Posted by: sandeepchrao | October 11, 2008

Voting Math

Elections are only a month away, and I think it’s a good time to start discussing voting systems and how the type of voting system greatly affects the outcome of votes. A voting system allows voters to choose between two options by defining the allowable set of votes, the tallying methods, and the algorithm for determining the outcomes. I am only going to discuss single winner methods, but there are many types of voting systems for multiple winner elections (

There are two main categories for single winner voting, Single/Sequential Voting Methods and Ranked Voting Methods.

Single Voting

Most people are familiar with this form of voting. Two of the more popular methods are plurality and elimination runoff elections.

In plurality election, each voter chooses one candidate on the ballot and the candidate with the most votes, regardless of majority, wins the election.

In elimination runoff elections, voting is done in the same way as in plurality elections. However, if no majority is established, the candidate with the least number of votes is eliminated and another round of election is conducted. The procedure is continued until two candidates are left and a majority is established.

Ranked Voting Methods

The implications of these voting methods are much more interesting than single voting methods. I general, ranked voting methods allow you to rank your preferences on the ballot instead of simply selecting one candidate. The most popular is called Instant-runoff voting (IRV)

In Instant-runoff voting a voter ranks the candidates by number in the order he prefers them. A voter can choose not to rank some candidates, but we will not consider this case.

Here is how the algorithm works.

1. Ballots are counted once for the top choice candidate listed.

2. If there is a majority for a candidate, he wins.

3. If there is no majority, the candidate with the least first place votes is eliminated, but the next highest non-eliminated candidate on the ballot receives an extra first place vote.

4. The counting process is repeated, and ballots are counted again (back to step 2).

Irish Presidential Election 1990

Here is one great case study of how the type of voting method has a large impact on the outcome of an election.

Brian Lenihan, of the dominant Fianna Fail party which had never lost a presidential election, ran against Austin Currie of the Fine Gael Party, and Mary Robinson of the Labour Party.

Here are the election results, with the listing of the first place votes of each candidate.

Labour Party: Mary Robinson     612,265 38.88%

Fianna Fail: Brian Lenihan 694,484 44.10%

Fine Gael: Austin Currie 267,902 17.01%

By single voting methods (without a runoff), Brian Lenihan would have won the election. However, following IRV rules, Austin Currie was eliminated and the second ranked candidate on all the eliminated ballots received an extra first place vote. Robinson was ranked 2nd on more than 80% of Austin Currie’s ballots. Therefore the second round of counting would have looked something like this.

Labour Party: Mary Robinson       826,586     52%

Fianna Fail: Brian Lenihan 748,064     48%

IRV helped David beat Goliath (or Mary beat Brian) in Ireland and drastically altered the election results. Anyway, I hope to maybe extend this voting analysis to the U.S. electoral college in my next posts.


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