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 theSort
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 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 captionTally
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 anyCriteria
Of the formal voting criteria, the ranked pairs method passes the majority criterion, the monotonicity criterion, theIndependence of irrelevant alternatives
Ranked pairs failsComparison table
The following table compares Ranked Pairs with other preferential single-winner election methods:Notes
References
External links