This is the second part of a paper concerning an iterative decentralized1 process designed to allocate resources optimally in decomposable environments that are possibly characterized by indivisibilities and other non convexities. Important steps of the process involve randomization. In Part I we presented the basic models and results, together with examples showing that certain assumptions can be satisfied in both classical and non convex cases. Part II goes further with such examples in showing that our process yields optimal allocations in environments in which the competitive mechanism fails, and also shows how abstract conditions used in Part I can be verified in terms of properties of preferences and production functions that are familiar to economists.