The structure of Nash equilibria in Poisson games

We show that many results on the structure and stability of equilibria in finite games extend to Poisson games. In particular, the set of Nash equilibria of a Poisson game consists of finitely many connected components and at least one of them contains a stable set (De Sinopoli et al., 2014). In a similar vein, we prove that the number of Nash equilibria in Poisson voting games under plurality, negative plurality, and (when there are at most three candidates) approval rule, as well as in Poisson coordination games, is generically finite. As in finite games, these results are obtained exploiting the geometric structure of the set of Nash equilibria which, in the case of Poisson games, is shown to be semianalytic.