I mean, it's not a paradox, but it loses predictiveness. I also don't agree with (a) and how it interacts with (b) because losses are unbounded on both sides and optional stopping is infinite for St. Petersburg? So I don't think you can have both (a) and (b).
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in practice nothing is infinite and you're never offered potentially infinitely many identical coinflips for all your money
and if you instead restrict to finite reasonable conditions you end up with "small % of huge payoff" which can in fact be worth a lot!
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> in practice nothing is infinite and you're never offered potentially infinitely many identical coinflips for all your money
Sure, I agree, but there are many other constructions of a similar flavor that have the same results...
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(with obvious silly ones being like \eps probability of winning 100/\eps amount and 1-\eps probability of losing 99, would be a positive EV bet. I guess if you would take such bets then sure, fine :)
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We’ve established this, SBF happily bites all those bullets :)
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There are plenty of people who make a good living making millions of such bets every day.
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Sorry, but I don't think most hedge funds use just the EV (with no risk adjustment) for pretty much anything other than HFT (i.e., with arbitrage and front-running being the obvious exceptions).
Certainly not in portfolio optimization.
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Seems like a big factor here is that edge is frequently a decreasing function of size, so if you are big enough, that probably stops linear utility from getting too out of hand.
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I guess the point here is that there is usually an explicit downside risk constraint in most basic portfolio optimization problems; not to mention there are usually leverage constraints, among many other things, which of course aren't captured by just vanilla EV.
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So what hedge funds really do here is:
1) estimate linear EV. This is always the first and most important thing to do.
2) estimate correlated volatility with your portfolio
3) if (1)/(2) is small then just do it
4) if it's short-term just do it
But honestly usually just (1).
There are exceptions, obviously! Times when the downside risk is significant.
But then you separately think about downside risk for it.
To use something other than linear EV (or similarly sharpe adjusting for correlation with portfolio) would get you laughed out of the room.
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Very few trades will be big enough where you'll want any nonlinear considerations; I think probably something like 0.01%?
And even on those 50% of what you talk about is linear EV (possibly divided by volatility), and the other 50% is risk limits.
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