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jimmy comments on Rational Romantic Relationships, Part 1: Relationship Styles and Attraction Basics - Less Wrong
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Obviously, I agree. But let me ask: for what values of X would you choose X$ with 15% chance instead of 1,000,000,000$ with 100% chance?
A quite extreme, but still somewhat defensible theoretical assumption is that utility is logarithmic in money. I once heard Bernoulli already worked with this assumption, many hundred years before Neumann-Morgenstern, and it is probably not so silly to assume this near the power-law tail of the wealth distribution. Not that I think it means anything, but from this admittedly extreme starting point, we get that lg(500) = 2.7 is three times as useful as 0.15*lg(1000000) = 0.9.
That assumes someone who initially has $1, and in that case it's certainly true. If on the other hand you initially have, say, $10k...
log(10.5k) - log(10k) ≈ 0.02
0.15 * (log(1.01M) - log(10k)) ≈ 0.3
The crossover point based on this system is $191. Less than that, and you do better with $500. More than that, and you'd try for the million.
That's not quite right in practice either. Even if you took all my money, I'd still take the 15% chance at $1M and maybe sell a 15% chance of $5k for $500.
Or if that is somehow not allowed, then I'd run into a bit of debt until my next pay check. Even if I really was spending all the money I make and averaging $0, $500 is a mere blip in the noise, not a factor of infinity more money.
It makes more sense to look at the total money in over whatever time scale you plan for.
Yeah, the prospect of other incomes makes a big difference. I neglected to include a requirement that the initial amount, whichever value it takes, is as much as you can come up with before you'll be needing money again.
The present value of my expected future income stream from normal labor, plus my current estimated net worth is what I use when I do these calculations for myself as a business owner considering highly risky investments.
For most people with decent social capital (almost anyone middle class in a rich country), the minimum base number in typical situations should be something >200kUS$ even for those near bankruptcy.
Obviously, this does not cover non-typical situations involving extremely important time-sensitive opportunities requiring more cash than you can raise on short notice (such as the classic life-saving medical treatment required).