  We study the semi-random graph process, and a variant process recently
suggested by Nick Wormald. We show that these two processes are asymptotically
equally fast in constructing a semi-random graph $G$ that has property
${\mathcal P}$, for the following examples of ${\mathcal P}$:
  - ${\mathcal P}$ is the set of graphs containing a $d$-degenerate subgraph,
where $d\ge 1$ is fixed;
  - ${\mathcal P}$ is the set of $k$-connected graphs, where $k\ge 1$ is fixed.
In particular, our result of the $k$-connectedness above settles the open case
$k=2$ of the original semi-random graph process.
  We also prove that there exist properties ${\mathcal P}$ where the two
semi-random graph processes do not construct a graph in ${\mathcal P}$
asymptotically equally fast. We further propose some conjectures on ${\mathcal
P}$ for which the two processes perform differently.
