Tragni's method: Difference between revisions
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Tragni's method is a [[Single Member system|Single-Winner]], invented by [[User:Aldo_Tragni|Aldo Tragni]]. |
Tragni's method is a [[Single Member system|Single-Winner System]], invented by [[User:Aldo_Tragni|Aldo Tragni]]. |
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The peculiarity of this method is the use of some non-cardinal values, and the use of multiplication to make the aggregation of votes. |
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==Procedure== |
==Procedure== |
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Voter score candidates using value {[worst],1,...,5,[best]}. |
Voter score candidates using value { [worst],1,...,5,[best] }. |
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* Make all head-to-heads, in which the candidate who is proportionally worse than the other loses (see Formula to calculate the proportionality). |
* Make all head-to-heads, in which the candidate who is proportionally worse than the other loses (see Formula to calculate the proportionality). |
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MAX indicates the highest value that can be used in the cardinal part of the vote. |
MAX indicates the highest value that can be used in the cardinal part of the vote. |
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''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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===P Table (boolean)=== |
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''This formula can also be used for cardinal systems (without [worst] and [best]).'' |
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''P Table contains all the P values, obtained with the Formula indicated above. Boolean P Table is a simplified version.'' |
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===P Table=== |
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''P Table (Proportions Table) contains all the P values, obtained with the Formula indicated above. Boolean P Table is a simplified version.'' |
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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. |
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. |
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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). |
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===Tie solutions=== |
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The procedure proposed by [[User:Aldo Tragni|Aldo Tragni]] is indicated below, but others may also be used: |
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* 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). |
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* If some candidates remain in tie, then the [[Tragni's method#P_TableFormula|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: |
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{| class="wikitable" style="text-align:center; margin: 0px 20px 0px 20px;" |
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|- style="font-size:100%;" |
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! | |
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! A |
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! B |
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! C |
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! D |
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! E |
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! Defeat Proportion |
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|- |
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|- style="font-size:100%; font-weight: inherit; background: white;" |
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! | A |
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| |
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| style="font-weight: bold; color: red;" | <math>P_{AB}^{*}</math> |
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| style="font-weight: bold; color: blue;" | <math>P_{AC}</math> |
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| style="font-weight: bold; color: green;" | <math>P_{AD}</math> |
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| | <math>P_{AE}^{*}</math> |
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| | <math>DP_{A} = P_{AB}\cdot P_{AE}</math> |
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|- |
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|- style="font-size:100%; font-weight: inherit; background: white;" |
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! | B |
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| style="font-weight: bold; color: red;" | <math>P_{BA}</math> |
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| |
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| | <math>P_{BC}^{*}</math> |
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| | <math>P_{BD}^{*}</math> |
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| | <math>P_{BE}</math> |
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| | <math>DP_{B} = P_{BC}\cdot P_{BD}</math> |
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|- |
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|- style="font-size:100%; font-weight: inherit; background: white;" |
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! | C |
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| style="font-weight: bold; color: blue;" | <math>P_{CA}^{*}</math> |
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| | <math>P_{CB}</math> |
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| |
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| | <math>P_{CD}^{*}</math> |
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| | <math>P_{CE}</math> |
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| | <math>DP_{C} = P_{CA}\cdot P_{CD}</math> |
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|- |
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|- style="font-size:100%; font-weight: inherit; background: white;" |
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! | D |
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| style="font-weight: bold; color: green;" | <math>P_{DA}^{*}</math> |
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| | <math>P_{DB}</math> |
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| | <math>P_{DC}</math> |
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| |
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| | <math>P_{DE}^{*}</math> |
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| | <math>DP_{D} = P_{DA}\cdot P_{DE}</math> |
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|- |
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|- style="font-size:100%; font-weight: inherit; background: white;" |
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! | E |
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| style="font-weight: bold; color: green;" | <math>P_{EA}</math> |
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| | <math>P_{EB}^{*}</math> |
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| | <math>P_{EC}^{*}</math> |
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| | <math>P_{ED}</math> |
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| |
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| | <math>DP_{E} = P_{EB}\cdot P_{EC}</math> |
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|- |
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|} |
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The candidate with the highest DP win. |
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* 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 *). |
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The candidate with the highest WP win. |
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* If some candidates remain in tie then, using the starting votes, the candidate who has the highest sum of points wins. |
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==Proportional head-to-head== |
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This method introduces the concept of proportional head-to-head (P-HtH), relating to cardinal systems, that is: |
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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). |
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The formula used to calculate this proportionality is indicated in the previous [[Tragni's method#Formula|Formula]] section. |
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===About [worst] and [best]=== |
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Given a range [1,3] with MAX = 3, and the following vote: |
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Vote: A[1] B[2] C[3] |
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[A-B] → 1/2 |
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[A-C] → <b>1/3</b> (worst) |
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[B-A] → 2 |
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[C-A] → <b>3</b> (best) |
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[B-C] → 2/3 |
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[C-B] → 3/2 |
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Proportions range [1/3,3] |
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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. |
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If you add 0 (worst value for the proportions), then you get: |
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Vote: A[1] B[2] C[3] D[0] E[D] |
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[A-D] → 1/0 = +inf → [MAX,+inf) = [3,+inf) → MAX |
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[D-A] → 0/1 = 0 → (0,1/MAX] = (0,1/3] → 1/MAX |
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[D-E] → 0/0 = ind → → 1 |
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[E-D] → 0/0 = ind → → 1 |
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If you add +inf (best value for the proportions), then you get: |
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Vote: A[1] B[2] C[3] D[+inf] E[+inf] |
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[A-D] → 1/+inf = 0 → (0,1/MAX] = [3,+inf) → 1/MAX |
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[D-A] → +inf/1 = +inf → [MAX,+inf] = (0,1/3] → MAX |
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[D-E] → +inf/+inf = ind → → 1 |
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[E-D] → +inf/+inf = ind → → 1 |
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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=== |
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It's assumed that the real appreciation (utility) of a voter can be converted into a linear range of appreciation like this [0,100]. |
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With MAX = 5, it can be assumed that the appreciation range is subject to this conversion, in the Tragni's method: |
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MAX = 5 |
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Real: [ < min | min | ... | 50 | ... | max | > max ] min * MAX = max | min = 100/(1+MAX) | max = 100/(1/MAX + 1) |
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Real: [ < 16.7 | 16.7 | ... | 50 | ... | 83.3 | > 83.3 ] 16.7 * 5 = 83.3 |
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Vote: [ [worst] | 1 | ... | 3 | ... | 5 | [best] ] |
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This means that MAX = 5 offers to the voter a good representation of his total true interests. |
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With others MAX values: |
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MAX = 2 |
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Real: [ < 33.3 | 33.3 | ... | 50 | ... | 66.7 | > 66.7 ] 33.3 * 2 = 66.7 |
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Vote: [ [worst] | 1 | ... | 1.5 | ... | 2 | [best] ] |
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MAX = 9 |
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Real: [ < 10 | 10 | ... | 50 | ... | 90 | > 90 ] 10 * 9 = 90 |
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Vote: [ [worst] | 1 | ... | 5 | ... | 9 | [best] ] |
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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. |
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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.'' |
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==Systems Variations== |
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===Cardinal Tragni's Method (CTM)=== |
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Instead of the proportions, the difference is used. |
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Obtain the D Table (Differences Table) using the following formula: |
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[[File:FormulaS.png|800px|frameless]] |
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The winner can be found in various ways, such as: |
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* in the D Table, the candidate who has the highest sum of points in his row wins. |
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* in the D Table, the candidate who has the most positive values in his row wins. |
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===SLE Tragni's Method (SLE-TM)=== |
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Sequential Loser-Elimination Tragni's Method (SLE-TM): |
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* use Tragni's method to find the candidate who loses the most times in P head-to-head. |
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* eliminate this candidate, normalizing the votes with Min-Max Normalization. If there are multiple losing candidates in a tie, delete them together. |
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These two procedures are repeated until only one candidate remains who will be the winner. |
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'''Min-Max Normalization''' |
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It's the normalization used in this method. Apply the following steps: |
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* 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]. |
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* 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]. |
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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. |
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''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.'' |
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===STAR Tragni's Method (STAR-TM)=== |
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* use Tragni's method to find the first 2 candidates who win the most times in P head-to-head. |
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* eliminate all other candidates, normalizing the votes with Min-Max Normalization. |
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* of the two remaining candidates, the one who wins in the P head-to-head wins the election. |
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==Systems Comparison== |
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===[[Copeland's method]]=== |
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Given these votes, with Tragni's method and MAX=5, A wins: |
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A[best] B[2] C[1] D[worst] E[1.5] |
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A[4] B[5] C[1] D[worst] E[best] |
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A[4] B[5] C[best] D[worst] E[2] |
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Using instead [[Copeland's method]], on the same votes, the winner would be B ([[Condorcet winner]]) |
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A > B > E > C > D |
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E > B > A > C > D |
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C > B > A > E > D |
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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). |
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Tragni's method cannot be considered [[Generalized_Condorcet_criterion|Smith-efficient]], unlike [[Copeland's method]] that it's. |
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===[[Distributed Voting]] ([[IRNR]])=== |
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[[Distributed Voting]] (specific variant of [[IRNR]]), can be considered a middle ground between [[Score Voting]] and Tragni's method, because: |
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* use the sum of the points, as in the [[Score Voting]], to determine which is the worst candidate. |
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* applies a proportional distribution of the points similar to the concept of proportion used into Tragni's method, and treats the value 0 of the range similarly to [worst]. |
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===[[Approval AV|Approval Voting]]=== |
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If the voters used only [worst] and [best], then the result would be equivalent to that of [[Approval AV|Approval Voting]]. |
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[[Category:Single-winner voting methods]] |
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[[Category:Cardinal voting methods]] |