  Motivated by recent results relating synchronizing DFAs and primitive sets,
we tackle the synchronization process and the related longstanding
\v{C}ern\'{y} conjecture by studying the primitivity phenomenon for sets of
nonnegative matrices having neither zero-rows nor zero-columns. We formulate
the primitivity process in the setting of a two-player probabilistic game and
we make use of convex optimization techniques to describe its behavior. We
develop a tool for approximating and upper bounding the exponent of any
primitive set and supported by numerical results we state a conjecture that, if
true, would imply a quadratic upper bound on the reset threshold of a new class
of automata.
