  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.
