ECON30290 - AUCTIONS AND MECHANISM DESIGN ASSIGNMENT 2

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ECON30290 - AUCTIONS AND MECHANISM DESIGN

ASSIGNMENT 2

Deadline: Noon, 26 April 2024

First Published: 12 March 2024

Online Answer Sheet Available: Noon, 22 April 2024

1. Instructions

a. No extensions will be given! Please set aside some time in advance to solve this assignment. You have been given over a month to do so.

b. Carefully read every problem.

c. Do not round answers while solving a problem. Only do so when you submit your answer.

d. To submit your answers, go to Blackboard and initiate the online assignment. There, you will receive an answer sheet, where you can supply your answers.

e. When you submit your final answer include up to 3 digits after the decimal point and round down. Furthermore, do not include redundant zeros. For example 1/3 should be 0.333, √ 7 = 2.645, and 1/4 should be 0.25 (not 0.250).

2. Mechanism Design

1. There are 2 buyers i = 1, 2. The valuations are given by the CDFs F1(x) = x in [0, 1] and F2(x) = 2/x in [0, 2]. The seller wishes to find the optimal selling mechanism.

1.1 The function ψi is given by ψi(x) = Aix+Bi . Find A1, B1, A2, B2. (30 marks)

1.2 The optimal allocation can be described as follows:

• x1 < C1 and x2 < C2 ⇒ item isn’t sold.

• x1 ≥ D1 and x1 − x2 > D2 ⇒ buyer 1 wins.

• x2 > E1 and x2 − x1 > E2 ⇒ buyer 2 wins.

Find C1, C2, D1, D2, E1, E2. (45 marks)

1.3 The optimal payment functions are

Find F, G, H, I, J, K. (25 marks)