  Until recently, multiple synthetic data sets were always released to
analysts, to allow valid inferences to be obtained. However, under certain
conditions - including when saturated count models are used to synthesize
categorical data - single imputation ($m=1$) is sufficient. Nevertheless,
increasing $m$ causes utility to improve, but at the expense of higher risk, an
example of the risk-utility trade-off. The question, therefore, is: which value
of $m$ is optimal with respect to the risk-utility trade-off? Moreover, the
paper considers two ways of analysing categorical data sets: as they have a
contingency table representation, multiple categorical data sets can be
averaged before being analysed, as opposed to the usual way of averaging
post-analysis. This paper also introduces a pair of metrics, $\tau_3(k,d)$ and
$\tau_4(k,d)$, that are suited for assessing disclosure risk in multiple
categorical synthetic data sets. Finally, the synthesis methods are
demonstrated empirically.

Methodology