  A solution to a problem of Erd\H{o}s, Rubin and Taylor is obtained by showing
that if a graph $G$ is $(a:b)$-choosable, and $c/d > a/b$, then $G$ is not
necessarily $(c:d)$-choosable. Applying probabilistic methods, an upper bound
for the $k^{th}$ choice number of a graph is given. We also prove that a
directed graph with maximum outdegree $d$ and no odd directed cycle is
$(k(d+1):k)$-choosable for every $k \geq 1$. Other results presented in this
article are related to the strong choice number of graphs (a generalization of
the strong chromatic number). We conclude with complexity analysis of some
decision problems related to graph choosability.
