-1990s and found that they top out at about four people: one speaking and three listening. When a fifth or sixth person joins, the conversation tends to split in two within minutes. Nobody decides this. It happens because past four, not everyone can follow and get a word in.

This is the size at which everyone still talks to everyone. A string quartet can play without a leader because each player can hear and watch the other three at once. A family at dinner, a startup's founders, and a squad of soldiers all work at about this size, and they share something important. Nobody needs a rule to know what the others are doing, and nobody can coast unnoticed.

That second point has a history of its own. Around 1913 a French agricultural engineer named Maximilien Ringelmann had men pull on a rope, alone and in groups, and measured the force. Each additional puller added less than the one before. In pairs, people pulled at a little over ninety percent of their solo effort. By eight, each was pulling at roughly half. Later researchers separated the causes: some of the loss was coordination, but much of it was what psychologists now call social loafing. When individual effort cannot be seen, it declines. At four or five, effort is still visible. Not long after that it starts to blur.

Seven to twelve: the committee, the jury, the tithing

Between seven and a dozen, a group can still meet around one table, but it can no longer run as a single conversation. Someone has to chair. Turns have to be taken. Things get written down. This is the size of the committee, and of the board, and of most bodies that have to make a decision rather than simply be together.

The cost of crossing into this range shows up in a quieter way too. In 1968 John Darley and Bibb Latané had students talk over an intercom, and staged an apparent seizure by one of the participants. Students who believed they were the only listener went to help 85 percent of the time before the fit ended. Students who believed four others were also listening helped 31 percent of the time. Responsibility diffuses. At this size, "someone else will handle it" becomes a reasonable guess, and therefore a common one.

C. Northcote Parkinson, half joking, put an upper limit on the committee in 1957. Studying the history of cabinets, he proposed a "coefficient of inefficiency": somewhere around twenty-one members, a governing body stops governing, and the real decisions move to a smaller inner group while the large one ratifies them. Anyone who has served on a large board will recognize the pattern, joke or not.

The jury sits at the top of this range, and the courts have fought over exactly where. In Williams v. Florida (1970) the Supreme Court allowed six-member criminal juries. In Ballew v. Georgia (1978) it refused to go to five, and the opinion leaned explicitly on social science about group size: smaller groups deliberated less thoroughly, remembered less of the evidence, and were less likely to contain anyone holding a minority view. That last point is Asch again. A jury of twelve is more likely than a jury of five to include the one person who sees it differently, and that person is more likely to have an ally.

The most interesting use of this cardinality may be one of the oldest. In medieval England, under the system called frankpledge, men were grouped into tithings of about ten, and each tithing was collectively responsible for the conduct of its members. If one was accused of a crime, the others had to produce him in court or pay. Ten is about the largest number of households that can genuinely vouch for one another, knowing each other's habits and comings and goings well enough to stand surety. The tithing put that knowledge to work as a system of order that needed no police.

Frankpledge is also a warning. In its developed form it was enforced by the Norman crown, which used the tithings to make neighbors police neighbors on the king's behalf, with the sheriff checking every year that every man was enrolled. The same small group that could be a form of self-government could be turned into an instrument of control. What made the difference was who the tithing answered to.

Fifteen to fifty: the band

The previous essay covered Dunbar's middle rings in detail, so a brief note is enough here. Groups of fifteen to fifty are about the size of a hunter-gatherer camp, a platoon, a congregation that meets in a house, or a large extended family at a holiday. Members can still recognize each other on sight and know something real about each one, but they cannot all coordinate directly. Some form of role or leadership appears.

Anthropologists call a group of this size a band, the first step in a sequence the anthropologist Elman Service laid out in 1962 that is still the field's starting vocabulary. A band is a few dozen people, usually mobile and living by foraging, and its leadership is informal and situational. Whoever knows the most about the matter at hand leads, for as long as it is at hand, and then the leadership lapses.

Situational leadership isn't only for foragers. The Orpheus Chamber Orchestra in New York has played since 1972 with no conductor, at about thirty musicians, a size at which nearly every other orchestra assumes one is required. For each piece, the players choose a concertmaster and a small core group to shape the interpretation, then bring it to the full ensemble, and the roles rotate from piece to piece. Moscow's Persimfans did something similar with a full symphony orchestra from 1922 to 1932. At this cardinality coordination needs structure, but the structure can be chosen by the members and handed around among them.

A hundred and fifty to a few thousand: the village

Somewhere past 150, members of a group stop knowing each other personally, but they can still know each other by reputation. Dunbar's outer layers put about 500 people at the edge of acquaintance and about 1,500 at the limit of faces a person can put a name to. In this range a group works through gossip, through friends of friends. Nobody knows everyone, but anyone can find out about anyone through someone they trust.

Kinship crosses this threshold in a revealing way. Anthropologists distinguish a lineage, whose members can trace their actual links to a known common ancestor, from a clan, whose members claim a common ancestor but cannot trace every link, and whose founder is often mythical. Much of the difference is size. A lineage is as large as memory can reach. When a kin group grows past what anyone can recount, it holds itself together with a story instead of a genealogy. That is the first appearance of something that becomes central at every larger size: membership held together by a shared belief rather than by personal knowledge.

Many peoples then joined clans into larger groupings: phratries of related clans, or moieties, the two complementary halves into which some societies divide themselves, each marrying into the other. These are groups of groups built out of kinship, and they let thousands of people keep obligations to each other without anyone ruling the whole. A collection of this kind, joined by kinship, language, and ritual but with no permanent central government, is what anthropologists mean by a tribe. Leadership comes from elders, or from "big men" whose influence lasts only as long as they keep giving and persuading. In everyday speech "tribe" gets used for any group that isn't a state, but in anthropology it names this particular middle scale, and the precision matters. A tribe is a federation that has no ruler.

This is the size of the village, the parish, and the small town, and it is the size where written rules first become necessary but can still be enforced by people who know whom they are enforcing them on. Elinor Ostrom's study of common-property institutions spent time on Törbel, a Swiss mountain village of a few hundred people whose residents have managed their shared alpine pastures under their own written rules for centuries. One of the old rules said that no one could send more cows to the summer pasture than the owner could feed through the winter. A rule like that can be enforced only where people can see each other's barns.

Philosophers noticed the limits of this range long before anyone measured it. Aristotle wrote that ten people cannot make a city, and that a hundred thousand are no longer one. His reasoning was practical: citizens have to know each other's characters well enough to elect officials and judge disputes, and past a certain size they cannot. Plato, in the Laws, fixed his ideal city at exactly 5,040 households. He chose that number because it can be divided evenly by every number from one to ten, which made it easy to split into tribes, districts, and work groups of any size the city might need. It is an odd choice and a revealing one. Plato wanted a number that could be cut cleanly into smaller groups.

Jefferson wanted the same thing for Virginia. In his letters of 1816, to Joseph Cabell and Samuel Kercheval, he proposed dividing every county into "wards" small enough that every citizen could attend, when called on, and act in person. Above the wards would come the counties, then the state, then the union, each handling only what the level below couldn't. Government, he wrote, should be divided and subdivided "until it ends in the administration of every man's farm by himself." The ward was his answer to the problem of cardinality: a unit small enough that self-government meant governing oneself in fact.

A version of the ward still meets. In two Swiss cantons, Appenzell Innerrhoden and Glarus, citizens gather once a year in the town square and vote on laws by raising their hands, in crowds of thousands. The Landsgemeinde works at that size because it is the top of a structure whose lower layers are much smaller, and because the agenda is prepared by bodies small enough to deliberate. In 2006 the Glarus assembly even voted to merge its twenty-five municipalities into three, a large group redrawing the cardinality of its own subunits by show of hands.

Chiefdom and state

Service's sequence continues past the tribe, and read as a list of sizes it is also a list of what holds a group together. A chiefdom runs from thousands to tens of thousands of people. It is ranked, with a hereditary chief at the top, and it works by redistribution: goods flow up to the chief and back out again, and the chief's standing depends on the flow. A state has a centralized government with a monopoly on legitimate force, a bureaucracy, taxation, and usually writing and codified law. An empire is a state that rules many peoples. Situational leadership in the band, kinship and persuasion in the tribe, rank and redistribution in the chiefdom, permanent force in the state: each type is what its size seemed to demand.

For a long time the sequence was read as a ladder that every society climbs, one rung at a time, toward the state. Its critics, most recently David Graeber and David Wengrow in The Dawn of Everything (2021), point out that many peoples moved up and down it on purpose. The anthropologist Robert Lowie described how several Plains nations, among them the Crow and the Cheyenne, came together each summer in great camps for the buffalo hunt. One hunter charging early could scatter the herd and starve everyone, so for the hunt they appointed a police force, often drawn in rotation from one of the men's societies, with real power to punish: to beat offenders, destroy their property, and confiscate their kill. When the hunt ended and the people dispersed into small bands for the winter, the police dissolved and their authority with it.

Coercion arrived with the large size and left with it. The people who lived this way treated the methods of a larger cardinality as a tool for a season, picked up for a task too big for bands and set down when the task was done. That is a different relationship to power than the one the modern state assumes, in which a coercive apparatus once built is permanent and finds new tasks to justify itself.

Beyond faces: the abstract group

Past a few thousand, membership stops being a matter of who we know or who we could find out about. It becomes a category. People belong to it because of where they live or what papers they hold, and they relate to it through symbols, officials, and the news, never through the other members.

The classifications from earlier name these groups. An ethnic group shares ancestry, language, or culture, real or claimed. A nation is a people claiming a shared political identity, and a nation-state is a government whose borders are drawn to match one. Each is held together the way a clan is, by a shared story, now stretched over millions. The other large group with no faces in it is the crowd, which can have the size of a city for an afternoon and almost none of its structure. A crowd has numbers without layers. That was exactly what worried the founders about large assemblies.

In Federalist No. 55, Madison wrote that "in all very numerous assemblies, of whatever character composed, passion never fails to wrest the sceptre from reason. Had every Athenian citizen been a Socrates, every Athenian assembly would still have been a mob." He took cardinality seriously: sixty or seventy men could be trusted with more than six or seven, he argued, but that didn't mean six or seven hundred would be better still, and at six or seven thousand "the whole reasoning ought to be reversed."

The Anti-Federalists worried about the other side of the ratio. Melancton Smith told the New York ratifying convention in 1788 that representatives ought to resemble those they represent and know their circumstances, and that a House of sixty-five could not possibly do that for three million people. The very first amendment Congress sent to the states in 1789, ahead of what became the Bill of Rights, would have fixed the size of districts so that the House grew with the population. It fell one state short of ratification. The House has been fixed at 435 since 1911, and after the 2020 census each member represents about 760,000 people. Whatever that relationship is, it isn't one in which the representative can know the represented, or even be known by more than a sliver of them.

The economist Mancur Olson gave the hard version of the problem in The Logic of Collective Action (1965). In a small group, each member's contribution to a shared goal is visible and makes a noticeable difference, so small groups often provide shared goods on their own. In a large group, each person's share is invisible and makes no difference, so everyone has reason to let others pay. Olson concluded that large groups get collective goods only through side benefits for those who contribute, or through coercion. It is Ringelmann's rope and Darley and Latané's intercom, scaled up to a nation. Olson's argument has been used ever since as a justification for the state: the only thing that can make forty million people contribute to what they share is a body that can compel them.

Three claims about size

The problem fits the three-claim form this collection uses for hard questions.

First: people can govern themselves, in the literal sense of making and enforcing their own rules together, only at the sizes where they know each other personally or by reputation. That means a few thousand at most.

Second: a prosperous and peaceful society requires cooperation among millions of people who will never meet.

Third: large groups can produce shared goods only by coercion.

Taken together, the three seem to say that freedom is possible only in villages, and that everyone who wants more than a village can provide has to accept a state.

The resolution lies in what the third claim is about. Olson's argument is correct for a single large group whose members each relate directly to the whole, as citizens relate to a national government or voters to a statewide ballot. It does not hold for a group of groups, where each layer is small enough to run by the methods that work at its own size. A ward of a few hundred can govern itself through reputation. A council of delegates from twenty wards is a committee and can deliberate. A council of delegates from twenty such councils is again a committee. Each member at each level knows the people they answer to. Venice ran its most important election through a cascade of bodies, none larger than forty-five. Clans joined into phratries and moieties, Jefferson's gradation of republics, the Landsgemeinde, and the affinity groups and circles of the previous essay all follow the same pattern. The Plains nations add a further lesson: even where a large group truly needed coercion, it could be limited to the task and the season that required it.

Markets are the other answer. Millions of people cooperate through prices every day without ever being a group at all. Each exchange is a dyad, the most consensual cardinality there is, and the coordination emerges from the sum of them.

So the first two claims were never in conflict. What conflicts with both of them is a hidden assumption: that cooperation among millions has to take place inside a single group of millions. That assumption is what turns Olson's observation into an argument for the state. The modern state governs forty million Californians as one group, at a cardinality where the only methods that work are abstraction and compulsion, and then cites the failures of that cardinality as proof that compulsion is needed.

Designing for the number

If the number of members changes what a group is, then the size of an institution is a moral choice and not only a practical one. A few principles follow.

Decide things at the smallest cardinality that includes everyone affected. A question about a street belongs to the people on the street. A question about a watershed belongs to a body that answers to the communities along it. Each step up the count costs some consent, so climb only as far as the question requires.

When a group outgrows its form, split it instead of adding management. A group that has grown past the size where its members know each other has two options. It can add rules, officers, and records until it works like a small bureaucracy, or it can divide into two groups that each work like the original did. The second option keeps self-government. The first slowly replaces it.

Keep the ratio between layers small enough to be personal. A delegate who answers to thirty people they know is a different kind of representative from one who answers to 760,000 they don't. The total number of people in a federation can be very large if each link in the chain stays at human scale.

Protect the three. Asch showed that a dissenter's independence depends on having one ally, and Simmel showed that power over a member first appears at three. A free society has to protect the smallest associations first: the freedom of two or three people to meet, agree, disagree with the majority, and act together. Large freedoms are built from small ones, and they are lost the same way.

Let people belong to many groups. In another chapter of Soziologie, translated as "The Web of Group-Affiliations," Simmel argued that modern individuality grows out of membership in many circles that meet in a single person. A member of a lineage society belonged to one nested set of groups, each inside the next. A person today can belong at once to a family, a firm, a congregation, a club, a neighborhood, and an online community, each of a different size and held together by a different kind of bond. Societies organized by age sets, in parts of East Africa, built the same protection in deliberately: men initiated in the same years formed a cohort that cut across every lineage, so that no quarrel between kin groups could split the whole people cleanly. Overlapping memberships mean no single group commands all of a person's loyalty. The state's characteristic claim is to be the one group that outranks all the others, and a free society keeps that claim in check partly by making sure people have many other groups to belong to.

The Venetians were not democrats, and their republic was an oligarchy of merchant families. But they understood something the modern state has mostly forgotten: that a body of nine and a body of forty-five are different instruments, and that whoever designs an institution is choosing among them. The cardinality of a group is part of its constitution, written or not. A society that wants to stay free has to keep most of life at the sizes where people can still know each other, and build everything larger out of those.

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Sources

• Georg Simmel, Soziologie (1908), ch. 2, "The Quantitative Determination of the Group," translated in Kurt H. Wolff, ed., The Sociology of Georg Simmel (1950).
• Georg Simmel, "The Web of Group-Affiliations," in Conflict and The Web of Group-Affiliations, trans. Reinhard Bendix (1955).
• Charles Horton Cooley, Social Organization (1909).
• Elman R. Service, Primitive Social Organization: An Evolutionary Perspective (1962).
• V. A. Graicunas, "Relationship in Organization," Bulletin of the International Management Institute (1933), reprinted in Gulick and Urwick, eds., Papers on the Science of Administration (1937).
• Solomon Asch, "Studies of Independence and Conformity," Psychological Monographs (1956).
• R. I. M. Dunbar, N. D. C. Duncan, and D. Nettle, "Size and Structure of Freely Forming Conversational Groups," Human Nature (1995).
• Maximilien Ringelmann, "Recherches sur les moteurs animés," Annales de l'Institut National Agronomique (1913), and D. A. Kravitz and B. Martin, "Ringelmann Rediscovered," Journal of Personality and Social Psychology (1986).
• J. M. Darley and B. Latané, "Bystander Intervention in Emergencies: Diffusion of Responsibility," Journal of Personality and Social Psychology (1968).
• C. Northcote Parkinson, Parkinson's Law (1957), "Directors and Councils, or Coefficient of Inefficiency".
• Williams v. Florida, 399 U.S. 78 (1970).
• Ballew v. Georgia, 435 U.S. 223 (1978).
• W. A. Morris, The Frankpledge System (1910).
• Orpheus Chamber Orchestra, organizational history.
• Elinor Ostrom, Governing the Commons (1990), and Robert McC. Netting, Balancing on an Alp (1981), on Törbel.
• Aristotle, Nicomachean Ethics IX.10 and Politics VII.4.
• Plato, Laws V, 737e–738b.
• Thomas Jefferson to Joseph C. Cabell, February 2, 1816, and to Samuel Kercheval, July 12, 1816.
• Canton of Glarus records on the Landsgemeinde vote to merge its municipalities (2006).
• The Federalist No. 55.
• Melancton Smith, speeches at the New York ratifying convention (June 1788).
• The Congressional Apportionment Amendment (proposed 1789).
• U.S. Census Bureau, 2020 apportionment results.
• Mancur Olson, The Logic of Collective Action (1965).
• Robert H. Lowie, "Some Aspects of Political Organization among the American Aborigines," Huxley Memorial Lecture (1948).
• David Graeber and David Wengrow, The Dawn of Everything (2021).
• Frederic C. Lane, Venice: A Maritime Republic (1973), on the ducal election of 1268.

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A note on how this piece was written: the subject, the questions it asks, and the positions it takes are mine. Much of the research, the examples, and the sentences were drafted by an AI model working from that direction, and then edited by hand. I'd rather say that plainly than have a reader guess at it.