Review: Examination I; Math 484; Mathematics of Fairness and Equity

Prepared by:

Joseph Malkevitch
Department of Mathematics and Computer Studies
York College (CUNY)
Jamaica, New York 11451


web page:

1. Describe briefly a mathematical model for studying voting and election situations.

2. Give 7 different contexts where voting or elections take place.

3. Discuss four different kinds of ballots that might be used in an election.

4. Consider the election diagram below:

a. How many voters preferred C to B?

b. How many voters preferred C to A?

c. How many voters ranked B lowest?

d. How many voters ranked C lowest?

e. How many voters rank C first?

f. How many voters ranked D is second place?

g. Who would win if each of the following methods was carried out?

(If there are ties, state this.)

i. Plurality

ii. Run-off

iii. Sequential run-off

iv. Borda Count

v. Condorcet

vi. Nanson

vii. Coombs

h. What is the ranking that is produced by the Borda count?

i. Construct a matrix whose rows and columns are labeled by the alternatives (candidates) and where the (i, j) entry is the number of voters who preferred candidate i to candidate j.

5. Give a brief discussion of the Arrow Theorem, and why it is important. As part of your discussion list some examples of fairness rules that one might want a good election method to obey.

6. Give a brief discussion of the Saitherwaite-Gibbard Theorem and why it is important.

7. For each of the bankruptcy situations below, determine the allocation of the estate using the different methods a. Total equality of gain; b. Maimonides gain; c. Total equality of loss d. Maimonides loss e. Proportionality; f. The contested garment rule (Talmudic method); g. Shapley Value. The claimants names are given with capital letters, and the estate size by E. Also draw a diagram to illustrate the situation, and the location of the the contested garment solution.

a. A = 50; B = 100; E = 120

b. A = 50; B = 200; E = 120

c. A = 100; B = 400; E = 200

d. A = 100; B = 400; E = 80

e. A = 200; B = 300; E = 150

8. For each of the bankruptcy problems below, apply a. Maimonides gain; b. Maimonides loss; c. The Shapley Value; d. The Aumann-Maschler extension of the Talmudic method to more than claimants:

a. A = 100; B = 200; C = 700; E = 360

b. A = 100; B = 200; C = 700; E = 800

c. A = 100; B = 200; C = 700; E = 500

d. A = 100; B = 100; C = 300; D = 500; E = 400

(note: for d. you can skip the Shapley Value, which would involve a lot of work without a computer!)

9. Give some examples of fairness axioms that might be appropriate for a solution method involving a bankruptcy problem.