Agents a₁,...,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 a1. 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 h₁. 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? Solution: Consider three agents with preferences a₂: h₂ az : h₂ 1₂ h₂ h₂ > h₂ ha (1) (2) (3) The allocation resulting from this mechanism is that as gets hi, a2 gets hs and as gets h2. This is inefficient: the allocation in which as gets hs, a2 gets hi, and as gets he is Pareto superior. (b) Is it strategyproof? Solution: For the same example as in the previous part, az is better off reporting that h₂> h₂.
Agents a₁,...,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 a1. 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 h₁. 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? Solution: Consider three agents with preferences a₂: h₂ az : h₂ 1₂ h₂ h₂ > h₂ ha (1) (2) (3) The allocation resulting from this mechanism is that as gets hi, a2 gets hs and as gets h2. This is inefficient: the allocation in which as gets hs, a2 gets hi, and as gets he is Pareto superior. (b) Is it strategyproof? Solution: For the same example as in the previous part, az is better off reporting that h₂> h₂.
Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
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Transcribed Image Text: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 a₁. 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 h₁. Then agent as either holds on to h3 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?
Solution: Consider three agents with preferences
@₁: h₂
a₂: h₂
a3: h₂
h₁ > h₂
h₁> h3
h₁h3
(1)
(2)
(3)
The allocation resulting from this mechanism is that as gets h₁, a2 gets h3 and as gets
h2. This is inefficient: the allocation in which ar gets hs, az gets h1, and as gets h2 is
Pareto superior.
(b) Is it strategyproof?
Solution: For the same example as in the previous part, az is better off reporting that
h₁> h₂.
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