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ciphergoth comments on Advancing Certainty - Less Wrong
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No, the jackpot is much more than a million times bigger than the stake.
EDIT: the expected utility might still be negative, because of the diminishing marginal utility of money.
Maybe I'm not understanding your point.
If the odds of winning are one in 100 million, you could very well expect to make a million statements of "I will not win the lottery" and not be right once.
As in the LHC example, the criterion is making a million statements with independent reasoning behind each. Predicting a non-win in a million independent lotteries isn't what ciphergoth was thinking, so much as making a million predictions in widely different areas, each of which you (or I) estimate has probability less than 10^-8.
Even ruling out fatigue as a factor by imagining Omega copies me a million times and asks each a different question, I believe my mind is so constituted that I'd be very overconfident in tens of thousands of cases, and that several of them would prove me wrong.
Everything is dependent on everything else. I can't make many independent statements.
That's certainly true given full rationality and arbitrary computing power, but there are certainly many individual things I could be wrong about without being able to immediately see how it contradicts other things I get right. I wouldn't put it past Omega to pull this off.
I'm not sure this properly represents what I was thinking. We all agree that any decision procedure that leads you to play the lottery is flawed. But the "million equivalent statement" test seems to indicate that you can't have sufficient confidence of not winning not to play given the payoffs. If you insist on independent reasoning, passing the million-statement test is even harder, and justifying not playing is therefore harder. It's a kind of real-life Pascal's mugging.
I don't have a solution to Pascal's mugging, but for the lottery, I'm inclined to think that I really can have 10^-8 confidence of not winning, that the flaw is with the million-statement test, and it's simply that there aren't a million disparate situations where you can have this kind of confidence, though there certainly are a million broadly similar situations in the reference class "we are actually in a strong position to calculate high-quality odds on this coming to pass".
I don't.
Can you please explain that further? Why not? Do you just mean that the pleasure of buying the ticket could be worth a dollar, even though you know you won't win?
Just reasoning based on a non linear relationship between money and utility.
Winning ten million dollars provides less than ten million times the utility of winning one dollar, because the richer you are, the less difference each additional dollar makes. That seems to argue against playing the lottery, though.
$5,000,000 debt. Bankruptcy laws.
Very clever! You're right; that is a situation where you might as well play the lottery.
This actually comes up in business, in terms of the types of investments that businesses make when they have a good chance of going bankrupt. They may not play the lottery, but they're likely to make riskier moves since they have very little to lose and a lot to gain.
It looks to me like the flaw is in calculating the expected utility after changing the probability estimate with the probability of error.
What alternative do you have in mind?
Well, in an abstract case it would be reasonable, but if you are considering (for example) the lottery, the rule of thumb "you won't win playing the lottery" outweighs any expectation of errors in your own calculations.
Potentially promising approach, but how does that translate into math?
Let A represent the event when the lottery under consideration is profitable (positive expected value from playing); let X represent the event in which your calculation of the lottery's value is correct. What is desired is P(A). Trivially:
From your calculations, you know P(A|X) - this is the arbitrarily-strong confidence komponisto described. What you need to estimate is P(X) and P(A|~X).
P(X) I cannot help you with. From my own experience, depending on whether I checked my work, I'd put it in the range {0.9,0.999}, but that's your business.
P(A|~X) I would put in the range {1e-10, 1e-4}.
In order to conclude that you should always play the lottery, you would have to put P(A|~X) close to unity.
Q.E.D.
Edit: The error I see is supposing that a wrong calculation gives positive information about the correct answer. That's practically false - if your calculation is wrong, the prior should be approximately correct.
I think this doesn't work, or at least is incomplete, because what is needed (under standard decision theory) to decide whether or not to play is not the probability of the lottery having a positive expected value, but the expected utility of the lottery, which I don't see how to compute from your P(A) (assuming that utility is linear in dollars).
ETA: In case the point isn't clear, suppose P(A)=1e-4, but the expected value of the lottery, conditional on A being true, is 1e5, then you should still play, right?
There's a non-negligible chance that you've been misinformed or are mistaken about the odds of winning.