# PRO-V

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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:

${\displaystyle $$C_{i}=\prod V_{ij}$$}$
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 :

${\displaystyle $$C_{i} =\prod \sqrt[n]{V_{ij}}$$}$
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.

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, like this:

[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.

## Proportional vote philosophy

Voters and candidates are treated as if they were groups of interests. In the case of the voter, interests are what he wants while in the case of candidates, interests are what they will actually do if they are elected (in the real world they can only be estimated by also evaluating the candidate's honesty, competence, sympathy, etc).

Voters and candidates are represented as points in the interest space, and a voter's appreciation of candidates depends on the distance between those points. The minimum distance between two points is 0, while the maximum distance between two points can be infinite.

In the philosophy of proportional voting, there cannot be a better candidate than the one who is 0 distance from the voter, while there can always be a worse candidate. This allows you to use the voter's position as a fixed point to derive proportions between candidates.

E.g. if a voter has these distances from candidates:

A[10] B[20] C[40] D[80] E[100] F[200] G[1000]

then the proportions between candidates will be:

A/A,  A/B,    A/C,    A/D,   A/E,    A/F,    A/G
A[1] B[0.5] C[0.25] D[1.25] E[0.1] F[0.05] G[0.001]

If the ratings to be used are [x1, x2, x4, x8] then the rating will be:

A[x8] B[x4] C[x2] D[x1] E[x1] F[x1] G[x1]

It's noted that the most liked candidates had good representation, while the worst candidates of D, all finished in 1x. However, it would be sufficient to add x16 as a possible rating to obtain a vote like this: A[x16] B[x8] C[x4] D[x2] E[x2] F[x1] G[x1] in which only G was misrepresented.

In the approval votes, instead, the range of distances (utility) is generally stretched until it becomes [0,5] obtaining as a vote:

A[5] B[5] C[5] D[5] E[5] F[4] G[0]

even increasing the range to [0,9] the vote would become:

A[9] B[9] C[9] D[9] E[9] F[8] G[0]

without actual improvements.

All this points out that, if the distances (utility) between voter and candidates aren't known, then it's not possible to convert an approval vote into a proportional vote, or vice versa.

### Normalization

If the proportions indicated in a proportional vote are to be kept unchanged, then there are no normalizations directly applicable to the vote.

Eg given this vote A[x8] B[x4] C[x2] D[x1] E[x1] if candidates D and E were removed, the vote could be normalized as follows A[x8] B[x2] C[x1] but the proportions with A would not be respected. If instead, candidate A were removed, the vote could be normalized as follows B[x8] C[x4] D[?] E[?] but it would be impossible to know whether candidates D and E deserve x1 or x2.

Even methods that change the weight/power of the vote, multiplying all ratings ​​by a certain value, don't change the proportions between candidates.

A type of indirect normalization applicable is the cumulative one, that is: each voter is assigned a certain amount of points (eg 100 points) which are distributed among the candidates according to the proportions indicated in the vote. By removing a candidate, the distribution of points changes, but the proportions don't change (DV uses this procedure).

## 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. One 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.