# Tragni's method

Tragni's method is a Single-Winner Symbolic voting systems that uses 2 non-cardinal symbols ( [worst] and [best] ), 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 ${\displaystyle \frac{\text{score of A}}{\text{score of B}}}$ 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).

With MAX = 5, the proportions range is [1/MAX, MAX] = [1/5, 5]. 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).

### Example

Given the following vote: A[worst] B[1] C[2] D[3] E[4] F[5] G[best] the respective complete P Table is obtained:

A [worst] B [1] C [2] D [3] E [4] F [5] G [best]
A [worst] 1 1/5 1/5 1/5 1/5 1/5 1/5
B [1] 5 1 1/2 1/3 1/4 1/5 1/5
C [2] 5 2 1 2/3 2/4 2/5 1/5
D [3] 5 3 3/2 1 3/4 3/5 1/5
E [4] 5 4 4/2 4/3 1 4/5 1/5
F [5] 5 5 5/2 5/3 5/4 1 1/5
G [best] 5 5 5 5 5 5 1

The propositions are all contained in [1/5, 5]. A [worst] always loses against everyone with 1/5, while G [best] always wins against everyone with 5. The voter will therefore be free to vote for his intermediate candidates without his vote changing the chances of victory of the [best] and [worst] candidates.

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

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 S-TM.

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.

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).

It's assumed that the 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 utility 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.

## Criteria

The following table refers to the Tragni's method which doesn't include tie case management processes, because tie case can be managed with different processes, each of which can cause different criteria to fail, making the analysis unnecessarily complex (also because ties are rare).

Majority Maj. loser Mutual maj. P.Cond. P.Cond. loser P.Smith IIA ILMP Clone proof Monotone Consistency Participation Rev. symmetry Later-no
Help
Later-no
Harm
Favorite
betrayal
Summable Strong Scale Inv.
Yes Yes Depends Yes Yes Yes Yes Yes Yes Yes Yes (half) Yes Yes Yes Depends Yes Yes Yes

### P criteria

There are 3 criteria among those indicated (concerning Condorcet) based on a head-to-head concept that considers only the order of the candidates, while Tragni's method also gives information on the distance (proportionality) that these candidates have among themselves in the votes.

The P criteria are the original criteria, which however use P-HtH as the definition of head-to-head, to make them suitable for a voting system that offers more information than just the order. No other criteria have been redefined other than these 3 (P Smith, P Cond. loser, P Cond.).

### IIA

1) Satisfy IIA if it's assumed that voters rate candidates individually and independently of knowing the available alternatives in the election, using their own absolute scale. If instead it's assumed that voters fully exploits their voting power, then case 2) applies.

2-1) Satisfy IIA if a candidate is added among those evaluated with cardinal values ​​[1,5], or if is added a candidate similar to those [best] or [worst].

2-2) Partially satisfies the IIA when a candidate in [worst] or in [best] moves among those rated cardinally (or vice versa), after adding a new candidate.

Example: given a vote in which (on average) all ratings are used, candidate H is added, who takes the place of G in [best] and moves G to the cardinal part of the vote. G, for the voter, is double better than F so the vote becomes like this:

A[worst] B[1] C[2] D[3] E[4] F[5] G[best]
A[worst] B[worst] C[1] D[1.5] E[2] F[2.5] G[5] H[best]


The addition of H caused the failure of the IIA only in candidate B and candidate G, while the others (6 out of 8 candidates, i.e. 75%) maintained their exact proportions, and didn't fail the IIA.

Also analyze how serious the failure was, looking at the proportions of B and G before and after adding H:

Before
B/A = 5 | B/C = 1/2 | B/D = 1/3 | B/E = 1/4   | B/F = 1/5 | B/G = 1/5
G/A = 5 | G/B = 5   | G/C = 5   | G/D = 5     | G/E = 5   | G/F = 5
B/A = 1 | B/C = 1/5 | B/D = 1/5 | B/E = 1/5   | B/F = 1/5 | B/G = 1/5
G/A = 5 | G/B = 5   | G/C = 5   | G/D = 5/1.5 | G/E = 5/2 | G/F = 5/2.5


Note that B remained unchanged compared to candidates with high ratings, while G remained unchanged compared to candidates with low ratings. Even in the case of B and G (the only candidates to undergo a change with the addition of H), however, the IIA is approximately 50% satisfied.

Conclusion: in case 1) and 2-1) the IIA is satisfied; in case 2-2), unless a negligible change in the proportions, the IIA can be considered overall satisfied, even if not perfectly.

### Other criteria

Majority: to support a candidate X more than any other candidate, that candidate would be rated with [best] by the voter, and in this case the criterion is met.

Majority loser: to support every other candidate over the candidate X, this candidate would be rated with [worst] by the voter, and in this case the criterion is met.

Mutual majority: if the set of candidates supported by the majority have all rated [best], then this criterion is met.

Clone proof: clones can change candidates in the P Smith set, so only in case of a tie, using P Smith-based procedures to solve them, this criterion fail.

Consistency: if the two separate elections give the same winner, then the union of the two electorates will give the same winner (meets the criterion). If in one of the two elections X is in tie with other candidates, while in the other X wins, then it's not said that with the union of the electorates X wins (it depends on what procedures are used to manage tie).

Reversal symmetry: if the candidates were rated only with [worst] and [best] (which are then reversed), then this criterion is met. If the candidates are rated even with the cardinal scores in [1,5] then the vote cannot be completely reversed, and the criterion isn't applicable. When it's applicable, it's always satisfied.

Later-no-harm: if the more-preferred candidate is rated [best], then the criterion is met.

## Strategies resistance

Min-maxing (benefit): a voter gives maximal support to some candidates and no support to all other candidates, to benefit those with maximal support (becomes Bullet voting if only 1 candidate has maximal support).

In Tragni's method the voter can give maximal support to a candidate, rating him [best]. The ratings given to the other candidates will not reduce the chances of victory for the rated [best] candidates, therefore the voter is free to show his/her true interests regarding those candidates.

Min-maxing (disadvantage): a voter gives maximal support to some candidates and no support to all other candidates, to disadvantage those with no support. It also includes the problem of candidates little known by the voter, who would receive additional support.

In Tragni's method the voter can give minimal support to a candidate, rating him [worst]. The ratings given to the other candidates will not increase the chances of victory for the rated [worst] candidates, therefore the voter is free to show his/her true interests regarding those candidates.

Regarding little-known candidates: if they aren't evaluated, they automatically receive [worst]; if they are evaluated, they will receive the lowest rating, but higher than [worst], that is 1. With this rating, it's practically impossible for a little known candidate to win.

Push-over: a voter rating an alternative lower in the hope of getting it elected, or rating an alternative higher in the hope of defeating it (concerns only methods that fail monotony).

Voting lesser of two evils: given 2 front runners, a voter gives maximal support to the best one and no support to the worst one.

In Tragni's method, assuming that the 2 front runners have different ratings in the non-strategic vote, then if the worst of the 2 is rated [worst], or if the best is rated [best], or they are rated 1 and 5 respectively, then the vote doesn't change (no strategy); otherwise, the worst of the 2 will be rated [worst]. The one described is the worst case.

If in the formula to manage the proportions of [worst] and [best], a higher value had been used instead of MAX (like MAX + 1 or MAX * 2), then in the case with the two front runners rated 1 and 5, the worst would have been put to [worst], increasing the damage of this strategy, even if only slightly.

## 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 (sC-TM): 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.

### Extended Tragni's method (E-TM)

It's Tragni's method in which [best] and [worst] are divided into 3 semi-cardinal symbols and MAX = 4. The range options are:

 [ 1w | 2w | 3w ]  | 1 | 2 | 3 | 4 |  [ 1b | 2b | 3b ]


The #w values will always be worst than the others. The #b values will always be best than the others. If two #w or #b values are to be considered, then they will be treated as cardinal values to make the proportion.

It offers a better representation of interests than Tragni's method, but it's more complex to understand how symbols work.

E-TM meets all the criteria satisfied by Tragni's method, replacing [worst] and [best] with the #w and #b values respectively.

It resists even more to the min-maxing tactic, because candidates who in the Tragni's method would all be put equally [worst] or [best], in E-TM can receive more precise ratings through the #w and #b values.

## 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 loser 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.