When this paradox gets talked about, people rarely bring up the caveat that to make the math nice you’re supposed to keep rejecting this first bet over a potentially broad range of wealth.
This is exactly the first thing I bring up when people talk about this.
But counter-caveat: you don’t actually need a range of $1,000,000,000. Betting $1000 against $5000, or $1000 against $10,000, still sounds appealing, but the benefit of the winnings is squished against the ceiling of seven hundred and sixty nine utilons all the same. The logic doesn’t require that the trend continues forever.
I don’t think so? The 769 limit is coming from never accepting the 100⁄110 bet at ANY wealth, which is a silly assumption
Suppose you’ll only reject the bet when your net worth is under $20,000, and you’ll accept it above. Can you see why, if you have a utility function, it’s still implied that up to $20,000, the positive dollars are worth less than 10⁄11 the negative dollars?
And then once you have that, does it make sense that the marginal utility of money is going down (at least) exponentially up to $20,000?
This is exactly the first thing I bring up when people talk about this.
I don’t think so? The 769 limit is coming from never accepting the 100⁄110 bet at ANY wealth, which is a silly assumption
Suppose you’ll only reject the bet when your net worth is under $20,000, and you’ll accept it above. Can you see why, if you have a utility function, it’s still implied that up to $20,000, the positive dollars are worth less than 10⁄11 the negative dollars?
And then once you have that, does it make sense that the marginal utility of money is going down (at least) exponentially up to $20,000?