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Continuous-Time Best-Response and Related Dynamics in Tullock Contests with Convex Costs

Edith Elkind, Abheek Ghosh, Paul W. Goldberg

TL;DR

It is shown that continuous-time best-response dynamics in Tullock contests with convex costs converges to the unique equilibrium using Lyapunov-style arguments, and convergence of related discrete-time dynamics is established, e.g., when the agents best-respond to the empirical average action of other agents.

Abstract

Tullock contests model real-life scenarios that range from competition among proof-of-work blockchain miners to rent-seeking and lobbying activities. We show that continuous-time best-response dynamics in Tullock contests with convex costs converges to the unique equilibrium using Lyapunov-style arguments. We then use this result to provide an algorithm for computing an approximate equilibrium. We also establish convergence of related discrete-time dynamics, e.g., when the agents best-respond to the empirical average action of other agents. These results indicate that the equilibrium is a reliable predictor of the agents' behavior in these games.

Continuous-Time Best-Response and Related Dynamics in Tullock Contests with Convex Costs

TL;DR

It is shown that continuous-time best-response dynamics in Tullock contests with convex costs converges to the unique equilibrium using Lyapunov-style arguments, and convergence of related discrete-time dynamics is established, e.g., when the agents best-respond to the empirical average action of other agents.

Abstract

Tullock contests model real-life scenarios that range from competition among proof-of-work blockchain miners to rent-seeking and lobbying activities. We show that continuous-time best-response dynamics in Tullock contests with convex costs converges to the unique equilibrium using Lyapunov-style arguments. We then use this result to provide an algorithm for computing an approximate equilibrium. We also establish convergence of related discrete-time dynamics, e.g., when the agents best-respond to the empirical average action of other agents. These results indicate that the equilibrium is a reliable predictor of the agents' behavior in these games.
Paper Structure (21 sections, 7 theorems, 64 equations, 1 figure, 1 table)

This paper contains 21 sections, 7 theorems, 64 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Related Work
  3. Preliminaries
  4. Best-Response
  5. Continuous-Time Best-Response Dynamics
  6. Equilibrium
  7. Convergence of Continuous-Time Best-Response Dynamics
  8. Warm-Up Phase
  9. Main Phase
  10. Discrete-Time Dynamics (Approximately Continuous BR and Fictitious Play) and Equilibrium Computation
  11. Algorithm
  12. Fictitious Play: Best-Response to Empirical Average
  13. Conclusion and Future Research
  14. Omitted Proofs
  15. Proof of Theorem \ref{['thm:contConverge']} (Lower Bound)
  16. ...and 6 more sections

Key Result

theorem 1

The continuous best-response dynamics $\bm{x}(t)$ in Tullock contests with weakly convex cost functions converges to an $\epsilon$-approximate pure-strategy Nash equilibrium in $O(\log(1/\epsilon))$ time. Further, there are instances where reaching an $\epsilon$-approximate equilibrium takes $\Omega

Figures (1)

  • Figure 1: Dependency of step-size on cost ratio.

Theorems & Definitions (18)

  • remark 1
  • definition 1: Pure-Strategy Nash Equilibrium
  • definition 2: Approximate Pure-Strategy Nash Equilibrium
  • theorem 1
  • proof : Theorem \ref{['thm:contConverge']} (Upper Bound)
  • lemma 1
  • proof
  • lemma 2
  • theorem 2
  • lemma 3
  • ...and 8 more