We investigate large-scale effects induced by external fields, phenomenologically interpreted as mass media, in multiagent models evolving with the microscopic dynamics of the binary naming game. In particular, we show that a single external field, broadcasting information at regular time intervals, can reverse the majority opinion of the population, provided the frequency and the effectiveness of the sent messages lie above well-defined thresholds. We study the phase structure of the model in the mean field approximation and in numerical simulations with several network topologies. We also investigate the influence on the agent dynamics of two competing external fields, periodically broadcasting different messages. In finite regions of the parameter space we observe periodic equilibrium states in which the average opinion densities are reversed with respect to naive expectations. Such equilibria occur in two cases: (i) when the frequencies of the competing messages are different but close to each other; (ii) when the frequencies are equal and the relative time shift of the messages does not exceed half a period. We interpret the observed phenomena as a result of the interplay between the external fields and the internal dynamics of the agents and conclude that, depending on the model parameters, the naming game is consistent with scenarios of first- or second-mover advantage (to borrow an expression from the jargon of business strategy).