So buying one ticket is “infinitely” better than buying no tickets.
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.
The one dollar spent on the ticket, in comparison, would have had a totally negligible impact.
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.
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.