The Computational Complexity of Variational Inequalities and Applications in Game Theory

TL;DR

This paper investigates the computational complexity of variational inequality problems, demonstrating PPAD-completeness, and applies these results to complex game theory concepts like resilient Nash equilibria and multi-leader-follower games.

Contribution

It establishes the PPAD-completeness of variational inequality problems and formulates complex game theory concepts within this framework, providing new insights into their computational difficulty.

Findings

Variational inequality problems are PPAD-complete.

Formulation of resilient Nash equilibrium as a variational inequality.

Multi-leader-follower games are shown to be PPAD-complete.

Abstract

We present a computational formulation for the approximate version of several variational inequality problems, investigating their computational complexity and establishing PPAD-completeness. Examining applications in computational game theory, we specifically focus on two key concepts: resilient Nash equilibrium, and multi-leader-follower games -- domains traditionally known for the absence of general solutions. In the presence of standard assumptions and relaxation techniques, we formulate problem versions for such games that are expressible in terms of variational inequalities, ultimately leading to proofs of PPAD-completeness.