Tragni's method

Tragni's method is a Single-Winner System, invented by Aldo Tragni.

Procedure

Voter score candidates using value { [worst],1,...,5,[best] }.

Make all head-to-heads, in which the candidate who is proportionally worse than the other loses (see Formula to calculate the proportionality).
The candidate who loses least times in head-to-head, wins the election.

Ballot

This method use ranges with values shown below:

    [worst] | 1 | 1.5 | 2 | 2.5 | 3 | 3.5 | 4 | 4.5 | 5 | [best]
or
    [worst] | 1 | 2 | 3 | 4 | 5 | [best]

The absence of evaluation is considered [worst]. The cardinal part of the vote is always included in the range [1,MAX] (positive, without 0). In this case MAX = 5. Different MAX values can generate different results.

Formula

Given the head-to-head [A-B], make for each vote ${\frac {\text{score of A}}{\text{score of B}}}$ and then multiply all the fractions between them. If the result is > 1 then wins A, if < 1 then wins B, if = 1 then both win (tie isn't a defeat).

Below is a more rigorous description, given the head-to-head [A-B]:

MAX indicates the highest value that can be used in the cardinal part of the vote.

If of the two candidates in head-to-head only the order is known, and not the proportion (as in the rankings), then the lesser is placed at [worst] and the greater one at [best], but this is not the context.

This formula can also be used for cardinal systems (without [worst] and [best]).

P Table

P Table (Proportions Table) contains all the P values, obtained with the Formula indicated above. Boolean P Table is a simplified version.

Boolean P Table initially has all values = 0. Put 1 in the candidates who win, and leave 0 in those who lose, for each head-to-head.

[A,B] →  B loses       [A,C] →  A loses      [A,D] →  tie (no one loses)
[B,C] →  B loses       [B,D] →  D loses      [C,D] →  D loses

	A	B	C	D
A		1	0	1
B	0		0	1
C	1	1		1
D	1	0	0

Wins C, the candidate who has least 0 (defeats) on his row (or the one that has most 1).

Tie solutions

The procedure proposed by Aldo Tragni is indicated below, but others may also be used:

If multiple candidates have the least defeats (tie), then eliminate all the other candidates, use the min-max normalization on the votes and repeat the process from the beginning (as long as there are candidates that can be eliminated).

If some candidates remain in tie, then the P Table is used. P with * ( $P^{*}$ ) indicates values less than 1. In this example, all candidates received two head-to-head defeats, so calculate the Defeat proportion (DP) for everyone:

	A	B	C	D	E	Defeat Proportion
A		$P_{AB}^{*}$	$P_{AC}$	$P_{AD}$	$P_{AE}^{*}$	$DP_{A}=P_{AB}\cdot P_{AE}$
B	$P_{BA}$		$P_{BC}^{*}$	$P_{BD}^{*}$	$P_{BE}$	$DP_{B}=P_{BC}\cdot P_{BD}$
C	$P_{CA}^{*}$	$P_{CB}$		$P_{CD}^{*}$	$P_{CE}$	$DP_{C}=P_{CA}\cdot P_{CD}$
D	$P_{DA}^{*}$	$P_{DB}$	$P_{DC}$		$P_{DE}^{*}$	$DP_{D}=P_{DA}\cdot P_{DE}$
E	$P_{EA}$	$P_{EB}^{*}$	$P_{EC}^{*}$	$P_{ED}$		$DP_{E}=P_{EB}\cdot P_{EC}$

The candidate with the highest DP win.

If some candidates remain in tie, then the Win Proportion (WP) is calculated for them by multiplying the P values greater than or equal to 1 for each candidate (the P values of the previous table, without *).

The candidate with the highest WP win.

If some candidates remain in tie then, using the starting votes, the candidate who has the highest sum of points wins.

Proportional head-to-head

This method introduces the concept of proportional head-to-head (P-HtH), relating to cardinal systems, that is:

In a proportional head-to-head between 2 candidates, the candidate who turns out to be the one proportionally greater than the other wins (both win in the ties).

The formula used to calculate this proportionality is indicated in the previous Formula section.

About [worst] and [best]

Given a range [1,3] with MAX = 3, and the following vote:

Vote: A[1] B[2] C[3]
[A-B] →  1/2
[A-C] →  1/3 (worst)
[B-A] →  2
[C-A] →  3 (best)
[B-C] →  2/3
[C-B] →  3/2
Proportions range [1/3,3]

note that in the best case one candidate is 3 times better than the other, and in the worst case it's 1/3 (3 times worse) than the other. This means that the [worst] values cannot be more than 1/3 [1/MAX] compared to the others, and the MAX values cannot be less than 3 (MAX) compared to the others.

If you add 0 (worst value for the proportions), then you get:

Vote: A[1] B[2] C[3] D[0] E[D]
[A-D] →  1/0 = +inf  →   [MAX,+inf) = [3,+inf)   →   MAX
[D-A] →  0/1 = 0     →   (0,1/MAX]  = (0,1/3]    →   1/MAX
[D-E] →  0/0 = ind   →                           →   1
[E-D] →  0/0 = ind   →                           →   1

If you add +inf (best value for the proportions), then you get:

Vote: A[1] B[2] C[3] D[+inf] E[+inf]
[A-D] →  1/+inf = 0        →   (0,1/MAX]   = [3,+inf)   →   1/MAX
[D-A] →  +inf/1 = +inf     →   [MAX,+inf]  = (0,1/3]    →   MAX
[D-E] →  +inf/+inf = ind   →                            →   1
[E-D] →  +inf/+inf = ind   →                            →   1

In Tragni's method, for the management of the [worst] and [best] symbols, values in [MAX,+inf) could be used instead of MAX, such as "MAX+1" or "MAX*2", but never lower values of MAX (which is the standard).

About MAX

It's assumed that the real appreciation (utility) of a voter can be converted into a linear range of appreciation like this [0,100].

With MAX = 5, it can be assumed that the appreciation range is subject to this conversion, in the Tragni's method:

  MAX = 5
Real:   [ < min   | min  | ... | 50 | ... | max  | > max  ]    min * MAX = max | min = 100/(1+MAX) | max = 100/(1/MAX + 1)
Real:   [ < 16.7  | 16.7 | ... | 50 | ... | 83.3 | > 83.3 ]    16.7 * 5 = 83.3
Vote:   [ [worst] | 1    | ... | 3  | ... | 5    | [best] ]

This means that MAX = 5 offers to the voter a good representation of his total true interests.

With others MAX values:

 MAX = 2
Real:   [ < 33.3  | 33.3 | ... | 50  | ... | 66.7 | > 66.7 ]    33.3 * 2 = 66.7
Vote:   [ [worst] | 1    | ... | 1.5 | ... | 2    | [best] ]

 MAX = 9
Real:   [ < 10    | 10   | ... | 50 | ... | 90   | > 90   ]    10 * 9 = 90
Vote:   [ [worst] | 1    | ... | 5  | ... | 9    | [best] ]

MAX = 9 is also not bad, but the voting system wants to offer ballot with both 5 and 9 cardinal options, and MAX = 5 is easier to use for this purpose. Using two different MAXs, depending on the amount of options, can change the result so MAX must always be the same.

Note that, knowing the votes with MAX = 5, it's possible to make a conversion to know the form of the votes with MAX = 2 (lower value), but not vice versa.

Systems Variations

Cardinal Tragni's Method (C-TM)

It's the Tragni's method which in the range doesn't have the [worst] and [best] symbols.

Semi-Cardinal Tragni's method: it's the Tragni's method which in the range doesn't have one of the symbols [worst] or [best].

Score Tragni's Method (S-TM)

Instead of the proportions, the difference is used.

Obtain the D Table (Differences Table) using the following formula:

The winner can be found in various ways, such as:

in the D Table, the candidate who has the highest sum of points in his row wins.
in the D Table, the candidate who has the most positive values in his row wins.

SLE Tragni's Method (SLE-TM)

Sequential Loser-Elimination Tragni's Method (SLE-TM):

use Tragni's method to find the candidate who loses the most times in P head-to-head.
eliminate this candidate, normalizing the votes with Min-Max Normalization. If there are multiple losing candidates in a tie, delete them together.

These two procedures are repeated until only one candidate remains who will be the winner.

Min-Max Normalization It's the normalization used in this method. Apply the following steps:

if there are no candidates in [best] then take all the candidates with the highest value among those in [1,5] and put them in [best].
if there are no candidates in [worst] then takes all the candidates with the lowest value among those in [1,5] and puts them in [worts].

The candidates left with values in [1,5] don't change. You can also have the "Max-normalization" or "Min-normalization" variant which performs only 1 of the points indicated above.

This normalization and variants can also be applied in Cardinal voting systems with range, replacing [worst] and [best] with the lower and higher value of the range.

STAR Tragni's Method (STAR-TM)

use Tragni's method to find the first 2 candidates who win the most times in P head-to-head.
eliminate all other candidates, normalizing the votes with Min-Max Normalization.
of the two remaining candidates, the one who wins in the P head-to-head wins the election.

Systems Comparison

Copeland's method

Given these votes, with Tragni's method and MAX=5, A wins:

A[best] B[2] E[1.5] C[1] D[worst]
E[best] B[5] A[4] C[1] D[worst]
C[best] B[5] A[4] E[1] D[worst]

Using instead Copeland's method, on the same votes, the winner would be B (Condorcet winner)

A > B > E > C > D
E > B > A > C > D
C > B > A > E > D

This depends on the fact that the values indicated in the P Table (used in Tragni's method) aren't to be confused with those that the candidates would have in the pairwise table comparison (used in Copeland).

Tragni's method cannot be considered Smith-efficient, unlike Copeland's method that it's.

Distributed Voting (IRNR)

Distributed Voting (specific variant of IRNR), can be considered a middle ground between Score Voting and Tragni's method, because:

use the sum of the points, as in the Score Voting, to determine which is the worst candidate.
applies a proportional distribution of the points similar to the concept of proportion used into Tragni's method, and the value 0 of the range works similarly to [worst].

Approval Voting

If the voters used only [worst] and [best], then the result would be equivalent to that of Approval Voting.