It is indeed possible that this is true; this is simply a claim that small but nonzero fee is generally optimal but we make no claim about its total utility. (Though the PDE and the explicit result we give depends on having a quadratic cost function.)
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I think that having a small but nonzero fee is only optimal *if* you have a nonlinear utility function; with a linear one I think infinite fees are optimal
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Perhaps; I'm afraid I haven't thought about this specific case carefully. Is there a simple argument that you have for this?
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(Also, is there a reason you would prefer linear utility directly, over, at least, some sort of mean-variance utility?)
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It's not clear to me that expected value wouldn't run straight into St. Petersburg type paradoxes.
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(more seriously:
a) st petersburg isn't a paradox, it's just highlighting how much people instinctively undervalue huge outcomes
b) the limit as t --> inf is irrelevant, and if you instead use realistic time scales it no longer gives crazy numbes
c) see twitter.com/SBF_Alameda/st)
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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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Otoh, yeah, I mean, if linear utility is all you're going for (and infinite downside losses are fine) then I too suspect that you get a 0-1-type law in the limit. (In the sense that either fee->0 or fee=1 is "optimal.")
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But I guess I'm just not fully clear why linear utilities are better than other basic models :)
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I would argue that linear utilities *are* the basic model, but that's a matter of taste :)
But, really, see twitter.com/SBF_Alameda/st and
This is fair!
I guess I meant slightly-more-complicated but still basic models (a la Markowitz, etc.) that reflect the some risk-aversion
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