We study in the laboratory a variant of the house allocation with existing tenants problem where subjects are partitioned into tiers with hierarchical privileges, and they know their position in the priority queue before making their decision. We evaluate the performance of the modified versions of three well‐known mechanisms: Top Trading Cycle (TTC), Gale‐Shapley, and Random Serial Dictatorship (RSD) with Squatting Rights. For all three mechanisms, we find low rates of participation (around 40%), high rates of truth‐telling conditional on participation (around 90%), high proportions of fair allocations (above 90%), and significant efficiency losses. We also observe differences across mechanisms: RSD is ranked highest in efficiency and TTC is ranked lowest in fairness. We then show that position in the queue has a positive and significant impact on participation whereas tier has little effect on behavior. Finally, the individual analysis reveals that the majority of subjects who do not play according to the theory still follow discernible patterns of participation and preference revelation.