Thread by @tarunchitra

⚠️ Paper Alert ⚠️

Remember the Twitter argument between @danrobinson and @SBF_Alameda?

Recall how it hinged on logarithmic vs. linear utility functions?

Using optimal control, we show utilities are a red herring

joint w/ @alexhevans @GuilleAngeris

stanford.edu/~guillean/pape…

Flip the problem upside down: LP returns are a function of how close the weights w (@BalancerLabs portfolio weights) are to the 'optimum' weight w*

Arbitrageurs can be viewed as a stochastic control mechanism that moves w around w*

Can you control |w-w*| as a function of fees?

Trad. Optimal Control: Robot/program has 100% deterministic control over the intervention (e.g. moving robot arm)

LP vs. Arbs: Stochastic control plus fees add a wrinkle — the fee interval. Arbs can never exactly get to w* because of fees, yet Martin/Dave show LPs are still 🤑🤑

Instead of looking at profits as a function of fees, drift, vol, we look at weights (controlled by arbitrageurs)

This gives you a Bellman equation (like those in reinforcement learning!) that only depends on your utility function indirectly (via ϕ)

2-assets: Solved w/ Wronskian

This paper is short and sweet, but gets to the heart of the debate between Dan & SBF: Why does utility matter?

Results intimate:
- When weights are very volatility you prefer linear
- When weights are static, you prefer logarithmic

Volatile weights ~ MMs frequently quoting

In Uniswap, you can replicate the volatile weights by adding and removing liquidity around orders (like sandwich attacks), which is _effectively_ the same as quoting a price schedule in an order book

c.f. @ProfessorJEY's nice little note

professorjey.com/assets/papers/…

By the way, @MartinTassy's discovery of the '0 fee phase transition' looks a lot more like conventional transitions in 1-dimension from the optimal control lens

This paper shows that it is more of a 2nd order transition than a 1st order one for @BalancerLabs/@UniswapProtocol

Timeline 🕰️

October: @alexhevans had a genius idea of viewing LP returns from the perspective of a stochastic process on weights

November: @MartinTassy / @_Dave__White_ result out, we were confused as to why the proof was so specific to Uniswap

December: Rough proof that the zero-fee phase transition from Martin/Dave is real, but is better viewed via the control theory lens

January: Rewrite proof in Reinforcement Learning language — describe Bellman equation for LP returns w/ fees

tl;dr: Given the immense interest in optimal control in DeFi, we found a way to rephrase the 'do you need fees? how much?' problem for AMMs in terms of a RL-like Bellman equation

Perhaps some of the predictions @htkao and I made will come true medium.com/gauntlet-netwo…

*very volatile