Gale’s Fixed Tax for Exchanging Houses

We consider taxation of exchanges among a set of agents in which each agent owns one object. Agents may have different valuations for the objects, and they need to pay taxes for exchanges. We show that, if a rule satisfies individual rationality, strategy-proofness, constrained efficiency, weak anonymity, and weak consistency, then it is either the no-trade rule or a fixed-tax core rule. For the latter rules, whenever any agent exchanges an object, the agent pays the same fixed tax (a lump sum payment that is identical for all agents) independently of which object the agent consumes. Gale’s top trading cycles algorithm finds the final assignment using the agents’ valuations adjusted with the fixed tax if the induced preferences are strict.