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"Popular vote" and "number of electoral votes" don't commute

I have dedicated several blog posts to the attitude of Trump critics to the elections. The biased media coverage and polls (yesterday, during a public debate in the bank, I noticed how much the anti-Trump and similar self-evidently unjustified biases are widespread even among top bank analysts etc.) and the emotionally immature reactions by the leftists were given some room.

But there's another unsurprising dimension of the leftists' denial of their loss: attacks on the mathematics. While Scott Aaronson sent thousands of dollars to sponsor a recount in three states (WI, MI, PA; leftists' dealing with the taxpayer money is even more wasteful when they get in front of the steering wheel), Brian Greene included himself among the embarrassingly sore losers by questioning the U.S. democracy itself:

Well, that's a painful tweet, Brian, one that shows how sick political positions are considered normal in the Western universities.

I have tried to explain to him that "democracy" generally means "the participation of the most general public at power" but the detailed implementation of this general concept requires additional laws and the U.S. implementation involves the electoral votes. There's nothing non-democratic about this recipe: the "demos" still rules by picking a sensible number of electoral votes etc. During the presidential elections, the U.S. democracy is defined by the rules involving the electoral votes. The rules involving the electoral votes aren't a "curious" flavor of democracy but the "U.S." flavor of democracy, perhaps the world's most celebrated flavor of democracy.

Leftists are used to bending and twisting the rules whenever they can (also changing the rules during the game) – e.g. when they are selectively hiring women or people of color or other privileged groups at the U.S. universities or when they harass conservatives in the Academia – and they seem to be shocked that the same dirty tricks can't be easily done after the presidential elections.

At any rate, Brian Greene and many others point out that Hillary Clinton ultimately won the popular vote by nearly 2 million votes. They implicitly claim that Hillary would have become the president if the popular vote were the quantity that mattered.

Except that this conclusion is unjustifiable. What mattered were the electoral votes and Trump won 306-to-232, a very solid victory (not surprising given his dominance in 30-against-20 of the U.S. states). I pointed out that Donald Trump's tweet about the same issue

was far more intelligent than Brian's tweet. You know, Donald Trump not only realizes that the outcome of the election depended on the rules. But he even knows what he would have done if the rules had been different. If the popular vote mattered, he wouldn't worry about getting swing states which would be irrelevant. Instead, he would organize rallies at places with lots of people and a great potential to make a difference in the popular vote. He quotes New York, Florida, and California as the three relevant states. Trump would still lose New York and California but it wouldn't matter and he would get many more votes from them.

In a sense, Donald Trump's political intelligence seems to be not one but two categories above Brian's. He not only realizes that the rules matter for the ideal campaigning and the result. He has also thought about the way how the campaigning should be adjusted assuming different rules to achieve the desired outcome.

You know, I can't be any certain that Trump would have won the elections if the popular vote were decisive. But I am certain that the assumption that Hillary would have won it – because she "won" it in the system where the popular vote isn't important – is simply logically invalid. It's equally justifiable or unjustifiable as the statement that the winner would probably be the same under any rules.

This is a physics blog and I can't resist to share an analogy. You may see that the measurement of "Trump's popular vote" and "Trump's electoral votes" interfere with each other. If Trump focuses on the former, he may get a different result than if he focused on the latter, and vice versa. This seems very analogous to the measurement of the position and momentum in quantum mechanics. If you measure the position, you unavoidably influence or change the particle's momentum, and vice versa. So you can't accurately measure the position and the momentum at the same time.

The position and momentum are complementary. Mathematically, the uncertainty principle applies to these two because\[

xp - px \neq 0.

\] In the analogy, Trump's popular vote \(N_P\) and his number of electoral votes \(N_{EV}\) may be said to be complementary as well. If we took the analogy seriously, we would have\[

N_P N_{EV} - N_{EV} N_P \neq 0.

\] If you want to have a contest where the popular vote matters, you must prepare a "measurement procedure" that includes the campaigning of both candidates who are trying to maximize the popular vote – or the difference between their popular vote and their opponent's popular vote. This campaigning changes the landscape – some people get persuaded at the rallies etc. – so you can no longer measure the number of electoral votes and claim that the result is the same as it would have been if no campaigns designed to increase the popular vote had ever taken place. And vice versa.

I would go further. I think that it's right to say that it's not just a metaphor. \(N_{P}\) and \(N_{EV}\) are literally complementary in the quantum mechanical sense and if you identify operators for these quantities that act on the Hilbert space, they will not commute with one another. If true, this statement means that \(N_{EV}\) is sensitive to the relative phases of the probability amplitudes in a basis of \(N_P\) eigenstates and vice versa. It may sound surprising that elections are "intrinsically quantum mechanical" but if you were able to define such operators uniquely (and I mean operators that measure the elections and allow the campaigning while the candidates know the rules), I think it would be unavoidable that they don't commute with each other.

At the end, in practice, I think that these nonzero commutators can't be made specific because you can't define the operators \(N_P\) and \(N_{EV}\) uniquely. And because the details you need to refine basically depend on the initial state, you can't really define any state-independent operators at all and the discussion about the nonzero commutators becomes mathematically meaningless.

At any rate, there's a sense in which Donald Trump has shown a better understanding of quantum mechanics than Brian Greene. More practically, Donald Trump and his campaign have shown better skills in the optimization of campaigning than Hillary and her comrades. They had a good idea which places had a chance and which places mattered – incidentally, their estimates were far more accurate than those of the would-be professional polling agencies and large newspapers – and they focused their activities on the right places.

It seems that Hillary and her comrades were completely ignorant about this whole concept of micro-management. They were hoping that they would win sort of everywhere, in a landslide, and they don't have to do much for the success. This is how Hillary has lived the bulk of her life. She has gotten almost anything she needed without any special efforts. But you know, this attitude often fails to work whenever some fair rules are enforced impartially and this is basically the case of the U.S. presidential election.

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