Equilibration Analysis and Control of Coordinating Decision-Making Populations

TL;DR

This paper introduces the concept of coordinating agents in decision-making populations, proves their almost sure convergence to equilibrium without needing potential functions, and proposes an incentive-based control algorithm applicable in general settings.

Contribution

It defines coordinating agents and populations, proves their equilibrium convergence without potential functions, and develops a practical control algorithm for desired outcomes.

Findings

Coordinating populations almost surely reach equilibrium.

Binary network games and coloring problems are coordinating.

The control algorithm performs near optimally without potential functions.

Abstract

Whether a population of decision-making individuals will reach a state of satisfactory decisions is a fundamental problem in studying collective behaviors. In the framework of evolutionary game theory and by means of potential functions, researchers have established equilibrium convergence under different update rules, including best-response and imitation, by imposing certain conditions on agents' utility functions. Then by using the proposed potential functions, they have been able to control these populations towards some desired equilibrium. Nevertheless, finding a potential function is often daunting, if not near impossible. We introduce the so-called coordinating agent who tends to switch to a decision only if at least another agent has done so. We prove that any population of coordinating agents, a coordinating population, almost surely equilibrates. Apparently, some binary network games that were proven to equilibrate using potential functions are coordinating, and some coloring problems can be solved using this notion. We additionally show that any mixed network of agents following best-response, imitation, or rational imitation, and associated with coordination payoff matrices is coordinating, and hence, equilibrates. As a second contribution, we provide an incentive-based control algorithm that leads coordinating populations to a desired equilibrium. The algorithm iteratively maximizes the ratio of the number of agents choosing the desired decision to the provided incentive. It performs near optimal and as well as specialized algorithms proposed for best-response and imitation; however, it does not require a potential function. Therefore, this control algorithm can be readily applied in general situations where no potential function is yet found for a given decision-making population.