Introduction

Abstract

Voting is a common way to resolve disagreements regarding policies to be adopted or candidates to be chosen for various positions and is therefore a necessary ingredient of democratic government. Yet there are numerous voting rules that differ from each other in processing the ballots into voting results. In other words, it is possible that for a given set of voters having a fixed distribution of preferences among the competing alternatives, one would obtain the election of a different alternative as a result of using a different voting rule. We focus on the most obvious desiderata associated with voting procedures, viz., the avoidance of paradoxical outcomes.

Appendices

Exercises

Problem 1.1

You are looking for a new bike and, on the basis of extensive study of relevant journals, three models stand out: Bike 1, Bike 2 and Bike 3. There are three bikes stores in your town: A, B and C. None of them has all these bike models. A has Bike 1 and Bike 2, B has Bike 2 and Bike 3 and C has Bike 1 and Bike 3. Suppose that you would prefer Bike 1 in store A, Bike 2 in store B and Bike 3 in store C. Does your choice behavior exhibit transitivity of underlying preferences? If it does, write down the ranking. If it doesn’t, which changes are needed to make it transitive?

Problem 1.2

Use now three criteria assumed to be of equal importance to you: price, weight, outlook. Suppose that in comparing any two bike models, your preference is determined by the respective ranking of these two models on a majority of criteria. Can you form a set of rankings over the three bike models with respect to the three criteria so that the resulting preference ranking is intransitive?

Problem 1.3

Construct a 3–voter, 3–alternative Condorcet Paradox. Switch the ranking of any two adjacent alternatives in one ranking. Analyze the ensuing profile: is there still an intransitive majority preference relation? Is there a Condorcet Winner?

Answers to Exercises

Problem 1.1

The answer to the first question: No, it doesn’t.

The answer to the second question: Choose Bike 1 in store C.

Problem 1.2

If the preference orderings for bikes (from top to bottom) in terms of the three criteria are as shown in the table below then the resulting ordering is intransitive.

Price	Weight	Outlook
Bike 1	Bike 2	Bike 3
Bike 2	Bike 3	Bike 1
Bike 3	Bike 1	Bike 2

Problem 1.3

Here’s a profile constituting a Condorcet Paradox in which the social preference ordering is intransitive (a $\succ$ b $\succ$ c $\succ$ a):

No. of voters	Preference ordering
1	a $\succ$ b $\succ$ c
1	b $\succ$ c $\succ$ a
1	c $\succ$ a $\succ$ b

Now switch a and b in the first ranking to get:

No. of voters	Preference ordering
1	b $\succ$ a $\succ$ c
1	b $\succ$ c $\succ$ a
1	c $\succ$ a $\succ$ b

Here the social preference ordering is transitive (b $\succ$ c $\succ$ a), i.e., b is the Condorcet Winner.