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5. Voting and Elections: Evaluating Election Systems

Voting and Elections: Evaluating Election Systems

Feature Column Archive

5. Evaluating Election Systems

There are many scales on which to evaluate voting and election methods. One can deal with questions of the following types: Given a ballot type, is it reasonable to assume that the average voter can complete the ballot properly? In the recent 2000 Presidential election most people were aware that George Bush, Albert Gore, and Ralph Nader were candidates and if you asked most voters to rank these three people without ties, they probably could do so. However, in some states such as New York, there were many other candidates on the Presidential ballot who certainly would not be recognized by people from Idaho; even many people from New York were not familiar with these candidates. If you forced people to use a ballot where all candidates are ranked, many voters would have to resort to listing names beyond those of Bush, Gore and Nader at random. This is not ideal. On the other hand, if a voter who is given a preferential ballot is allowed to rank a subset of the candidates rather than ranking them all, then one is in essence dealing with a very different problem in designing a system to use these ballots as inputs to an election system. Using mathematics in the real world is often very complex because one can not assume away unpleasant aspects of the problem. (For example, another problem that mathematicians have looked at is that of determining as fair a way as possible to use census data to determine how many seats each state should get in the United States House of Representatives. Some aspects of this problem might be easier but for the Constitutional requirement that each state must have at least one seat, even though one might in some solutions be tempted to argue that some states with a very small population do not deserve even a single seat. However, no matter how elegant an answer we might find to this problem where one need not give each state one seat, this solution is irrelevant to the problem that the constitution poses for us.)

In investigating voting procedures and elections mathematical researchers have different goals. They strive to understand how to formulate ideas about fairness that translate into finding which methods obey or do not obey these fairness axioms that they devise. Another very practical aspect of elections is the issue of having the voters be straightforward in their expression of preferences for candidates. When there are three major candidates running for office, as in the election of 2000 (i.e. George Bush, Albert Gore, and Ralph Nader), some voters may face the dilemma of voting for their second choice over their first choice rather than throw their votes away. Thus, voters who are given a preference ballet would have voted: Nader preferred to Gore preferred to Bush, chose on a ballot where they were asked for only a single name to vote for Gore. Similarly, someone who with a preference ballot of Nader preferred to Bush preferred to Gore, and who had reason to believe that Bush was the candidate to beat, would vote for Nader, Gore, Bush under some systems so as to promote the interests of their top ranked candidate rather than give an additional assist to their second ranked candidate.

The question that mathematical thinking can raise is whether or not this type of reasoning can be avoided by a suitable choice of method. More specifically, is there a method of conducting elections in which it never pays for a voter to vote for anything other than the ballot that most represents his or her preferences?

It turns out that two researchers into decision procedure methods showed that there is no strategy-proof election method other than dictatorship when there are three or more candidates. This result is due to Mark Satterthwaite and independently to Allan Gibbard.

The theoreticians of democracy base their support for this system on many grounds. One argument is that if voters are sufficiently educated, having many independent individuals reach conclusions about candidates and then holding free elections will lead to a stable effective society. However, as democracies have evolved it has become possible to obtain reliable information about other citizens' opinions. Taking surveys and polls makes it possible to attempt to put information about other citizens' opinions to use to achieve one's own goals more effectively. Thus, rather than being straightforward (sincere) in expressing one's opinions, one tries to use the information gotten strategically. Though it may be difficult to prevent polls and the information from them from being distributed, there are indications that this information may have a destabilizing effect on a democratic society. In choosing ballots and election decision systems one criterion one might pick is making it difficult for people with information about other citizens' preferences to take advantage of this information in a way that helps them at the expense of others.

Another thread of interest in studying elections is to see the consequences to different countries in using different election methods. For example, it has been proposed that one reason that countries which use plurality voting tend to have two-party systems is that the very choice of the plurality voting system has made it hard for more than two significant parties to develop. Some people believe this is a good thing since many such countries have very stable societies. Some consider this unfortunate, because it means that the voters are not given as rich a range of options for getting done what they would like to see done. One of the people to make the connection between two-party countries and the use of plurality voting was the French sociologist Maurice Duverger. Many statistical and mathematical arguments have been given to study the empirical relation between party structure and the choice of electoral system.

There is also the perspective of computational complexity to take into account. There are methods that one can propose to conduct elections which are computationally very difficult to carry out. Thus, it might be that one would feel that some particular method was truly the best but be faced with the prospect that in a large sized election one could not compute the winner with this system! Some have suggested the use of the internet to conduct elections. Such proposals raise important questions about running secure and honest elections over a computer network.

The work that has been done by mathematicians and workers in other fields using mathematical methods in the area of elections and voting clearly indicates the tension that exists between pursuing mathematics for its own sake versus using mathematical methods to get insight into practical problems. One may get a nice theorem from the analysis of election systems where voters are forced to rank strictly all alternatives from most preferred to least preferred, but psychologists may advise one that voters will be unable to carry out this task, perhaps for even as many as 5 candidates in a consistent manner, and political scientists will point out that in a typical election at least 2 of the 5 candidates may be totally unknown to the voters.
Those who examine the behavior of voters with actual ballots see that many ballots are not filled out as required.

Mathematics continues to grow because its practitioners attempt to get insights into problems that initially arise from real world situations. Even in cases where these initially formulated problems move in directions where the mathematical results are no longer of interest or importance in the settings from which they arose, mathematics can continue to benefit from the ideas and tools that studying such problems provides for solving problems in other domains.

As far as making democracy work better, we are still seeking additional insights.

Joseph Malkevitch
York College (CUNY)


  1. Introduction
  2. Ballots
  3. Election Decision Methods
  4. Enter Kenneth Arrow
  5. Evaluating Election Systems
  6. References