Formation of committees under constraints through random voting rules

We consider the problem of choosing a committee from a set of available candidates through a randomized social choice function when there are bounds on the size (the number of members) of the committee to be formed. We show that for any (non-vacuous) restriction on the size of the committee, a random social choice function (RSCF) is onto and strategy-proof if and only if it is a range-restricted random dictatorial rule. Next, we consider the situation where an “undesirable committee” can be chosen with positive probability only if everyone in the society wants it as his best committee. We call this property strong unanimity. We characterize all strongly unanimous and strategy-proof RSCFs when there is exactly one undesirable committee. A common situation where a single committee is undesirable is one where the null committee is not allowed to be formed. We further show that there is no RSCF satisfying strong unanimity and strategy-proofness when there are more than one undesirable committees. Finally, we extend all our results when strategy-proofness is strengthened with group strategy-proofness.