Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Complexity of Sparse Win-Lose Bimatrix Games

Eleni Batziou, John Fearnley, Abheek Ghosh, Rahul Savani

Abstract

We prove that computing an $ε$-approximate Nash equilibrium of a win-lose bimatrix game with constant sparsity is PPAD-hard for inverse-polynomial $ε$. Our result holds for 3-sparse games, which is tight given that 2-sparse win-lose bimatrix games can be solved in polynomial time.

The Complexity of Sparse Win-Lose Bimatrix Games

Abstract

We prove that computing an -approximate Nash equilibrium of a win-lose bimatrix game with constant sparsity is PPAD-hard for inverse-polynomial . Our result holds for 3-sparse games, which is tight given that 2-sparse win-lose bimatrix games can be solved in polynomial time.
Paper Structure (43 sections, 44 theorems, 57 equations, 2 tables)

This paper contains 43 sections, 44 theorems, 57 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Our contribution.
  3. Related Work
  4. Preliminaries
  5. Nash equilibrium and approximations.
  6. Sparsity.
  7. Win--lose and $S$-valued games.
  8. Technical Overview
  9. $\{ 0, 1, 2, 8 \}$-Valued Bimatrix Games
  10. Column Simulation
  11. First-type single column simulations.
  12. Applying column simulations.
  13. Second-type single column simulations.
  14. Dual column simulations.
  15. Polynomial approximations.
  16. ...and 28 more sections

Key Result

Lemma 1

ChenDT09 In a bimatrix game $(A, B)$ with $A, B \in [0, 1]^{n \times m}$, given any $(\varepsilon^2/8)$-NE, we can find an $\varepsilon$-WSNE in polynomial time.

Theorems & Definitions (84)

  • Lemma 1
  • Definition 5.1: $\varepsilon$-RePolymatrix
  • Theorem 1
  • Definition 5.2: $\varepsilon$-ReBimatrix$(m)$
  • Theorem 2
  • proof
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Definition 6.1: Stage 1 game
  • ...and 74 more