On the Efficiency of Nash Equilibria in Aggregative Charging Games

TL;DR

This paper analyzes the efficiency of Nash equilibria in aggregative charging games for electric vehicles, showing that under certain conditions, the inefficiency diminishes as the population grows, with bounds provided for finite populations.

Contribution

It provides a theoretical analysis of the price of anarchy in aggregative charging games, including convergence results and bounds for various classes of price functions.

Findings

Price of anarchy converges to one for linear price functions as population grows.

Price of anarchy converges to one for demand-dependent monomial price functions.

Bounds on the price of anarchy are established for general price functions and finite populations.

Abstract

Several works have recently suggested to model the problem of coordinating the charging needs of a fleet of electric vehicles as a game, and have proposed distributed algorithms to coordinate the vehicles towards a Nash equilibrium of such game. However, Nash equilibria have been shown to posses desirable system-level properties only in simplified cases. In this work, we use the concept of price of anarchy to analyze the inefficiency of Nash equilibria when compared to the social optimum solution. More precisely, we show that i) for linear price functions depending on all the charging instants, the price of anarchy converges to one as the population of vehicles grows; ii) for price functions that depend only on the instantaneous demand, the price of anarchy converges to one if the price function takes the form of a positive pure monomial; iii) for general classes of price functions, the asymptotic price of anarchy can be bounded. For finite populations, we additionaly provide a bound on the price of anarchy as a function of the number vehicles in the system. We support the theoretical findings by means of numerical simulations.