they interact with. F = 2 (dashed line): agents take note of individual differences in the quality of information other agents offer. F = 3 (solid line): agents take note of individual differences between other agents and rely on third party information about each other received from other agents. Increasing calculation capacity allows larger groups in each of the three strategies for evaluating the quality of potential collaborators. However, each strategy has an upper limit (glass ceiling), and if groups need larger group size they have to switch to more sophisticated methods of information acquisition, and that necessitates an increase in the computational costs (i.e. brain size). (Parameters used: k = 4, = 20, = 11. The results are robust to these parameters.) 4. Discussion It is important to note that the more complex strategies are highly disadvantageous when group size is small: indeed, the more complex the strategy, the more disadvantageous it is. Thus, the evolution of communicative and cognitive complexity is explicitly dependent on an ecological demand for large social groups: it is only when there is a need for large groups that the selection pressure will be sufficient to warrant the costs involved. This is driven by the demands of social coordination. If there is no requirement for coordination (in terms of the present model, optimal group size n* 1 because organisms do not need to cooperate), then there is insufficient selection pressure to motivate either complex communication or investment in the large brains required to support the requisite cognitive abilities. This suggests that complex communicative abilities, large social groups that involve social coordination and large brains will all be equally rare, as indeed seems to be the case [13]. Although we have used a very simple (and computationally easy to implement) objective function in the model (achieving synchrony in compass direction), it is important to be clear that our model is not limited to this particular context. Rather, our device stands as an abstraction for any behaviour that requires synchrony or coordination in order to maximize biological fitness, whether the fitness payoff is a direct or an indirect consequence of coordination along stable, expansive dyadic network edges. Direct fitness payoffs may arise from coordinated foraging or hunting (as in some social carnivores [36]), cooperative defence against predators or rival conspecifics (e.g. group territorial defence; as in many primates) or any other ecologically relevant behaviour that requires group members to synchronize or coordinate their behaviour in some way. However, in species characterized by multi-level social systems (e.g. elephants, some cetaceans, most primates [37] and humans [27,28]) where the higher level of organization functions to solve a collective action problem, the payoff may be indirect and mechanisms are needed to facilitate cooperation at group level. In these cases, solving a collective action problem is a two-step process: willingness to collaborate is established before the need to collaborate [38], and is often (but not necessarily always) signalled by some marker (or 'tag') of group membership. Our model not only shows how cognition (processing power) could limit group size, but also sheds