## General information

With DEMOCRATIX, you can evaluate preference profiles with respect to various voting rules. We currently support strict-order, complete preference profiles, that is, profiles where each voter provides a strict order over all choices (candidates). We are currently working hard on extending the tool and the examples on this webpage.

### Voting

Here are some basic notions and notations that we will need for the examples.

We are given a set of candidates C={A,B,C,...} and a set of voters V={1,2,3,...}. Let us assume that |C|=m and |V|=n. A preference relation (or preference) > is a strict total order over C. The top-ranked candidate of > is at position 1, the successor at position 2, . . ., and the last-ranked candidate is at position m. Voter i prefers a candidate A over a candidate B if A >i B. The vote of voter i ∈ V is the preference relation >i. A preference profile is a collection of votes P=(>1,...,>n).

An election is given by a triple E=(C,V,P). A voting rule F is a mapping from an election E to a non-empty set W ⊆ C, the set of winners of the election. In other words, the winners of election E with respect to F are the candidates in set W. (Actually, what we call for sake of simplicity a "voting rule" here is often called a "voting correspondence" in the literature.)

### Input format

Assume that the following preference profile (election) is given:

Suppose that you and your three friends want to decide on the fruit to be added to your fruit salad. Apples, bananas and cherries are available. Each of you gives an order on all the possible choices:

1: > > 2: > > 3: > > 4: > > For example, your choice of preferring apples over bananas over cherries is represented in the first preference relation.

Input for Democratix is specified in the PrefLib format. The preference profile from the example above is represented as follows:

3 1,Apple 2,Banana 3,Cherry 4,4,3 2,1,2,3 1,2,1,3 1,2,3,1

The lines represent the following (explanation based on PrefLib description):

• Overall number of candidates
• 1, candidate name
• 2, candidate name
• ...
• Number of voters, sum of vote count, number of unique preference relations
• count (number of votes for the preference relation), preference relation (strict total order over candidates)
• ...

Preference relations are sorted by the number of votes (in descending order). The "number of voters" field is the number of actual ballots that were cast. The "sum of vote count" should (normally) simply correspond to "number of voters".

For the example above we have 3 candidates, 4 voters, 4 "vote counts", and 3 unique preference relations (voter 1 and 2 gave the same preference relation).