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Minimizing Rosenthal's Potential in Monotone Congestion Games

Vittorio Bilò, Angelo Fanelli, Laurent Gourvès, Christos Tsoufis, Cosimo Vinci

TL;DR

This work explores the problem of computing a state of minimum potential inCongestion games, using the maximum number of resources that a player can use at a time, and the possible symmetry in the players' strategy spaces to settle the complexity of the problem.

Abstract

Congestion games are attractive because they can model many concrete situations where some competing entities interact through the use of some shared resources, and also because they always admit pure Nash equilibria which correspond to the local minima of a potential function. We explore the problem of computing a state of minimum potential in this setting. Using the maximum number of resources that a player can use at a time, and the possible symmetry in the players' strategy spaces, we settle the complexity of the problem for instances having monotone (i.e., either non-decreasing or non-increasing) latency functions on their resources. The picture, delineating polynomial and NP-hard cases, is complemented with tight approximation algorithms.

Minimizing Rosenthal's Potential in Monotone Congestion Games

TL;DR

This work explores the problem of computing a state of minimum potential inCongestion games, using the maximum number of resources that a player can use at a time, and the possible symmetry in the players' strategy spaces to settle the complexity of the problem.

Abstract

Congestion games are attractive because they can model many concrete situations where some competing entities interact through the use of some shared resources, and also because they always admit pure Nash equilibria which correspond to the local minima of a potential function. We explore the problem of computing a state of minimum potential in this setting. Using the maximum number of resources that a player can use at a time, and the possible symmetry in the players' strategy spaces, we settle the complexity of the problem for instances having monotone (i.e., either non-decreasing or non-increasing) latency functions on their resources. The picture, delineating polynomial and NP-hard cases, is complemented with tight approximation algorithms.
Paper Structure (25 sections, 14 theorems, 29 equations, 3 tables)

This paper contains 25 sections, 14 theorems, 29 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Complexity of Min Potential with non-decreasing latencies
  4. Approximating Min Potential with polynomial latencies
  5. Approximation algorithm for Min Social Cost: a quick overview
  6. Min Potential versus Min Social Cost
  7. Characterization of the approximation factor for polynomial latencies
  8. Min Potential with non-increasing latencies
  9. Conclusion
  10. Appendix
  11. Proposition \ref{['prop:sym:size1']} does not extend
  12. Missing proofs of Section \ref{['sec4']}
  13. Proof of Claim \ref{['claim:symm:3']}
  14. Proof of Claim \ref{['claim:symm:2']}
  15. Proof of Theorem \ref{['thm:asym:size2']}
  16. ...and 10 more sections

Key Result

Proposition 1

All pure Nash equilibria of a symmetric congestion game $\cal G$ with ${\tt size}({\cal G}) = 1$ have the same potential.

Theorems & Definitions (37)

  • Proposition 1
  • proof
  • Corollary 1
  • proof
  • Theorem 2
  • proof : Proof sketch.
  • Claim 1
  • Claim 2
  • Theorem 3
  • Theorem 4
  • ...and 27 more