The real solution is to drop out unless you are clearly a superstar or you are extremely close to finishing your PHD. No need to deny the truth.
I say this as someone who wasted three years in a PHD program. Luckily I got out when I did before I wasted more years or started a postdoc. If you are not a superstar start planning your exit.
There are several alternatives available to US residents. If you are willing to bet using crypto you can get large sizes and much lower fees than predictit. If you did not make substantial amounts of money (relative to your net worth) from the US election you made a mistake.
dirac_x(A) is a function of A, x is fixed.
Since this is lesswrong let me try to explain with probability.
If you have a random variable X then you have an associated measure:
measure(A) = probability that X is in A.
If X has a probability density function f then:
measure(A) = integral over A of f = probability that X is in A
If you measure has an associated density then you can visualize it by visualizing the probability density function.
You get the Dirac measure if X is constant. Your random variable X always returns the same result. The associated pdf is the Dirac delta 'function'. Sometimes people visualize the Dirac delta as an infinitely tall and infinitely thin Gaussian.
Does this make sense?
Something about your explanation of the Dirac measure feels off to me. "The dirac measure for whether you're standing in a shadow is something that outputs 1 when you're standing in a shadow, and 0 if you're not." What you seem to be describing is the indicator function of the shadowed region bounded in green. A measure takes a set as an input, not a point.
Maybe I am the one misunderstanding. But at the very least you are explaining the direct measure in a non-standard way. The way I think of the Dirac measure is that all the 'mass' is concentrated at a single point. This is not the same as an indicator function.
Indicator_function (point) = 1 iff the point is in the set
Direct_measure (set) = 1 iff the set contains the special point
I bet around 22K and will win around 11K. I had some bets that look like they will not work out (Dems win Senate). I would have put in more but by the time I decided to trust catnip.exchange I did not have time to get the crypto. A number of my friends won way more 11K. It was somewhat harder to make money if you were American but ways existed. At least I got out with 11K.
We are in a scenario Nate Silver explicitly described:
But what’s tricky about this race is that — because of Trump’s Electoral College advantage, which he largely carries over from 2016 — it wouldn’t take that big of a polling error in Trump’s favor to make the election interesting. Importantly, interesting isn’t the same thing as a likely Trump win; instead, the probable result of a 2016-style polling error would be a Biden victory but one that took some time to resolve and which could imperil Democrats’ chances of taking over the Senate. On the flip side, it wouldn’t take much of a polling error in Biden’s favor to turn 2020 into a historic landslide against Trump.
As of writing this comment Biden has several paths to victory and is heavily favored to hit at least one. I think he is about 75% to win at this point, despite all the bad news. We are in a world where Biden might win despite losing all of PA FL OH NC. Very little had to go right for Biden to win. Neither myself nor Nate Silver made any statements saying the election would definitely be a landslide. I told people to bet on Biden to win not to bet on Biden winning in a landslide. I will say there were some people pushing more aggressive Bets (Rainbow Jeremy comes to mind) and we should definitely update against their point of view. But Nate remains credible.
Thanks for giving details of the disagreements you are considering. The only community with norms even remotely conducive to large bets of this type is probably the poker community? There is a lot of big betting on 'props'. I don't know how often they bet on factual disagreements. But there have been a lot of big bets on random stuff like 'can X person do Y pushups' or 'does Scott think God exists. For a certain well known player doing pushups and a well known Scott both bets are real.
I don't really understand the conclusion. If your betting fund is 100$ we are back to small stakes betting. For many rationalists, a 'real deal' Kelly fund would need to be five or six figures. I do think you should consider fractional Kelly betting in some situations and am in favor of your worldview strongly influencing your investment portfolio. But even 25% Kelly betting your savings is a huge amount of work, you need to think you have a pretty good edge.
I am honestly unsure what the right norms are for 'doing this for real'.
If you live outside of the USA there are places you can still place a ton of money on pop-vote at good odds (~1.15). Biden is probably over 99% to win the popular vote. You can bet on Biden and place a larger bet on pop-vote to 'hedge'. I know some people with over 100K on pop-vote.