Elections Tidbit (09/13/2003)

Prepared by:

Joseph Malkevitch
Mathematics and Computing Department
York College (CUNY)
Jamaica, NY 11451

Email: malkevitch@york.cuny.edu (for additions, suggestions, and corrections)

web page: www.york.cuny.edu/~malk

A pillar of American Democracy is the regular elections that are held for public officials at many levels of government. Elections or voting is also used in a wide variety of other settings: union officials, club officers, student government prom king and queen, food choice for the club picnic, and hunk of the month. The methods of election used vary considerably. Thus, for President of the United States we use a system based on the Electoral College while for Mayor of New York City we have a system where an initial election is held, and if no candidate gets a plurality of at least 40 percent of the vote, a run-off election is held at a later time between the two candidates who had the largest number of votes. But elections are not sufficient to guarantee that one is living in a democratic environment. There were elections in Stalinist Russia and in Nazi Germany. By far the most common election method is to have the voters pick a single choice from the list of candidates and the winner is that person having the largest number of first place votes (plurality voting or "first past the post." However, in a democracy, which of the the different methods for conducting elections is best and which is fairest, and why are so many different methods employed?

In constructing a mathematical model for an election system the key components are:

1. Voters.

2. Alternatives to choose among.

3. Ballots for the voters to express their preferences among the alternatives.

4. A method of selecting a winner or winners based on the ballots.

Furthermore, in judging the success of the election system one needs to look at various choices for criteria for what one wishes to achieve. One's goal may be maximizing voter turnout, fairness, stability, minimization of cost, or other objectives.

To give some of the flavor of these issues, let us consider the issue of ballots first. For concreteness and simplicity we will consider only elections where a single individual is to be selected by the group of voters. Historically, the ballot nearly always used in American elections consisted of asking a voter to select one candidate from a list of candidates. This type of ballot will be referred to as the standard ballot. In recent years there has been considerable discussion of whether or not this type of ballot allows voters to express their preferences (opinions) in as full a way as possible. For example, instead of employing the standard ballot one can require each voter to rank all of the candidates on a preferential ballot where a voter is not allowed to be indifferent between choices or alternatives. (We will deal with objections to using this procedure later.) With this in mind one can use the symbolism below to represent a typical example of how a voter could express his/her views on the candidates A, B, C, and D:

The interpretation of this ballot, called the ordinal or preferential ballot, is that D is preferred strictly to C, B, and A, while C is strictly preferred to B and A, and B is preferred to A.

Note that in this environment we are not allowing a voter to express the view that under no circumstance would he/she be happy to have candidate A and B serves for office. All that is permitted is to allow the voter to compare his/her opinions of the candidates with respect to each other. Furthermore, the voter is not "allowed" to truncate his/her ballot and vote for only a subset of the candidates. (Of course, one can generalize what is discussed here by allowing truncation of ballots and/or indifference between candidates. One reason that is suggested why truncation should be allowed is that when there is a large list of candidates (e.g. the California Governor recall election) it is not realistic that voters will know much about all the candidates who are running.) Clearly, there are many, many approaches to ballet design. One comparatively recent new ballot innovation is the idea of an approval ballot. The idea is to have each voter indicate those candidates they approve to hold office. The candidate with the largest number of approval votes would win.

Suppose we have decided on a type of ballot to use, and suppose this type of ballot is the preferential ballot without ties allowed and no truncations allowed.

Given the use of a ballot of this kind, who do you think should win the election below, in which 55 ballots where cast:

Many people are surprised that using different reasonable voting systems different candidates win. We like to think that the mere act of voting selects the best or wisest candidate. The same ballots can result in different winners depending on the method used to decide the winner. In fact, the election above was designed to show that there could be five different winners when the following five popular or seemingly reasonable methods are employed:

1. Plurality (The winner is that candidate who gets the largest number of first place votes.)

2. Run-off (If no candidate gets a majority of first place votes, eliminate all but the two candidates who got the largest number of first place votes and hold an election between these two candidates.)

3. Sequential run-off (If no candidate has a majority, eliminate the candidate who currently has the lowest number of first place votes, and hold a new election. Repeat this procedure until one candidate gets a majority.)

4. Borda Count (For each ballot and candidate X on that ballot, give the candidate a number of points equal to the number other candidates below X. The candidate with the largest number of points wins.)

5. Condorcet (That candidate wins (if there is such a candidate) who can beat all the other candidates in a two way race.

One response to the fact that there are many different seemingly reasonable methods is to create a list of fairness conditions which one would like to have an election system obey and then use election methods that meet these fairness conditions. This is the approach adopted by Kenneth Arrow, an economist who majored in Mathematics at City College (now a branch of CUNY). Arrow proved the surprising result that under very reasonable conditions there is no election method (when 3 or more candidates are running) which obeys a short list of fairness conditions. His theorem does not mean that all election methods are equally bad. It does means that depending on one's view about what are the essential fairness features one wants an election system to obey, one must select specific systems systems. Few who have learned about alternative elections methods would argue that plurality voting is the best. Perhaps some of the disturbing recent situations involving voting will bring about true election reform.


Arrow, K., Social Choice and Individual Values, Wiley, New York, 1963.

Black, D., Theory of Committees and Elections, Cambridge U. Press, Cambridge, l968.

Brams, S., Paradoxes in Politics, Free Press, New York, l976.

Brams, S., Voting Systems, in Handbook of Game Theory, Volume 2, ed. R. Aumann and S. Hart, Elsevier Science, New York, 1994.

Brams, S., and W. Lucas, P Straffin, (eds.), Political and Related Models, Springer-Verlag, New York, 1983.

Brams, S., and P. Fishburn, Approval Voting, Birkhauser, Boston, 1983.

Fishburn, P., and S. Brams, Paradoxes of preferential voting, Mathematics Magazine, 56 (1983) 207-214.

Malkevitch, J. et al., For All Practical Purposes, 6th edition, W.H. Freeman, New York, 2003.

Malkevitch, J., Mathematical Theory of Elections, in Mathematical Vistas, J. Malkevitch and D. McCarthy (eds.), Annal 607, New York Academy of Sciences, 1990.

McLean, I., and A. Urken (eds.), Classics of Social Choice, U. Michigan Press, Ann Arbor, 1995.

Straffin, P., Topics in the Theory of Voting, Birkhauser, Boston, l980.

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