Agents a,,ay each have strict preferences over houses h₁,...,hy. Houses are to be allocated using the following mechanism. All agents submit their preferences over houses to the matching authority. The matching authority then goes through the agents in order, starting with a tentative allocation of hi to a. If a2 prefers h2 to h1, she is tentatively assigned h₂; otherwise, she and a swap, with a tentatively assigned h₂ and a2 tentatively assigned hy. Then agent as either holds on to ha or swaps it with the agent holding his more preferred house between hy and h2, and so on, until agent ay chooses either to hold on to hy or swap it with a house held by any one of the previous agents. After agent an makes her choice, the tentative assignment becomes the final allocation. For each of the following properties, either give a clear explanation of why it is satisfied (a formal proof is not necessary), or provide a counterexample. (a) Is this mechanism Pareto efficient?

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