The arbitrageurs make money, ergo the LP could just decide to run the arbitrage against themselves. If they don't, it's because of risk considerations, ergo utility factors in for your argument. Not a red herring.
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Log vs. Linear is the red herring. Any concave, cadlag utility function will work (c.f. how we use phi)
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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 $)
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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Of course, it's totally unenlightening; might as well just hold that value in your wallet.
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