Essays on Random Assignment Problems

This dissertation studies random assignment problems and investigates the scope of designing a desirable random assignment rule. Specifically, I ask the following two questions:

1. Is there a reasonably restricted domain of preferences on which there exists an sd-strategy-proof, sd-efficient and sd-envy-free or equal-treatment-of-equals rule?

2. Moreover, if the answer is in the affirmative, what is that rule?

We characterize the possible connected domains as the class of restricted tier domains and characterize the unique desirable rule on these domains as the probabilistic serial (PS) rule.

We then go beyond the connected domains and provide a class of maximal domains for the PS rule to be sd-strategy-proof.

Our findings significantly extend the scope of designing a desirable rule indicated by the existing literature and support the PS rule against the random priority rule unambiguously.