Benefit of forming groups

One of the main benefits of forming a group is to multiply a collective output by making better use of members’ skills and resources through cooperation. We assume the output of a group of n persons is where c > 0 is a proportional constant. We call an exponent α ≥ 0 the group efficiency, which represents the degree of effectiveness of the group size in output. For α = 1, the performance of the group is simply proportional to its size. As such, the group size essentially has the null effect on the group production. If α > 1, n^α exhibits increasing returns to scale and the size of each group has a greater effect on its share than for α = 1. This gives an motivation to organize as a large group as possible, since the members mean share increase accordingly. On the contrary, the managing cost for large groups can grow faster than the group size, for example, due to bureaucratic inefficiency. If 0 ≤ α < 1, forming a group is not an attractive choice to people as their mean share reduces with the group size, unless there is further motivation. From here on, we assume that the proportional coefficient c is one for convenience. We additionally assume that there is a group income tax paid for the central maintenance and service. A group of n persons pays the tax which is proportional to the group output n^α with the tax rate r_T, 0 ≤ r_T ≤ 1. The tax in the model represents a fixed-rate expenditure proportional to an absolute group size.

Conflict-induced wealth redistribution

Wealth redistribution between groups, which is induced by group-group conflicts, contrasts with the tax and depends on a relative group size. Suppose N people forms K exclusive groups, n₁, n₂, ⋯, n_K with ∑_{k = 1} n_k = N ≥ 3. We consider a situation where groups directly confronts one another and a part of their group products, are exposed to mutual appropriation. That is, here r_C represents the ratio of the assets subject to appropriation, reflecting intensity of conflicts.

We assume that, in conflicts, a group’s gain(loss) against another group depends on their size difference. To be more specific, suppose that group i and group j confront each other and group i is larger than group j. We assume that group i deprives group j of its exposed asset in proportion of their relative size difference (n_i − n_j)/N. Hence the amount of asset that group i takes away from group j is The total asset of group i is now obtained by summing up the above values: the original product minus the income tax , the gain from the small groups , and the loss to the larger groups . This can be formulated as (1) (2) Note that the amounts of the tax and the appropriation do not directly affect each other. While the amount of the tax is determined from the absolute size of the group, what determines the appropriation is the relative size of the group.

Now we define the individual payoff of a member in group i as π_i = g_i/n_i, assuming that the wealth in the group is equally shared. That is, (3) (4) Note that introducing group conflicts and appropriation may twist the group size effect. For example, we confirmed above that no group spontaneously forms with the small group-size effect, 0 ≤ α < 1. This is no more true with conflicts existing, since their smallness makes groups more vulnerable to appropriation. Even when forming a group does not promote efficient productions in output, people can make an alliance to protect their assets.

If r_C is low, conflicts between groups are not severe, and a relatively large part of the group production can be secured for each group. On the contrary, a high value of r_C indicates that the society is driven by intensive conflicts and the large portion of the assets is subject to appropriation by other groups. Especially when r_C > 1 − r_T, the society is driven by intensive conflicts and the situation becomes very harsh to minor groups. Most of their properties is at risk of being taken by larger groups. What is even worse is that they have to pay out the fixed-rate tax, regardless of how much they lost in the conflicts. This implies that some small groups unavoidably end up with a negative payoff.

Group dynamics

People change or create their groups to increase their individual payoff. We assume that an individual can deviate from one group and get in another, only with the consent of the people in the group that he/she wants to join. This implies that transfer can occur only when it leads to increase in payoffs of both the new comer and existing members. In addition, an individual can escape from a group and create a singleton, if staying alone is more beneficial than remaining in the group. Group dynamics is a serial combination of such individual movements. We are interested in searching Nash equilibriums of group distributions where no individual has incentive to change their groups any more.

Society with neutral group efficiency (α = 1)

In case that the group product linearly increases with the group size, all possible group distributions are equally attractive and people are indifferent in changing their groups. However, the presence of group conflicts and central taxation makes the situation more complicated. People further have to consider possible appropriation between them when forming groups. As a result, there are two groups left in the end, of which size ratio may vary.

Theorem 1 Suppose α = 1. Let n⁰ be the size of the largest group in the initial group distribution. Then the group distribution converges to a two-group formation, (n*, N − n*), where .

Here ⌈x⌉ denotes the smallest integer greater than or equal to x. According to Theorem 1, distributions with more than three groups are unstable and bound to merge into a two-group formation. The size of the larger group is or greater. Once there appears a group larger than 70% of the population, the members have no incentive to accept a new comer. That is, unless the initial group formation includes a larger group from the beginning, the distribution ends up with the 70:30 formation. It is notable that the asymmetric size ratio depends only on the initial condition n⁰, not on either r_T or r_C, although the result cannot be achieved without the presence of group conflicts and central taxation,

Society with high group efficiency (α > 1)

The desire to unify grows with the group efficiency. For an intermediate value of α > 1, people tend to create a large group to maximize the group output. This motivation competes with a tendency to leave out small groups outside so that they can appropriate the small group’s production. However, with high values of α, the benefit of unifying groups overwhelms the potential appropriation.

Theorem 2 For sufficiently large α > 1, the only Nash equilibrium is single grand group distribution.

Low conflict society with low group efficiency (α < 1, r_C ≤ 1 − r_T)

When the group efficiency is less than 1, α < 1, the situation becomes more complex. Although people basically want to stand alone due to low group efficiency, still in the presence of conflicts, they further have to seek security against appropriation. This motivation leads to two completely different situations, depending the intensity of conflict r_C.

If conflicts between group are rather weak, multiple groups commonly appear. As α decreases below 1, two-group formations in Theorem 1 are bound to break, making more various group formations possible. For example, if N = 100, r_T = r_C = 4/9 and α = 4/5, the group distributions such as (83, 17), (74, 20, 6), (63, 29, 7, 1) are all possible Nash equilibriums. Still, in this regime of mild competition, r_C ≤ 1 − r_T, an extremely low group efficiency makes people scatter, giving up grouping:

Theorem 3 Suppose r_C ≤ 1 − r_T. For sufficiently small α < 1, the groups break into all singletons.

Conflict-driven society with low group efficiency (α < 1, r_C > 1 − r_T)

When α is small, the group formation depends on r_C and drastically changes across the value r_C = 1 − r_T. If r_C > 1 − r_T, appropriation of other groups becomes a dominant part of the group members’ gain. In the appropriation, since the group efficiency is low, more significant factor is how much bigger their group is than their opponents. They concern the size difference, rather than opponent’s absolute size and the corresponding output. That is, the situation becomes close to a pure power game. The group distribution ends up with an asymmetric formation that consists of one large group and several minor groups. For example, (80, 2, 2, 1, …1) is one typical equilibrium.

As α decreases, the size gap gets widened. Especially in the limit of α → 0, with a little strong condition as , all small groups are eventually merging to the largest group, leaving one unfortunate individual as an outcast.

Theorem 4 Suppose r_C > 1 − r_T. For sufficiently small α < 1, there is at most one group of which size is greater than 1. Moreover, suppose the initial distribution is not the single grand group and suppose a slightly stronger condition holds. Then the only Nash equilibrium is (N − 1, 1).

This indicates occurrence of the social rejection, like (99, 1). The absolute portion of the profits that the members of the larger group share is from exploitation of the isolated individual. Theorem 4 implies that people form a large group and isolate an outcast when they find no intrinsic benefits in forming a group.

When a large portion of their asset is exposed to severe competition, people start to use the exclusion and isolation strategy. They aggregate not because they can produce more, but because they can deprive more. The model indeed reveals a subtle point regarding social exclusion: in a predatory economy, a welfare gap between being a part of a larger group and being an isolated individual becomes extremely large. One is made to cooperate to exclude someone else, in fear of others victimizing him or her first.