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

TL;DR

This paper establishes new conditions for the existence of solutions in non-monotone variational inequalities, particularly in game theory contexts, using normal mapping properties and P-matrix conditions.

Contribution

It introduces novel sufficient conditions for solution existence in non-monotone VIs and applies these to game theory to ensure Nash equilibria under broader circumstances.

Findings

Normal mapping norm coercivity implies VI solution existence.

Full rank generalized Jacobian conditions ensure solutions.

P-matrix conditions provide sufficient criteria for Nash equilibria.

Abstract

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 $F_K^{\rm nor}(\cdot)$ is norm coercive over a set $K$, and the generalized Jacobian of the normal mapping has a full rank at points $x$ where $F_K^{\rm nor}(x)\ne0$, 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$_\Upsilon$-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$_\Upsilon$-matrix condition cannot be employed.