  We consider the degree distributions of preferential attachment random graph
models with choice similar to those considered in recent work by Malyshkin and
Paquette and Krapivsky and Redner. In these models a new vertex chooses $r$
vertices according to a preferential rule and connects to the vertex in the
selection with the $s$th highest degree. For meek choice, where $s>1$, we show
that both double exponential decay of the degree distribution and
condensation-like behaviour are possible, and provide a criterion to
distinguish between them. For greedy choice, where $s=1$, we confirm that the
degree distribution asympotically follows a power law with logarithmic
correction when $r=2$ and shows condensation-like behaviour when $r>2$.
