  We introduce a model, based on the Evolutionary Game Theory, for studying the
dynamics of group formation. The latter constitutes a relevant phenomenon
observed in different animal species, whose individuals tend to cluster
together forming groups of different size. Results of previous investigations
suggest that this phenomenon might have similar reasons across different
species, such as improving the individual safety (e.g. from predators), and
increasing the probability to get food resources. Remarkably, the group size
might strongly vary from species to species, and sometimes even within the same
species. In the proposed model, an agent population tries to form homogeneous
groups. The homogeneity of a group is computed according to a spin vector, that
characterizes each agent, and represents a set a features (e.g. physical
traits). We analyze the formation of groups of different size, on varying a
parameter named 'individual payoff'. The latter represents the gain one agent
would receive acting individually. In particular, the agents choose whether to
form a group (receiving a 'group payoff'), or if to play individually
(receiving an 'individual payoff'). The phase diagram representing the
equilibria of our population shows a sharp transition between the 'group phase'
and the 'individual phase', in correspondence of a critical 'individual
payoff'. In addition, we found that forming (homogeneous) small groups is
easier than forming big groups. To conclude, we deem that our model and the
related results supports the hypothesis that the phenomenon of group formation
has evolutionary roots.
