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Convergence rates of random-order best-response dynamics in public good games on networks

Wojciech Misiak, Marcin Dziubiński

TL;DR

This work studies convergence rates of random-order best-response dynamics in games on networks with linear best responses and strategic substitutes, and identifies structural properties of graphs which make such phenomena more likely.

Abstract

We study convergence rates of random-order best-response dynamics in games on networks with linear best responses and strategic substitutes. Combining formal analysis with numerical simulations we identify phenomena that lead to slow convergence. One of the key such phenomena is convergence to stable strategy profiles in parts of the network neighboring sets of nodes which remain inactive until the dynamics is close to converging and then switch to activity, initiating convergence to profiles with a new set of active agents and possibly leading to another iteration of such behavior. We identify structural properties of graphs which make such phenomena more likely. These properties go beyond the spectrum of a graph, which we demonstrate analyzing convergence rates on co-spectral mates.

Convergence rates of random-order best-response dynamics in public good games on networks

TL;DR

This work studies convergence rates of random-order best-response dynamics in games on networks with linear best responses and strategic substitutes, and identifies structural properties of graphs which make such phenomena more likely.

Abstract

We study convergence rates of random-order best-response dynamics in games on networks with linear best responses and strategic substitutes. Combining formal analysis with numerical simulations we identify phenomena that lead to slow convergence. One of the key such phenomena is convergence to stable strategy profiles in parts of the network neighboring sets of nodes which remain inactive until the dynamics is close to converging and then switch to activity, initiating convergence to profiles with a new set of active agents and possibly leading to another iteration of such behavior. We identify structural properties of graphs which make such phenomena more likely. These properties go beyond the spectrum of a graph, which we demonstrate analyzing convergence rates on co-spectral mates.
Paper Structure (29 sections, 12 theorems, 52 equations, 39 figures, 1 table)

This paper contains 29 sections, 12 theorems, 52 equations, 39 figures, 1 table.

Table of Contents

  1. Introduction
  2. Related literature
  3. Our contribution
  4. Preliminaries
  5. The model
  6. Best-response dynamics and stability
  7. The criterion for convergence
  8. Analysis
  9. Path graphs
  10. Structure of equilibria of path graphs
  11. Small path dynamics
  12. Large paths
  13. General graphs
  14. Structure of equilibria general graphs
  15. Deterministic graphs
  16. ...and 14 more sections

Key Result

Theorem 1

Equilibrium exists for any $\bm{G} \in \{0,1\}^{N\times N}$ and $\delta \in [0,1]$. If $\delta < 1/|\lambda_{\min}(\bm{G})|$ then equilibrium is unique. If $\delta > 1/|\lambda_{\min}(\bm{G})|$ then there exists an equilibrium with at least one inactive agent.

Figures (39)

  • Figure 1: An example of cospectral graphs
  • Figure 2: Convergence times for cospectral mates.
  • Figure 3: The path graph over $n=7$ vertices.
  • Figure 4: The configuration $1-1-1-2$ on the path graph of length $8$. Inactive agents are marked red.
  • Figure 5: Convergence times $T$ for the path graphs over $n = 2$ nodes.
  • ...and 34 more figures

Theorems & Definitions (27)

  • Theorem 1: Bra14
  • Definition 2
  • Theorem 3
  • Definition 4
  • Proposition 5
  • Theorem 6
  • Proposition 7
  • Theorem 9
  • Theorem 10
  • Theorem 11
  • ...and 17 more