note that linear isn't concave :)
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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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Because we had this argument last time: In the current setup, the result should hold trivially for a linear utility function, if we can write w*=1 or w*=0 i.e. "no rebalancing."
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+++
(all rebalances are + EV for the arb and so - EV for the other side if you're just doing straight linear EV of $)
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If we can write w*=1 or 0, there is no arb to speak of — that's what I mean by "holds trivially" :)
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er also though if w is equally valuable for any w in [0,1]
If you have linear utility then optimal weight for the risky asset is either 0 (if mu is negative) or 1 (if mu is positive) or doesn’t matter (if mu is 0). Most likely it has positive mu (since even if you don’t care about risk others do) so it’s probably 1
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