Problem Set 3, Suggested Solutions
1
Kidney Exchange
Let there be five patients, each with a spouse who is a potential donor. The blood types of the patients and their prospective donors are: A-AB, B-A, A-AB, O-A, and AB-O (the first letter is the blood type of the patient and the second letter is the blood type of the donor).1 Suppose there are no problems of tissue type incompatibility, and that all blood-type acceptable transplants are equally likely to be successful (so all patients have “dichotomous” preferences (i.e., each patient is indifferent between all kidneys that are blood-type compatible, and prefers every blood-type compatible kidney to the outside option to blood-type incompatible kidney)).
1. What is the most transplants in principle that could be done in this group? For computing this, assume that the only constraint is the physical avail-ability of kidneys. So, for instance, allow for possibilities that some pa-tients whose donors donate a kidney to someone may not receive a kidney.
AnswerIn principle, you could make four transplants, by linking patients and donors as follows: A-A, A-A, AB-AB, and either B-O or O-O. One AB donor cannot be used,2 and either the B or the O patient do not get a transplant.
2. What’s the largest number of kidney transplants that can be done if donors will donate only if their spouse receives a transplant, and what options can achieve this maximum?
AnswerIf donors only participate conditional on spouses getting a kidney, the most that can be done is three transplants: a three-way exchange involving the AB-O pair, one A-AB pair, and either the O-A pair or the B-A pair, as follows (for the grading purpose, it is fine to find one example):
(a) Couples 1 or 3,2,5: A-A, B-O, AB-AB (1 or 3, and 4 don’t trade)
(b) Couples 1 or 3,4,5: A-A, O-O, AB-AB (2, and 1 or 3 don’t trade)
3. Suppose donors prefer to donate to their spouse and can opt out of an exchange to do this. Will this affect your answer to the last question? How? (Still assume, as in question (2), that donors will donate only if their spouse receives a transplantation)
AnswerCouple 5 is already compatible, and given the chance will choose to opt out. Without this couple, there are two A donors and two A kidneys, so two feasible transplants, but no way to give the spouses of the donors successful transplants, so no exchanges are possible with that constraint.
1Recall that an AB patient can accept any kidney, an A patient can accept an A or O
kidney, a B patient can accept a B or O, and an O patient can accept only an O.
2Because there are two AB donors while there is only one AB patient. This, in particular,
2
School Choice
Consider the following school choice problem: Let 1, 2, and 3 be students, andA, B, C be schools. Assume that each school has one seat. Preferences of students are
1:B, C, A,
2:B, A, C,
3:C, B, A.
Schools’ priorities over students are
A: 1,2,3,
B: 3,2,1,
C: 1,3,2.
1. Compute the result of the student-proposing DA.
AnswerStudent Proposing DA algorithm leads to{(1, C),(2, A),(3, B)}. 2. Is the result you obtained in the last question the only stable matching? Is it Pareto efficient for students? (Note that, for school choice, Pareto efficiency only takes into account student preferences, and not school pref-erences).
Answer School Proposing DA algorithm leads to {(1, C),(2, A),(3, B)}.
So the student-proposing DA and school proposing DA coincide. An im-plication of this fact is that there is only one stable matching. How-ever, it is not Pareto efficient for students: an alternative matching, {(1, B),(2, A),(3, C)}, is strictly preferred by students 1 and 3, while 2 is indifferent.
3. Use the TTC algorithm to find a matching that is Pareto efficient for students. Is the matching you found stable? (Don’t just say “stable” or “unstable”. Explain your answers and prove your assertion).
Answer TTC algorithm leads to a matching that is Pareto efficient (for students), {(1, B),(2, A),(3, C)}. This match is not stable because(2, B) is a blocking pair.
4. Are there other assignments that are Pareto efficient for students?
AnswerThere are other matches that are Pareto efficient for the students: for instance{(1, C),(2, B),(3, A)}.
3
Advanced Problem in School Choice
1. A mechanism ϕ is said to be non-wasteful if, for every school a and student i, a i ϕ()(i) implies |ϕ()(a)| =qa. In words, a mechanism
is non-wasteful if there is no situation such that there is an unallocated seat at a certain schoolawhile some studentiprefersato her matching,
ϕ()(i).
In the setting of the setting by APR, show if the following is true: If a mechanism is non-wasteful and proof, then there is no strategy-proof mechanism which dominates it (since the student-proposing DA is non-wastefulness, this claim would be a generalization of Theorem 1 of APR).
Answer Yes, the claim is true. We can see this by verification. At each point of the proof, they only use non-wastefulness, not full stability.
2. Is there any mechanism that is non-wasteful and strategy-proof, but is different from the student-proposing DA with a tie-breaking?
Answer Yes, there is such a mechanism. One example is the top trading cycles (as defined for school choice by Abdulkadiroglu and Sonmez 2003 AER). Another example is the so-called “sequential dictatorship”: An ex-ogenously fixed agent obtains her stated most preferred object; then, an agent is chosen (depending only on the object that the first agent obtained, and independently of everything else, such as reported preferences of all other agents) who receives his stated most preferred remaining object, and so on. Clearly this mechanism is strategy-proof and non-wasteful (in fact, it is Pareto efficient), but this is different from any DA algorithm.
The above is a perfectly legitimate answer, but I am curious to know whether there are mechanisms that is non-wasteful and strategy-proof, but is bad in terms of efficiency or not.
4
Random Assignment
Consider the random assignment problem as in the class. Let there be two real goods a, b with one unit each, plus the outside option ∅ with infinite supply. There are 4 agents,N={1,2,3,4}, and
1 likes a, b,∅ (in this order), 2 likes a,∅,
3,4 like b,∅.
Answer The result of the PS mechanism is: Good a Good b Good ∅
Agents 1 and 2 1/2 0 1/2
Agents 3 and 4 0 1/2 1/2
Or, in the matrix notation which we introduced in class,
1/2 0 1/2 1/2 0 1/2 0 1/2 1/2 0 1/2 1/2
.
(Either of these notations are fine for the purpose of grading). To show that this random assignment is ordinally efficient, let P be this random assignment, which is the result of the PS, and assume for contradiction that there exists another random assignmentP0which first-order stochas-tically dominatesP. Then, this in particular means thatP1a0 ≥P1a = 1/2. Similarly, P2a0 ≥ P2a = 1/2. Since there is only one unit of a, we have
P1a0 +P2a0 ≤1, so P1a0 =P2a0 = 1/2. By a symmetric argument we also obtain P1b0 = P2b0 = 1/2. Then, since P1a0 +P2a0 +P3a0 +P4a0 = 1, we obtainP3a0 =P4a0 = 0, soP30∅ =P40∅ = 1/2. By a symmetric argument, we obtainP1b0 =P2b0 = 0andP10∅ =P20∅= 1/2. Thus P0 =P, which is a contradiction. Thus we have proved thatP is ordinally efficient.
2. Compute the result of the PS mechanism when 1 mis-reports her prefer-ences, claiming that she prefersbtoato∅(while the true preferences are still the same as before).
Answer By computation,
Good a Good b Good ∅
Agent 1 1/3 1/3 1/3
Agent 2 2/3 0 1/3
Agents 3 and 4 0 1/3 2/3
3. Let agent1be endowed with the standard expected utility function, that is, her utility from a random assignment is given by an expected value of the utility from the assigned object. Assume that her utility from a, b, and∅are1,u, and0, respectively (assume thatuis a number in an open interval(0,1)). Characterize the values ofusuch that a misreporting by the agent1in the last question gives strictly higher expected utility to her than truthtelling does.
Answer By the calculations from the previous questions, the expected utility for agent 1 from reporting the true preferences is 12, while the expected utility for her from the misreporting is 13+ 13u. Therefore, the agent obtains strictly higher expected utility under the misreporting if and only if
1 2 <
1 3 +
1
4. Find a lottery over feasible allocation that implements the random assign-ment you found in question (??). Hint: Birkhoff- von Neumann theorem guarantees that, for the case of allocatingndistinct objects tondistinct agents, any random assignment can be implemented. In the case at hand, there are only 2 real goods a, b, but note that the outside option ∅ has infinite supply. So an easy modification should work.
Answer Let us use the matrix notation such that the row icorresponds to agent i, while the first, second, and the third column corresponds to
a,b, and∅. The random assignment matrix for the PS outcome found in question (??) is:
1/3 1/3 1/3 2/3 0 1/3 0 1/3 2/3 0 1/3 2/3
.
This matrix can be written as the following convex combination:
1/3 1/3 1/3 2/3 0 1/3 0 1/3 2/3 0 1/3 2/3
=1 6
1 0 0 0 0 1 0 1 0 0 0 1
+1 6
1 0 0 0 0 1 0 0 1 0 1 0
+1 3
0 1 0 1 0 0 0 0 1 0 0 1
+1 6
0 0 1 1 0 0 0 1 0 0 0 1
+1 6
0 0 1 1 0 0 0 0 1 0 1 0
