⚠️ Paper Alert ⚠️
Remember the Twitter argument between and ?
Recall how it hinged on logarithmic vs. linear utility functions?
Using optimal control, we show utilities are a red herring
joint w/
stanford.edu/~guillean/pape
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You still have a variance penalty. The point is that at the margin you should have linear utility, especially if you can rebalance to other strategies.
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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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(so, yeah, I remain confused about how linear vs log was a red herring; it seems to me that with linear it's always bad to LP)
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Yeah, possibly true! But the explicit removal of utility from the problem statement makes it a lot cleaner to generalize past Uniswap
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Replying to @SBF_FTX @SBF_Alameda and 4 others
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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yeah agreed, and I do think this probably holds for all strictly convex utility functions you'll be able to get some results, though obv. the closer to linear the utility function is the less these approximations hold
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