  We study the problem of designing group-strategyproof cost-sharing
mechanisms. The players report their bids for getting serviced and the
mechanism decides which players are going to be serviced and how much each one
of them is going to pay. We determine three conditions: \emph{Fence
Monotonicity}, \emph{Stability} of the allocation and \emph{Validity} of the
tie-breaking rule that are necessary and sufficient for
group-strategyproofness, regardless of the cost function. Fence Monotonicity
puts restrictions only on the payments of the mechanism and stability only on
the allocation. Consequently Fence Monotonicity characterizes
group-strategyproof cost-sharing schemes. Finally, we use our results to prove
that there exist families of cost functions, where any group-strategyproof
mechanism has unbounded approximation ratio.
