On two mechanisms in job rotation problems

We study a class of priority-based allocation problems, the general job rotation problems first proposed by Yu and Zhang (2020), in which each position is occupied by at most one agent, and each agent occupies at most one position. The priority structure is that at each occupied position, its occupant has the lowest priority while all other agents have equal highest priority; at other positions, all agents have equal priority. This priority structure is the “opposite” to that in house allocation with existing tenants (HET) problem proposed by Abdulkadiroğlu and Sönmez (1999). We propose two constrained efficient mechanisms: one is the modified top trading cycles mechanism, the other is the you request my position, I get your turn mechanism. Both of the two mechanisms are adapted from mechanisms in the HET model. In a special setup that there are equal numbers of agents and positions, each agent occupies exactly one position, and all positions are acceptable for all agents, these two mechanisms yield the same outcome, which is also the outcome of the backward-induction top trading cycles mechanism proposed by Yu and Zhang (2020). In such a setup, our two mechanisms are also weakly group strategy-proof.