Calculating Electoral Votes
August 17, 2004 6:18 PM   Subscribe

DiscreteMathFilter: Colorado will vote this November on a proposal to abandon its "winner-take-all" system in presidential elections, and award its electoral votes based on the percentage of votes cast for each candidate. I tried to model this based on the 2000 election, and the concept fell apart. [more inside]

In the 2000 presidential election, 1,741,368 votes were cast in Colorado (8 electoral votes).
50.75% went for Bush.
42.39% went for Gore.
5.25% went for Nader.
1.61% went for other candidates.

Under the new system, Colorado's 8 electoral votes would be assigned as follows:
Bush: 8 * 50.75% = 4.06, rounded to 4 electoral votes.
Gore: 8 * 42.39% = 3.39, rounded to 3 electoral votes.
Nader: 8 * 5.25% = 0.42, rounded to 0 electoral votes.

Under the proportional system, what happens to the eighth vote?
Would you award it to Nader, because he has the highest remainder? Or would you award it to Bush, because he got the most votes overall?

Here's another, slightly different example:

In the 2000 presidential election, 10,965,850 votes were cast in California (54 electoral votes).
Bush: 54 * 41.65% = 22.49, rounded to 22 electoral votes.
Gore: 54 * 53.45% = 28.86, rounded to 29 electoral votes.
Nader: 54 * 3.82% = 2.06, rounded to 2 electoral votes.
Other: 54 * 1.08% = 0.58, rounded to 1 electoral vote.

You can't give an electoral vote to "Other," since "Other" is five different people. So where does the 54th vote go? Bush, because he has the highest remainder? Or Gore, because he won the state?

(Note: I'm not trying to ascertain the best public policy. I only want to know, from a purely mathematical perspective, how to most efficiently and fairly assign a fixed number of items to an awkward set of populations.)
posted by PrinceValium to Law & Government (13 answers total)
Response by poster: Data source here, by the way.
posted by PrinceValium at 6:20 PM on August 17, 2004

In that single instance there really is no "fair" way to do it since the small number of things will never match the proportions.

However, if you are asking about how to assign it fairly over a large number of instances or events, then just use probability to effectively "lottery" the items (electoral votes) under consideration. That is, using your first example, you'd devise some random event where Bush has a 50.75% chance of getting that last vote.

This would ensure that if Nader for example was reliably getting 5% of the vote from state to state, he may get more or less than that many electoral votes in a particular state, but if you totaled them up they would probably be pretty close to 5% of the electoral votes available. And that would, in the long run, be pretty fair.
posted by vacapinta at 6:43 PM on August 17, 2004

What does the text of the amendment say? It must address the issue.
posted by kenko at 6:46 PM on August 17, 2004

Section 4c of the amendment says that the ticket getting the greatest number of ballots cast gets any leftover electoral votes.

The amendment text as pdf.
posted by spacehug at 7:12 PM on August 17, 2004

The best way to do it is to assign the electoral votes such that the sum of the differences between the way they were allocated and the ideal is as small as possible.

In other words, "4.06, rounded to 4 electoral votes" would mean a 0.06 discrepency. Add up the discrepancy for each candidate. If the "other" always gets zero votes, it doesn't matter if you include it or not.

If my arithmetic is right, it adds up to Nader getting his one vote being very slightly closer to the ideal, in your first example. The "highest remainder" thing should probably always work, I think.

On preview: As usual, it's obvious that the people who write the law are not mathematicians.
posted by sfenders at 7:21 PM on August 17, 2004

Well, I think it is not quite the one with the highest remainder, but the one whose remainder is closest to 0.5. You want to change the vote that was most affected by rounding to minimize the error. i.e., if you need to add a vote, add to the one with the greatest remainder < 0.5; if you need to subtract subtract from the with the least remainder>= 0.5.
posted by mcguirk at 7:58 PM on August 17, 2004

....properly, the algorithm should probably be: always round down, then assign remaining votes starting with highest remainder.
posted by sfenders at 8:11 PM on August 17, 2004

properly in the sense of easiest to implement, that is. I ain't no mathematician, just a programmer.
posted by sfenders at 8:12 PM on August 17, 2004

Yes, that should be equivalent to what I said, but you stated it much more simply.

you'd devise some random event where Bush has a 50.75% chance of getting that last vote.

I think that would be wrong because it doesn't take into account how much each candidate is already ahead or behind on the rounding. It would be fair (in the long run) if you did all the votes that way, but if you only do the last one you have to factor in the prior rounding.
posted by mcguirk at 8:27 PM on August 17, 2004

To properly round numbers, you need to alternate the "half" case (0.5 candidates/whatever) between rounding-up and rounding-down. IIRC.

In real life, I'm certain the rounding is always done in the roundee's favour.
posted by five fresh fish at 8:35 PM on August 17, 2004

They should just give out fractions of electoral votes. Then do that for the rest of the country. And make the fractions precise enough to count every vote.
posted by Space Coyote at 8:37 PM on August 17, 2004

Section 4c of the amendment says that the ticket getting the greatest number of ballots cast gets any leftover electoral votes.

This may be a desired result -- it keeps some of the "relief" (topographically speaking) of election results. All-or-nothing is probably too sharp of a relief. Maybe an even split isn't enough. Space Coyote's suggestion is very egalatarian, but it has some practical problems that the relief might help avoid.

I wonder, though, if this decreases the strategic advantage to doing this. Spliting the votes means that even the state's underdog candidates have reason to court voters. But the sharper the relief mentioned above, the less an underdog takes away from an election.

But anyway, this problem reminds me of this riddle from The Gift of Asher Lev:

"An old Hasid called his sons together and told them that he wished them to divide his property in a certain way after he departed for the True World. The oldest son was to take one-half; the next son, one-third; the youngest son, one-ninth. Soon afterward the old Hasid was called by the Master of the Universe to his eternal rest. The sons wished to obey their father, but they discovered that their father's property consisted of seventeen goats, and seventeen cannot be divided by one half or one third or one ninth. They didn't want to kill any of the goats and divide it, because each goat was much more valuable alive than dead. And so they went to the Rebbe, and in his wisdom the Rebbe immediately solved their problem. What did the Rebbe tell them?"

A equitable solution, as I understand it, is arrived at by solving

(1/2)x + (1/3)x + (1/9)x = 17, which yields x=18, and so you give one son 9 goats, another 6, and another 2.

I wondered if you could apply something like that to the results above, but it doesn't seem to work, which makes me think the goat puzzle is contrived.

(883,748/1,741,368)x + (738,227/1,741,368)x + (91,334/1,741,368)x = 8 ...

posted by weston at 9:09 PM on August 17, 2004

Response by poster: Thanks for all the responses! I had been googling for the text of the amendment, and couldn't find it.

Here's a quick analysis using electoral vote allocations and votes from the 2000 election. If anyone wants to check my math, feel free.

If the entire country followed the Colorado method, Gore gets 269, Bush gets 263, and Nader gets 6. The election goes to the House of Representatives.

If the entire country went proportional, but allocated extra votes to the candidate(s) with the highest remainder(s), Bush gets 263, Gore gets 262, and Nader gets 13. The election goes to the House.

Now suppose the remainder system is used, but two electoral votes automatically go to the overall winner in the state. For example, of California's 54 votes, 2 would go directly to Gore and the other 52 would be split based on proportional analysis. In this case, Bush wins a majority with 272, compared to 258 for Gore and 8 for Nader.

Using either method, third parties can accumulate electoral votes fairly easily. This makes it a lot more difficult for a candidate to get the required 270 votes. The Colorado system attempts to bandage this by requiring that the lowest vote-getter lose his electoral vote if the rounding creates more votes than is allocated to the state. But in the 2000 analysis, this would only affect Minnesota (Nader loses his vote, and Bush and Gore each would have 5.) As you can see, Nader would still throw the election to the House.

By contrast, the Maine/Nebraska system, which assigns one vote per congressional district, would protect two-party rule. The election would still be ridiculously close, but Nader would not have received any electoral votes.
posted by PrinceValium at 9:28 PM on August 17, 2004

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