On Games with Constant Nash Sum

A class of games in strategic form with the following property is identified: for every $\mathbf{n} \in E$, i.e. Nash equilibrium, the (Nash) sum $\sum_l n^l$ is constant. For such a game sufficient conditions for $E$ to be polyhedral and semi-uniqueness (i.e. $\# E \leq 1$) are given. The abstract results are illustrated by applying them to a class of games that covers various types of Cournot oligopoly and transboundary pollution games. The way of obtaining the results is by analysing so-called left and right marginal reductions.