ECO00055M Design and Analysis of Mechanisms and Institutions 2019-0
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ECO00055M
MSc Degree Examinations 2019-0
Design and Analysis of Mechanisms and Institutions
1. Mechanism Design [50 marks] Answer the following questions.
(a) Consider a kidney exchange problem with five patient-donor pairs. The compatibility matrix and the priority order are given below.
|
╱ 0 1
． C = ．(．) 1 0 ． ．(．) 0 1 ． 0 1 |
1 1 0 1 1 |
0 1 1 0 1 |
0 、 0 ．(．) ． 0 ．(．) , ． 1 ．(．) 0 ． |
╱ 、 ╱ 3 、 ．(．) π(2) ．(．) ．(．) 2 ．(．) ． ． ． ． ．(．) π(3) ．(．) = ．(．) 5 ．(．) ． ． ． ． ．(．) π(4) ．(．) ．(．) 4 ．(．) ．π(5) ． ．1 ． |
Draw the compatibility graph and find all priority matchings. In cases when there are multiple priority matchings, they are often said to be “welfare equivalent”. Explain how this relates to the assumption of dichotomous preferences. [10 marks]
(b) Suppose that three bidders 1, 2, 3 compete for three complementary items (A, B , C). Valuations of every bidder and the reserve values of the seller are given on the table. Compute all efficient allocations. [10 marks]
Table 1: Bidders’ and seller’s values over items.
|
|
0 |
A |
B |
C |
AB |
AC |
BC |
ABC |
|
Bidder 1 |
0 |
2 |
2 |
0 |
7 |
3 |
4 |
|
|
Bidder 2 |
0 |
2 |
0 |
2 |
3 |
6 |
3 |
|
|
Bidder 3 |
0 |
0 |
2 |
2 |
4 |
3 |
6 |
|
|
|
0 |
1 |
1 |
1 |
2 |
2 |
2 |
|
(c) Apply the dynamic auction for complements to this example by starting from the initial prices p0 (A) = p0 (B) = p0 (C) = 1, p0 (AB) = p0 (AC) = p0 (BC) = 2, and p0 (ABC) = 3. Show every step (prices, demands and supplies) of the auction. [30 marks]
2. Stability of Marriage [50 marks] Consider a society where each man has preferences over women and each woman has preferences over men. A man and a woman can marry by their
mutual consent. Everyone wishes to find a best partner if she or he can, or remains single otherwise.
(a) Formulate the concept of stable marriage matching and explain it in detail. [10 marks]
(b) Describe the deferred acceptance mechanism in detail with women mak- ing proposals. [10 marks]
(c) Find stable matchings of the following example for two cases: First, men make proposals; Second, women make proposals. Compare and discuss the two matchings. In the example, men are called Abel, Basil, Charlie, and Dan; women are Ashley, Emily, Julia and Mia. Their preferences are shown in the table below. [10 marks]
Preferences of men and women.
|
Abel |
Ashley |
Emily |
Julia |
Mia |
|
Basil |
Emily |
Ashley |
Mia |
Julia |
|
Charlie |
Julia |
Mia |
Ashley |
Emily |
|
Dan |
Mia |
Julia |
Emily |
Ashley |
|
Ashley |
Dan |
Charlie |
Basil |
Abel |
|
Emily |
Charlie |
Dan |
Abel |
Basil |
|
Julia |
Basil |
Abel |
Dan |
Charlie |
|
Mia |
Abel |
Basil |
Charlie |
Dan |
(d) Prove that the matching found by the deferred acceptance mechanism in the general case must be a stable marriage matching. [20 marks]
PART 2:
3. Mechanism Design [50 marks] Answer the following questions.
(a) In the kidney exchange model, state the definitions of maximal match- ings and maximum matchings and explain in words how they are differ- ent. Give an example of a maximal matching that is not a maximum matching and explain your answer. Under what conditions are maximal matchings guaranteed to be a maximum matching? [10 marks]
(b) Calculate the following two-person bargaining problem. The disagree- ment alternative is (2, 1). Player 1’s bargaining power is 0.6 and player 2’s bargaining power is 0.4. The set of feasible alternatives X is given by X = {(x1 , x2 )|x1 + x2 ≤ 18, x1 ∈ R, x2 ∈ R}. Calculate the gen- eralized Nash bargaining solution to this problem. Moreover, design a non-cooperative game such that its unique stationary subgame perfect equilibrium outcome coincides with the above solution. [40 marks]
4. Bargaining [50 marks]
Answer the following questions.
(a) Critically comment on the Nash Programme in the context of the bar- gaining problem. [20 marks]
(b) Consider the paper “Efficiency and Compromise: A Bid-Offer-Counteroffer Mechanism with Two Players” (Ju, 2013) that claims to have con- structed an “adequate” strategic mechanism. What problem is this mechanism used to solve? And what are the requirements for a mech- anism to be “adequate” in solving this problem? [30 marks]
2022-05-24