Pareto optimal , strategy proof , and non-bossy matching mechanisms

The set of all Pareto optimal, strategy proof and non-bossy mechanisms is characterized as the set of trading and braiding mechanisms. Fix a matching problem with more than three houses and a profile of preferences. At the start of a trading and braiding mechanism at most one house is brokered; all other houses are owned. In the first trading round, owners point to their most preferred houses, the broker if there is one points to his most preferred owned house, and houses point to the agents who control them. At least one cycle forms. The agents in such a cycle are matched to the houses they point to. The process is repeated with the remainder. Once there are only three houses left the mechanism might turn into a braid, a device that avoids a particular matching.