Voting Theory

Voting Theory

Voting Theory studies how to take individual rankings of voters and aggregate them to form the global ranking.

Examples:

• Votes for a president of a company/country, etc. All voters communicate their results and based on that the president is chosen
• Search engines: there are many results, how to show them?

Notation and Relations

• let $A = {a, b, c, …}$ be the set of candidates
• there are $N$ voters
• each voter can express his preference on the basis of a ‘‘total order’’
  • i.e. he has to rank all the candidates

For this notation we define the following relations (Voting Theory Relations)

• Weak and Strong Preference
• Indifference

Voting Mechanisms and Principles

A ‘‘voting mechanism’’ (or ‘‘voting procedure’’ or ‘‘voting method’’) takes a collection of votes (individual preferences of the candidates from set $A$) and forms the global ranking. Usually it choses a single candidate from the set $A$.

There are several voting procedures:

• Plurality Voting
• Two-Round Voting
• Borda’s Rule
• Condorcet’s Rule

Criteria

How to characterize “good” voting methods?

There are several criteria

• Monotonicity
• Independence to Third Alternatives
• Condorcet Fairness Criterion
• Solution Existence
• Separability

PV

2PV

Borda

Cond.

Monotonicity

Solution Existence

Manipulation

Separability

Condorcet Fairness

Other principles:

• Unanimity

Theorems

• May’s Theorem
• Arrow’s Impossibility Theorem

Examples and Exercises

• Voting Theory Exercises
• Voting Theory Examples

Misc.

• Banzhaf Power Index - shows how strong a party is
• Parliamentary Allocation - how to allocate seats between parties in a parliament

Links

• Mathematics of Voting - slides [http://www.ms.uky.edu/~lee/ma111fa09/slides01.pdf]
• Criteria [http://www.ctl.ua.edu/math103/voting/whatdowe.htm]
• EC228 Voting Theory Lecture Notes [http://www2.warwick.ac.uk/fac/soc/economics/current/modules/ec228/details/lecturenotes/lecturenotesbook.pdf]
• Social Choice Theory and Multicriteria Decision Aiding [http://www-desir.lip6.fr/publications/pub_1389_1_BouyssouMarchantPerny_soc_choice.pdf]
• Book: Voting, Arbitration, and Fair Division [http://xaravve.trentu.ca/pivato/Teaching/voting.pdf]
• Methods vs Voting Criteria [http://en.wikipedia.org/wiki/Voting_system_criterion]

Sources

• Decision Engineering (ULB)
• The mathematics of voting and elections: Paradox, deception, and chaos [http://xaravve.trentu.ca/pivato/Teaching/votingslides.pdf]