The Hong Kong University of Science and Technology
Mathematical Models in Economics and Finance
MATH 4824

Spring 2014
Mathematical Models in Economics and Finance
Homework Two
Course instructor: Prof. Y.K. Kwok
1. Two simple games are equivalent (or isomorphic) if the players can be labeled in such
a way that the winning coalitions are the same in both games. Show that t
The Hong Kong University of Science and Technology
Mathematical Models in Economics and Finance
MATH 4824

Spring 2014
Mathematical Models in Economics and Finance
Topic 1 Proportional representation and Apportionment
Schemes
1.1 General issues of apportionment of legislature seats
1.2 Quota Method of the Greatest Remainder (Hamiltons method)
and paradoxes
1.3 Geometric c
The Hong Kong University of Science and Technology
Mathematical Models in Economics and Finance
MATH 4824

Spring 2014
Mathematical Models in Economics and Finance
Topic 2 Analysis of powers in voting systems
2.1 Weighted voting systems and yesno voting systems
2.2 Power indexes: ShapleyShubik index and Banzhaf index
2.3 Case studies of power indexes calculations
2.4 Pr
The Hong Kong University of Science and Technology
Mathematical Models in Economics and Finance
MATH 4824

Spring 2014
Mathematical Models in Economics and Finance
Topic 3 Fundamental theorem of asset pricing
3.1 Law of one price and Arrow securities
3.2 Noarbitrage theory and risk neutral probability measure
3.3 Valuation of contingent claims
3.4 Binomial option pricing
The Hong Kong University of Science and Technology
Mathematical Models in Economics and Finance
MATH 4824

Spring 2014
Mathematical Models in Economics and Finance
Solution to Homework Two
Course instructor: Prof. Y.K. Kwok
1. To show the equivalence, it suces to consider the set of winning coalitions for each of the
voting systems and check whether they are the same. It
The Hong Kong University of Science and Technology
Mathematical Models in Economics and Finance
MATH 4824

Spring 2014
Mathematical Models in Economics and Finance
Homework Three
Course Instructor: Prof. Y.K. Kwok
1. Show that a dominant trading strategy exists if and only if there exists a trading strategy satisfying V0 < 0
and V1 () 0 for all .
Hint: Consider the domina
The Hong Kong University of Science and Technology
Mathematical Models in Economics and Finance
MATH 4824

Spring 2014
Mathematical Models in Economics and Finance
Solution to Homework One
Course instructor: Prof. Y.K. Kwok
1. (a) When S = 2, due to the satisfaction of the quota property, one state is rounded
up while the other state is rounded down (the very special case
The Hong Kong University of Science and Technology
Mathematical Models in Economics and Finance
MATH 4824

Spring 2014
Mathematical Models in Economics and Finance
Homework One
Course instructor: Prof. Y.K. Kwok
1. (a) Show that the Alabama Paradox does not occur in Hamiltons method if the number
of states S = 2.
(b) Is it possible that an increase in house size h can cau
The Hong Kong University of Science and Technology
Mathematical Models in Economics and Finance
MATH 4824

Spring 2014
Mathematical Models in Economics and Finance
Solution to Homework Three
Course Instructor: Prof. Y.K. Kwok
1. part: The trading strategy H with V0 < 0 and V1 () 0, , dominates the zeroholding trading strategy H = (0 0 0)T . The zeroholding strategy gives