Ranked Pairs
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Ranked pairs (sometimes abbreviated "RP") or the Tideman method is an
electoral system An electoral system or voting system is a set of rules that determine how elections and Referendum, referendums are conducted and how their results are determined. Electoral systems are used in politics to elect governments, while non-political ...
developed in 1987 by
Nicolaus Tideman Thorwald Nicolaus Tideman (, not ; born August 11, 1943 in Chicago, Illinois) is a Georgist economist and professor at Virginia Tech. He received his Bachelor of Arts in economics and mathematics from Reed College in 1965 and his PhD in economics ...
that selects a single winner using votes that express
preferences In psychology, economics and philosophy, preference is a technical term usually used in relation to choosing between alternatives. For example, someone prefers A over B if they would rather choose A than B. Preferences are central to decision the ...
. The ranked-pairs procedure can also be used to create a sorted list of winners. If there is a candidate who is preferred over the other candidates, when compared in turn with each of the others, the ranked-pairs procedure guarantees that candidate will win. Because of this property, the ranked-pairs procedure complies with the
Condorcet winner criterion An electoral system satisfies the Condorcet winner criterion () if it always chooses the Condorcet winner when one exists. The candidate who wins a majority of the vote in every head-to-head election against each of the other candidatesthat is, a ...
(and is a
Condorcet method A Condorcet method (; ) is an election method that elects the candidate who wins a majority of the vote in every head-to-head election against each of the other candidates, that is, a candidate preferred by more voters than any others, whenever ...
).


Procedure

The ranked-pairs procedure operates as follows: # Tally the vote count comparing each pair of candidates, and determine the winner of each pair (provided there is not a tie) # Sort (rank) each pair, by strength of victory, from largest first to smallest last.In fact, there are different ways how the ''strength of a victory'' is measured. This article uses Tideman's original method based on margins of victory. # "Lock in" each pair, starting with the one with the largest strength of victory and, continuing through the sorted pairs, add each one in turn to a graph if it does not create a cycle in the graph with the existing locked in pairs. The completed graph shows the winner. The procedure can be illustrated using a simple example. Suppose that there are 27 voters and 4 candidates ''w'', ''x'', ''y'' and ''z'' such that the votes are cast as shown in the table of ballots.


Tally

The vote tally can be expressed as a table in which the (''w'',''x'') entry is the number of ballots in which ''w'' comes higher than ''x'' minus the number in which ''x'' comes higher than ''w''. In the example ''w'' comes higher than ''x'' in the first two rows and the last two rows of the ballot table (total 18 ballots) while ''x'' comes higher than ''w'' in the middle two rows (total 9), so the entry in the (''w'',''x'') cell is 18–9 = 9. Notice the
skew symmetry In mathematics, particularly in linear algebra, a skew-symmetric (or antisymmetric or antimetric) matrix is a square matrix whose transpose equals its negative. That is, it satisfies the condition In terms of the entries of the matrix, if ...
of the table.


Sort

The positive majorities are then sorted in decreasing order of magnitude:


Lock

The next stage is to examine the majorities in turn to determine which pairs to "lock in". This can be done by building up a matrix in which the (''x'',''y'') entry is initially 0, and is set to 1 if we decide that ''x'' is preferred to ''y'' and to –1 if we decide that ''y'' is preferred to ''x''. These preferences are decided by the list of sorted majorities, simply skipping over any which are inconsistent with previous decisions. The first two majorities tell us that ''x'' is preferred to ''z'' and ''w'' to ''x'', from which it follows by transitivity that ''w'' is preferred to ''z''. Once these facts have been incorporated into the table it takes the form as shown after two steps. Notice again the skew symmetry. The third majority tells us that ''z'' is preferred to ''w'', but since we have already decided that ''w'' is preferred to ''z'' we ignore it. The fourth majority tells us that ''x'' is preferred to ''y'', and since we know that ''w'' is preferred to ''x'' we infer that ''w'' is preferred to ''y'', giving us the table after 4 steps. The fifth majority tells us that ''y'' is preferred to ''z'', and this completes the table.


Winner

In the resulting graph for the locked pairs, the
source Source may refer to: Research * Historical document * Historical source * Source (intelligence) or sub source, typically a confidential provider of non open-source intelligence * Source (journalism), a person, publication, publishing institute o ...
corresponds to the winner. In this case ''w'' is preferred to all other candidates and is therefore identified as the winner.


Tied majorities

In the example the majorities are all different, and this is what will usually happen when the number of voters is large. If ties are unlikely, then it does not matter much how they are resolved, so a random choice can be made. However this is not Tideman's procedure, which is considerably more complicated. See his paper for details.


An example


The situation

The results would be tabulated as follows: * indicates voters who preferred the candidate listed in the column caption to the candidate listed in the row caption * indicates voters who preferred the candidate listed in the row caption to the candidate listed in the column caption


Tally

First, list every pair, and determine the winner: Note that absolute counts of votes can be used, or percentages of the total number of votes; it makes no difference since it is the ratio of votes between two candidates that matters.


Sort

The votes are then sorted. The largest majority is "Chattanooga over Knoxville"; 83% of the voters prefer Chattanooga. Thus, the pairs from above would be sorted this way:


Lock

The pairs are then locked in order, skipping any pairs that would create a cycle: * Lock Chattanooga over Knoxville. * Lock Nashville over Knoxville. * Lock Nashville over Chattanooga. * Lock Nashville over Memphis. * Lock Chattanooga over Memphis. * Lock Knoxville over Memphis. In this case, no cycles are created by any of the pairs, so every single one is locked in. Every "lock in" would add another arrow to the graph showing the relationship between the candidates. Here is the final graph (where arrows point away from the winner). In this example, Nashville is the winner using the ranked-pairs procedure. Nashville is followed by Chattanooga, Knoxville, and Memphis in second, third, and fourth places respectively.


Summary

In the example election, the winner is Nashville. This would be true for any
Condorcet method A Condorcet method (; ) is an election method that elects the candidate who wins a majority of the vote in every head-to-head election against each of the other candidates, that is, a candidate preferred by more voters than any others, whenever ...
. Using the
First-past-the-post voting In a first-past-the-post electoral system (FPTP or FPP), formally called single-member plurality voting (SMP) when used in single-member districts or informally choose-one voting in contrast to ranked voting, or score voting, voters cast thei ...
and some other systems, Memphis would have won the election by having the most people, even though Nashville won every simulated pairwise election outright. Using
Instant-runoff voting Instant-runoff voting (IRV) is a type of ranked preferential voting method. It uses a majority voting rule in single-winner elections where there are more than two candidates. It is commonly referred to as ranked-choice voting (RCV) in the Un ...
in this example would result in Knoxville winning even though more people preferred Nashville over Knoxville.


Criteria

Of the formal voting criteria, the ranked pairs method passes the
majority criterion The majority criterion is a single-winner voting system criterion, used to compare such systems. The criterion states that "if one candidate is ranked first by a majority (more than 50%) of voters, then that candidate must win". Some methods that ...
, the
monotonicity criterion The monotonicity criterion is a voting system criterion used to evaluate both single and multiple winner ranked voting systems. A ranked voting system is monotonic if it is neither possible to prevent the election of a candidate by ranking them h ...
, the
Smith criterion The Smith criterion (sometimes generalized Condorcet criterion, but this can have other meanings) is a voting systems criterion defined such that it's satisfied when a voting system always elects a candidate that is in the Smith set, which is the ...
(which implies the
Condorcet criterion An electoral system satisfies the Condorcet winner criterion () if it always chooses the Condorcet winner when one exists. The candidate who wins a majority of the vote in every head-to-head election against each of the other candidatesthat is, a ...
), the
Condorcet loser criterion In single-winner voting system theory, the Condorcet loser criterion (CLC) is a measure for differentiating voting systems. It implies the majority loser criterion but does not imply the Condorcet winner criterion. A voting system complying wi ...
, and the
independence of clones criterion In voting systems theory, the independence of clones criterion measures an election method's robustness to strategic nomination. Nicolaus Tideman was the first to formulate this criterion, which states that the winner must not change due to the ...
. Ranked pairs fails the
consistency criterion A voting system is consistent if, whenever the electorate is divided (arbitrarily) into several parts and elections in those parts garner the same result, then an election of the entire electorate also garners that result. Smith calls this property ...
and the
participation criterion The participation criterion is a voting system criterion. Voting systems that fail the participation criterion are said to exhibit the no show paradox and allow a particularly unusual strategy of tactical voting: abstaining from an election can he ...
. While ranked pairs is not fully independent of irrelevant alternatives, it still satisfies local independence of irrelevant alternatives.


Independence of irrelevant alternatives

Ranked pairs fails
independence of irrelevant alternatives The independence of irrelevant alternatives (IIA), also known as binary independence or the independence axiom, is an axiom of decision theory and various social sciences. The term is used in different connotation in several contexts. Although it ...
. However, the method adheres to a less strict property, sometimes called
independence of Smith-dominated alternatives Independence of Smith-dominated alternatives (ISDA, also known as Smith- IIA or Weak independence of irrelevant alternatives) is a voting system criterion defined such that its satisfaction by a voting system occurs when the selection of the win ...
(ISDA). It says that if one candidate (X) wins an election, and a new alternative (Y) is added, X will win the election if Y is not in the
Smith set In voting systems, the Smith set, named after John H. Smith, but also known as the top cycle, or as Generalized Top-Choice Assumption (GETCHA), is the smallest non-empty set of candidates in a particular election such that each member defeats ever ...
. ISDA implies the Condorcet criterion.


Comparison table

The following table compares Ranked Pairs with other preferential single-winner election methods:


Notes


References


External links


Descriptions of ranked-ballot voting methods
by Rob LeGrand
Example JS implementation
by Asaf Haddad
Pair Ranking Ruby Gem
by Bala Paranj
A margin-based PHP Implementation of Tideman's Ranked Pairs

Rust implementation of Ranked Pairs
by Cory Dickson {{voting systems Electoral systems Monotonic Condorcet methods Single-winner electoral systems