Tragni's method: Difference between revisions
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Tragni's method is a
==Procedure==
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[[File:Formula Tragni's method.png|700px|frameless]]
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.''
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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.
<span style="color:red">[A
[B
{| class="wikitable" style="text-align:center; margin: 0px 20px 0px 20px;"
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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:
{| class="wikitable" style="text-align:center;"
|-
!
! 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===
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The candidate with the highest WP win.
* If some candidates remain in tie then, using
==Proportional head-to-head==
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Vote: A[1] B[2] C[3]
[A
[A
[B
[C
[B
[C
Proportions range [1/3,3]
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Vote: A[1] B[2] C[3] D[0] E[D]
[A
[D
[D
[E
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
[D
[E
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).
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===About MAX===
It's assumed that the
With MAX = 5, it can be assumed that the
MAX = 5
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''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 [[Tragni's method#Tie_solutions|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).
{| class="wikitable" style="text-align:center"
|- style="font-size:80%;"
! style="border-left: 3px solid #a0a0a0;" | [[Majority criterion|Majority]]
! | [[Majority loser criterion|Maj. loser]]
! | [[Mutual majority criterion|Mutual maj.]]
! style="border-left: 3px solid #a0a0a0;" | P.[[Condorcet criterion|Cond.]]
! | P.[[Condorcet loser criterion|Cond. loser]]
! | P.[[Smith criterion|Smith]]
! style="border-left: 3px solid #a0a0a0;" | [[Independence of irrelevant alternatives|IIA]]
! | [[Independence of the least/most preferred|ILMP]]
! | [[Clone independence|Clone proof]]
! style="border-left: 3px solid #a0a0a0;" | [[Monotonicity criterion|Monotone]]
! | [[Consistency criterion|Consistency]]
! | [[Participation criterion|Participation]]
! | [[Reversal symmetry|Rev. symmetry]]
! style="border-left: 3px solid #a0a0a0;" | [[Later-no-help criterion|Later-no<br>Help]]
! | [[Later-no-harm criterion|Later-no<br>Harm]]
! | [[Favorite betrayal criterion|Favorite<br>betrayal]]
! style="border-left: 3px solid #a0a0a0;" | [[Summability criterion|Summable]]
! style="border-right: 3px solid #a0a0a0;" | [[Scale invariance|Strong Scale Inv.]]
|- style="font-size:80%;"
<!-- Color: (yes) #1aff19 , #b9ffb9 , #f8ffbc , #ffc7c7 , #ff4847 (no) -->
|-
! style="background: #1aff19; font-weight: inherit; border-left: 3px solid #a0a0a0;" | Yes
! style="background: #1aff19; font-weight: inherit;" | Yes
! style="background: #b9ffb9; font-weight: inherit;" | Depends
! style="background: #1aff19; font-weight: inherit; border-left: 3px solid #a0a0a0;" | Yes
! style="background: #1aff19; font-weight: inherit;" | Yes
! style="background: #1aff19; font-weight: inherit;" | Yes
! style="background: #1aff19; font-weight: inherit; border-left: 3px solid #a0a0a0;" | Yes
! style="background: #1aff19; font-weight: inherit;" | Yes
! style="background: #1aff19; font-weight: inherit;" | Yes
! style="background: #1aff19; font-weight: inherit; border-left: 3px solid #a0a0a0;" | Yes
! style="background: #b9ffb9; font-weight: inherit;" | Yes (half)
! style="background: #1aff19; font-weight: inherit;" | Yes
! style="background: #1aff19; font-weight: inherit;" | Yes
! style="background: #1aff19; font-weight: inherit; border-left: 3px solid #a0a0a0;" | Yes
! style="background: #b9ffb9; font-weight: inherit;" | Depends
! style="background: #1aff19; font-weight: inherit;" | Yes
! style="background: #1aff19; font-weight: inherit; border-left: 3px solid #a0a0a0;" | Yes
! style="background: #1aff19; font-weight: inherit; border-right: 3px solid #a0a0a0;" | 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 [[Tragni's method#Proportional_head-to-head|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 criterion|Smith]], P [[Condorcet loser criterion|Cond. loser]], P [[Condorcet criterion|Cond.]]).
===IIA===
[https://en.wikipedia.org/wiki/Comparison_of_electoral_systems#cite_note-IIA_rating_methods-8 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]
H is added:
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
After addition of H
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 criterion|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 criterion|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 criterion|Mutual majority]]''': if the set of candidates supported by the majority have all rated [best], then this criterion is met.
'''[[Clone independence|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 criterion|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 criterion|Later-no-harm]]''': if the more-preferred candidate is rated [best], then the criterion is met.
==Strategies resistance==
<!-- Color: (resistance) #00e800, #a5fa00, #ffff00, #ffff00, #ffc000, #ff8837, #ff0000 (no resistance) -->
{| class="wikitable" style="text-align:center"
|- style="font-size:80%;"
! style="width: 150px;" |[[Tactical voting#Definitions|Min-maxing (benefit)]]
! style="width: 150px;" |[[Tactical voting#Definitions|Min-maxing (disadvantage)]]
! style="width: 150px;" |[[Tactical voting#Pushover|Push-over]]
! style="width: 150px;" |[[Tactical voting#Voting_for_the_lesser_of_two_evils|Voting lesser of two evils]]
|- style="font-size:80%;"
|-
! style="background: #00e800; font-weight: inherit; height: 16px;" |
! style="background: #a5fa00; font-weight: inherit; height: 16px;" |
! style="background: #00e800; font-weight: inherit; height: 16px;" |
! style="background: #00e800; font-weight: inherit; height: 16px;" |
|}
'''[[Tactical voting#Definitions|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.
'''[[Tactical voting#Definitions|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.
'''[[Tactical voting#Pushover|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).
'''[[Tactical voting#Voting_for_the_lesser_of_two_evils|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 [[Tragni's method#Formula|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==
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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)===
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* 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==
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[[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
* 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].
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