Consider the following housing market: There are four agents a1, a2, a3, a4, and four houses...

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Consider the following housing market: There are four agents a1, a2, a3, a4, and four houses h, h2, h3, h4, where a; is the initial owner of house h; for each i = 1, 2, 3, 4, and preferences of agents are given as: a1 h h3 h h4 a2 a3 h h h h3 h3 h4 h3 hs h h4 a4 h h This way of representing preferences here is different from the approach we used in class. The literature has used both. So do not get confused. For instance, in this problem, a's preference is that she prefers h to h3 to h to h4. 1. Compute the allocation resulting from Gale's TTC algorithm under the above preference profile. 2. The result of TTC which you computed in the previous question should be Pareto efficient, from the general property of TTC. Verify that the resulting allocation you found is indeed Pareto efficient. 3. Suppose that agent a misreports her preferences by declaring that she prefers h3 to h to h to h4, although her true preference is still given as in the above table. Compute the result of TTC under this misreporting (while all other agents still report their true preferences). Is agent 1 made better off with her misreporting? Is the resulting allocation Pareto efficient (with respect to the true preferences)? 4. Consider an allocation outcome {(a, h3), (a2, h), (a3, h4), (a, h)}. Show that this outcome is also Pareto efficient. 5. Show that the allocation in the previous question is not in the core (we actually proved in class that the result of TTC is the unique allocation in the core, but don't rely on that fact. Instead, point out which group of agents can block this outcome and why). Consider the following housing market: There are four agents a1, a2, a3, a4, and four houses h, h2, h3, h4, where a; is the initial owner of house h; for each i = 1, 2, 3, 4, and preferences of agents are given as: a1 h h3 h h4 a2 a3 h h h h3 h3 h4 h3 hs h h4 a4 h h This way of representing preferences here is different from the approach we used in class. The literature has used both. So do not get confused. For instance, in this problem, a's preference is that she prefers h to h3 to h to h4. 1. Compute the allocation resulting from Gale's TTC algorithm under the above preference profile. 2. The result of TTC which you computed in the previous question should be Pareto efficient, from the general property of TTC. Verify that the resulting allocation you found is indeed Pareto efficient. 3. Suppose that agent a misreports her preferences by declaring that she prefers h3 to h to h to h4, although her true preference is still given as in the above table. Compute the result of TTC under this misreporting (while all other agents still report their true preferences). Is agent 1 made better off with her misreporting? Is the resulting allocation Pareto efficient (with respect to the true preferences)? 4. Consider an allocation outcome {(a, h3), (a2, h), (a3, h4), (a, h)}. Show that this outcome is also Pareto efficient. 5. Show that the allocation in the previous question is not in the core (we actually proved in class that the result of TTC is the unique allocation in the core, but don't rely on that fact. Instead, point out which group of agents can block this outcome and why).

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