MEV and the Limits of Arbitrage with Lucas Baker

foreign [Music] great all right testing testing great hello everyone I'm Lucas Baker from jump crypto and I'm here today to speak to you about two things the first is every researcher's favorite path Pastime which is redefining Mev and the second is an interesting theoretical result that we have about the invariance of certain types of Mev so without further ado first I'll be uh covering the difference between ordering and Signal Mev which is a distinction that's been covered in a recent post by Frontier research second I'll go through the mechanics of the setup that we have in order to prove some invariance results about Mev or essentially uh signal Mev and finally we will go through the results themselves and their uh and their implications so at a high level there are a couple of ways to think about Mev the normal way is instantaneous value extractable from a transaction by reordering inserting or removing in other words it's what you can do by swapping things around taking them in or out and that is the perfect way to think of the internal frame of reference in other words if you're looking at pgas or contention within a block it's what you'd naturally arrive at in a you know on-chain each denominated environment however it is not the only way uh specifically and uh you know jump is a is a trading firm so as a traditional Finance firm our reference point typically has some aspect of the outside world of reference prices and when you think about you know cross-domain non-atomic value we've had a lot of conversations where somebody says cross domain Mev is is large it's important and uh you know how important uh well I don't know it's it's very difficult to measure it's uh and I think that part of this discrepancy is because it hasn't been very well defined thus far uh all of all of the things that people tend to focus on in Mev tend to be within the internal frame of reference so the external frame of reference would be something that we'd want to consider when centralized exchanges or other chains or any other source of true or reference prices involved so up here are a few of the dashboards with which you're likely familiar uh there's eigenfi there's mevboost.pics there's the flashbots so this is all internal Mev now the post that I alluded to by Frontier is uh defining the difference between ordering and Signal Mev so ordering is what we're used to it's the atomic Arbitrage it's sandwiching it's liquidations including for example uniswap jet liquidations EV signal on the other hand would involve some external frame of reference some notion of Truth and that would be what we get if we were for example performing c5d5 Arbitrage or if there were any notion of proprietary order flow so why uh why belabor the definition well it is uh it's quite different in terms of the consequences of the Mev as well as the actual type so EV ordering more of a relative frame of reference it's about redistribution between participants signal Mev you can think of as a pie that is really there no matter who claims it it will be claimed by someone in an efficient market eventually all signal-based strategies or gaps converge and so it's going to be there one way or another finally Mev for this talk so we'll mostly be speaking about EV signal the reason it's not just labeled EV signals that we're also going to be considering Atomic Arbitrage but essentially the main point to absorb here is that we're looking at all transacting addresses rather than a subset of let's say informed Flow versus uninformed flow and as a result of this it's essentially non-extractive specifically there's no notion of a victim transaction there is just a discrepancy in price that is eventually closed and yes finally as as covered it's mostly Arbitrage just putting those gaps together Okay so with that we can ask some interesting questions about block design specifically the size of the signal pi for example if we change around the block time does it affect the amount of EV signal that we have what about the transaction ordering mechanism you know if we have fifo if we have Fair ordering what uh what does that have what impact does that have on the signal EV or the external frame of reference specifically now our priori it is unclear whether this is true or false so it's easy to generate hypotheses under which it's true for example you let's say that we have a price that moves around a certain reference you know you you might be able to arbit down arbit up and if you have more frequent block times you might imagine that you'd get to do that several times before the price converges back to where it started and our results are going to prove that no neither the ordering mechanism nor the shorter block times increase the EV signal and here's a visual representation of what we're analyzing so we're looking at as I described the external price if it goes up then you can imagine the uh imagine for example that we were granted the ability to confirm a certain transaction or that you know the block ended earlier in that case we could simply take the unit swap price put it up to the external price and then put it down put it up put it down so that that would be the Arbitrage that the system would collectively be performing question is you know if this happens if block times are more frequent if we change the ordering mechanism does it affect this of course in this case it does but there are other possible trajectories of price which is uh what we'll be turning to now okay so uh consider if you will frictionless sphere D5 the uh platonically perfect world where we have one amm one risky asset one numerator uh you know a base asset in which everything is denoted and uh the risky asset this can generalize to multiple risky assets but we'll just consider one for the time being now price is a semi Martingale uh you can you can think of it as a Martingale for our purposes there will be small technical distinctions based on the fees or not but uh basically we have a price that moves around semi-randomly we'll model the liquidity pool as a state machine so this is the states the admissible actions the are the set of actions the admissible actions some transitions some payoff and some initial State just as you'd expect okay we have uh three steps of setup to go through and then we can present our results step one liquidity pools so we can define a no Arbitrage State as a situation where the pool is going to converge to some external Price p at that point any Divergence from that state will lose money uh frictionless there is a unique no Arbitrage state for any given reference price and this would describe most amm's most uni swap style amms without trading fees also we have a notion of efficiency so that's uh where every state is no Arbitrage for some price these properties will come in later when we use them to distinguish between different types of Mev and variants and finally we will consider pools with fees but we will always consider them with reference to a fiction or a frictionless no fee version as the underlying pool all right step two the blockchain market so this is the consensus process as you think of it there is an ordering mechanism you know some theoretical sequence of actions T not necessarily admissible like we can't necessarily do it and then according to the probability distribution some actual admissible permutation of these transactions will be selected so in the end we'll get a block and that's our T Prime we assume for our purposes that it's deterministic block time so this is the the State of Affairs in ethereum um self-financing strategies that's just a way of saying we can't Source external liquidity so anything that a strategy has has to be fueled by its prior positions uh we have Arbitrage strategies which end up holding only the numerator and we have a simple arbitrage strategy that uh is executing a single trade so specifically the simple arbitrage strategy is going to you can think of it as wait until the end of the block and execute the single most profitable trade now we have a four more Concepts to get through concurrent so disjoint uh competing for all the same liquidity covering that means if you sum the strategies together we get all the trades complete if you sum the strategies together that they produce an arbitrage strategy and competitive so the block ends in a no Arbitrage State and all strategies participate sometimes what does sometimes mean well they all participate at least some non-zero number of times per strategy and there are at least two that you know expected infinitely many times okay now set up on the Mev and the p l we're going to define a p l as a stochastic notion uh stochastic function of T there's also a p l star and that is the uncontested p l for a complete set of strategies and remember that's just uh you know it adds up to all of the transactions now we have three Notions of Mev and part of the reason why we've established all of these mechanics is so that we can give a complete formalization here in a rigorous way that can be tested and proved so we have the pathwise competitive Mev this is the best Mev that's achievable under competitive circumstances when you have a condition of path Independence so that's uh you know lowest upper bound of the sum of all the P L's of the strategies you also have the competitive Mev so this uh this will be useful as we introduce the notion of fees we can no longer think about path Independence you can think that for example if you went up you went down you went back again if you incur fees three times it's worse than incurring them once so no longer path independent but we still have a competitive Mev which means that the the strategies are still going to try to ARB off any opportunity when they can and now we just have the best expected value finally we have a notion of a non-competitive Mev where we no longer require all of the strategies to be competitive what this means in practice is that you can wait a little bit if you'd want to for example to perform an Arbitrage and you don't have to worry about someone else sniping you okay finally uh our results in the frictionless case so the pathwise competitive Mev is simply equal to the uncontested p l of the simple arbitrage strategy we also can prove that the competitive Mev so that's uh in the in the fee case mostly is equal to the expected p l of the simple arbitrage strategy and finally if the pool is path independent and price is fully a Martingale then you know we have no Alpha about the price then we have that the Mev star so the non-competitive Mev is equal to the competitive Mev is equal to the expected p l uncontested p l of the arbitrage strategy this sums up to it doesn't matter how you slice it the Mev opportunity from signal is the same either way okay now let's add in some fees uh this makes things a little bit harder because we can no longer guarantee path Independence but we can have reference to an underlying pool that is efficient in path independent in which case we can say that the competitive Mev when p is a Martingale is the expected value of the uncontested strategy Etc as before okay but now when we introduce fees as you might imagine block time does play a role so consider instead of s0 which is our simple arbitrage strategy in St that doesn't submit at the end of every block instead it just waits as long as it can until the end of the the teeth block and it submits one trade that captures all of the optimal Arbitrage there then we have that the non-competitive Mev equals the expected p l of VST strategy and finally results for Block times we have no fees so if we have a b Prime remember blockchain Market is basically just how we construct the block so consider a b Prime blockchain Market that's just like B but subdivided in that case we can say that the simple ARB strategy performs equally well for both competitive and non-competitive mevs so in the in the no fees case it doesn't matter how you slice it in the fees case if we have again an efficient path independent underlying Pisa Martingale then the non-competitive Mev is constant because we can have a strategy that Waits while all the others don't execute but the uh the competitive Mev May decrease because if you try to pull that in a competitive context then you're you're going to be out competed or there's going to be an ARB at the end of every block so what have we got in here well first of all I'd like you to come away with signal Mev matters so we ought to be able to think in both the internal and the external frame of reference ordering Mev is mostly about a redistribution signal Mev is really more of the public good public goods fund the good sort of Mev that people want it's a revenue stream that is available to all agents in the transaction pipeline it is mostly design invariant so of course Fair ordering uh or any notion thereof affects how value is distributed between participants which is again what we've been you know what most work has been concerned with but in terms of the signal value it's independent of the block time without fees it gets actually smaller as the block time decreases with fees and the ordering mechanism doesn't matter at all finally we should be thinking cross domain uh you know C5 Defy is only one instance of cross domain but everybody has had the notion for a long time that Mev outside of the bear context of you know in protocol ethereum matters no one has really even made an effort to quantify it as far as we're aware so this is uh is still extremely early and overall low visibility area where we think that some of the theoretical research from traditional finance and some of the general knowledge and expertise can be leveraged to have a fuller knowledge of the expanded system and that's all I got thank you [Music] here is a convergence point and there's only a single one and the scene converges at all so I was wondering whether those assumptions are realistic as you know a dynamic system like the blockchain is very chaotic and just like any other machine learning real world problem it doesn't really converge to a single point in a lot of cases doesn't even converge so how does that affect your result and when special spikes happen like when there's an event a popular board ape thing that got released and there's a spike so the environment changes it's no longer stationary and as a result still hold when the environment changes and at what rate uh thank you for that that's the uh the question was about the changing of uh environment and whether that affects our results uh I think the general flexibility of our results is uh the the most important condition is the price being under a constant conditions it's a Martingale so that uh if the if the regime changes or you know we have uh we have conditions that don't fit our frictionless sphere defy that uh that may have a limited impact on the results but as long as the price movement is consistent I think most of it holds and you can you can look into the details when we publish on archive shortly all right thank you very much [Music]