Feature Column from the AMS

2. Voting and Elections: Ballots

Voting and Elections: Ballots

2. Ballots

Perhaps the first, and rather surprising, insight that mathematical approaches have yielded is the complexity of the ballot/decision method choice. Most elections that we participate in involve the election of a single candidate (alternative) from a slate of two candidates. In this case if one votes for one's favorite candidate, one of the candidates must get a majority (except in the unlikely case of a tie when an even number of votes are recorded) and there is little quibble about the result. The importance of ties or near ties has recently made the news. The probability is not so high that an exact tie will occur but when an election is truly close, there will enough noise in how the votes are counted that there will be considerable controversy concerning the winner. (An interesting topic for mathematical investigation has been to estimate how likely it is, depending on the closeness of a vote, that additional information in the form of recounts, etc. will affect the results of the election.)

When there are three or more candidates (alternatives) and a single choice must be made, then the ballot form becomes rather important. Among the types of ballots that one might use are:

a. Choose one (so-called standard ballot).

b. Rank the candidates favorite to least favorite; indifference is not permitted.

As an example of such a ballot, consider how one voter might rank the three major candidates in the last Presidential election:

This clever symbolism means that Gore is preferred to either Nader or Bush and that Nader is preferred to Bush. My first introduction to this notation was in Duncan Black's book. This ballot is called an ordinal ballet or a preferential ballot.

c. Rank the candidates favorite to least favorite; indifference is permitted.

d. Choose all candidates one is willing to have serve.

This ballot is known as an approval voting ballot.

e. For each candidate vote yes or no.

f. Give a list of candidates one is not willing to have serve and a ranking of the remaining ones, with or without indifference.

g. Distribute 100 points among the candidates as one sees fit.

h. Distribute 100 points among the candidates that one is willing to have serve, as one sees fit.

Until recently, only types a. - c. were studied, and many new ideas have emerged from the observation that there are a wide variety of other ways that information about voter preferences can be obtained. However, the discussion of exotic ballets must proceed in the context of theoretical studies and political realities. There may be nice decision methods that arise if voters are willing or able to rank all 12 candidates running in a certain way, yet it may not be realistic to assume that such a system of voting can actually be adopted, given the political realities of the world.

Here, I will concentrate on the type of ballot that requires each voter to rank all the candidates and does not allow the voters to be indifferent between candidates. Of course, this is a very artificial type of requirement but it does raise an interesting question of voter behavior. No matter how simple the rules are for completing a ballot there will always be voters who get it wrong. If one is instructed to put an X next to the candidate whom one wants to vote for and instead the voter puts a circle around the person's name, should the vote not be counted? The ballot we are describing is not that simple, especially when there are lots of candidates. The voter may not know a lot of the names on the ballot and may prefer not to list all the candidates. If, however, the law is that a valid ballot requires certain actions, then presumably ballots that do not meet the required conditions will not be counted.

From a mathematical point of view there are a variety of reasons to make certain assumptions about a type of ballot. One reason to make these assumptions might be that one is trying to describe what is actually done in practice and selects a mathematical environment that closely resembles what is done. The other reason might be to study something that might be done instead of what is currently done and deduce some consequences. Another reason might be that using these particular assumptions one can prove facts about a voting system that are interesting. Using different assumptions perhaps the problem becomes to hard to solve.

Assuming that voters are required to use a ballot where they rank all the candidates, without being indifferent between any candidates, what can one now do with these ballots to decide a winner?