- Dunbar's number
Are we sure we can groom beyond Dunbar’s number?
This post is part of a Webinar debating Dunbar's Number.
When I heard about Kanai et al.'s paper [1] announcing a correlation between grey matter density in several brain areas and online social network size, I immediately updated my Facebook status, and touted the size of my brain in the eyes of my 500 friends. Well, Kanai et al.’s findings were not at all about its size; they were limited to very specific parts of it. Yet I must admit that I maliciously took advantage of the ignorance of some of my mates. Their reactions to my childish post were various: some of them made fun of me, some pretended to be impressed. There was inevitably a bunch of them to criticize the findings as well as my own 'interpretation' of the results. Eventually, there were also a significant number of people who threatened my concept of 'friendship'. After all, how could I have 500 real friends when, as everybody knows, one can only maintain about 150 social bonds? Even Britney Spears knows that. Rumor has it that she chose to join the social network Path where the number of interactants is limited to 150. Why 150? Path’s cofounder Dave Morin justifies this limit by quoting Robin Dunbar's work on the social brain hypothesis: this is our 'social ceiling'. One cannot maintain approximatively more than 150 trusted relationships.
Here is the rationale behind this claim.
The high neocortex ratio we find in primates (compared to other mammals), and its critical value in humans, would be the outcome of selective pressures related to social competition. Primates have larger brains (correcting for body size) because they live in highly social worlds. Our evolutionary history thus shaped the amount of neural matter we can dedicate to social affairs, limiting our social skills and eventually the size of our networks. In 2010, support was provided to the evolutionary scenario: studying various mammal fossils, Schultz and Dunbar [2] showed that encephalization has not been a universal trend among mammals throughout evolutionary history. Instead, it was associated with sociality. More recently, some evidence was found for the neurocognitive aspect of the hypothesis: Powell and colleagues (2012) [3] have shown that a positive correlation between cortical volume of brain bits (here, the orbital prefrontal cortex which is involved in the monitoring of social cognitive demands) and the offline social network is not direct but mediated by mentalizing capacities, thus matching the actual predictions of Dunbar's hypothesis.
The fact that Path and others (e.g., the Swedish administration, when it reorganized tax collectors in 2007) use the social brain hypothesis as a basis for technical choices is not only a sign that Dunbar's number has become very popular. It also suggests that it is no longer in the hands of scientists: the move from 'observation' to 'normativity' has begun. Soon, the idea that we do have a social ceiling beyond which we cannot maintain trustful relationships will spread and contaminate policy makers. As there might no longer be any possible backward move in the future, de Ruiter, Weston & Lyons have summed up the points of view of the discontents in a recent paper published in American Anthropologist [4]. Neocortex size, they claim, should not be given the primacy in accounting for human group size and structure. This post exposes and discusses their arguments.
Can we groom beyond Dunbar's number, and therefore, beyond our neocortical capacity? Our elaborated cultural systems, the authors argued, can help us do so.
First, it would be incorrect to limit our cognitive apparatus to the borders of the cranial skull; de Ruiter and colleagues take the extended mind hypothesis seriously. They contend that the distinction we typically draw between the mind and the world often falls. Numerous mnemic systems could allow one to make use of external resources so as to groom beyond the 150 ceiling at low intracranial cost. Kinship algebras are taken as a good example of how one can compute a very complex set of relationships between oneself and others and determine one's social position and duties without having to hold all this information in memory. External resources would therefore be essential in making our social abilities sophisticated without depleting the brain.
Second, the authors argue that the choice of 'grooming'-type behaviors as the basis for investigating social bonding processes has detrimental consequences on how we understand the notion of 'social bond'. This choice (which is itself a consequence of a "phylogenetic continuity" constraint, i.e., finding a social bonding mechanism not too complex to give an account of how early hominins came to bond in an evolutionary 'modern' way) would narrow the definition of what a social bond is, and eventually exclude a range of other significant relationships which cannot be fostered by means of grooming-like behaviors and which would nonetheless count as true bonds from a subjective perspective. Thus, enmities or ancestor worship would not be termed social bonds, even though they are of critical importance for one's life. This would make Dunbar's definition 'tautological': if someone has more than 150 bonds, those extra bonds would not count as genuine ones.
Third, technological augmentation would provide cognitively economic ways of fostering our social bonds. Mobile phones or social-network websites would allow us to push against the 150 social ceiling. They would be much more efficient in this than are conversation and laughter (which, as for Dunbar, would have replaced non-human primate-like grooming [5]). Updating a Facebook status would just be like being able to groom dozens of individuals at one time, with minimal cognitive cost.
Thus, it would follow from these 3 main arguments than Dunbar's number can be outreached by extracranial means: there would virtually be no cognitive limit imposed by the neural volume.
Do these arguments undermine Dunbar's number? I do not think so, for at least 3 reasons.
First, the choice of grooming as the prototypical transactional mechanism shall not be considered as an indiscriminate choice. Evolutionary significant relationships are those than can bring coalitionary support to both individuals. Support is a commitment and requires trust. Because of their psychopharmacological effects and the "neuroendocrine cascades" they trigger, grooming-like behaviors create a psychological environment that makes people feel comfortable together. Grooming-like behaviors are, in this respect, trust-builders. It follows that transactional behaviors that fail to bring these psychopharmacological effects would be useless in the fostering of social bonds. In this respect, Pollet, Roberts and Dunbar (2011) [6] have showed that Facebook does not bring any significant change in the number of bonds that one consider to be significant, probably because it fails to provide the relevant psychopharmacological effects, unless personal conversations and more classical social bonding behaviors are involved (such as conversation, which restricts the interaction to a lower number of individuals). Does technological augmentation help in building trust? Robin Dunbar suggests it does not [7]. Our personal experience in web-based social networking would probably convince us that, if Facebook is useful in enjoying a collection of virtual mates, it does not improve our social skills nor extend them.
Conversely, could kinship (or other) algebras significantly improve our social skills? Kinship algebras may be very useful in computing complex relationships and remembering our social duties towards others, but I am afraid they would not be of any help for extending our mentalizing capabilities. This brings us back to the actual rationale of the social brain hypothesis: what is important for an individual embedded in a social network is not just being able to count the number of bonds that she has, or to compute which duty she is committed to, given her social position and the social position that bond of hers holds. This would simply depend on her memory and simple reasoning capacities (being able to compute logical relationships). It would not be very different, in terms of cognitive demands, from a non-social task, such as remembering a set of relationships between different cars and their brands. Rather, successfully navigating in a human social world requires being able to adjust one's behavior to that of others, inferring their mental states, desires and goals. In this respect, Powell et al. (2012) [3] have found no relationship between the volume of OFC (which positively correlates with online social network size) and short-term memory capacity. This suggests that successfully navigating into the social world requires much more than simply being able to compute a set of relationships. Dealing with the social world goes with mentalizing, not only with remembering [5].
Third, it is not clear at all how external mechanisms can allow for a more efficient maintenance of social bonds, given their obvious lack of "emotional" flavor. The tightening of evolutionarily relevant social bonds (bonds that can bring coalitional support) requires personal involvement and emotional closeness. This is definitively out of the scope of external resources such as kinship algebras (which are based on cold and logical genealogical relationships between kins) or web-based social networks (which do not extend the number of people we feel emotionally close to [6&7]).
Should we offer more scrutiny to popular scientific hypotheses that have a similar normativity potential? For sure, we should. But this is not specific to Dunbar's number, and more importantly, this should not undermine the scientific legitimacy that Dunbar's version of the social brain hypothesis has acquired through numerous data collections.
Sources
[1] Kanai, R., Bahrami, B., Roylance, R., & Rees, G. (2012). Online social network size is reflected in human brain structure. Proceedings of the Royal Society B: Biological Sciences, 279(1732), 1327–1334.
[2] Shultz, S., & Dunbar, R.I.M. (2010). Encephalization is not a universal macroevolutionary phenomenon in mammals but is associated with sociality. Proceedings of the National Academy of Sciences, 107(50), 21582–21586.
[3] Powell, J., Lewis, P.A., Roberts, N., García-Fiñana, M., & Dunbar, R.I.M. (2012). Orbital prefrontal cortex volume predicts social network size: an imaging study of individual differences in humans. Proceedings of the Royal Society B: Biological Sciences.
[4] deRuiter, J., Weston, G., & Lyon, S.M. (2011). Dunbar’s Number: Group Size and Brain Physiology in Humans Reexamined. American Anthropologist, 113(4), 557–568.
[5] Dunbar, R.I.M. (2012). Bridging the bonding gap: the transition from primates to humans. Phil. Trans. R. Soc. Lond. 367B: 1837-1846.
[6] Pollet, T. V., Roberts, S. G. B., & Dunbar, R. I. M. (2011). Use of social network sites and instant messaging does not lead to increased offline social network size, or to emotionally closer relationships with offline network members. Cyberpsychology, Behavior and Social Networking, 14(4).
[7] Dunbar, R.I.M. (in press). Social Cognition on the Internet: Testing Constraints on Social Network Size. Phil. Trans. R. Soc. Lond.
Denis Tatone 4 June 2012 (13:12)
Thanks Guillame for this splendid post!
I can't agree more with the arguments contra de Ruiter and colleagues. I believe that the debate about the Dunbar's number partly results from a misunderstanding of what this number is really about.
I do believe that the cognitive constraint hypothesized by Dunbar does not simply constitute a limit of the individual-specific information (name, identity, kin) we can memorize and recall, but it concerns specifically the access to and use of knowledge relevant for maintaining a stable relationship with another individual. This may include specific economic partnerships, patterns of previous exchanges, entitlements, rights, and the like. The retrieval and manipulation of this information is crucial for a social relationship to persist.
However, I also agree with your conclusions about the perils of granting immediate and full citizenship to the Dunbar's number into the pop-culture circles. The decision of Path's founder to limit your friend bag to 150 is an eloquent example. The Dunbar's number does not indicate the maximum number of people which a cohesive group can sustain, but the maximum number of people an individual can develop a social bond with. Given the fact that an individual belongs to many other social networks, I can't see the point of limiting the number of Path friends to 150. Unless your friends on Path are exactly the same of your off-line life (what a nightmarish scenario), you will end up having an indefinite number of non-socially cohesive relationships with your digital friends anyway.
Jayarava Attwood 4 June 2012 (13:29)
The one of Dunbar's numbers that is 150 is an average that results from observations of groups in group sizes (in fact his original paper observes many different species). That average comes with a standard deviation I can't recall off the top of my head because no one ever cites it. That is to say it's not a theoretical limit, but an empirical observation. Dunbar's contribution was to conjecture that this number was correlated with neocortex size, ad to gather data to show that there is a correlation. I'm not sure that the causal link between the two has been established beyond doubt has it? It's certainly plausible though. But perhaps more socialising causes the neocortex to expand?
There are other Dunbar numbers (both smaller and larger) for groups of different function or with different degrees of bonding. Very large groups of loosely bonded individuals are possible. Surely Facebook is one of these.
My experience is that online interactions fail to stimulate empathy, which is the glue of social monkey relationships. Interactions without empathy are stressful. Indeed lots of contact with strangers is stressful for us social monkeys. This may help to explain why there is so much hostility on the web.
Anyway thanks for the last 2 references. I will follow them up with interest.
Guillaume Dezecache 5 June 2012 (11:36)
Denis, I think you're true in emphasizing that the kind of cognitive processes which is involved in social affairs is qualitatively very different from classical mnesic demands. I'd be glad deRuiter et al.'s comment more on this aspect...
Michael: there's a recent article which might be of interest to examine the direction of causality (Sallet et al., 2011): http://www.sciencemag.org/content/334/6056/697.full.pdf
Dan Sperber 7 June 2012 (11:56)
Great post, and great debate! Thanks to Guillaume and Olivier.
Now, however novel, powerful and impressive, today's Facebook uses a very limited technology. Communication is mostly through written text and still picture and - Guillaume is right - this usually fails to activate the proper brain chemistry that bonds. Still, this is not the final word on the capacity of communication technology to allow us to bond easily with a much larger number of people that we ordinarily do - and I agree with Olivier in his post, there are so many outliers that calling them so, or speaking of what is 'ordinary' is misleading.
To take a minimal technological effect, I was struck the other day when looking at the personal page of a new Facebook 'friend' to see that her picture changed expression when you passed the cursor over it. What impressed me was not this already old technology, but the effect it had on me: let me say I felt groomed! When you have a video chat with people you had never met, the 'psychopharmaceutical' effect can be much stronger than with a mere written chat. It would not be that surprising therefore if grooming networks much beyond the Dunbar number became a 'normal' thing in the not too distant future. Or is the argument that the brain would not allow this?
Guillaume Dezecache 7 June 2012 (15:42)
Thanks Dan for your suggestion!
You're right in saying that there might soon be technological improvements that would make grooming easier in terms of computational demands. However, I'm afraid we would face another constraint: the time-budget. Indeed, there seems to be an upper limit of that time-budget that we can dedicate to social activities. This is true of baboons (see http://www.springerlink.com/content/tr25600383r51368/), other primates (http://eprints.bournemouth.ac.uk/900/1/grooming_final_No_highlight_2nd_rev_Anim_Behav.pdf) and most probably of modern humans (http://rstb.royalsocietypublishing.org/content/367/1597/1837.abstract).
If we think of video chat for instance, there'll still be a constraint on the number of people that can be groomed at a same time: holding a conversation with more than 3 people will still be difficult (http://www.springerlink.com/content/n50138642w32882w/).
I'm afraid that other technological advancements (such as replacing the current way of updating one's own facebook status -which consists in typing a text- by the recording of a video) would not allow for an critical expansion of the number of people we can groom at a same time. The reason is that other factors that are important for social bonding processes seems to require the use of "live" interactions. One of these factors is synchrony (in this respect, see that very interesting study by Emma Cohen and colleagues: http://rsbl.royalsocietypublishing.org/content/6/1/106.full) and I don't see which technological change would allow for an increase in broadcast size while keeping interactions lively enough to maintain synchrony between the interactants...
Stephen Lyon 9 June 2012 (16:54)
Like Gavin (in his comment linked to Oliveri's piece), I certainly enjoyed the Pop Dunbar idea, though I think I still probably have some qualms about the number as well as the implentation of it as a normative concept. It's hard to know which piece to attach this comment to, but since my comment is perhaps less compromising than Gavin's I thought it best to submit it to the more critical of the two initial comments.
Dwight Read, in his recent book, How Culture Makes Us Human: Primate Social Evolution and the Formation of Human Societies (2011, Walnut Creek, Ca.: Left Coast Press.), provides data which bring into question the number of 150 for humans, even if the relationship between neocortex and group size were to be accepted in the way proposed. For hunter-gatherer groups, the number 150 doesn't really appear anywhere. Residence groups of people who interact on a daily basis, number around 30. H-G societies number 500-800.
In addition, the number comes from a regression of non-human primate neocortex size and group size extrapolated to humans. There is, of course a great deal of heterogeneity in non-human primate groups. Read shows that when one partitions the primate data according to differences in troop structure, then the regression gives a group size of 90 for humans, not 150.
All of that notwithstanding, once cogntive reckoning systems, including kinship terminologies and others, are present, the constraining feature of group size is no longer number of people, but complexity of the cultural system. I take the point about affective psychological responses made in some of the comments in this webinar, which are not necessarily directly discernible from the kin terms, but I'm not exactly sure how a specific number (either 90 or 150) can be reliably determined if the key aspect of relationship is feelings of emotional solidarity. Kinship terminologies, on the other hand, might provide a pretty interesting way of determining some maximal group sizes. Read shows that the kinship terminologies of Kalahari hunter-gather groups, with residential groups of around 30, can integrate a group of up to about 800. Clearly, that is insufficient for larger scale peasant or industrial societies, which gives a pretty convincing argument for why kinship isn't the principle organising idiom for larger scale societies-- the complexity of the algebraic system isn't able to integrate a large enough number of people. When kinship is combined with other cultural systems (those underpinning the notions of tribe or caste, for example) the resulting hybrid system of integration can create groups number in the 10s and 100s of thousands.
So, yes, I am with the majority voice in this webinar that Professor Robin Dunbar is substantially more interesting and useful than Pop Dunbar. And I will say that the Number is not uninteresting nor is it a waste ot time. But I am perhaps more critical of the application to humans than most people on this list and perhaps even more critical than my co-authors.
Olivier Morin 11 June 2012 (11:11)
I think it's fair to say that the number is a theoretical speculation first and foremost. You take the relation between brain size and group size in other primates and then extrapolate. What does this extrapolation actually predict? It is hard to tell.
If we try to make the prediction strict and easily refutable, we should say that human groups should never be much above (say, 2 standard deviations above) 150, with a rather tight grouping around the mean. This is obviously false, even with hopelessly convoluted definitions of what counts as a "human group".
Another strategy (Dunbar's) consists simply in saying that there is such a thing as a level of social organization numbering 150 people. He cites clans in populations of hunter-gatherers and horticulturalists, Hutterite communities, and the number of people the average English person sends Christmas Cards to (120). Of course, horticulturalist villages, for instance, are much more populous, but Dunbar rests his claim on the fact that there is such a thing as functioning groups numbering around 100-200 people.
The problem is, this "test" is arguably too charitable. Nobody would have denied (or would have been surprised to learn) that groups numbering around 100-200 exist in many places, that there are huge variations, that 150 is neither a limit nor a "glass ceiling", and that some societies may do without this scale of organization. Does Dunbar's number predict nothing more than this?
Thom Scott-Phillips 11 June 2012 (16:14)
How about this prediction? That once human groups reach the limit*, there is a step-change in the nature of the relationships: from relationships that are built on human values such as trust, reciprocity, etc., to relationships that are more transactional, and perhaps require more formal regulation (laws, currency, etc.) if they are to be reliable and stable. This is because we have a cognitive limit of the number of informal relationships we can keep track of; above that limit, we have to reply on formal institutions. Then the claims about group sizes in organizations become claims along the following lines: once the groupings get too big, formal structures have to come into place. To maintain the nature or ethos of the group, it has to subdivide. That all seems quite plausible to me, but I agree that a more precise statement of the prediction would be welcome, to facilitate rigourous testing. As for facebook, many of us (myself included) use it maintain relationships that are, ultimately, more transactional than informal in nature. So numbers of friends there don't tell us much.
* On the number itself: it's important to remember that it's not 150: it's 150 +/- 50. The margin of error is part of the number, and it's a pretty big margin too. Also, it's worth remembering that 150 is a mean, not a limit. So some people will be able to manage >150 informal relationships. The prediction is just that the mean will be 150.
Olivier Morin 11 June 2012 (17:14)
Hello Thom, nice to read from you here.
In the original paper, Dunbar's prediction is specifically about the size of groups, not about the size of groups of friends. Grooming cliques bound by affective ties should not, in Dunbar's model, become larger as brain size and group size increase. After a certain threshold their size should stay constant (if I understand Kudo and Dunbar 2003 well). After a certain threshold, bigger societies do not require bigger groups of friends, in Dunbar's own admission.
(On the +/- 50 clause: you're right, but I don't think this gets the Number out of trouble. It would if we were sure that the distribution around the mean is normal, with a proper standard deviation that would justify the +/- 50. And judging from, a.o., the data in Hill and DUnbar 2003, we can't really do that.)
Olivier Morin 12 June 2012 (19:28)
About the quote you mention: "Kudo, Lowen and Dunbar have shown that grooming clique size, a surrogate variable that indexes alliance size, correlates rather tightly with relative neocortex and social group size in primates, including humans."
This paper, 'Neocortex size and social network size in primates', eventually became Kudo and Dunbar 2001 (unless I am mistaken). In this paper, they write: "The data suggest that larger groups are created not simply by having larger coalitions (coalition size does not increase isometrically with group size), but rather by welding together more coalitions." Indeed, the paper insists that grooming clique size in primates stops increasing with group and brain size after a certain threshold. After that threshold, the correlation is not "tight": it does not exist.
Kudo and Dunbar were writing about non-human primates. The question is, are humans below or above the threshold they describe? Dunbar occasionally wrote, and I tend to agree, that we are above. We have big groups, but the size of our groups is not constrained by the size of our cliques — at least, not "tightly".
Which is why I have so much trouble understanding how and why Dunbar's number could be a prediction about groups of friends...
Aniko Sebesteny 14 June 2012 (06:16)
Thank you very much for this "Webinar", Guillaume and Olivier!
And great debate!
I agree with Tom's concise formulation of what seems to be an implication of Dunbar's findings:
>
I think there is great truth in this, and it does seem to fit well what Dunbar describes: people who somehow belong together, who meet in rituals, as in the case of the clan of Aboriginal Australian, who celebrate rites of passage together, and have stories that are told about why they belong together. Ritual probably fosters a level of emotional togetherness that grounds respective trust and help. This creates coalition groups were, as Dunbar writes in the presented article, "the level of competition and harassment is minimized", "creating relations that can be relied on". (p. 186) So it seems predictable that trust and reciprocity will work inside these groups. But not outside.
I feel Olivier's last comment opens up a new direction for debate:
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I think the concept of "friend" is quite misleading here, and as a good anthropologist who's important role is to point out that concepts vary from culture to culture, I would say that the occidental definition of the very close emotional bond designed by the term "friend" does maybe not account for what Dunbar is writing about. But he does state, p. 185-186, that "Kudo, Lowen and Dunbar have shown that grooming clique size, a surrogate variable that indexes alliance size, correlates rather tightly with relative neocortex and social group size in primates, including humans." So while what Tom is writing about, a group of people sharing reciprocal trust, may not fit Olivier's definition of "friend", I think it does fit Dunbar's definition of the bond between people from the same coalition group, that corresponds in function to the primates' grooming group.
Dunbar also talks about other kinds of groups, talking about a 500 and a 2000, this last one being the upper threshold of how many faces humans can remember and attribute a name to.
As societies grow bigger than +/- 150, other forms of organization appear. The coalition groups of around 150 seem to remain, but other kinds of interactions are required with people outside of the coalition group. Inside the group, reciprocity works or is made to work, while interactions with people from other groups require other rules: more formal regulations, laws, as Tom writes.
I can relate to this through various anthropologic accounts, and I can also relate it to the society I am working on. Balinese cities are organized based on "banjars", that do have around the critical number of participants. People in the banjar help each other for important rituals, funerals, they worship together the protecting divinity of the banjar, and there is a high level of cooperation amongst them. This does contrast with the way the central Indonesian administration works in Bali.
Guillaume Dezecache 18 June 2012 (17:26)
Thanks all for these very interesting comments, and sorry for being that late to reply. A lot of very relevant points have been made since I last commented but I would like to focus on Stephen Lyon's criticisms.
I am afraid I don't understand why kinship algebras could provide good estimators of group size. I guess we probably agree (don't we?) on the fact that they don't do social bonding. A natural consequence of that is that if we are to determine the likely maximal size of a group whose cohesion crucially depends on intense social bonding activities between its members, we need measures that take into account the costs of conventional (conversation, laughter, touch, dance etc.) as well as "non-conventional" (web-based social networking) grooming behaviors. Kinship algebras can't do the job. If determining a maximal grouping size using measures of emotional solidarity is indeed tricky, it is probably the most appropriate measure (and it's actually done quite well through questionnaires...).
What kinship algebras can probably do, however, is to estimate the size of higher levels of social organizations, as well as to make interactions between groups-of-150 possible (through the regulation of transactional activities between these groups). As said by Thom, there's a change-step in the nature of the relationships when you go beyond 150 (and actually, there are also different layers of intimacy before we get to 150: support clique, sympathy group etc. That's also why talking of "150 friends" is probably abusive): what 150 accounts for is the size of an intermediate-level of social organization which lies between two other layers of social organization: a smaller one which includes people you interact with on a daily basis and which size is around 30-50 (and which perfectly corresponds to Read's residential groups, overnight camps in Dunbar, 1993 http://www.cogsci.ucsd.edu/~johnson/COGS184/3Dunbar93.pdf) and an upper and much wider grouping level (the tribal group) that may correspond to Read's H-G societies (actually, the number you give are close to Dunbar's). In your paper, you largely ignore (though you mention it) that intermediate-level of grouping and give various numbers of community sizes without telling which layer of social organization the numbers you give reflect.
As argued in Dunbar 1993, there can be no obvious physical manifestation of this intermediate level of grouping but this doesn't mean that it does not exist neither that it does not play a role in scaling the size of human communities.
Dwight Read 17 July 2012 (09:30)
The figure of 150 was determined through a regression analysis and not from first principles, hence what it represents depends on whether the data used in the regression analysis are homogneous; that is, all primate groups in the regression analysis show the same regression relation. However there is heterogeneity in the relationship between the neocortical ratio and group size among the prosimians and the OW monkeys and when this heterogeneity is taken into account the regression analysis predicts a group size of 90, not 150, for humans. So trying to use 150 as a "magic number" is not justified by the primate data. (It is possible that with more complete data, the revised figure of 90 will need to be revised further; such is the nature of statistical regularities based on sample data.)
For hunter-gatherers, the only "magic numbers" that have been identified are the mean residence group size of around 30 persons and the size of a simple hunter-gatherer group with a mean size of around 600 - 800 persons. (The latter has been disputed, but by reference to demographic data on all hunter-gatherer societies. Complex hunter-gatherer societies need to be excluded since they have forms of political organization that transcend the limitation on group size imposed by co-society members being kin to each other.) The mean size of simple hunter-gatherer societies is both a statistical regularity and justified from first principles as I show in my book How Culture Makes Us Human referenced by Steve Lyon.
Simple hunter-gatherer societies are not "tribal" societies as they do not have the descent-group or political structure often associated with tribal societies. Simple hunter-gatherer societies are bounded in size by the fact that social interaction only takes place among those who can recognize each other as kin, which is where the kinship algebra argument mentioned by Lyon comes into play -- persons who are not part of one's current kin network are identified as kin through being able to compute, using the kinship terminology, whether the person in question is one's kin. The computation depends upon the kinship terminology having an algebra-like structure (which, in fact, is the case) and the bound on the size of the group is the bound on the size of a cohort in which kin term computations can be made to determine if someone is one's kin. This is where the upper bound of around 600- 800 persons comes into play.
With tribal societies based on descent groups, the lineage is defined by tracing back to a reference ancestor (or ancestress). For patrilineal paramount states, the size of the smallest social unit (e.g., the folks living in a town or village) varies from 12 - 60, mean of 46, and in matrilineal paramount states from 180 - 1080, mean of 587 (data from Wright 2006 Atlas of chiefdoms and early states. Sturct Dyn eJ Anthropol Relat Scie 1; see also Read 2010 Agent-based and multi-agent simulations. Comput Math Organ Theory 16:329-347). The units in the patrilineal states are typically based on single lineages and the group sizes, with a stable population size, correspond to lineages tracing back 5 generations (this is not to say that all of these paramount states have lineages with 5 generations of depth, but just to give some rough approximations). The matrilineal states typically have units that include several lineages and have the hierarchical structure that makes it possible to have larger, stable unit. Neither corresponds to Dunbar's figure of 150 and only approximately, for a single descent group, to my revised figure of 90.
That there are "magic numbers" for the size of social units is undoubtedly the case, but with the magnitudes depending on how the social unit is internally structured. In my view, it is more useful to identify how social units are organized and structured, how this imposes bounds on the size of social units and what are likely values for the magnitude of those bounds. This is the approach Gregory Johnson (in an article that appeared in the 1970's) used to argue for why hunter-gatherer residence groups have a size of around 30 persons and the importance of hierarchical structure for forming larger social units where the hierarchical structure keeps the size of those who interact intensively to cognitively manageable sizes.
Trying to argue from statistical regularities is hazardous and subject to unwarranted reification as with Dunbar's figure of 150. It may be that in human societies today, 150 is a "magic number" for certain kinds of interactions, but this should be shown directly from relevant data and the reasons for the bound of around 150 identified, not by extrapolating from primate data through regression analysis and assuming this corresponds to a universal cognitive constraint for human societies, especially when the original regression analysis does not correct for heterogeneity in the data used in the regression analysis.