Mathematics of Voting: 2. Electing a candidate, simple plurality

Simple plurality is one of the simplest voting systems, easy to understand, easy to administer, easy to vote, and easy to tally. For those reasons it is very popular, and is used worldwide. It is also a very good system for a two candidate election.

Each voter simply votes for his or her preferred candidate, and the winner is the candidate with the most votes. There is no requirement that the winner obtain a majority of votes, just more votes that any other candidate. Suppose there is an election with 100,000 voters and four candidates: Alfa, Bravo, Charlie and Delta. Suppose the votes are tallied as:

Candidate: Alfa Bravo Charlie Delta
Votes: 30,000 20,000 35,000 15,000

With 35,000 votes – more than any other candidate, Charlie wins the election, and everybody (certainly his supporters) go to bed happy.

Or do they? It might be (and often is) argued that the result is monstrously unfair: in this case 65,000 voters – a very large majority – have a winner who they didn’t vote for. There is no way for voters to express preferences. It may be that many voters who voted for Alfa would have been almost as happy with Bravo. But there’s no way to express that preference on a simple plurality ballot – a single vote for just one candidate is all you’re allowed.

Simple plurality works fine, of course, if there’s just one vote and only two candidates. To win, a candidate must amass a majority of votes. However, the situation is murkier if there’s a number of voting populations all voting independently, and the results of those independent votes must be tallied. Suppose Alfa and Bravo are the two candidates for Supreme Bottle-Washer in a country of 1,000,000 voters, each in a district (constituency, electorate) of 100,000 voters each. Suppose that in six of the 10 districts Alfa wins with 60,000 to 40,000 votes, and in the other four districts Bravo wins with 90,000 votes to 10,000. Since Alfa has won in a majority of the districts, he wins the election. But in fact the total number of votes cast for Alfa is

6\times 60000 + 4\times 10000=400000

and the total number of votes cast for Bravo is

6\times 40000 + 4\times 90000=600000

showing a greater support across the whole country for Bravo.

This situation is not merely an academic exercise. Something similar can (and does) happen in American Presidential elections, where the President is elected not by the citizens, but by a 538-strong “electoral college”. The electors who comprise the college are apportioned to the states roughly by population, with each state having at least three electors. A vote for a particular candidate then, is really an instruction to the electors in that state on how to vote. In most states, electoral votes are cast on a “winner takes all” method: all votes in that state go to the candidate who has amassed the most votes. In just two states: Maine and Nebraska, are the votes apportioned by district. The effect of this two stage process: popular vote followed by the electoral vote, is that the electoral vote tends to magnify the results of the popular vote. For example, in the 1960 Kennedy/Nixon election, Kennedy won 49.7% of the popular vote to Nixon’s 49.6% – a very small plurality – but the electoral college votes fell 303 to 219.

What’s worse is that the electoral votes might choose a President who didn’t win the popular vote, as happened in 2000, with Gore winning 48.4% of the popular vote to Bush’s 47.9% (a greater plurality than won by Kennedy in 1960). However, the electoral college handed the Presidency to Bush with 271 electoral votes to 266.

Simple plurality is also vulnerable to “spoilers”: a candidate who may not amass enough popular votes to obtain any electoral votes, but who might take away votes from a stronger candidate. Again, in 2000, Nader (who won a handsome 2.7% of the popular vote), probably took some of the votes from Gore. At any rate, many citizens who voted for Nader would have preferred Gore over Bush. It has been estimated that spoilers have influenced the results of at least 5 elections, and according to William Poundstone: “we’ve elected the wrong president eleven percent of the time”.

For a more concrete example, suppose that election is between Alfa, a member of the Rotten Party, and Bravo and Charlie, candidates for the Excellent Party. Most people want an Excellent Party candidate to win, but suppose the votes (out of 100,000) are:

Candidate: Alfa Bravo Charlie
Votes: 40,000 30,000 30,000

The Excellent Party candidates have in total amassed more votes than the Rotten Party candidate certainly, but because the Rotten candidate has a plurality of votes, he wins. Another example is given by the 1912 Presidential Election, where Roosevelt and Taft basically split the Republican vote, allowing the Democrat Wilson to win the Presidency.

Simple plurality is considered by voting theorists to be a very poor method when there are more then two candidates. For that reason, there are many variations. One is “runoff voting”. This is really a number of different systems, but in all of them some of the candidates are eliminated, and a new ballot is held for the other candidates. There may be two rounds (the two top candidates only proceed to the next ballot), or an exhaustion process, where at each ballot the candidate with the fewest votes is eliminated. The trouble with these systems is the repeated ballots – time-consuming and expensive to manage on a large scale, and unnecessarily tiring for the voters. For this reason political scientists and social theorists would prefer a system which implements the effects of runoff voting, but on a single ballot.

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