We prove that for generic network-formation games where players incur a strictly positive cost to propose links the number of Nash equilibria is finite. Furthermore all Nash equilibria are regular and, therefore, stable sets.

This paper proves the generic determinacy of Nash equilibrium in network-formation games: for a generic assignment of utilities to networks, the set of probability distributions on networks induced by Nash equilibria is finite.

In games with population uncertainty some perfect equilibria are in dominated strategies. We prove that every Poisson game has at least one perfect equilibrium in undominated strategies.