We define balance games, which describe the formation of friendships and enmity in social networks. We show that if the agents give high priority to future profits over short term gains, all Pareto optimal strategies will eventually result in a balanced network. If, on the other hand, agents prioritize short term gains over the long term, every Nash equilibrium eventually results in a network that is stable but that might not be balanced.

A commonly used dynamic epistemic logic is one obtained by adding commonknowledge and public announcements to a basic epistemic logic. It is known from Kooi (2007) that adding public substitutions to such a logic adds expressivity over the class K of models. Here I show that substitutions also add expressivity over the classes KD45, S4 and S5 of models. Since the combination of common knowledge, public announcements and substitutions, was shown in Kooi (2007) to be equally expressive to relativized common (...) knowledge these results also show that relativized common knowledge is more expressive than common knowledge and public announcements over KD45, S4 and S5. These results therefore extend the result from van Benthem et al. (2006) that shows that relativized common knowledge is more expressive than common knowledge and public announcements over K. (shrink)