On Adaptive-Gain Control of Replicator Dynamics in Population Games

Abstract

Controlling evolutionary game-theoretic dynamics is a problem of paramount importance for the systems and control community, with several applications spanning from social science to engineering. Here, we study a population of individuals who play a generic 2-action matrix game, and whose actions evolve according to a replicator equation -- a nonlinear ordinary differential equation that captures salient features of the collective behavior of the population. Our objective is to steer such a population to a specified equilibrium that represents a desired collective behavior -- e.g., to promote cooperation in the prisoner's dilemma. To this aim, we devise an adaptive-gain controller, which regulates the system dynamics by adaptively changing the entries of the payoff matrix of the game. The adaptive-gain controller is tailored according to distinctive features of the game, and conditions to guarantee global convergence to the desired equilibrium are established.

Figures (2)

• Figure 1: Trajectories of the (uncontrolled) replicator equation in Eq. (\ref{['eq:replicator']}) for (a) a pure coordination game with $a=d=1$; (b) a prisoner's dilemma with $c=0$, $a=1$, $d=2$, and $b=3$; and (c) a minority game with $b=c=1$.
• Figure 2: Trajectories of Eq. (\ref{['eq:controlled_replicator']}) for (a) a coordination game ($k=1$, $h=0.4$); (b) a prisoner's dilemma ($k=2$, $h=1$); and (c) a minority game ($k=0.1$, $h=1$). Common parameters are $\alpha=\beta=1$ and $x(0)=0.99$.

Theorems & Definitions (14)

• Proposition 1
• Example 1: Pure coordination game
• Example 2: Prisoner's dilemma
• Example 3: Minority game
• Proposition 2
• Remark 1
• Lemma 1
• proof
• Theorem 1
• proof