Assigning a Small Agreeable Set of Indivisible Items to Multiple Players

We consider an assignment problem that has aspects of fair division as well as social choice. In particular, we investigate the problem of assigning a small subset from a set of indivisible items to multiple players so that the chosen subset is \emph{agreeable} to all players, i.e., every player weakly prefers the chosen subset to any subset of its complement. For an arbitrary number of players, we derive tight upper bounds on the size for which a subset of that size that is agreeable to all players always exists when preferences are monotonic. We then present polynomial-time algorithms that find an agreeable subset of approximately half of the items when there are two or three players and preferences are responsive. Our results translate to a 2-approximation on the individual welfare of every player when preferences are subadditive.