This paper studies a bi-parametric family of decision rules, so-called restricted distinguished chairman rules, which contains several one-parameter classes of rules considered previously in the literature. Roughly speaking, these rules apply to a variety of situations where the original committee appoints a subcommittee. Moreover, the chairman of the subcommittee, who is supposed to be the most competent committee member, may have more voting power than other jurors. Under the assumption of exponentially distributed decision skills, we obtain an analytic formula for the probability of any restricted distinguished chairman rule being optimal. We also study, for arbitrary fixed voting power of the chairman, the connection between the probability of the rule being optimal and the size of the subcommittee.