ABSTRACT
In order to allow for each individual agent’s payoff function to depend on other agents actions, we define Uk: → . We write Uk(a)= Uk(ak, a-k), in order to distinguish the dependence on the different variables (those chosen by the kth agent, and those chosen by the others). The kth agent’s rational behaviour can now be described by a correspondence µk: → Ak given by:
A social equilibrium for a social system , is a point a* such that, for all k=1, 2,…, h, one has:
In other words, a* is a social equilibrium if and only if for all k. Then, by letting µ: → be defined by , a* turns out to be a social equilibrium if and only if a*µ(a*) (i.e., if and only if a* is a fixpoint of the correspondence µ). Under suitable convexity and continuity conditions, a social equilibrium can be shown to exist.
