Ranked pairs
Ranked Pairs (also Tideman , after Nicolaus Tideman ) is an election and voting process developed in 1987 in which the voter specifies several preferences.
If there is a candidate that voters prefer to pair over all others, Ranked Pairs ensures that candidate wins. Because of this property, Ranked Pairs is by definition a Condorcet method . This is where Ranked Pairs differs from other preferential voting processes such as Borda voting and instant runoff voting .
Ranked Pairs resolves the Condorcet paradox by disregarding the pair with the lowest number of winning votes in the case of circular conclusions when determining the ranking.
Procedure
The votes are counted as follows:
- The number of votes is noted for all pair comparisons ; the winner is determined for each pair (provided there is no tie).
- All pairs are sorted according to the number of votes that the stronger of the two candidates received.
- The pairs are determined : the names of all candidates are recorded on a chart. The pair that has the largest number of votes of the stronger candidate is “determined” and entered on the graph. This is done by an arrow from the winner to the loser of the pair comparison. Then the next pair comparison is entered and determined according to the list created under 2. The other pair comparisons are entered according to the order of the list, provided that this does not result in a circle. The winner is the candidate to whom no arrow is pointing.
Ranked Pairs can also be used to create a complete ranking of candidates. To do this, the winner is first determined. To determine the runner-up, the first winner will be removed from the list of candidates and a new winner will be selected from the remaining candidates. To determine the third-placed winner, the second winner is also deleted, etc.
Counting the pair comparisons
All voter preferences are taken into account when counting votes. If a voter z. B. states “A> B> C” (A is better than B, and B is better than C), then one vote is added for A when comparing A: B, and one vote for A and B for A: C as well : C one vote for B. The voter can also express indifference (e.g. A = B). If a voter does not give a preference for one or more candidates, these candidates are rated worse than the others and equal to each other.
The majorities can then be determined. If “Vxy” is the number of votes that rate x higher than y, then “x” wins if Vxy> Vyx; and "y" wins if Vyx> Vxy.
Sorting the pairs
The winning pairs, called “majorities”, are then ranked from the largest to the smallest majority. A majority for x over y takes precedence over a majority for z over w if and only if at least one of the following conditions is met:
- Vxy> Vzw. In other words, the majority with the greater support comes first.
- Vxy = Vzw and Vwz> Vyx. If the majorities are equal, the majority comes first where the minority opposed is the smaller.
An example
The situation
Let's imagine a vote on the capital of Tennessee . The US state has an east-west extension of over 800 km, but only a north-south extension of 170 km. Let's say the candidates for the capital are Memphis (at the far west end), Nashville (in the middle), Chattanooga (a good 200 km southeast of Nashville), and Knoxville (far to the east, a good 180 km northeast of Chattanooga). The population of the catchment areas of these cities is distributed as follows:
- Memphis (Shelby County): 826 330
- Nashville (Davidson County): 510 784
- Chattanooga, Hamilton County: 285 536
- Knoxville (Knox County): 335 749
Let's say voters vote based on their geographic proximity. If we also assume that the population distribution of the rest of Tennessee corresponds to that of the population centers, then one can imagine a vote with the following distribution:
| 42% of voters (Near Memphis) |
26% of voters (Near Nashville) |
15% of voters (near Chattanooga) |
17% of voters (Near Knoxville) |
|---|---|---|---|
|
|
|
|
The results would be as follows:
| A. | |||||
|---|---|---|---|---|---|
| Memphis | Nashville | Chattanooga | Knoxville | ||
| B. | Memphis | [A] 58% [B] 42% |
[A] 58% [B] 42% |
[A] 58% [B] 42% |
|
| Nashville | [A] 42% [B] 58% |
[A] 32% [B] 68% |
[A] 32% [B] 68% |
||
| Chattanooga | [A] 42% [B] 58% |
[A] 68% [B] 32% |
[A] 17% [B] 83% |
||
| Knoxville | [A] 42% [B] 58% |
[A] 68% [B] 32% |
[A] 83% [B] 17% |
||
| Paired election results (won-lost-draw): | 0-3-0 | 3-0-0 | 2-1-0 | 1-2-0 | |
| Votes against in worst pairwise defeat: | 58% | N / A | 68% | 83% | |
- [A] stands for voters who prefer the candidate who is in the column heading over the candidate who is in the row heading
- [B] stands for voters who prefer the candidate who is in the row heading over the candidate who is in the column heading
- [NP] stands for voters who did not indicate a preference between the candidates concerned
Counting
First, each pair is listed and the winner is determined:
| Pair | winner |
|---|---|
| Memphis (42%) vs. Nashville (58%) | Nashville 58% |
| Memphis (42%) vs. Chattanooga (58%) | Chattanooga 58% |
| Memphis (42%) vs. Knoxville (58%) | Knoxville 58% |
| Nashville (68%) vs. Chattanooga (32%) | Nashville 68% |
| Nashville (68%) vs. Knoxville (32%) | Nashville 68% |
| Chattanooga (83%) vs. Knoxville (17%) | Chattanooga: 83% |
Note that both absolute values and percentages of the total number of votes can be used; it makes no difference.
sort by
Then the votes are sorted. The largest majority are "Chattanooga via Knoxville"; 83% of voters prefer Chattanooga. Nashville (68%) beats both Chattanooga and Knoxville with 68% over 32% (an exact tie that is actually unlikely with so many voters). Since Chattanooga> Knoxville, and they are the losers, Nashville vs. Knoxville added, and then Nashville vs. Chattanooga.
So the pairs would be sorted as follows:
| Pair | winner |
|---|---|
| Chattanooga (83%) vs. Knoxville (17%) | Chattanooga 83% |
| Nashville (68%) vs. Knoxville (32%) | Nashville 68% |
| Nashville (68%) vs. Chattanooga (32%) | Nashville 68% |
| Memphis (42%) vs. Nashville (58%) | Nashville 58% |
| Memphis (42%) vs. Chattanooga (58%) | Chattanooga 58% |
| Memphis (42%) vs. Knoxville (58%) | Knoxville 58% |
Fix
The pairs are then set in turn. All of those pairs that would result in a circle are left out:
- Chattanooga fixed via Knoxville.
- Nashville priced via Knoxville.
- Nashville priced via Chattanooga.
- Nashville priced via Memphis.
- Chattanooga fixed over Memphis.
- Knoxville priced via Memphis.
In this case, neither pair creates a circle. Hence each is fixed.
Each “fix” adds another arrow to the diagram showing the relationship between the candidates. This is what the diagram looks like at the end. (The arrows go out from the winner of the pair.)
In this example, Nashville is the winner when using Ranked Pairs.
Example for the resolution of ambiguities
Let's say there is an ambiguity, e.g. B. a situation with candidates A, B and C.
- A> B 68%
- B> C 72%
- C> A 52%
In this situation we set the majorities and start with the largest.
- Set B> C
- Set A> B
- We do not stipulate C> A because it would create an ambiguity or a circle.
Hence, A is the winner.
Summary
In the sample voting, Nashville is the winner. The same would apply to any other Condorcet method .
Using relative majority voting and some other techniques, Memphis would have won since it has the largest population, although Nashville would have easily won the pairwise vote.
Using instant runoff voting , in this example, Knoxville would have won, although more people prefer Nashville over Knoxville.