A cost-sharing mechanism defines how to share the cost of a service among serviced customers. It solicits bids from potential customers and selects a subset of customers to serve and a price to charge each of them. The mechanism is group-strategyproof if no subset of customers can gain by lying about their values. There is a rich literature that designs group-strategyproof cost-sharing mechanisms using schemes that satisfy a property called cross-monotonicity. Unfortunately, Immorlica et al. showed that for many services, cross-monotonic schemes are provably not budget-balanced, i.e., they can recover only a fraction of the cost. While cross-monotonicity is a sufficient condition for designing group-strategyproof mechanisms, it is not necessary. Pountourakis and Vidali recently provided a _complete characterization of group-strategyproof mechanisms. We construct a fully budget-balanced cost-sharing mechanism for the edge-cover problem that is not cross-monotonic and we apply their characterization to show that it is group-strategyproof. This improves upon the cross-monotonic approach which can recover only half the cost, and provides a proof-of-concept as to the usefulness of the complete characterization. This raises the question of whether all "natural" problems have budget-balanced group-strategyproof mechanisms. We answer this question in the negative by designing a set-cover instance in which no group-strategyproof mechanism can recover more than a (18/19)-fraction of the cost.
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