A log utility function says you should play this game over and over again, but you will most definitely lose all of your money by playing it (with the slim potential for wonderful returns).
No, one thing that absolutely will not happen is losing all your money. You literally can’t.
Though yes, the conclusion that utility is not logarithmic in money without restriction does hold. If on the second gamble you “win” 10^2174 dollars (as the formula implies), what value does that have? At best you’re now going to jail for violating some very serious currency laws, at worst you get it in gold or something and destroy the universe with its gravitational pull. Somewhere in the middle, you destroy the world economy and make a lot of people very unhappy, and probably murderously angry at you.
In no circumstance are you actually going to be able to benefit from “winning” 10^2174 dollars. Even if you somehow just won complete control over the total economic activity of Earth, that’s probably not worth more than 10^14 dollars and so you should reject this bet.
But since this a ridiculous hypothetical in the first place, what if it’s actually some God-based currency that just happens to coincide with Earth currency for small amounts, and larger quantities do in fact somehow continue with unbounded utility? To the extent where a win on the second bet gives you some extrapolation of the benefits of 10^2160 planets worth of economic output, and wins on the later bets are indescribably better still? Absolutely take those bets!
No, one thing that absolutely will not happen is losing all your money. You literally can’t.
Though yes, the conclusion that utility is not logarithmic in money without restriction does hold. If on the second gamble you “win” 10^2174 dollars (as the formula implies), what value does that have? At best you’re now going to jail for violating some very serious currency laws, at worst you get it in gold or something and destroy the universe with its gravitational pull. Somewhere in the middle, you destroy the world economy and make a lot of people very unhappy, and probably murderously angry at you.
In no circumstance are you actually going to be able to benefit from “winning” 10^2174 dollars. Even if you somehow just won complete control over the total economic activity of Earth, that’s probably not worth more than 10^14 dollars and so you should reject this bet.
But since this a ridiculous hypothetical in the first place, what if it’s actually some God-based currency that just happens to coincide with Earth currency for small amounts, and larger quantities do in fact somehow continue with unbounded utility? To the extent where a win on the second bet gives you some extrapolation of the benefits of 10^2160 planets worth of economic output, and wins on the later bets are indescribably better still? Absolutely take those bets!