Tragni's method: Difference between revisions
Added criteria table
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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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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 (doesn't meet the criterion).
'''[[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.
==Systems Variations==
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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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