Group formation under limited resources: narrow basin of equality

The formation of groups in competition and the aggressive interactions between them are ubiquitous phenomena in society. These include student activities in the classroom, election races between political parties, and intensifying trade wars between countries. Why do individuals form themselves into groups? What is the optimal size of groups? And how does the group size distribution affect resource allocations? These questions have been the subjects of intense research in economics, political science, sociology, and ethology. In this study, we explore the group-size effects on the formation of groups and resource allocations from an economic standpoint. While being in a large group is generally advantageous in competition, an increase in the management costs would set an upper bound to the individual benefit of members. Under such counteracting size effects, we consider the dynamics of group formation in which people seek a conservative measure to reduce their possible maximum loss. We are especially interested in the effects of group size on social inequalities at both group and individual level in resource allocation. Our findings show that the low positive size-effect and the high negative size-effect result in different types of social inequalities. We conclude, from the relation between the inequality measures and group distributions predicted within the model, that overall social equality only can be achieved within a narrow region where two counteracting size-effects are balanced.