 Given the way in which the primary candidacy of Donald Trump has been ripping apart the Republican Party, I'm surprised that there has been very little discussion in the press of alternative ways of counting votes. Trump has won primary after primary in part because he's had multiple opponents who have divided the anti-Trump vote. In the voting system that is universally used in the United States, called "plurality voting", the winner is simply the candidate who gets the most votes. If there are only two candidates, the vote is a simple head-to-head competition, and a candidate can only win by getting more than 50 percent of the votes that are cast in favor of one of the candidates (ignoring ballots in which the voter abstains from making a selection). But if there's a third candidate, someone can win without obtaining a majority. And that, in most primaries, is how Donald Trump won. He seldom got a majority of the votes, but with multiple opponents, he often got the greatest number. Some electoral systems deal with this issue by scheduling a run-off election between the two top vote-getters in an initial election. This is seldom done in the United States, and would almost certainly be considered excessive in a primary contest. But there's a simple way to accomplish the same thing without having two separate votes. It's called "preferential voting". We used to use it in our student government at MIT (I was Secretary of the Baker House dormitory). See below what a typical ballot might have looked like with three candidates for a single position. In the example, I've used the candidates for President who are still in the Republican Party primary race as of the posting of this entry in early April, 2016:
Counting the ballots is done as follows:
The effect is the same as holding a run-off election if no candidate gets a majority, except without the necessity of actually voting a second time. Voters are instructed to "Give your '2nd' place vote to the person you would vote for if your first choice were to be eliminated." I might note that if step 2 were to be eliminated in the above counting procedure, you might get a different winner. But the method as shown replicates the usual procedure in a run-off election: if a candidate gets a majority in the initial election, no run-off is held. This method of voting can be extended to more than three candidates. For example, if the most recent Republican primary candidate to drop out, Marco Rubio, were still in the race, the ballot might look like this:
To count, simply add an additional elimination round. Note 1 Of course, this has the possibly undesirable effect of giving the least popular candidate's voters the first shot at putting another candidate over the top. Admittedly, this can get a bit difficult if there are lots of candidates. The New Hampshire Republican primary in 2016 had nine candidates on the ballot! Alphabetically, these were Bush, Carson, Christie, Cruz, Fiorina, Gilmore, Kasich, Rubio, and Trump. One approach is to never ask voters to select more than a first and second choice. All the candidates in third or lower place are then removed (all at once) in the first elimination step, and their second-place votes are distributed. If their second-place votes go to another eliminated candidate, they are lost. But in plurality voting, all the votes to other than the two top candidates are also lost, in the sense that they don't contribute to choosing between the two front-runners. Would this system have made a difference in the 2016 Republican primaries that were won by Donald Trump? We don't have enough information to know for sure, since we don't know the second-place and third-place choices of the voters for candidates who would have been dropped from consideration in the elimination rounds during counting. Although Trump got less than 50% of the votes in many of the primaries, he was close enough to 50% that second and third-choice votes might have put him over the top anyway. Why isn't this system used? Back when votes were counted by hand, counting the votes in a preferential election was a bit more cumbersome than plurality voting. We did it by hand back in Baker House, but the total number of voters wasn't all that high. However, in these days of electronic voting machines (or electronic machines counting mark-sense ballots), that's hardly an issue. The system is a bit more complex for the voters. Voting might take a bit longer, and some voters might find voting confusing. The voters who used it in Baker House didn't find it confusing at all, but they were all MIT students. Every voting system has its strengths and weaknesses, and different ways of counting can produce different results. Preferential voting probably has a tendency to elect more compromise candidates, who may not carry the field as a first choice, but who are at least acceptable to a large number of voters. And that point touches upon what is probably the real reason that preferential voting has never made any inroads in the United States. Although the US Constitution doesn't mention political parties, our political system has been taken over by only two large political parties. Starting a third party is virtually impossible, because in the early days of a third party, voters are reluctant to throw away their votes on a candidate who doesn't have a realistic chance. But preferential voting removes that obstacle! Voters can feel good about voting for a third-party candidate secure in the knowledge that if he or she is eliminated, their second-place vote will then count. This would allow a third party to get stronger and stronger until one day it triumphs. Neither of our two political parties want that to be able to happen, so they will never approve a change to preferential voting. According to an article called Preferential Voting: Another Democracy Lesson from Australia (in the Huffington Post), Australia has been using preferential voting for years. I haven't researched the details of their system, so I don't know how they handle races with four or more candidates.
  Note 1: If you're a computer programmer, here's the algorithm for the general case. As candidates are eliminated, to "count the votes" means to go through the ballots, and count the highest choice that goes to a candidate still remaining. As an example, if a voter's first and second choices have both been eliminated, then that vote goes to his/her third choice candidate. The procedure in the general case is then quite simple. In essence, the votes are counted and re-counted after each elimination round, until someone gets a majority:
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