This paper studies efficient resource allocation in a team consisting of a large number of firms and a resource allocator. We examine procedures based on a single demand message from the firms calculated using local information only. Our main result shows that for appropriately calculated demands, if firms are given (or "Grab") exactly what they demand until resources are exhausted and thereafter nothing, the per firm output converges, as the number of firms increases, to the maximal output obtainable using any decision rule including fully optimal ones requiring a complete exchange of all information. A similar result is also shown under stronger convexity conditions for demand messages defined by profit-maximizing under a (generally non-equilibrium) price system.