University of Delhi 2011 M.A Economics winter semester 104- game theory- ii (admissions od 1999 & onwards) - Question Paper
This question paper contains 1 printed page.
Your Roll No. A
2235
M.A. Winter Semester ECONOMICS
Course 104 - Game Theory - II (Admissions of 1999 & onwards)
Time: 2-1/2 hours Maximum Marks : 70
(Write your Roll No. on the top of immediately on receipt of this question paper). Attempt as many as you want.
l.
a) Define a TU game and an allocation of a TU game. (3] .
b) Define Symmetry, Dummy and Additivity in this contcxt.' [3j
c) Define the Shapley value. [3]
d) Show that the Shapley value satisfies the above properties. (5j
. e) Propose an allocation which satisfies Symmetry and Additivity but violates Dummy. [3]
/) Define a convex game. {3]
g) A game (N, v) is called superadditive if for all .$,T such that S'P\T = 0, UT) > i;(.s) -f v(T). Show that convex games are also superadditive. [5]
h) Give an example of a superadditive game which is not convex. [5]
2. Consider an auction setting. Suppose there are just two bidders and their valuations are independently drawn from a uniform distribution on (0, l). Consider a second price auction.
a) Show that truth telling is a dominant strategy? (2]
b) Show that truth telling is a Bayes-Nash equilibrium? (4]
c) Find the sellers expected revenue. {4]
d) Define a second price auction with a reserve price. (2]
e) Show that truth telling is still a Bayes-Nash equilibrium? [2]
/) Find the sellers expected revenue. [4]
g) Can you use the revenue equivalence principle' to compare (c) and (/)? Why or why not? {2j
3. Consider a bilateral trade setting in which buyers and sellers valuations {Ob and Os respectively) of an indivisible object are drawn independently from the uniform distribution on [0,1]. Utility of buyer and seller are as follows, u, = transfer to i, where 1* = \ it i gets the object and 0 otherwise. Suppose that 0* is private information of agent i-
a) Model this situation as a mechanism design problem (specify the Bayesian game) [3]
b) Define cflicieny, individual rationality and budget balanced-ness in this context? [3]
6) Find the pivotal mechanism. (3]
c) Is pivotal mechanism individually rational? . [3]
d) Find a mechanism which is Bayesian incentive compatible (where truth telling is Bayesian-Nash equilibrium), efficient and individually rational. [5]
e) Show that if a mechanism is Bayesian incentive compatible, efficient and individually rational then the sum of the buyers and sellers expected utilities can not be less than [5]
/) Show that if a mechanism is budget balanced then the sum of the buyers and sellers cxpected utilities can not exceed . (5j
g) Show that there does not exist any Bayesian incentive compatible, efficient, individually rational and budget balanced mechanism in bilateral trading. [3]
200
|
Attachment: |
| Earning: Approval pending. |
