The components of the mean square error (MSE) of estimates from surveys with multiplicity, where households report on events which occurred in them and also in other households (linked to them according to well-specified counting rules), can be estimated on the basis of a specially designed evaluation survey, under certain simplifying assumptions. The empirical results from such a survey, designed to estimate births and marriages, indicate that a wide counting rule is more efficient than a conventional rule, since the slightly higher response bias and variance are more than offset by the reduction in sampling variance.