Equilibrium Selection in Replicator Equations Using Adaptive-Gain Control

TL;DR

This work tackles equilibrium selection in replicator dynamics for symmetric two-action population games by introducing a closed-loop adaptive-gain controller that modulates a single payoff entry via $\boldsymbol{A}(t)=\boldsymbol{\hat{A}}+\boldsymbol{G}g(t)$ with $\dot g = \phi(x)g$. The authors prove easy-to-check sufficient conditions on the adaptation rate $\phi$ that enable convergence under three problem manifestations: consensus reaching (g→0), consensus stabilization (g→$\bar g$), and set-point regulation (target $\bar x$ in (0,1) with g→$\bar g$). They distinguish conformity versus innovation gain controllers, derive dedicated results for each, and illustrate with simulations that hybrid strategies can balance speed and control effort. The framework works with limited a priori information on the payoff structure and is applicable to promoting social change, cooperation, and congestion management, providing a flexible tool for robust, population-level interventions in socio-technical systems.

Abstract

In this paper, we deal with the equilibrium selection problem, which amounts to steering a population of individuals engaged in strategic game-theoretic interactions to a desired collective behavior. In the literature, this problem has been typically tackled by means of open-loop strategies, whose applicability is however limited by the need of accurate a priori information on the game and scarce robustness to uncertainty and noise. Here, we overcome these limitations by adopting a closed-loop approach using an adaptive-gain control scheme within a replicator equation -a nonlinear ordinary differential equation that models the evolution of the collective behavior of the population. For most classes of 2-action matrix games we establish sufficient conditions to design a controller that guarantees convergence of the replicator equation to the desired equilibrium, requiring limited a-priori information on the game. Numerical simulations corroborate and expand our theoretical findings.

Figures (6)

• Figure 1: Schematic of the controlled evolutionary game-theoretic dynamics.
• Figure 2: Trajectories of the (uncontrolled) replicator equation in Eq. (\ref{['eq:replicator']}) for (a) a pure coordination game (Example \ref{['ex:pure']}) $a=d=1$; (b) a prisoner's dilemma (Example \ref{['ex:pd']}) with $c=0$, $a=1$, $d=2$, and $b=3$; and (c) a minority game (Example \ref{['ex:min']}) with $b=c=1$. Different colors denote different initial conditions.
• Figure 3: Trajectories of the controlled pure coordination game with $a=d=1$ and $x(0)=0.7$ using (a) conformity gain control ($q=1$ and $p=7$) and (b) innovation gain control ($q=p=1$).
• Figure 4: Total control effort and peak gain of the innovation gain control for a pure coordination game ($a=d=1$) using adaptation rate Eq. (\ref{['eq:affine']}) with different values of $p$ and $q$.
• Figure 5: Controlled prisoner's dilemma with $a=1$, $b=3$, $c=0$, $d=2$ and $x(0)=0.99$, using conformity gain control with $\phi$ from Eq. (\ref{['eq:power']}). In (a), we report a trajectory of the controlled dynamics (with $p=0.4$ and $q=1$); in (b) the final gain $\bar{g}$, for different choices of the parameters $p$ and $q$.

Theorems & Definitions (34)

• Definition 1: Nash equilibrium
• Proposition 1
• Example 1: Pure coordination game
• Example 2: Prisoner's dilemma
• Example 3: Minority game
• Proposition 2
• proof
• Remark 1
• Remark 2
• Definition 2