Tragni's method: Difference between revisions

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Tragni's method is a [[Single Member system|Single-Winner]] [[Cardinal voting systems]], invented by [[User:Aldo_Tragni|Aldo Tragni]].
Tragni's method is a [[Single Member system|Single-Winner]], invented by [[User:Aldo_Tragni|Aldo Tragni]].


The peculiarity of this method is the use of multiplication to make the aggregation of votes.
The peculiarity of this method is the use of some non-cardinal values, and the use of multiplication to make the aggregation of votes.


==Procedure==
==Procedure==


Voter score candidates using value {0,1,...,5}.
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 for proportionality).
* 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.
* The candidate who loses least times in head-to-head, wins the election.


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This method use ranges with values shown below:
This method use ranges with values shown below:


| 0 | | 1 | 1.5 | 2 | 2.5 | 3 | 3.5 | 4 | 4.5 | 5 |
[worst] | 1 | 1.5 | 2 | 2.5 | 3 | 3.5 | 4 | 4.5 | 5 | [best]
or
or
| 0 | | 1 | 2 | 3 | 4 | 5 |
[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.
These ranges have MAX = 5. The absence of evaluation is considered 0.

Different MAX values can generate different results.


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


[[File:Formula Tragni's method.png|500px|frameless]]
[[File:Formula Tragni's method.png|700px|frameless]]


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


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

===Tie solutions===

In case of a tie in head-to-head, there are various ways to try to find a winner anyway.
The procedure proposed by [[User:Aldo Tragni|Aldo Tragni]] is indicated below, but others may also be used:

* The P Table is used. P with * (<math>P^{*}</math>) indicates values ​​less than 1. In this example, all candidates received two head-to-head defeats, so calculate the Defeat proportion (DP) for everyone:

{| class="wikitable" style="text-align:center; margin: 0px 20px 0px 20px;"
|- style="font-size:100%;"
! |
! A
! B
! C
! D
! E
! Defeat Proportion
|-

|- style="font-size:100%; font-weight: inherit; background: white;"
! | A
|
| style="font-weight: bold; color: red;" | <math>P_{AB}^{*}</math>
| style="font-weight: bold; color: blue;" | <math>P_{AC}</math>
| style="font-weight: bold; color: green;" | <math>P_{AD}</math>
| | <math>P_{AE}^{*}</math>
| | <math>DP_{A} = P_{AB}\cdot P_{AE}</math>
|-

|- style="font-size:100%; font-weight: inherit; background: white;"
! | B
| style="font-weight: bold; color: red;" | <math>P_{BA}</math>
|
| | <math>P_{BC}^{*}</math>
| | <math>P_{BD}^{*}</math>
| | <math>P_{BE}</math>
| | <math>DP_{B} = P_{BC}\cdot P_{BD}</math>
|-

|- style="font-size:100%; font-weight: inherit; background: white;"
! | C
| style="font-weight: bold; color: blue;" | <math>P_{CA}^{*}</math>
| | <math>P_{CB}</math>
|
| | <math>P_{CD}^{*}</math>
| | <math>P_{CE}</math>
| | <math>DP_{C} = P_{CA}\cdot P_{CD}</math>
|-

|- style="font-size:100%; font-weight: inherit; background: white;"
! | D
| style="font-weight: bold; color: green;" | <math>P_{DA}^{*}</math>
| | <math>P_{DB}</math>
| | <math>P_{DC}</math>
|
| | <math>P_{DE}^{*}</math>
| | <math>DP_{D} = P_{DA}\cdot P_{DE}</math>
|-

|- style="font-size:100%; font-weight: inherit; background: white;"
! | E
| style="font-weight: bold; color: green;" | <math>P_{EA}</math>
| | <math>P_{EB}^{*}</math>
| | <math>P_{EC}^{*}</math>
| | <math>P_{ED}</math>
|
| | <math>DP_{E} = P_{EB}\cdot P_{EC}</math>
|-

|}

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 [[Tragni's method#Formula|Formula]] section.

===About 0 value===

Given a [[Cardinal voting systems|Cardinal systems]] that uses range [1,3] with MAX = 3, and the following vote:

Vote: A[1] B[2] C[3]
[A-B] → 1/2
[A-C] → <b>1/3</b> (worst)
[B-A] → 2
[C-A] → <b>3</b> (best)
[B-C] → 2/3
[C-B] → 3/2

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.

If you add 0 as value of the range, 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 → (0,1] → 1
[E-D] → 0/0 = ind → (0,1] → 1

+inf and 0 cannot be accepted. Since the amplitude of the ranges, within which it's possible to choose sensible values, ​​depends on MAX, then the value chosen must also (proportionally) depend on MAX. The simplest thing to do is to multiply MAX by 1 (leaving it unchanged) and then choose respectively MAX and 1/MAX as values.

In the case that 2 candidates have 0, the only thing that matters is that they don't favor each other to win, so their proportion cannot return a value greater than 1. Having both received the same score, the simplest thing is consider them proportionally equal, so a comparison of them will return 1.

===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, the appreciation range is subject to this conversion, in the Tragni's method:

MAX = 5
Real: [0 |10 |20|40|60|80|100]
Vote: [0 |0.5|1 |2 |3 |4 |5 ]
Round vote: [0 |1 |1 |2 |3 |4 |5 ]

All candidates with a real appreciation of less than 10, would have 0 points in an honest vote.

This means that MAX = 5 offers to the voter a good representation of his total true interests, requiring that only the most hated candidates be put to 0. This conversion is practically the same as that of [[Cardinal voting systems|Cardinal systems]], which use range and sum of the points.

With others MAX values:

MAX = 2
Real: [0 |50|...|100]
Vote: [0 |1 |...|2 ]
MAX = 10
Real: [0 |10|...|100]
Vote: [0 |1 |...|10 ]

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.

===About normalization===

MAX doesn't have to change, because it's used to indicate how much at most one candidate can be considered better than another (so that all the votes have the same weight).

Consider the following context:

Vote: A[0] B[1] C[5] Proportions range [1/5, 5] → <b>MAX = 5</b>
New candidate D is added, considered by the voter double best of C:
Normalized vote: A[0] B[0.5] C[2.5] D[5]
[B-D] → 1/10
[D-B] → 10
New proportions range [1/10, 10] → <b>MAX = 10</b>

To prevent MAX from changing after this type of normalization, you can rounded to integer the values ​​in [0,1].

==Systems Comparison==

===[[Copeland's method]]===

Given these votes, with Tragni's method, A wins

A[5] B[2] C[1] D[0]
A[4] B[5] C[1] D[0]
A[4] B[5] C[1] D[0]

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

A > B > C > D
B > A > C > D
B > A > C > 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 isn't [[Generalized_Condorcet_criterion|Smith-efficient]], unlike [[Copeland's method]].

===[[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, and treats the value 0 of the range in a similar way to the Tragni's method.


[[Category:Single-winner voting methods]]
[[Category:Cardinal voting methods]]

Revision as of 07:30, 28 July 2020

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

The peculiarity of this method is the use of some non-cardinal values, and the use of multiplication to make the aggregation of votes.

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 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.

P Table (boolean)

P 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).

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