Mathematical Analysis of the Electoral College

Let us consider the electoral college voting system used to elect the American president. I have a few questions from the point of view of decision-making/gaming theory. My ultimate goal is to vote in such a way as to minimize the probability of the republicans winning. As I see it, there are three different strategies for me. I can either not vote at all. I can vote for democrats. Or I can vote for a third party candidate.

1. What is the best strategy among these three? I don't care which party wins as long as the republicans don't win. We also assume that the electors are faithful and vote for whichever party they have pledged to vote for. Otherwise, it'll be completely out of the voters' hands. The peculiarities of the electoral college with its uneven voting power among residents of different states and the winner-take-all approach makes its analysis a bit tricky.

2. What is the best strategy if I want to achieve the same goal (don't want the republicans to win) but in a popular election such as for the members of the congress?

3. I also don't understand exactly how it is possible for someone to win the popular vote but lose the electoral college. What is the mechanism/explanation behind this? I know various numerical examples but it still doesn't make sense to me. Does anyone have a good explanation of why that happens? Something intuitive perhaps? The way I see it, in each state if you win the popular vote then you get all of the electoral votes. So if you have a majority of the electoral votes then you must have a majority of the popular votes. Is this related to a well-known paradox (like Simpson's paradox or something) perhaps? How would you explain this to a six year old? Is this an artifact of the electoral college itself or the winner-take-all system? If winner-take-all system was eliminated, would this paradox be eliminated as well?

4. Why was the winner-take-all system implemented? Why not proportional voting? Is there a mathematical reasoning/advantage/disadvantage to doing things this way? The electoral college itself makes sense that it protects minorities to some extent by given them a bit more voting power. But what is the point of winner-take-all?

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"in each state if you win the popular vote then you get all of the electoral votes." This is somewhat exaggerated. The Constitution of the United States says that state legislatures decide how to elect electors. Last I heard, two states, Maine and Nebraska, don't follow the winner-take-all system. And some states have recently passed laws saying all of their electoral votes will go to the winner of the popular vote throughout the whole country, provided a sufficiently large number of other states pass essentially the same law. – Michael Hardy Oct 12 '12 at 21:39
It's quite appropriate that you call this "gaming theory" -- the mathematical theory is called "game theory"; what you're doing is gaming the election :-) – joriki Oct 12 '12 at 21:54
I know but I was ignoring the two states. This phenomenon still exists if we assume that all states use winner-take-all system which is why (see question 4) I suspect that this phenomenon may be an artifact of the winner-take-all system instead of the unequal representation itself. – Fixed Point Oct 12 '12 at 21:54
Thinking about what I just commented above, if some people have more voting power than others, then it is possible for those with higher voting power to elect someone even though the majority voted for another candidate. The majority's choice didn't win because they didn't have enough voting power. Hmmmmmm talk about an epiphany! Is this right? – Fixed Point Oct 12 '12 at 21:57

for 1 and 2, you should vote for the most popular candidate other than the one you want to lose.

The way you can lose the popular vote and win the electoral college is to win states very narrowly and lose big in the ones you lose. If all the states had the same number of electoral votes, you would have to win 26. It is possible to only build up a plurality of 26 votes doing that. Then you can lose the other 24 states by millions. In total, you would lose the popular vote by millions less 26. You have to have a majority of the popular vote in the states you win but there is no restriction on what happens in the ones you lose.

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I second what Ross Millikan says except for one point: One need not have a "majority" of the popular vote; only a plurality. Although, in many states, such a plurality turns out to be a majority.

As for why the WTA system was adopted, the state Legislators over time found They could increase the influence of the state and the individual Voters of Their state by using such a system. Professor Alan Natapoff of MIT worked out the mathematics showing how, even if, say, a Romney Voter in New Obamaton were to cast a vote in that state, said Voter's influence on electing the President, on average, is much larger than if a straight popular vote is used.

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