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The Computational Complexity of Variational Inequalities and Applications in Game Theory

Bruce M. Kapron, Koosha Samieefar

TL;DR

A computational formulation for the approximate version of several variational inequality problems is presented, investigating their computational complexity and establishing PPAD-completeness, ultimately leading to proofs of PPAD-completeness.

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.

The Computational Complexity of Variational Inequalities and Applications in Game Theory

TL;DR

A computational formulation for the approximate version of several variational inequality problems is presented, investigating their computational complexity and establishing PPAD-completeness, ultimately leading to proofs of PPAD-completeness.

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.

Paper Structure

This paper contains 53 sections, 47 theorems, 140 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Some Elementary Definitions
  4. Semi-Continuity and Convexity
  5. Kakutani's Fixed Point Theorem
  6. Basic Game Theory
  7. Variational Inequalities
  8. Equilibria and Coalitions
  9. Multi-Leader-Follower Games
  10. Some Computational Complexity Definitions
  11. Computational Complexity of Variational Inequalities
  12. Computational Generalized Variational Inequalities (GQVI)
  13. Applications In Game Theory
  14. Computational Complexity of Multi-Leader-Follower Games
  15. Computational Complexity of Resilient Nash Equilibrium
  16. ...and 38 more sections

Key Result

Theorem 1.1

Finding an approximate solution to computational variants of GQVI, QVI and VI is PPAD-complete where:

Theorems & Definitions (126)

  • Theorem 1.1: informal
  • Theorem 1.2: informal
  • Theorem 1.3: informal
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • ...and 116 more