Continuity and sensitivity analysis of parameterized Nash games

In this paper we consider continuity of the set of Nash equilibria and approximate Nash equilibria for parameterized games. For parameterized games with unique Nash equilibria, the continuity of this equilibrium mapping is well-known. However, when the equilibria need not be unique, there may exist discontinuities in the equilibrium mapping. Because the parameters of a game need to be estimated in practice, these discontinuities can result in an incorrect understanding of the Nash equilibria and, thus also, the decision-making of the players. The focus of this work is to summarize continuity properties for parameterized Nash equilibria and prove continuity via the approximate Nash game with uniformly continuous objective functions over potentially non-compact strategy spaces. That is, we consider a robust representation for the set of Nash equilibria so that small perturbations in the parameters lead to small changes in the set of Nash equilibria.