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Voting for Committees in Agreeable Societies

We examine the following voting situation. A committee of $k$ people is to be formed from a pool of n candidates. The voters selecting the committee will submit a list of $j$ candidates that they would prefer to be on the committee. We assume that $j \leq k < n$. For a chosen committee, a given voter is said to be satisfied by that committee if her submitted list of $j$ candidates is a subset of that committee. We examine how popular is the most popular committee. In particular, we show there is always a committee that satisfies a certain fraction of the voters and examine what characteristics of the voter data will increase that fraction.