  We introduce and study a novel semi-random multigraph process, described as
follows. The process starts with an empty graph on $n$ vertices. In every round
of the process, one vertex $v$ of the graph is picked uniformly at random and
independently of all previous rounds. We then choose an additional vertex
(according to a strategy of our choice) and connect it by an edge to $v$.
  For various natural monotone increasing graph properties $P$, we prove tight
upper and lower bounds on the minimum (extended over the set of all possible
strategies) number of rounds required by the process to obtain, with high
probability, a graph that satisfies $P$. Along the way, we show that the
process is general enough to approximate (using suitable strategies) several
well-studied random graph models.
