Flexibility in House Allocation and Housing Markets

A desirable house allocation rule is flexible in its response to changes in agents' preferences. We propose a specific notion of this flexibility. An agent is said to be `swap-sovereign' over a pair of houses at a profile of preferences if the rule assigns her one of the houses at that profile and assigns her the other house when she instead reports preferences that simply swap the positions of the two houses. A pair of agents is said to be `mutually swap-sovereign' over their assignments at a profile if the rule exchanges their assignments when they together report such `swap preferences'. An allocation rule is `individually swap-flexible' if any pair of houses has a swap-sovereign agent, and is `mutually swap-flexible' if any pair of houses has either a swap-sovereign agent or mutually swap-sovereign agents. We show for housing markets that the top-trading-cycles rule is the unique strategy-proof, individually rational and mutually swap-flexible rule. In house allocation problems, we show that queue-based priority rules are uniquely strategy-proof, individually swap-flexible and envy non-bossy. Varying the strength of non-bossiness, we characterise the important subclasses of sequential priority rules (additionally non-bossy) and serial priority rules (additionally pair-non-bossy and pair-sovereign).