  We consider a model of language development, known as the naming game, in
which agents invent, share and then select descriptive words for a single
object, in such a way as to promote local consensus. When formulated on a
finite and connected graph, a global consensus eventually emerges in which all
agents use a common unique word. Previous numerical studies of the model on the
complete graph with $n$ agents suggest that when no words initially exist, the
time to consensus is of order $n^{1/2}$, assuming each agent speaks at a
constant rate. We show rigorously that the time to consensus is at least
$n^{1/2-o(1)}$, and that it is at most constant times $\log n$ when only two
words remain. In order to do so we develop sample path estimates for quasi-left
continuous semimartingales with bounded jumps.
