Fairness and Equity: Notes 7
Department of Mathematics and Computer Studies
York College (CUNY)
Jamaica, New York 11451
Theory of Elections
To run an election or a decision making mechanism one must have:
2. Alternatives to choose from
3. A ballot (a way for voters to express their opinions about the alternatives)
4. A method of translating the ballots cast into a "societal" choice
In American national elections:
(a) Voters are those US citizens who are 18 years of age or older who are not felons. The number of felons who are ineligible to vote is growing increasingly large and some have called for a change in the law that would permit felons who have served their penalty to vote.
(c) The alternatives in a Presidential election consist of the Republican and Democratic candidates but in recent years there has also been a 3rd party candidate of some national strength (Nader, Perrot) who ran. There are typically many other candidates who vary from state to state, since many of these candidates fail to get on the ballot in all states.
(d) In the Presidential election the ballot requests that one provide the name of one person (alternative). This is sometimes called the standard ballot.
In some elections where several candidates are running, the voters are sometimes instructed to vote for no more than a certain number of candidates (usually the number of seats to be filled). This method is sometimes used for filling the position of member of a committee. It is common for professional society elections.
(e) The most common election decision method is plurality. The votes are counted for each candidate (which is all the information the standard ballot provides) and whoever gets the largest number of votes, wins. There is no guarantee that a candidate gets a majority.
In the recent recall election for governor in California, approximately 100 candidates ran, and a standard ballot was used. It was possible that a candidate could have become governor of California with a little more than 1% of the vote! This does not seem to be a wise approach in many people's view to get either stable or good government.
What are the alternatives?
One can use a more complex ballot or one can use a two-stage procedure with a standard ballot. The latter idea involves setting a requirement that if a candidate does not receive a sufficiently large plurality, a "run-off" election be held. This is traditionally done by taking two candidates with the largest number of votes in the initial election and having a new vote just between them at a later time. Not only is this procedure quite expensive, but almost invariably the number of people who vote in the run-off election is much smaller than in the original election. Another approach to deciding a winner for an election is to give candidates credit for how high up on a ballot they are ranked by the voters. Getting a first-place vote should count more for that candidate than getting a 2nd or third place vote. This idea is due to Jean de Charles Borda, who proposed it during the period of the French Revolution. Borda's method is used for some voting methods in sports and in entertainment.
Among the methods that have been discussed over the years are:
c. Sequential Run-off (Hare system or the single transferable vote; used in Ireland and Australia and until recently in NYC school board elections.
e. Borda count
Fifty-five voters have produced the following collection of ballots on 5 choices. Who do you think should win this election and why?
What properties should a good election method have?