Apologies for being a bit late to the party. Thanks for the paper @rfritsch et al! Some comments for you.
What I’m not a fan of
- (pg. 3) What are the units for liquidity L1 and L2? Hopefully not TVL, since the analysis is on at least one concentrated AMM. I worry this detail might be especially relevant, since Uniswap’s competitors have different AMM mechanisms. You justify this by saying that V3 is a CPMM for small price ranges, but it’s important to take into account that the ratio of funds-to-depth is incomparable.
- I worry that the two-AMM model is a bit unfair in assuming L1+L2 is a constant, since there’s obviously capital flight in times when yields on exchange 1 and 2 go down. Maybe this could be resolved by an alternative liquidity sink L3 whose return profile can be approximated as not depending on the amount of capital deployed to it, like some kind of DeFi risk-free-rate, and then assume that L1+L2+L3 is constant.
- Similar to (2), assumption that V is constant is surely false, however it may be fine if you constrained your analysis to talking about market share rather than market size, which seems to be what you’re getting at here.
In aggregate, your assumptions are systematically biased in the direction of underestimating liquidity flight and volume flight that would occur upon introduction of a taker rate.
What I’m a fan of
- Interestingly, due to the fact that each assumption seems to be in the direction of over-estimating optimal take rates, you might be able to use your results as an upper bound of optimal take rates.
- Your conclusion was perfectly modest, and it’s worded so as to encourage future work. Love this. And the suggestion you give regarding game theory and the repeated taker rate game is excellent. While my intuition is that taker rates are an obvious race-to-the-bottom if there’s 0 sticky liquidity, it’s also obvious that 0 fees is not a Nash equilibrium when sticky liquidity does exist (no matter fees charged, multiply taker rate by the sticky volume and you’re guaranteed to have more exchange revenue than with 0 taker rate). Would love to see (or do?) more research on the stage game and multi-game equilibria here.
- Beautiful graphs.
Overall
I think it would be fruitful to reframe the conclusions to provide an upper bound on the exchange-revenue-maximizing taker rate, and I think there’s a straightforward yet illuminating analysis that could be done on the game theory behind multi-stage exchange taker rate wars.