Constrained dictatorial rules are subject to variable-population paradoxes

In the context of classical exchange economies, we study four ways in which agents can strategically take advantage of allocation rules by affecting who participates and on what terms (Thomson in Soc Choice Welf 42:289–311, 2014). (1) An agent transfers their endowment to someone else and withdraws. The two of them may end up controlling resources that allow them to simultaneously reach higher welfare levels than they otherwise would. (2) An agent invites someone in and let their guest use some of their (the host’s) endowment. The guest transfers back to them what they are assigned over their endowment. The host may benefit. (3) An agent withdraws with their endowment. As in (1), they and someone who stays may end up controlling resources that allow the two of them to simultaneously reach higher welfare levels than they otherwise would. (4) An agent pre-delivers to someone else the net trade that the rule would assign to that agent had the agent participated. The second agent withdraws. The first agent participates with a modified endowment. The first agent may benefit. We ask whether “the constrained priority rules”, defined by maximizing the welfare of a particular agent subject to each of the others finding their assignment at least as desirable as their endowment satisfy these various requirements. The answers are all negative. Because these types of rules are often better behaved than rules that attempt some fairness in distributing gains from trade, these results strengthen the negative conclusions reached in Thomson (2014), and they may provide the key to identifying circumstances in which rules exist that satisfy the axioms, or to proving general impossibility results.