Problems for Part III.1
Except where mentioned otherwise all concepts and notation have been explained in class.
Please refer to your notes or the slides in case of doubt. If you have any questions or comments, please send them to awest@uni-bonn.de.
Exercise 1: In this exercise we consider the men-optimal stable matching mechanism fM. (i) Show that if jMj 2 and jW j 1 then fM is not dictatorial.
(ii) Suppose it is commonly known that all women always submit preferences truthfully. In this case, we know from the lecture that fM is strategyproof.
Which assumptions of the Gibbard-Satterthwaite Theorem are not satised in the mar- riage model?
Exercise 2:
(i) Give an example of a marriage market with jW j = jMj = 2 that has at least two stable matchings.
(ii) Show that your example from part (i) implies that no stable matching mechanism is strategyproof if minfjW j; jMjg 2.
Exercise 3: Consider a matching problem with two rms f; f0 and two workers w; w0. Firm f can hire two workers and rm f0 can hire at most one worker.
Suppose it is commonly known that w and w0 are complements for f, i.e. f wants to hire w and w0 when they are both available, but is not willing to hire either w when w0 is not available, or w0 when w is not available. More formally, assume that f's preferences are given by Rf = fw; w0g; ;; fwg; fw0g.
Say that a matching is stable, if it is not blocked by a coalition consisting of one of the two rms and one or both of the workers.
Can you nd preferences for f0; w; w0 such that no stable matching exists?
Exercise 4: This is an exercise on the school choice problem with strict priorities. Assume for simplicity that no school can admit more than one student and that all students nd all schools acceptable. Consider the following algorithm:
1
Round 1:
Each school points to the student who has the highest priority for it.
Each student points to his most preferred school.
There is at least one cycle C = (s1; i1; : : : ; sn; in) (i.e. for all k n, sk points to ik and ik points to sk+1, where we set n + 1 := 1). For all k n, assign ik to school sk+1. Remove all students and schools involved in C and proceed to round 2.
Round k:
Each school points to the student who has the highest priority for it among all students remaining after rounds 1 through k 1.
Each student points to his most preferred school among all schools remaining after rounds 1 through k 1.
There is at least one cycle C = (s1; i1; : : : ; sn; in). For all k n, assign ik to school sk+1. Remove all students and schools involved in C and proceed to round k + 1.
Let fT(R; ) denote the outcome chosen by this algorithm given the preference prole R and the priority structure .
(i) Calculate fT(R; ) for the following problem
Ri1 Ri2 Ri3 s1 s2 s3
s1 s1 s2 i3 i1 i1
s3 s2 s1 i2 i2 i2 s2 s3 s3 i1 i3 i3
:
(ii) Fix some strict priority structure . Show that
(ii.1) for any preference prole R, fT(R; ) is ecient, and (ii.2) fT(; ) is strategyproof for students. (Harder)
[Hint: Fix an arbitrary preference prole R and let t denote the round in which i is removed from the procedure when the prole of submitted preferences is R. Let R0i be any possible alternative report for i and let t0 be the round in which i is removed from the procedure when the prole of submitted preferences is R0 = (R0i; R i).
Distinguish the cases t t0 and t0 > t.]
(iii) Construct a school choice problem (R; ) such that fT(R; ) is not fair.
2
Exercise 6: Consider the following school choice problem in which some students have iden- tical priorities for the same school:
Ri1 Ri2 Ri3 s1 s2 s3 s2 s2 s1 i1 i3 i3 s3 s3 s3 i2 i1; i2 i2
s1 s1 s2 i3 i1 :
Assume that all schools can admit at most one student.
Given a preference prole ~R, a matching is
fair, if there is no student-school pair (i; s) such that s ~Pi(i), (i0) = s, and i si0. non-wasteful, if s ~Pi(i) implies that there is some student i0 such that (i0) = s.
stable, if non-wasteful and fair.
constrained ecient, if it is stable and if there is no other stable matching 0 that Pareto dominates with respect to ~R.
(i) Calculate all constrained ecient matchings for the above example.
(ii) Let f be some constrained ecient matching mechanism. Show that no matter which constrained ecient matching f chooses at the above preference prole R, at least one student has a strict incentive to manipulate f at R.
3
