# Normalizations list

LEGEND

• MIN = lowest possible rating of a range, or the worst (last) position in the ranking.
• MAX = highest possible rating of a range, or the best (first) position in the ranking.
• min = lowest rating used in a range vote, or the worst position used in a ranking.
• max = highest rating used in a range vote, or the worst position used in a ranking.
• C_old = rating of candidate C in the vote, before the normalization.
• C_new = rating of C, after the normalization.
• C_i = ratings of all candidates in the vote, before the normalization.

MIN-MAX NORM

• 0/1 Norm
all to 0/1.

• Min Norm
min to MIN.

• Max Norm
max to MAX.

• Bmin Norm (Bullet Min Norm)
min to 0, the others to 1.
MIN,MAX can be used instead of 0,1.

• Bmax Norm (Bullet Min Norm)
max to 1, the others to 0.
MIN,MAX can be used instead of 0,1.

SCALARS NORM

• Borda Norm
Candidates are sorted in a vote, and then assigned a value based on position.

When using this norm it’s necessary to indicate exactly how the values to be assigned are calculated.
Ex. starting from the last position, the values are assigned in this way: 0,1,2,3,4,… (+1, starting from 0).

CONTINUOUS NORM

• Str Norm (Stretch Norm)
The following formula is used:

Interesting variants of this formula are: MIN = 0, MIN = min, MIN = 0 and min = 0 .

• Ln Norm
The following formula is used:

Formula for L1 Norm, with range [0,MAX]:

COMPOSED NORM

The vote is normalized to a matrix in which the results of all head-to-head among candidates are indicated.
In HtH the pair of candidates can be normalized in various ways (usually 1 is assigned to the best of the two, and 0 to the worst).

• AV Norm
Choose the threshold th in (0,1). Calculate T with the following formula:
T = (min - max) * th + min
A norm is applied to values greater than or equal to T (by default, 1 Norm), while another norm applies to the others (by default, the 0 Norm).

• MV Norm (Multi-Voting Norm)
Given one vote, it is normalized on each subset of candidates, obtaining more normalized votes which are then aggregated into one (generally, through the sum).

• nV Norm (n Voting Norm)
The candidates (in each vote) are grouped into sets, based on the rating.
One norm applies to candidates in the greater set (by default, the 1 Norm), while another norm applies to the others (by default, the 0 Norm).
If there is only 1 set containing all the ratings, then it’s subdivided into subsets.

When using this norm it’s necessary to indicate exactly how the ratings are divided into sets.
Ex. 2V Norm:
[0,1,2] < [3,4,5]
[0] < [1] < [2] , [3] < [4] < [5]
Ex. 3V Norm:
[0,1] < [2,3] < [4,5]
[0] < [1] , …
Ex. 2-2V Norm:
[0,1,2,3] < [4,5,6,7]
[0,1] < [2,3] , [4,5] < [6,7]
[0] < [1] , …

VOTING SYSTEMS

They are mainly used in the AR (Automatic Runoff), SLE and SWLE methods.

• AR-Norm_name: indicates an AR method which uses the “Norm_name” norm.
• SLE-Norm_name: indicates an SLE method which uses the “Norm_name” norm.
• SWLE-Norm_name: indicates a SWLE method that uses the “Norm_name” norm.

Some sensible voting systems:
AR-Bmin / AR-Bmax (STAR)
AR-Str (STLR MIN=0 and min=0)

SLE-Bmax (IRV, IRV-AV)
SLE-Bmin (FAIR-V) *
SLE-AV *
SLE-Borda (Baldwin)
SLE-Str (Cardinal Baldwin MIN=0, Stretch Baldwin MIN=min)
SLE-Ln (IRNR)
SLE-L1 (Distributed Voting)

SWLE-Bmax (WIRV) *
SWLE-Bmin *
SWLE-AV *

In this Codepen you can test some of this systems like: SLE-AV, SLE-Bmin, SWLE-AV, SWLE-Bmin, W-IRV.

P.S.
The system names indicated do not replace the originals; they serve only to indicate the connection between the various systems. Ex. in simulations it may be easier to notice that the systems that use Bmin, are all subject to a certain problem that the others do not have.
The systems with the asterisk (and their respective norms) are new voting systems designed by me; if any of them have already been invented with another name, please let me know so I update the post.
If there is any error or you know other norms, write it in reply.

We should not rename things. It will just cause confusion. Stretch Norm is Baldwin.

The norm I call “Stretch Norm” is used in image processing (see here). Baldwin is this norm applied to the SLE method, in fact I said “SLE-Str = Cardinal Baldwin” and not “Stretch Norm = Baldwin”.

Specifically, this normalization is used in image processing to perform Contrast Stretching, which is why I called it Stretch. By using that norm, in fact, a range stretching is actually applied.
It seems fair enough to me.

However I added at the end of the post that the names indicated in any case do not replace the original ones of the systems.