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Problem Set. Voting and Social Choice, Part 1.

In the lab you can experiment with different voting schemes discussed in the lectures.

The most important part in these questions is to clearly explain your answers and show how you worked these out - you will be graded on this!

Problem 1. Favorite subjects.

Last year 32 Princeton students were quizzed on what their favorite courses were at graduate school. The results are below:

10 people ranked Math Alive as their favorite, followed by English Language, with Archeology For Beginners as their least favorite.
10 people ranked in the order English, Archeology, Math Alive.
12 people ranked in the order Archeology, Math Alive, English.

(a) Given these results, how many students preferred Math Alive to English? Express this as a percentage of the total number of students questioned.
(b) How many students preferred English to Archeology? And as a percentage?
(c) Based on these two observations, what subject would you place as the most popular?
(d) How many people rank Archeology over Math Alive? In the light of this result does this change your opinion made in (c)? Is there a clear 'most popular' subject?
(e) Using what we have found here, can you describe how statistics may be manipulated to fit a desired outcome?

Problem 2. Condorcet winners.

Explain what is meant by a Condorcet winner. Given an example scenario in which there is a Condorcet winner. Why is it advantageous to have a Condorcet winner? Is there a Condorcet winner in the example given in Question 1? Explain your answer.

Problem 3. Setting a drinks preference schedule.

In the lab you are asked to determine the winner according to each of the different voting schemes, starting from a given preference schedule, that is, a table that gives you the numbers of voters who ranked the candidates in several possible orders.

Here is a different preference schedule for the same five beverages. The numbers in bold type across the top are the numbers of voters that ranked the five alternatives as listed in the columns below. There are six groups of preference rankings.

15 11 10 8 5 3 1 5 5 5 1 1 4 1 2 4 5 4 3 2 3 3 3 2 5 4 1 2 2 5 2 3 4 1 4 3

Work out (and make sure you clearly show the steps in your working out) the winner according to each of the five different voting schemes below (described in the lab):

• Plurality
• Plurality with run-off.
• Sequential run-off.
• Borda-count.
• Condorcet winner.

Problem 4. A new name for APC 199?

We considered a name change for APC 199 and votes were collected on three other possibilities.

a) The preference schedule after 28 people were asked was

7 6 5 4 3 3
Math at Work 3 1 1 3 2 2
Practical Mathematics 2 2 3 1 1 3
The Unreasonable Effectiveness of Mathematics 1 3 2 2 3 1

Give the winner and show all the steps in your working out for:

• plurality.
• plurality with run-off.
• sequential run-off.
• Borda-count.
• Condorcet-winner.

Problem 5. Sincere and insincere voting.

Sincere voting is voting according to one's own preferences and beliefs, while insincere voting is voting in an effort to get a particular overall outcome rather than specifically according to preferences and beliefs.

Describe a recent voting or decision-making situation (personal or general) in which voting sincerely would have led to a different outcome than voting insincerely.

Problem 6. You might also like to buy...

In class we talked about an instance in which the algorithms used to decide what other items a shopper might like to buy backfired. Explain why this caused complications and how a store might work to avoid such problems arising.