So here’s a simple algorithm for winning the lottery:
Buy a ticket. Suspend your computer program just before the lottery drawing—which should of course be a quantum lottery, so that every ticket wins somewhere. Program your computational environment to, if you win, make a trillion copies of yourself, and wake them up for ten seconds, long enough to experience winning the lottery. Then suspend the programs, merge them again, and start the result. If you don’t win the lottery, then just wake up automatically.
The odds of winning the lottery are ordinarily a billion to one. But now the branch in which you win has your “measure”, your “amount of experience”, temporarily multiplied by a trillion. So with the brief expenditure of a little extra computing power, you can subjectively win the lottery—be reasonably sure that when next you open your eyes, you will see a computer screen flashing “You won!” As for what happens ten seconds after that, you have no way of knowing how many processors you run on, so you shouldn’t feel a thing.
Yes, there is something like paradox on whether you should anticipate winning or losing the lottery. So let’s taboo “anticipate”:
Postulate 1. Anticipating (expecting) something is only relevant to decision making (for instance, expected utility calculation).
So to measure the expectation one needs to run a prediction market. The price of an outcome share is equal to the probability of its outcome, so that expected payoff of each share is exactly zero. For sake of argument, let’s assume that the market is unmovable.
Once the branch when you won was split into trillion copies, you can expect winning the lottery at odds 1000:1. So your 10^12 + 10^9 copies (trillion who have won the lottery, billion who have lost) will each buy 1 “WIN” share at price of 1000/1001.
Now, let’s consider what should happen during merge for such expectation to be sound. If your trillion copies are merged into one who has bought 1 “WIN” share, then the summary profit is 1 - (10^9 + 1) * 1000/1001, which is clearly negative. Seems like something went wrong and you shouldn’t have anticipated 1000:1 odds of winning?
The different merging procedure makes everything right. In it, you need to sum all purchased “WIN” shares, so that your winning copy has 10^12 “WIN” shares and each of 10^9 losing copies has one share. The summary profit is 10^12 - (10^9 + 10^12) * 1000 / 1001 = 0!
That can be translated as “if merging procedure will multiply your utility of an outcome by number of merged copies, then you can multiply the odds by copies count”. On the other hand, if during merge all copies except one are going to be ignored, then the right probability to expect is 1:10^9 - totally unchanged. And if you don’t know the merging procedure, you’ll have to use some priors and calculate the probabilities based on that.
Why doesn’t this affect real life much?
I guess that splitting rate (creation of new Boltzmann brains, for example) and merging rate are approximately equal so all the possible updates cancel each other.
Anthropics—starting from scratch
I’m trying to derive coherent scheme of anthropic reasoning from scratch, posting my thoughts here as I go. Pointing flaws out is welcome!
The anthropic trilemma (https://www.lesswrong.com/posts/y7jZ9BLEeuNTzgAE5/the-anthropic-trilemma)
Yes, there is something like paradox on whether you should anticipate winning or losing the lottery. So let’s taboo “anticipate”:
Postulate 1. Anticipating (expecting) something is only relevant to decision making (for instance, expected utility calculation).
So to measure the expectation one needs to run a prediction market. The price of an outcome share is equal to the probability of its outcome, so that expected payoff of each share is exactly zero. For sake of argument, let’s assume that the market is unmovable.
Once the branch when you won was split into trillion copies, you can expect winning the lottery at odds 1000:1. So your 10^12 + 10^9 copies (trillion who have won the lottery, billion who have lost) will each buy 1 “WIN” share at price of 1000/1001.
Now, let’s consider what should happen during merge for such expectation to be sound. If your trillion copies are merged into one who has bought 1 “WIN” share, then the summary profit is 1 - (10^9 + 1) * 1000/1001, which is clearly negative. Seems like something went wrong and you shouldn’t have anticipated 1000:1 odds of winning?
The different merging procedure makes everything right. In it, you need to sum all purchased “WIN” shares, so that your winning copy has 10^12 “WIN” shares and each of 10^9 losing copies has one share. The summary profit is 10^12 - (10^9 + 10^12) * 1000 / 1001 = 0!
That can be translated as “if merging procedure will multiply your utility of an outcome by number of merged copies, then you can multiply the odds by copies count”. On the other hand, if during merge all copies except one are going to be ignored, then the right probability to expect is 1:10^9 - totally unchanged. And if you don’t know the merging procedure, you’ll have to use some priors and calculate the probabilities based on that.
Why doesn’t this affect real life much?
I guess that splitting rate (creation of new Boltzmann brains, for example) and merging rate are approximately equal so all the possible updates cancel each other.