A Simple Model of Group Size

A simple model can help to frame the issue of group size in the presence or absence of an internal organization. Organized groups can cooperate more easily, but the organization entails costs. Consider n individuals engaged in harvesting a common property resource who can freely form one or more groups and can freely organize themselves (20). In an unorganized group, everyone makes harvesting decisions independently, and efficiency losses will likely increase along with group size (2). This relation is grounded in theoretical models of decentralized interactions (1) and in empirical evidence (31). Cooperation in larger groups may be easier when engaging in contributions toward a public good rather than in harvesting resources from a common pool, which we are considering here. Rivalry in benefits is absent from the former but present in the latter. For example, the aggregate benefits of one unit of effort in keeping the air clean increase with group size. In contrast, the benefits of limiting overfishing are constant, regardless of the group size; as the group grows, the individual shares of the aggregate benefit will be reduced (8, 29, 32). Both theoretical and empirical perspectives on common property resources suggest full efficiency with individual ownership and a rapid decline in efficiency already occurring when the groups reach eight members, with losses above 50% and approaching 100% for large sizes (19) (Supporting Information).

In our model, such efficiency losses can be entirely avoided by building an organization through a charter. While restoring full efficiency, the organization also involves transaction costs related to bargaining, monitoring, coordinating, solving conflicts, and so forth (18, 20). Fig. 1 illustrates the choice of groups between experiencing efficiency losses when unorganized (dashed line) and paying organizational costs (solid line). The organization entails fixed setup costs (FC) and variable running costs (VC) so that the total costs (TC) can be modeled as (18): TC(n) = FC + VC(n). We assume running costs increase with group size, VC′ > 0, and increase more than proportionally, VC′′ > 0.

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Fig. 1.

Simple model of group size.

An implication is that small groups will be unorganized, which is in line with the available evidence (18). Organization will only benefit groups larger than n, which is defined in Fig. 1 as the threshold that occurs at the intercept of the lines representing efficiency losses and organizational costs. Another implication is that large groups will be subject to fissions. An organized group will benefit from splitting into two separate entities when it passes a threshold ñ in size because of savings in organizational costs. The definition of ñ is given in Fig. 1 as the crossing point between the two organizational cost curves. If organizational costs increase proportional to group size, no reasons exist for fissions. Instead, if costs increase more than proportionally, as assumed in the model, group size will have an upper bound.

For simplicity, the model assumes constant, or declining, returns from cooperation in group size, at least beyond some point. Evidence from forest management shows increasing returns from cooperation at very small group sizes and then decreasing returns as the group size becomes larger (8). This evidence is based on exogenously imposed group sizes, and what we present next is instead based on groups that decide their size endogenously.

Average Group Size Has Been Stable for Centuries

The Trentino communities that successfully managed common resources remained remarkably similar in group size over time. The mean number of resource appropriators in organized groups stayed stable at around 176 over the very long run (Fig. 2). The time trend over about six centuries is flat (Table 1, column 1). The median number of resource appropriators is also stable, at around 140. When the time interval of 1249–1801 is divided into six subintervals with the same number of observations, the median group sizes show no statistically significant difference (Kruskal–Wallis rank test: χ² = 1.354, df = 5, P = 0.929, n = 248). We note that this conclusion, and the following ones, are based on groups that succeeded in averting the tragedy of the commons. Owing to data limitations, we cannot observe cases of failed adaptation or collapse. Recall that the model does not define an optimal group size, even when all groups have an identical cost structure, but rather a range of possible values for organized groups. The data exhibit some dispersion with group size between 10 and 1,476, which may be compatible with the above prediction, especially when allowing for heterogeneity in bargaining, monitoring, and conflict resolution costs across communities (12, 18).

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Fig. 2.

Group size over six centuries. The analysis of 248 documents with listed assembly attendants shows that group size has a constant trend during 1249–1801. The trend estimation (red line) is shown in Table 1 (column 1). The observations refer to 156 different communities.

The observed long-term stability in the average group size of appropriators stands in sharp contrast to the patterns of population growth in Trentino. The regional population grew by about 275% from 1312 to 1810. During the same period, the general Italian population also exhibited similarly substantial growth (Supporting Information). Larger groups pose a greater challenge for cooperation and for reaching a durable consensus, as exemplified by anecdotal evidence regarding the citation from a charter highlighting the difficulties of face-to-face debates (Supporting Information). The documents from several communities show that groups enacted fissions along geographical lines (23, 33). One example is the community composed of Coredo, Smarano, and Sfruz, which jointly managed its commons in 1437. The community split into two in 1582 and subsequently into three in 1696. A fission usually divided both the common land and the group into two parts, with no apparent production specialization across subgroups. Fissions allowed maintaining a constant group size despite population growth.

Another piece of evidence comes from the following dynamic analysis of group size. We focus on a subset of observations with repeated recordings for the same community (n = 92). One can exploit the direction of change in group size (up or down) to identify an attraction point, which is an evolutionary stable point for the size of the group. Single groups experienced variations in size over time (the median absolute variation was of 61.5 appropriators over a 32.5-y interval) due to a variety of external shocks and internal population dynamics. One of the possible forces at play was also the need to maintain a group size suitable for the successful management of the commons. The evidence presented below is compatible with this factor having played a detectable role in shaping group size. To estimate this attraction point, we locate the threshold at which the direction of change reverses: Groups below the threshold tend to increase in size, while those above the threshold tend to decrease in size. First, we ordered groups from smallest to largest in terms of initial size, irrespective of their geographical location. We then computed the share of groups that remained constant or grew in size (Fig. 3). When the sample is split in half, this share is lower for large groups than for small ones (34% vs. 73%; χ² test, z = 14.19, P < 0.001, n₁ = 46, n₂ = 46). One can refine the analysis by constructing a moving average of groups that are the most similar in size. In fact, single groups are likely affected by both idiosyncratic and common random shocks. By averaging across groups, the idiosyncratic shocks will wash out, and thus cleaner evidence results.

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Fig. 3.

Attraction point in group size. Each mark (red circle or black triangle) represents a group that we observe repeatedly over time (n = 92). The type of mark denotes the direction of change in group size: downward (red circle) or upward/unchanged (black triangle), respectively. The solid line builds upon the sequence of circles and triangles and indicates the share of triangles; more specifically, it computes the moving average of 31 adjacent observations (left axis). The balance between groups declining and growing in size at 0.5 determines the attraction point of 154.

The moving average tends to decline as group size increases (solid line in Fig. 3). Its crossing point at 50% estimates the location of the attraction point because it signals a balance between the groups that have grown in size and those that have become smaller. The estimated attraction point is 154. When performing a similar dynamic analysis, including the magnitude of the change in addition to its direction, we obtain an estimated attraction point of 150 (Supporting Information). These estimates are between the median and mean group size that we obtained from the cross-sectional analysis. Hence, the stability of the average group size over time is in line with evidence showing (i) the time invariance of mean and median sizes over the centuries, (ii) the active engagement in fissions to counterbalance internal population growth, and (iii) the existence of an attraction point in the evolution of group sizes.

Ecological Versus Social Determinants

A central question is whether the forces shaping group size are ecological or social (34). The observed size could depend on the specific type of natural resource under management, or it could be a feature of how humans generally interact in groups. We exploited variability across communities to disentangle the two forces. Direct and indirect evidence shows the predominance of social factors over ecological factors in determining group size.

Common resource endowments varied considerably across communities: Some had a prevalence of forest (n = 95), others were mainly endowed with pasture (n = 81), while others had limited or mixed common resources (n = 72) (Supporting Information). Forestland can differ substantially from pastureland in terms of yields, monitoring costs, timing of harvest, and optimal scale. Trees can take several decades to mature, while grass takes only a few months. In Trentino, the most valuable tree (red fir) reaches its maximum productivity between 60 and 90 y of age (35). Our results show that group size is not significantly affected by the diversity in resource endowment (Mann–Whitney test on forest-rich vs. pasture-rich communities: z = −0.111, P = 0.914, n₁ = 95, n₂ = 81). Panel data regression analyses offer similar results (Table 1, column 2), which are robust to other measures of resource endowment, such as the ratio of forest over pasture land (Supporting Information).

By contrast, social factors are strongly linked with group size. We measured group diversity in terms of the surnames of assembly participants. Larger group sizes are empirically associated with wider surname diversity (Table 1, column 3). Because this result is somewhat expected due to statistical laws, we also present other types of empirical analysis. The origin of surnames is largely independent from the objective of this study, and hence provides a useful indication of partitioning within a group. A classification of surnames cited in Trentino documents over four centuries (1156–1595) refers to occupation (17%); bodily parts/qualities; moral values; objects, such as weapons, clothes, or instruments; places; animals; plants; food; women-related; and others (Supporting Information). Surnames have been used in anthropology, biology, and genetics to study, among other variables, migrations, genetic isolation, and distances between populations (36, 37). Our surname diversity index h = 1 − Σ_j(N_j/N)² is analogous to the Herfindahl index and refers to an assembly of N individuals in which N_j is the number of people with a given surname j (38). It measures the probability that two randomly selected individuals from an assembly come from different “surname groups.” The index h ranges from 0, when all surnames are equal, to 1 − 1/N, when they are all different.

Indirect evidence about ecological and social factors comes from the study of institutions (39). Group organization may become increasingly unmanageable as a group encounters more obstacles to the ability to function. One may expect larger communities to be more complex, which may eventually trigger a fission or lead to the collapse of cooperation. We focus on institutional complexity, which is proxied using the number of institutional roles, and employ the stated qualifications of assembly attendees, such as consul, guard, or controller, to understand the governance structure of the community. Our proxy of institutional complexity positively correlates with group size (Table 2). Social factors also play a significant role in shaping institutional complexity, but ecological factors do not. Communities with high surname diversity, those above the median of the fractionalization index, are more complex than those with low diversity (Mann–Whitney test: z = −3.699, P = 0.0003, n = 236). Regression analyses of panel data also provide a similar result after controlling for group size (Table 2). In regressions, both group size and surname diversity compete with one another in explaining institutional complexity, which is reassuring, given their known correlation (Table 1, column 3). In contrast, the effect of the type of natural resources on the number of institutional roles is not statistically significant (Mann–Whitney test: z = −0.700, P = 0.4840, n₁ = 90, n₂ = 77). We also performed a robustness check using charter length as a proxy of institutional complexity and obtained similar results (Supporting Information).

In summary, our finding is that social factors shape group size and institutions more than ecological factors. A caveat applies to this conclusion. The dataset is a sample of communities that span altitudes between 76 and 1,579 m above sea level. Some communities grow olive trees, others grow vineyards, while others border perennial glaciers. Some are in remote locations, while others are along main trading routes. Only a few rely on lakes and rivers for their activities.

Nevertheless, we are aware that common property resources around the world encompass a much wider variety than is described here. For this reason, we looked for evidence on group sizes in common property resources that cover a broader range of societies and resources types. Elinor Ostrom and her collaborators developed a database, currently maintained by the Center for Behavior, Institutions and the Environment at Arizona State University, with cases from 27 different countries and comprising social-ecological systems, such as common forests, fisheries, and irrigation systems (n = 84). Data on the mean (μ = 235), median (M = 109), and empirical distribution of group sizes of appropriators are in line with the evidence from Trentino showing a prevalence of small group sizes (Supporting Information).

The SBH and CCH

Human societies often solve ecological problems socially through bargaining and explicit agreements, but social interactions can present obstacles that limit the size of groups. Here, we focus on two existing views of social-ecological systems, the SBH and the CCH, as tentative interpretations of our empirical evidence. Although other interpretations are possible, these perspectives have been extensively debated in the literature and resonate with some key patterns observed in Trentino. Both hypotheses refer to factors that can shape the size of well-functioning groups, which are social rather than ecological.

The SBH, which was developed within the fields of neuroscience and anthropology, states that biological limitations in human cognition impose a ceiling on group size owing to constraints on memory and social skills for managing relationships (24⇓–26). This limitation refers to “the largest group of people who know everyone in the group as individuals at the level of personal relationships” (26). It also applies to the number of users that harvest a common property resource with whom one can easily relate and keep track of. This restriction could interfere with peer monitoring and sanctioning of resource users and could hamper the ability to sustain cooperation (18, 20). The SBH assumes that humans exhibit limits in communication, strategic thinking, and social bonding that raise the transaction costs of social and economic exchanges and may restrict group size.

The CCH instead refers to obstacles to the coherent functioning of the group in terms of decision making. When preferences over outcomes are heterogeneous, merging individual wants and needs into a collective will can be problematic (27). If unanimity is required, a diverse group may be incapable of coming to an agreement, for instance, in approving a new charter. If a simple majority is required, the collective decision making could be volatile and subject to endless revisions because of repeated reversals in voting outcomes (28). Already in 1785, Condorcet (40) showed that incoherencies in majority voting could emerge even within groups as small as three members. These results are consolidated within economics and political science, although no agreed-upon quantitative predictions about group size have emerged.

According to this line of interpretation, to function well, the Trentino groups should respect size limitations originating from both the SBH and CCH. Our evidence on group size from Trentino offers partial support for the SBH. The 95% confidence interval of our mean group size estimate for the users of collective pasture and forest is 157–196 individuals (n = 248). Dunbar (41) estimates a range of 100.2–231.1 individuals, and Killworth and coworkers (25) estimate a mean of 291. Large shares of the Trentino groups are below these levels: 76% and 85%, respectively. Groups also actively engage in fissions, which is a typical strategy within the SBH, aiming to maintain group size within a manageable cognitive load (23, 33). Some groups appear to be too large to be compatible with the SBH. We will later discuss the possible role of institutional strategies in these large groups.

However, a careful examination of the data reveals the explanatory power of the CCH, which focuses on the number of people with voting rights and on their similarity in interests and preferences. On the one hand, the CCH redefines what constitutes a group in terms of the assembly participants instead of the resource users. On the other hand, it highlights heterogeneity, which the SBH does not explicitly mention.

Two pieces of evidence specifically point toward a role for the CCH: the empirical relevance of group heterogeneity and the type of organizational strategies adopted. As already noted, larger groups are more heterogeneous than smaller ones. However, after controlling for group size, we find that higher heterogeneity in terms of surnames consistently correlates with more complex institutions (Table 2). Moreover, under a nominal unity, some groups adopted organizational strategies other than fission to cope with the challenges of increasing group size: a multiple-tiers or group-clustering strategy.

Under a multiple-tiers strategy, resource appropriators met face to face in smaller subgroups. In those meetings, they selected representatives, who would later meet with representatives of other subgroups. For instance, in the year 1544, Comun Comunale, a community with 1,476 appropriators, approved new regulations in a plenary assembly (Fig. 1). In 1611, appropriators structured themselves into four subgroups and sent only representatives to the general assembly.

Large communities could also follow a group-clustering strategy, which entailed partitioning the common land into multiple commons. Each individual appropriator was assigned to only one of the commons. For instance, in 1480, the community of Fiemme organized into four clusters. These institutional changes attempted to reduce the size of the group that interacted face to face and to account for the presence of groups that persistently remained nominally large.

Although both strategies make sense under the CCH, only group clustering is in line with the SBH. A multiple-tiers strategy reduces the assembly size but not the number of resource users.