Mechanism Design and Social Choice, summer ’09 Prof. Dr. Benny Moldovanu
Wirtschaftstheorie II Lenn´estr. 37, 53113 Bonn
Konrad Mierendorff Office hours: Montag 11:45 - 12:45 Lenn´estraße 37, 2. Stock mierendorff@uni-bonn.de
Homework Assignment 6
This exercise sheet will be discussed June 18.
Please prepare solutions for the exercise session.
1. Marriage Markets I. Assume that men and women have strict preferences. LetM denote the set of men, and W the set of women. For two stable matchings µ and µ0 we write µ >M µ0 if for all m ∈ M, µ(m) ≥m µ0(m) and for at least one m ∈ M, µ(m) >m µ0(m). Similarly we writeµ >W µ0 if for all w∈W,µ(w)≥w µ0(w) and for at least onew∈W,µ(w)>w µ0(w).
Show that µ >M µ0 if and only if µ0>W µ!
(Hint: Proof by contradiction. Assume µ >M µ0 but not µ0 >W µ. Show that there existsw∈W who strictly prefers µ overµ0. Then show thatw and her partner under µ, blockµ0.)
2. Marriage Markets II. Again assume that men and woman have strict preferences.
For two stable matchingsµ andµ0 we define the function λas follows. For allm∈M λ(m) =
(µ(m), ifµ(m)>m µ0(m), µ0(m), otherwise.
For allw∈W
λ(w) =
(µ(w), ifµ(w)<w µ0(w), µ0(w), otherwise.
(a) Show that λ(m) =w ⇒ λ(w) =m.
(b) (more difficult) Show thatλ(w) =m ⇒ λ(m) =w.
(Hint: LetM0 be the set of men who are not unmatched according toλand W0 the set of women who are not unmatched according to λ. Show that |W0|= |µ(W0)|,
|λ(M0)|=|M0| and |M0| ≥ |µ(W0)|. Now show that |W0|=|λ(M0)|. Use this and (a) to complete the proof.)
(c) Show that λis a stable matching. (Note: (a) and (b) imply thatλis a matching.) 3. Marriage Markets III. LetM ={m1, m2, m3, m4} and W ={w1, w2, w3, w4}. Pre-
ferences are given by:
m1:w2, w3, w4, w1 m2:w1, w4, w3, w2 m3:w3, w4, w2, w1 m4:w1, w3, w4, w2
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and
w1 :m4, m2, m3, m1 w2 :m3, m1, m2, m4 w3 :m2, m4, m1, m3 w4 :m1, m4, m3, m2
Find the set of stable matchings!
4. Assignment Game. Consider a market with three buyersb1, b2, b3 and three sellers s1, s2, s3. Valuations are given by the following matrix (each buyer wants at most one object, sellers have values of zero.)
4 8 10 8 6 12 9 7 6
The aijth entry of this matrix is the value that buyeri attaches to objectj.
(i) What is the largest (smallest) payoff that a buyer can obtain in a core allocation?
(ii) Suppose a fourth potential buyer arrives at the market who values the objects at 4, 5 and 6, respectively. How does this change your answer for part (i)?
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