PRO-V

Product Voting (PRO-V) is a Single-Winner Cardinal voting systems developed by Aldo Tragni.

The objectives of this voting system is the balance between simplicity, resistance to strategies, elect utilitarian winner and provide the voter with a good representation of interests (range with 5 ratings).

Procedure

Ballots: voter score each candidates with bonus in [x1,x2,x3,x4,x5] and the absence of evaluation is equivalent to x1. Candidates at the beginning all have 1 point.

Counting: bonuses are applied to each candidate (eg x3 means multiplying the candidate's points by 3), and the one with the highest score in the end wins.

Formula

The formula used in the count is the following:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{equation}  C_{i}=\prod V_{ij}  \end{equation}}

Ci  = final score of a candidate Ci.
Vij = values of candidate Ci, obtained from the ballots.

However, this formula can return very large results, difficult to manage.

In computer systems the following formula can be used :

Failed to parse (unknown function "\begin{equation}"): {\displaystyle \begin{equation}  C_{i} =\prod \sqrt[n]{V_{ij}}  \end{equation}}

n = total number of votes.

If you have paper ballots then, before counting, you can eliminate ratings x1 and those ratings that appear at least once on all candidates, even in different votes.

Eg given these 3 votes:

A[x1] B[x2] C[x3] D[x3] E[x5]
A[x3] B[x5] C[x1] D[x2] E[x3]
A[x5] B[x3] C[x2] D[x4] E[x1]

delete the ratings x1, and x3 that appears at least once on all candidates, making the votes like this:

      B[x2]             E[x5]
      B[x5]       D[x2]
A[x5]       C[x2] D[x4]

so there is less multiplication to do.

Ratings scale

The minimum value of the range is always x1.

In a context with very different options (such as the electoral context) it's better to use an exponential scale of this type:

[x1, x2, x4, x8, x16]

while in contexts with options not very far from each other (such as satisfaction surveys) it's better to use a linear scale of this type:

[x1, x2, x3, x4, x5]

Adapting the scale to the context allows the voter to represent their interests well, maintaining simplicity in the vote (which always has only 5 ratings) and also more resistance to strategies.

Name derivation

PRO-nV: the PRO-V procedure works with ranges of different sizes and n indicates the amount of ratings used in the range.

PRO-3V: uses 3 ratings.
PRO-V: is the default definition, with 5 ratings.
FAIR-9V: uses 9 ratings.

Voting systems comparison

FAIR-V

The proportional ratings of the PRO-V make the intermediate ratings more used by the voter, because eg. adding these two votes A[x1] B[x2] (B is worth double A) and A[x4] B[x2] (A is worth double B), candidates A and B are equal, unlike the methods that add up the scores. A single point can make a lot of difference. Taking this characteristic of the PRO-V into consideration, and following the analysis of the resistance to strategies in FAIR-V, it can be seen that PRO-V is also resistant to strategies, but not as strong as FAIR-V which uses a range [0,2].

However, the PRO-V procedure is easier to understand than the FAIR-V one, and also offers a wider range of ratings to the voter.