Money is only valuable to me prior to the point of no return, so the value to me of a bet that pays off after that point is reached is approximately zero. In fact it’s not just money that has this property. This means that no matter how good the odds are that you offer me, and even if you pay up front, I’m better off just taking out a low-interest loan instead.
As I understand it, you’re arguing that if Eliezer wants $100 now that he doesn’t need to pay off for 13 years, it would be cheaper to take it in loan interest than to make a bet with Bryan.
Using this loan calculator with the default 6% interest over 13 years compounded annually, Eliezer would owe $213.29 when his loan matured, rather than the $200 he’d owe Bryan if he loses the bet.
Eliezer enjoys $100 now, whether he bets with Bryan or takes out a loan.
If Eliezer loses the bet, his outcome is approximately the same whether he took out the loan or bet with Bryan: he owes $200. If he wins the bet, his outcome is also the same whether he took out the loan or bet with Bryan. Bryan’s outcomes are also identical whether he lends out $100 at 6% annually compounded interest for 13 years, or bets with Eliezer. He’s out $100 now, makes back the same either way if he wins, and isn’t worried about money anymore if he loses.
Edit: I no longer think you can adjust by inflating the odds. If Bryan offers less favorable betting odds than Eliezer could get at the bank, then Eliezer could just take out the biggest loan he can get and ignore Bryan’s offer to bet. I no longer think you can extract information on people’s confidence about the end of the world based on a bet, unless you assume they’re both acting as if they didn’t understand basic finance.
Edit 2: However, the limiting factor here is the opportunity cost for Eliezer. The opportunity to take any loan at all, including an equivalent bet with Bryan, should look attractive to Eliezer (if we ignore the dollar vs. utility objection). Hence, if he were unwilling to take a bet with Bryan, or wanted to keep it small, then this should still be some evidence that he’s not as confident in his claim as he’s projecting. An apocalypticist who won’t take out massive loans on the expectation he’ll never have to pay them pack is not behaving in a manner consistent with his statements.
It seems like we could deal with this by inflating the odds. For example, if Eliezer bet Bryan at 9:1 odds, then Eliezer would get $100 now, and Bryan would make back $1000 if he wins, an $800 surplus over what he’d have gotten loaning his money out. Likewise, if Eliezer loses the bet, he would lose much more money paying back Bryan than he’d have lost taking out a 13-year loan.
So it seems like we can deal with this problem with an adjustment for opportunity cost. Eliezer and Bryan’s bet is very close to a refusal to bet at all, since there is no different in outcome for either party whether they loan or bet, and no matter who wins. The real stakes for such a bet is something like the odds beyond adjustments for opportunity cost. In this case, if Eliezer was paid $100 up front by Bryan, and had to pay back about $400 if he lost the bet in 13 years, this would seem to me to be actually equivalent to a bet at 2:1 odds.
In general, the formula to calculate the “true odds” of the bet would be:
([Payment if Eliezer loses the bet] - [Interest of equivalent loan])/[Up-front payment by Bryan]
From your linked post:
As I understand it, you’re arguing that if Eliezer wants $100 now that he doesn’t need to pay off for 13 years, it would be cheaper to take it in loan interest than to make a bet with Bryan.
Using this loan calculator with the default 6% interest over 13 years compounded annually, Eliezer would owe $213.29 when his loan matured, rather than the $200 he’d owe Bryan if he loses the bet.
Eliezer enjoys $100 now, whether he bets with Bryan or takes out a loan.
If Eliezer loses the bet, his outcome is approximately the same whether he took out the loan or bet with Bryan: he owes $200. If he wins the bet, his outcome is also the same whether he took out the loan or bet with Bryan. Bryan’s outcomes are also identical whether he lends out $100 at 6% annually compounded interest for 13 years, or bets with Eliezer. He’s out $100 now, makes back the same either way if he wins, and isn’t worried about money anymore if he loses.
Edit: I no longer think you can adjust by inflating the odds. If Bryan offers less favorable betting odds than Eliezer could get at the bank, then Eliezer could just take out the biggest loan he can get and ignore Bryan’s offer to bet. I no longer think you can extract information on people’s confidence about the end of the world based on a bet, unless you assume they’re both acting as if they didn’t understand basic finance.
Edit 2: However, the limiting factor here is the opportunity cost for Eliezer. The opportunity to take any loan at all, including an equivalent bet with Bryan, should look attractive to Eliezer (if we ignore the dollar vs. utility objection). Hence, if he were unwilling to take a bet with Bryan, or wanted to keep it small, then this should still be some evidence that he’s not as confident in his claim as he’s projecting. An apocalypticist who won’t take out massive loans on the expectation he’ll never have to pay them pack is not behaving in a manner consistent with his statements.
It seems like we could deal with this by inflating the odds. For example, if Eliezer bet Bryan at 9:1 odds, then Eliezer would get $100 now, and Bryan would make back $1000 if he wins, an $800 surplus over what he’d have gotten loaning his money out. Likewise, if Eliezer loses the bet, he would lose much more money paying back Bryan than he’d have lost taking out a 13-year loan.So it seems like wecandeal with this problem with an adjustment for opportunity cost. Eliezer and Bryan’s bet is very close to a refusal to bet at all, since there is no different in outcome for either party whether they loan or bet, and no matter who wins. The real stakes for such a bet is something like the oddsbeyondadjustments for opportunity cost. In this case, if Eliezer was paid $100 up front by Bryan, and had to pay back about $400 if he lost the bet in 13 years, this would seem to me to be actually equivalent to a bet at 2:1 odds.In general, the formula to calculate the “true odds” of the bet would be:([Payment if Eliezer loses the bet] - [Interest of equivalent loan])/[Up-front payment by Bryan]