**Problem Set 3: (Election Methods)
**

Prepared by:

Joseph Malkevitch

Department of Mathematics and Computer Studies

York College (CUNY)

Jamaica, New York

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1. The information below describes an election with preference also known as ordinal ballots where there are 4 candidates being ranked (no ties, no truncations). The notation being used is that the number at the start is the number of votes for this ordinal ballot, and the further to the left a candidate is the more preferred that candidate is. The voters voted in two different districts - the North and South Districts.

North District

8:B>C>A>D

6:C>D>A>B

4:D>B>C>A

South District

8:B>A>D>C

2:B>C>D>A

6:C>A>D>B

2:C>B>D>A

2:D>B>C>A

Question 1

a.

i. Construct a matrix whose entries are the number of voters who prefer a particular candidate in a two-way race, using a *separate* matrix for each district by itself. Thus, the entry in row A and column C would be the number of voters who prefer A to C and the entry in row C and column A would be the number of voters who prefer C to A based on the ballots shown (for each district).

ii. Construct the matrix as above when the ballots for the voters in each district are lumped together.

iii. Do you notice anything interesting about the matrix in ii as compared with the two matrices in i?

b.. Using the methods:

i. Plurality ii. Borda iii. Sequential run-off iv. Coombs, v. Condorcet vi. Bucklin vii. Baldwin

Who wins in each of the districts?

c. Lump the ballots and consider the election for all the voters as one group. Using the methods:

i. Plurality ii. Borda iii. Sequential run-off iv. Coombs, v. Condorcet vi. Bucklin vii. Baldwin

Who wins when all of the voters are lumped together?

Question 2

Do you notice any pattern above? Can you formulate what you think is a "desirable" feature of an election system in light of what you notice?

Comment: We are trying to understand what makes one election method better or worse than another. Since often the winner is different for the same set of ballots as the election method changes, knowing which "nice" properties each system obeys is of great interest. Insight grows when one can identify different properties that distinguish "better" from "worse" election methods.