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?

- 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

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:

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:

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

- Mathematics of Voting - slides [1]
- Criteria [2]
- EC228 Voting Theory Lecture Notes [3]
- Social Choice Theory and Multicriteria Decision Aiding [4]
- Book: Voting, Arbitration, and Fair Division [5]
- Methods vs Voting Criteria [6]

- Decision Engineering (ULB)
- The mathematics of voting and elections: Paradox, deception, and chaos [7]