Positive Invariance of Subset of Positive Externalities

The notion of subset of positive externalities is explored. The subset of positive externalities for games by two pseudo-gradient agents with quadratic payoffs is characterized. It is shown that only a certain subset of the subset of positive externalities can be a positively invariant set of the pseudo-gradient dynamics. A necessary and sufficient condition such that this subset is a positively invariant set of the pseudo-gradient dynamics is given. The results are illustrated with an example.