Existence of Solutions for Non-monotone VIs and Implications for Games

In this paper, we study the existence of solutions in non-monotone variational inequalities (VIs) through the normal mapping properties. In particular, we show that when the normal mapping $F_K^{\mathrm{n}\mathrm{o}\mathrm{r}}(\cdot )$ is norm coercive over a set $K$, and the generalized Jacobian of the normal mapping has a full rank at points $x$ where $F_K^{\mathrm{n}\mathrm{o}\mathrm{r}}(x)\mathrm{\ne }0$, then the VI$(K,F)$ has a solution. We then investigate conditions on the mapping $F(\cdot )$ and its Jacobian that imply the full rank condition for the generalized Jacobian, such as the uniform P-function and the uniform P-matrix condition. Subsequently, we focus on VIs arising from games and interpret our main result in a game setting. Based on the P${}_{\mathrm{{\rm Y}}}$-matrix condition, we provide a sufficient condition for a game to have a Nash equilibrium. Additionally, through examples we show that our sufficient conditions can be used to assert the existence of a solution to a VI, or a quasi-Nash in a game, while the existing results relying on the uniform P-function property or the P${}_{\mathrm{{\rm Y}}}$-matrix condition cannot be employed.