The Nevada Democratic caucus results are out - sort of. The pundits have all crowned Bernie Sanders a runaway winner. (Some outlets called the race as early as 4-percent of precincts in.)

If you've been following the pundits' logic, as I have on this blog (link), you might be scratching your head here.

After New Hampshire, they told us that Sanders did really poorly since his vote share this year was merely half of what he got in New Hampshire against Hillary Clinton four years ago.

I checked Nevada 2016. Clinton beat Sanders by 53% to 47%. As of Sunday afternoon, with 70 percent of precincts reporting, Sanders took 47% of the votes this year. So, **according to the logic previously embraced by these pundits, Sanders did equally well this Saturday as he did in 2016**.

Wait a minute - but the 47% which resulted in a loss in 2016 is described as a landslide in 2020.

So what went wrong with that logic? You simply cannot judge the vote shares in an N-person contest (N>2) as if they came from a 2-person contest. I've been pounding this point the last two weeks. Finally, these pundits might see the light.

***

If you can't compare a 2-person contest to a 7-person contest (what Nevada 2020 boiled down to) by directly comparing vote shares, then how can you compare performance across elections with different numbers of contestants?

I've a series of posts that address this question (1, 2, 3). In the last post, I presented a method to turn any vote share in an N-person contest (N>2) to a vote share equivalent in a 2-person contest.

I quickly ran the numbers for the Nevada result. The vote share table of (46%, 20%, 15%, 10%, 5%, 4%, 0.2%) is summarized by the DFER metric, which is the distance of this distribution of votes from the hypothetical even race, in which each candidate wins one-seventh of the vote. The DFER is 0.155. The maximum value is when one candidate wins all 100% of the vote, with DFER = 0.378. So, the result from this election is about 41% of the way between an even race and an extreme race.

A 2-person race that has SDFER of 41% is just above 70/30. So, the Nevada result in 2020 (as of now) is equivalent to a 70%-30% vote split in a 2-person contest, like the one in 2016.

***

With this model of the N-person contest, we can make some general comments about adding candidates to elections.

First, here is a visual analysis of a 3-person contest, compared to a 2-person contest.

First, think about this question: **which is the better performance: winning 50% of the votes in a 3-person contest or winning 50% of the votes in a 2-person contest?**

This simple question puts the lie to all the nonsense we're hearing about how Bernie Sanders failed to match his 60% vote share in this year's 11-candidate contest relative to the 2-candidate contest in 2016.

**Winning 50% in a 2-person contest does not even win the contest but winning 50% in a 3-person contest is a pretty dominant victory.** We need math to quantify this advantage. In essence, we want to adjust the 50% vote share upwards to reflect its equivalence in a 2-person contest.

We use the 2-person contest as a reference because it is so easy to understand. The entire contest can be summarized by one number, the vote share of the winner.

The method for translating an N-person contest to a 2-person contest deals with this head-on. As shown below, a crucial step is to turn the set of N vote shares into a single distance metric that can be ordered. Here is a simple diagram of the method:

Back to the pie charts. As you move from left to right on the diagram, the votes of the runners-up are made more even. In the second row, the 50% is split 40-10 while in the other case, it is split 25-25. Is a vote distribution of 50-40-10 more even thatn 50-25-25? Or more one-sided?

This illustrates why when you have more than two candidates, it's not simple to rank competitiveness. According to my model, the 50-25-25 vote distribution is (slightly) less even i.e. more one-sided than the 50-40-10 vote distribution. The way to understand it is that the 50-25-25 distribution has one clear winner while the 50-40-10 has one winner and a close second.

Another reveal from the pie charts. As you look down a column, the winning vote share increases. This increases the SDFER metric, moving closer to the extreme, one-sided case. The impact of the winner taking more share is greater than the impact of whether the runners-up are evenly matched or not.

Thus, if you are the leading candidate, the vanity candidates who each take a percent share of the total are like mosquitoes that you want to swat away. They are small but their stings hurt.

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