Abstract
We study a vector-valued game with uncertainty in the pay-off functions. We reduce the notion of Nash equilibrium to a robust set optimization problem and we define accordingly the notions of robust Nash equilibria and weak robust Nash equilibria. Existence results for the latter are proved and a comparison between the former and the analogous notion in Yu and Liu (J Optim Theory Appl 159:272–280, 2013) is shown with an example. The proposed definition of weak robust Nash equilibrium is weaker than that already introduced in Yu and Liu (2013). On the contrary, the robust Nash equilibrium we introduce is not comparable with the notion of robust equilibrium in Yu and Liu (2013), that is defined componentwise. Nevertheless, by means of an example, we show that our notion has some advantages, avoiding some pitfalls that occurs with the other.