How to compare a two-person and a six-person election?

The last post left hanging the question of how to assess the 6-person race. The two-person race is easy because one person's gain is the other person's loss. When you have four more contestants, the math gets a lot more complex.

It took me a while to figure out how to do it. The following table gives the main story:

Recall from my last post, I'm treating the New Hampshire Democratic primary as a six-person race by lumping all the minor candidates into one invented person.

The above table says that if the winner in New Hampshire gets 17% of the votes, the contest is evenly matched so that is equivalent to 50/50 in a two-person race.

If the winner in NH gets 20%, the contest is equivalent to a 56/44 split in a two-person race.

Using this table, I estimate that if the race this year is similar to that in 2016, when Sanders got 60% and Clinton got 38%, then we should see the winner this year get 25% of the votes.

Yes, that is only 25% of the votes. That means if the winner gets 40% of what Sanders got in 2016, s/he would have done just as well!

This is all due to the expansion of the field. The key thing to bear in mind is this: in a two-person race, if the winner gets 5 percent more votes, the loser gets 5 percent less, widening the margin by 10 percent. This is not the case in a 6-person race.

If the leader gets 5 percent more votes, the bottom 5 competitors split the 5-percent loss. On average, each suffers a one-percent loss, so that the margin between the top two closes by 6 points. (In practice, it will likely expand by fewer than 6 points, because the less competitive candidates will take a larger loss than the runner-up.)

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How did I come up with the table of numbers?

I use the metric of winner vote share divided by the average vote share of the other candidates. In a 2-person race, there is only one other candidate so it's just the vote share of the winner vs vote share of the loser.

In the 6-person race, the denominator is the average vote share of five candidates. So, in the case in which the winner earned 25% of the votes, the other 5 shared 75%, which is 15% on average. The ratio is 25/15 = 1.67. In a 2-person race, 63/38 has a ratio of 1.67.

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Let me know what you think of this, and if you know of other ways of addressing the question.

In the meantime, please stop the media from saying silly things like "[the winner] got half of what Sanders got in 2016" or "the winning margin was much smaller than in 2016".

 

P.S. I'm writing this note with about 84 percent precincts reporting and Sanders is leading with a vote share of 26%. If it stays this way, then using my metric, he did just about the same as in 2016.

Comments

[Dave C.]

Another interesting question is: In a six-person race, what percent would you use as a rule for whether the candidate is viable, or if they should drop out?

[Kaiser]

DC: We'd need to refine the question. The current race has to be representative of the general election, or else we need to know how predictive it is of the general election. The easier problem is if we can assume the current race is representative, then you eliminate anyone who isn't within a margin of error of the leader. But of course, that assumption isn't realistic.