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From Discrete to Continuous Binary Best-Response Dynamics: Discrete Fluctuations Almost Surely Vanish with Population Size

Azadeh Aghaeeyan, Pouria Ramazi

TL;DR

The paper addresses the asymptotic behavior of well-mixed, heterogeneous populations evolving under binary best-response dynamics. It develops continuous-time mean dynamics as a good upper semicontinuous differential inclusion and shows that the finite, discrete (Markov) population dynamics form a generalized stochastic approximation process (GSAP) for this inclusion. The analysis characterizes the equilibrium structure (clean-cut, anticoordinator-driven, and coordinator-driven) and proves global stability properties and the concentration of invariant measures on the Birkhoff center as $\mathsf{N} \to \infty$, implying perpetual finite-population fluctuations vanish in the limit. Practically, this justifies studying the simpler continuous-time mean dynamics to infer the asymptotic behavior of large finite populations and provides a rigorous link between discrete and continuous descriptions of coordination/anti-coordination phenomena.

Abstract

In binary decision-making, individuals often choose either the rare or the common action. In the framework of evolutionary game theory, the best-response update rule can be used to model this dichotomy. Those who prefer the common action are called \emph{coordinators}, and those who prefer the rare one are called \emph{anticoordinators}. A finite mixed population of the two types may undergo perpetual fluctuations, the characterization of which appears to be challenging. It is particularly unknown whether the fluctuations persist as population size grows. To fill this gap, we approximate the discrete finite population dynamics of coordinators and anticoordinators with the associated mean dynamics in the form of differential inclusions. We show that the family of the state sequences of the discrete dynamics for increasing population sizes forms a generalized stochastic approximation process for the differential inclusion. On the other hand, we show that the differential inclusions always converge to an equilibrium. This implies that the reported perpetual fluctuations in the finite discrete dynamics of coordinators and anticoordinators almost surely vanish with population size. The results encourage to first analyze the often simpler {continuous-time} mean dynamics of the discrete population dynamics as the continuous-time dynamics partly reveal the asymptotic behavior of the discrete dynamics.

From Discrete to Continuous Binary Best-Response Dynamics: Discrete Fluctuations Almost Surely Vanish with Population Size

TL;DR

The paper addresses the asymptotic behavior of well-mixed, heterogeneous populations evolving under binary best-response dynamics. It develops continuous-time mean dynamics as a good upper semicontinuous differential inclusion and shows that the finite, discrete (Markov) population dynamics form a generalized stochastic approximation process (GSAP) for this inclusion. The analysis characterizes the equilibrium structure (clean-cut, anticoordinator-driven, and coordinator-driven) and proves global stability properties and the concentration of invariant measures on the Birkhoff center as , implying perpetual finite-population fluctuations vanish in the limit. Practically, this justifies studying the simpler continuous-time mean dynamics to infer the asymptotic behavior of large finite populations and provides a rigorous link between discrete and continuous descriptions of coordination/anti-coordination phenomena.

Abstract

In binary decision-making, individuals often choose either the rare or the common action. In the framework of evolutionary game theory, the best-response update rule can be used to model this dichotomy. Those who prefer the common action are called \emph{coordinators}, and those who prefer the rare one are called \emph{anticoordinators}. A finite mixed population of the two types may undergo perpetual fluctuations, the characterization of which appears to be challenging. It is particularly unknown whether the fluctuations persist as population size grows. To fill this gap, we approximate the discrete finite population dynamics of coordinators and anticoordinators with the associated mean dynamics in the form of differential inclusions. We show that the family of the state sequences of the discrete dynamics for increasing population sizes forms a generalized stochastic approximation process for the differential inclusion. On the other hand, we show that the differential inclusions always converge to an equilibrium. This implies that the reported perpetual fluctuations in the finite discrete dynamics of coordinators and anticoordinators almost surely vanish with population size. The results encourage to first analyze the often simpler {continuous-time} mean dynamics of the discrete population dynamics as the continuous-time dynamics partly reveal the asymptotic behavior of the discrete dynamics.
Paper Structure (24 sections, 17 theorems, 38 equations, 5 figures)

This paper contains 24 sections, 17 theorems, 38 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Background
  4. The link to the continuous-time dynamics
  5. The analysis of the continuous-time dynamics
  6. Intuition and example
  7. The abstract dynamics.
  8. Equilibrium points
  9. Global stability analysis
  10. The Birkhoff center of the continuous-time population dynamics
  11. The asymptotic behavior of the discrete population dynamics
  12. Concluding Remarks
  13. Proof of \ref{['thm:implicationOFSandholm']}
  14. Proof of \ref{['propMarkov']}
  15. Proof of \ref{['lemGSAPS']}
  16. ...and 9 more sections

Key Result

Theorem 1

For a vanishing sequence $\langle \epsilon_n \rangle_{n=0}^{\infty}$ let $\langle \langle \mathbf{X}^{\epsilon_n}_k\rangle{_{k=0}^\infty}\rangle_{n}$ be GSAPs for a good upper semicontinuous differential inclusion $\dot{\bm{x}} \in \bm{\mathcal{V}}(\bm{x})$. Assume that for each $\epsilon_n$, $\lang

Figures (5)

  • Figure 1: The long-term fluctuations of the population proportion of strategy-$\mathtt{A}$ players for varying population sizes. The circles and crosses represent the maximum and minimum of the population proportion of $\mathtt{A}$-players, respectively. As the population size increases, these values get closer.
  • Figure 2: How does the asymptotic behavior of the finite population of interacting agents \ref{['populationDynamicsDiscrete']} evolve as the population size approaches infinity? A finite population of all coordinators and a finite population of all anticoordinators each equilibrates in the long run, ramazi2020convergenceramazi2017asynchronous. In the latter case, for some specific configurations, the population may admit a two-state cycle. The proportion of $\mathtt{A}$-players in a mixed finite population may undergo perpetual fluctuations roohi.
  • Figure 2: [Revisit]-The connection between the asymptotic behavior of finite populations of interacting agents evolving based on \ref{['populationDynamicsDiscrete']} and their associated continuous-time population dynamics \ref{['eq: type_mixed']} based on the stochastic approximation theory. As population size approaches infinity, the amplitude of the fluctuations in the population proportion of strategy-$\mathtt{A}$ players vanishes with probability $1$.
  • Figure 3: The associated abstract state with the discrete population dynamics described in Example \ref{['exampleFinite']} approaches $\tau_1$. The black solid line represents the evolution of the abstract state over time. The circles and crosses represent the upper and lower bounds of the invariant sets for different population sizes.
  • Figure 4: Example \ref{['exampleSpecific']}. The solution of the continuous-time population dynamics for two different initial conditions. The upper panel shows the evolution of the population state which converges to clean-cut equilibrium point $\mathbf{c}^{22} = (\textcolor{red}{2,3,0},{0,0,3,3})/28$ and the lower one depicts the evolution of the population state converging to the anticoordinator-driven equilibrium point $\mathbf{a}^{24} = (\textcolor{red}{2,2,0},{6,8,3,3})/28$, where the corresponding population proportion of $\mathtt{A}$-players equals $\tau_2 = 6/7$. The observed non-smoothness in the evolution of some subpopulations is due to the change in their preferred strategy. For example, in the upper panel, at time zero, the abstract state is below $\tau'_2$ (resp. $\tau_3$), resulting in $\dot{x}'_2 <0$ ($\dot{x}_3 >0$). As abstract state increases and exceeds $\tau'_2$ (resp. $\tau_3$), the preferred strategy of coordinating (resp. anticoordinating) subpopulation $2$ (resp. $3$) changes to $\mathtt{A}$ (resp. $\mathtt{B}$) yielding $\dot{x}'_2 >0$ (resp. $\dot{x}_3 <0$).

Theorems & Definitions (48)

  • Remark 1
  • Definition 1
  • Example 1
  • Definition 2
  • Example 2
  • Definition 3: roth2013stochastic
  • Definition 4
  • Theorem 1
  • Proposition 1
  • Definition 5
  • ...and 38 more