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DanielLC comments on What are your contrarian views? - Less Wrong Discussion
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At the risk of looking even more like an idiot: Buying one $1 lottery ticket earns you a tiny chance - 1 in 175,000,000 for the Powerball - of becoming absurdly wealthy. The Powerball gets as high as $590,500,000 pretax. NOT buying that one ticket gives you a chance of zero. So buying one ticket is "infinitely" better than buying no tickets. Buying more than one ticket, comparably, doesn't make a difference.
I like to play with the following scenario. A LessWrong reader buys a lottery ticket. They almost certainly don't win. They have one dollar less to donate to MIRI and because they're not wealthy they may not have enough wealth to psychologically justify donating anything to MIRI anyway. However, in at least one worldline, somewhere, they win a half a billion dollars and maybe donate $100,000,000 to MIRI. So from a global humanity perspective, buying that lottery ticket made the difference between getting FAI built and not getting it built. The one dollar spent on the ticket, in comparison, would have had a totally negligible impact.
I fully realize that the number of universes (or whatever) where the LessWrong reader wins the lottery is so small that they would be "better off" keeping their dollar according to basic economics, but the marginal utility of one extra dollar is basically zero.
edit: Digging myself in even deeper, let me attempt to simplify the argument.
You want to buy a Widget. The difference in net utility, to you, between owning a Widget and not owning a Widget is 3^3^3^3 utilons. Widgets cost $100,000,000. You have no realistic means of getting $100,000,000 through your own efforts because you are stuck in a corporate drone job and you have lots of bills and a family relying on you. So the only way you have of ever getting a Widget is by spending negligible amounts of money buying "bad" investments like lottery tickets. It is trivial to show that buying a lottery ticket is rational in this scenario: (Tiny chance) x (Absurdly, unquantifiably vast utility) > (Certain chance) x ($1).
Replace Widget with FAI and the argument may feel more plausible.
So your utility function is nonlinear with respect to probability. You don't use expected utility. It results in certain inconsistencies. This is discussed in the article the allais paradox, but I'll give a lottery example here.
Suppose I offer you a choice between paying one dollar and getting a one in a million chance of winning $500,000, and paying two dollars and getting a one in one million chance of winning $500,000 and a one in two million chance of winning $500,001. You figure that what's basically a 0.00015% chance of winning vs. a 0.0001% chance isn't worth paying another dollar for, so you just pay the one dollar.
On the other hand, suppose I only offer you the first option, but, once you see if you've won, you get another chance. If you win, you don't really want another lottery ticket, since it's not a big deal anymore. So you buy a ticket, and if you lose, you buy another ticket. This results in a 0.0001% chance of ending up with $499,999, a 0.00005% chance of ending up with $499,998, and a 99.99985% chance of ending up with -2$. This is exactly the same set of probabilities as you had for the second option before.
No it would not. Or at least, it's highly unlikely for you to know that.
Suppose MIRI has their probability of success increased by 50 percentage points if they get a 100 million dollar donation. This means that, if 100 million people all donate a dollar, their probability of success goes up by 50 percentage points. Each successive one will change the probability by a different amount, but on average, each donation will increase the chance of success by one in 200 million. Furthermore, it's expected that the earlier donations would make a bigger difference, due to the law if diminishing returns. This means that donating one dollar improves MIRI's probability of success by more than one in 200 million, and is therefore better than getting a one in 100 million chance of donating 100 million dollars.
Even if MIRI does end up needing a minimum amount of money or something and becomes an exception to the law of diminishing returns, they know more about their financial situation, and since they're dealing with large amounts of money all at once, they can be more efficient about it. They can make a bet precisely tailored to their interests and with odds that are more fair.