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 letter, we use the concept of price of anarchy (PoA) to analyze the inefficiency of Nash equilibria when compared to the social optimum solution. More precisely, we show that: 1) for linear price functions depending on all the charging instants, the PoA converges to one as the population of vehicles grows; 2) for price functions that depend only on the instantaneous demand, the PoA converges to one if the price function takes the form of a positive pure monomial; and 3) for general classes of price functions, the asymptotic PoA can be bounded. For finite populations, we additionally provide a bound on the PoA as a function of the number vehicles in the system. We support the theoretical findings by means of numerical simulations.