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Suppose we have an election held over multiple districts (e.g., a first-past-the-post legislative election, or a US-presidential style electoral college). There are several ways you could measure how close the election is:

  1. Overall popular vote margin (e.g., Biden is currently up by about 4.3 million votes nationwide)
  2. Tipping-point margin (e.g., the current tipping-point state is PA, in which Biden is up by about 43,000 votes)
  3. The cumulative margin: the total number of votes required to flip the election if those votes are optimally distributed (e.g., at the moment Trump could win the election by gaining 43,000 votes in PA, 19,000 votes in AZ, and 10,000 votes in GA for a total of about 72,000 votes). Usually seen in the wild in arguments that an election was closer than it appeared.

Or, of course, percentage-based versions of the popular vote and tipping point margins (the correct denominator for the cumulative margin is a little unclear).

Has there been any study of how these measures are likely to relate to each other?

I'm specifically curious about how the cumulative margin relates to the other two. My intuition is that popular and tipping point margins should be roughly linear functions of each other, while the cumulative margin is a non-linear (probably quadratic?) function of the other two. I would be happy with an answer based on 538-style models rather than directly on actual election data.

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  • Is your election system fixed or not? This is an important distinction as overall popular vote does not really mean anything in a "winner takes it all" system. One can easily construct cases where the loser has a huge majority in overall votes but loses the election in terms of seats. By the way, this case already contradicts any linear dependency of the different measures
    – Manziel
    Nov 9, 2020 at 8:44
  • @Manziel: Obviously you can construct artificial scenarios where these measures are all totally uncorrelated. I'm curious about attempts to answer what is actually likely to happen: that is, if all you knew about an election was a single one of these measures, what would be your best guess at the other two?
    – Micah
    Nov 9, 2020 at 17:36
  • Constructing scenarios makes this more clear but even using real data these measures do not correlate. Bush won by 500 votes in Florida with a negative overall margin. 2016 was the same but different. The system simply has such a strong non-linearity that makes it impossible to use any linear approach
    – Manziel
    Nov 10, 2020 at 7:40
  • @Manziel A lack of a linear relationship isn't an absence of correlation. Non-linear correlations are a thing to. emathzone.com/tutorials/basic-statistics/….
    – ohwilleke
    Nov 9, 2021 at 20:37

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