heh true but more generally linear doesn't imply that there's any impetus to rebalance
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so I think it's still strictly bad to put $ in an AMM if you have linear utility
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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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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!
> 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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