# Redistributing LVR to AMM LPs with dynamic fees and first-access auctions: empirical analysis

[Juuso Ahlroos](https://x.com/juusoah), [Antero Eloranta](https://x.com/antsae_) - November 29, 2024

## Abstract
This study examines methods to redistribute Loss-Versus-Rebalancing (LVR), or arbitrage profit, back to Automated Market Maker (AMM) Liquidity Providers (LPs) using dynamic fees and first-access auctions. Analyzing data from Uniswap V2 and V3 pools on Ethereum and CoW AMM between April 2023 and July 2024, we evaluate various dynamic fee functions and their impact on LVR, average fees, and mispricing. Our findings indicate that short-term volatility and momentum-based fee functions offer significantly better trade-offs between LVR reduction and fee increases for certain pools, particularly USDC-WETH. However, results vary across pools, with UNI-WETH showing no improvement over constant fees. First-access auctions show promise in redistributing LVR to LPs, but only when price differences exceed 1\%. While dynamic fees and first-access auctions can improve LVR redistribution in some cases, their effectiveness varies by pool and market conditions, suggesting the need for tailored approaches.

## 1 Introduction
Loss-versus-rebalancing (LVR) is a metric quantifying the cost of adverse selection on automated market makers' (AMM) liquidity providers (LPs). AMM LPs are prone to adverse selection as there are arbitrageurs with an information advantage in the form of knowledge of current market prices of assets in different venues. They can leverage this information advantage to arbitrage price differences between AMMs and other venues such as centralized exchanges caused by stale AMM prices. Any profit made by an arbitrageur incurs a loss of similar magnitude to LPs as the interactions between the two are a zero-sum game.

Besides arbitrageurs incurring losses to LPs, their actions also carry other consequences. Block builders can gain a competitive advantage by vertically integrating with an arbitrageur as vertical integration between arbitrageur and block builder decreases the time it takes to process a transaction allowing vertically integrated arbitrageurs to submit their transactions later than an arbitrageur that is not vertically integrated. This latency advantage results in an information asymmetry between block builders and allows vertically integrated builders to produce better blocks compared to non-integrated builders. For example, vertical integration of Symbolic Capital Partners and beaverbuilder and Wintermute and rsync-builder previously led to a situation where beaverbuilder and rsync-builder were producing over half of the Ethereum blocks. Wahrstätter [2024a], Winnie [2024]

Similar behavior can be noticed among validators. If a validator intentionally delays their block proposal they can increase their MEV rewards. This gives block builders more time to propose blocks meaning arbitrageurs have more time to submit transactions to builders. This has led to a situation where staking service providers intentionally delaying their block proposals are capable of offering higher returns to their stakers leading to potential centralization of validators. Wahrstätter [2024b]
There have been multiple proposals to address these issues. Different MEV-Burn proposals, such as Thiery [2024], have been discussed that attempt to capture value from CEX DEX arbitrage and burn the captured value resulting in a transfer of wealth from LPs to the network. However, MEV-Burn is currently not planned to be included in Pectra Upgrade happening early 2025 meaning MEV-Burn will not improve the centralization issue at least before 2026. In addition, several solutions are trying to fix the problem with either first access auctions, such as CoW AMM, or dynamic fees, such as Ambient finance. However, these solutions have failed to gain significant market share, together accounting for less than 1\% of Uniswap’s volume Steimetz [2024]. While MEV-Burn could potentially help solve the centralization issue arising from arbitrages, it would fail to distribute value from captured arbitrage to LPs providing the service of on-chain liquidity. We believe rewarding LPs for providing liquidity would be a fairer way of distributing value arising from arbitrage than distributing most of the value to the network.

In this study, we examine different options for redistributing the arbitrage profits back to AMM LPs with dynamic fees and first-access auctions. Using different inputs such as the momentum of the pool price, the gas price of the underlying blockchain, or off-chain inputs such as CEX volatility, we backtest and compare different dynamic fee functions and parameters and their effect on LVR. To compare the results, we optimize for the best trade-off between minimizing LVR while keeping the increase in average fee as low as possible to capture as much LVR as possible while not significantly affecting the experience of the average uninformed trader. In addition, we optimize for as small mispricing between AMM and the venue where price discovery happens in as possible. For first-access auctions, we perform empirical analysis on the CoW AMM WETH-USDC pool to analyze whether first-access auctions are an efficient in capturing arbitrages.

The findings suggest that dynamic fees derived from either pool volatility or momentum can be used to decrease LVR with a more efficient trade-off between captured LVR and increase in average fees in the most liquid Uniswap V2 and V3 pools (WETH-USDC). However, they do not outperform the constant fee in more illiquid pool (WETH-UNI). In addition, the findings suggest that first-access auctions can be an efficient way of decreasing LVR but only if the price discrepancy is significant enough.
The rest of the study is structured as follows. Section 2 covers the theoretical background related to the study. Section 3 goes through the data and methodology used in the study. Section 4 displays the results of our analyses. Section 5 discusses the findings. Finally, section 6 summarizes and concludes the study.

## 2 Theoretical background

This section reviews the most relevant literature, starting with looking at market making in its traditional finance sense. The section then examines automated market makers, followed by a discussion of Loss-versus-rebalancing literature and different proposed solutions on minimizing impact of loss-versus-rebalancing. Finally, the section concludes with a comparison of solutions aiming to minimize loss-versus-rebalancing that have already been implemented.

### 2.1 Market making

Market making has been studied extensively. It is typically viewed as a stochastic optimal control problem where market makers try to determine optimal bid and ask quotes for a market, like Ho and Stoll [1981]. Different market-making models try to find the optimal balance between profit-making opportunities and inventory risk management by making bid and ask quotes based on various parameters. Volatility is typically a core component in most market-making models, as it helps to gauge the level of risk in the market. Market makers use volatility to adjust their quotes, typically by increasing spread between their bid and ask quotes in high-volatility environments to compensate for increased risk and decreasing the spread in low-volatility environments to remain competitive.

In addition to volatility, market makers often incorporate other parameters to fine-tune their models, such as order flow imbalance, market depth, trading volume, time of day, macroeconomic indicators, and correlation with other assets. These additional factors allow for more sophisticated models that can adapt to various market conditions, improving risk management and profitability.

For example, order flow imbalance can provide insights into the net buying and selling pressure in the market, which can impact the optimal bid-ask spread. Market depth and trading volume can also impact the risk faced by market makers. The time of day can impact trading activity and volatility, leading market makers to adjust their quotes accordingly. Correlations between the asset and other related assets can be used to better understand and hedge the market maker's overall risk exposure.

As an example of a market making model, in the widely referred Avellaneda-Stoikov model proposed by Avellaneda and Stoikov [2008], volatility is directly incorporated into the optimal pricing formula, impacting the size of the spread between bid and ask quotes. In addition to volatility the model considers the market maker's current inventory and risk aversion. The model first determines indifference price $r$ based on current inventory, risk aversion, and volatility where $s$ is the current price, $\sigma$ is volatility, $T-t$ is time remaining until terminal time, $q$ is current inventory, $\gamma$ is the risk aversion parameter and $\kappa$ is the order arrival rate parameter

\begin{equation}
\label{eq:as}
r = s - q \gamma \sigma^2(T-t)
\end{equation}

Indifference price represents the “mid-price for the market maker”, which they are indifferent to both buying or selling at if they have no inventory. After determining the mid-price the model calculates spread

\begin{equation}
\label{eq:as_spread}
\delta = \delta^a + \delta^b = \gamma \sigma^2 q + \frac{2}{\gamma}ln(1+\frac{\gamma}{\kappa})
\end{equation}

Spread $\delta$ represents how wide around the mid-price market maker wants to make their bid and ask quotes and $\delta^a$ and $\delta^b$ represent ask and bid quote specific spreads. If market maker does not hold an inventory ask and bid quote specific spreads are same size. However, if the market maker holds an inventory ask and bid quote spreads different from one another. This is due to skewing the spread to be bigger on the side with existing inventory meaning that the probability of market makers getting rid of their existing inventory is higher than the probability of accumulating further inventory.

Based on this, the model expresses the optimal bid and ask prices as:
\begin{equation}
\label{eq:bid}
\text{bid} = r - q \gamma \sigma^2 - \frac{1}{\gamma}ln(1+\frac{\gamma}{\kappa})
% \text{bid} = r - \frac{\delta}{2}
\end{equation}

\begin{equation}
\label{eq:ask}
\text{ask} = r - q \gamma \sigma^2 + \frac{1}{\gamma}ln(1+\frac{\gamma}{\kappa})
%\text{ask} = r + \frac{\delta}{2}
\end{equation}

### 2.2 Automated Market Makers
Automated market makers (AMMs) are a subset of market-making that can be used to facilitate trading on blockchains where data throughput is limited. Initially popularized by Uniswap, different AMM designs are now capturing most of the on-chain trading volume. The major difference between AMMs and traditional limit order book (LOB) markets is that price in AMMs develops independently of the liquidity providers’ (LPs) actions, whereas in LOB markets the price is determined by the LPs decisions to post limit orders. This means that each LP can passively deposit funds, and the price of the venue is determined by the trades that interact with the pool without LPs active participation on the venue. This cuts down the operations required to achieve an efficient market significantly but also takes away the opportunity for LPs to control their quotes for purposes such as market making. This means that price is determined only by actions of traders which means the might become stale. The key advantage of AMMs is that they enable permissionless, automated trading without the need for a centralized matching engine to match orders.

AMMs mostly differ by the function that determines their pricing. Uniswap initially started with a constant function $k = xy$ but then improved in V3 by introducing concentrated liquidity, that allows LPs to provide their liquidity in a specific range for more capital efficient solution. This is a very useful solution for assets whose prices are correlated, like pair of two stablecoins. Curve is another AMM whose design notably differs from Uniswap. Curve's main focus is offering an AMM algorithm that is well-suited for stablecoins and other correlated asset pairs with deep liquidity.

### 2.3 Loss-Versus-Rebalancing
Initially gaining wide attention among the Ethereum community after Milionis et al. [2022] Loss-Versus-Rebalancing (LVR) is a metric quantifying the cost of adverse selection on automated market maker (AMM) liquidity providers (LPs). AMM LPs are prone to adverse selection as there are arbitrageurs with an information advantage in the form of knowledge of current market prices of assets among multiple venues. They can take advantage of this information advantage to arbitrage price differences between AMMs and other venues such as centralized exchanges. Any profit made by an arbitrageur incurs a loss of similar magnitude to LPs as the interactions between the two are a zero-sum game. The authors state that LVR can be interpreted either as arbitrage profits due to stale AMM prices or as the hypothetical cost of hedging the AMM LP position against centralized exchange. LVR has become important, providing a quantitative measure for LPs to assess the potential risks and rewards of participating in different AMM pools.

Rao and Shah [2023] propose triangle fees to balance the trade-off between price accuracy and arbitrage profits by dynamically adjusting marginal fees based on trading volume. They argue that AMMs with linear fees, or fees where doubling the trade size also doubles the paid fees, can never reach a perfect accurate price but will always deviate by at least by the fee amount, which both makes them inefficient in terms of pricing but also deprive LPs' of the fees they could earn on the margin. Under the triangle fees introduced in the study the marginal fees progressively decrease as the trading volume increases. This fee model incentivizes arbitrageurs to correct price deviations more accurately resulting in less stale prices and allowing AMMs to collect fees without compromising on price accuracy. Authors present a simulation framework for calculating triangle fees for an AMM model, demonstrating they improve the raito between LP losses and price deviation compared to AMM model outperforming fixed fee models. Implementing this model would benefit larger traders the most who would effectively receive "volume discounts" while small traders may face higher relative costs.

Milionis et al. [2023], expand on their previous research by introducing fees when measuring arbitrage profit. When fees are included, arbitrage is only present in scenarios where the price moves outside of the “no-trade region”, which means that the price discrepancy between decentralized exchange and centralized venue is higher than the trading fees and which never happens in continuous time. However, blockchains with discrete time are prone to be arbitraged due to the incapability to continuously settle trades. Authors find that “fees simply scale down arbitrage profits by the fraction of time that an arriving arbitrageur finds a profitable trade”. The study shows that arbitrage profits are generally proportional to the square root of block time and it does not negatively impact LP fee income from noise trading, making decreasing block time one of the most efficient solutions. The study highlight the importance of blockchain infrastructure the AMM is built on top of, suggesting that improving the underlying blockchain, would have significant positive impact on AMM efficiency and LP profitability.

Adams et al. [2024] expands on Milionis et al. [2023] by introducing an Auction-Managed AMM by adding a new pool manager mechanism to an AMM. This design involves an on-chain auction held once in a while where the highest bidder becomes a temporary pool manager that is able to set swap fee rate, collect all fees from the pool's swaps and trade against the pool with zero fees. The pool manager can adjust fees dynamically based on market conditions and traders' sensitivity to fee changes. At the same time pool manager is able to take advantage of zero fees to capitalize on small arbitrage opportunities. Liquidity providers are free to enter or exit the AMM pool if they want. This AMM design should incentivize higher liquidity compared to static fee AMMs as compared to static fee AMMs individual LPs are outsourcing the decision making of at which fee to provide liquidity to a more sophisticated pool manager. However, pool manager trading without fees could expose the AMM pool to sandwich attacks. Furthermore, running such auctions could lead to further centralization in block building supply network as sophisticated entities having better access to private order flow could dominate the auction process due to the information advantage they have.

Fritsch and Canidio [2024] expands on Milionis et al. [2022] by measuring historical earned fees and simulated arbitrage losses an LP would have incurred on actual data. Their results show that for most large Uniswap V3 liquidity pools, the earned fees by liquidity providers were smaller than the losses to arbitrageurs. Because of this, they question the LPs' decision to contribute to these pools at all and the results highlight the need to design a system to compensate the LPs from the arbitrage profit better. On the other hand, authors found that Uniswap V2 pools were more profitable for passive liquidity providers than their V3 counterparts, with fees earned by the pools being three times larger than arbitrage losses in 2023. The study also investigated how block time impacts arbitrage losses and found that decreasing Ethereum's block time from 12 seconds to 100 ms would decrease arbitrage losses of pools by 20-70\% depending on the LP pool.

In a LOB exchange, market-making profits are fully allocated to market makers as they constantly cross the bid-ask spread to buy low and sell slightly higher. Market makers take on the inventory risk and are rewarded for carrying it as well as the liquidity they provide to the market. However, in assets where the price discovery happens off-chain, AMMs require arbitrageurs on top of the LPs to keep the price in line with other venues. Arbitrageurs trade against the LPs, which means that a profit for arbitrageurs incurs a loss of same size for the LP. We aim to contribute to this growing literature by empirically examining best ways to redistribute arbitrage profits back to LPs. This difference in profit allocation between LOB market-making and AMM-based design represents a key challenge in the design of decentralized exchanges.

Currently, the value arising from arbitrage opportunities is mainly split between arbitrageurs, block builders, and validators, while LPs are only rewarded with a constant trading fee the pool charges for each executed transaction, whether arbitrage or not. Redistributing this arbitrage profit to LPs has two main questions, how to target the arbitrageurs and without negatively affecting the average uninformed traders and how should the value arising out of the arbitrage be split between LPs and arbitrageurs for the split to be fair for both parties.

The balance between rewarding LPs and rewarding arbitrageurs is a fine line. If the split is too skewed for the LPs, the price will not efficiently follow the price of off-chain venues. If it is skewed towards the arbitrageurs, it can disincentivize LPs from providing liquidity, which also leads to a worse experience for on-chain traders due to higher slippage while trading. In an optimal scenario as much value as possible arising from arbitrage would be distributed to LPs while also offering incentives for arbitrageurs to keep the price in line with other venues' prices.

There has been multiple proposals on how to more efficiently redistribute bigger part of the arbitrage profits to LPs. However, these solutions such as CoW AMM and Ambient Finance have yet to gain widespread adoption.

### 2.4 Limit order book and LVR
LOB market makers are not passive market makers like AMM liquidity providers as they actively update their quotes based on information such as market volatility, their inventory and prices of highly correlated assets. As adverse selection happens due to stale information, LOB market makers want to react to new information as quickly as possible to minimize losses incurred due to processing new information slower than others. The markets also settle trades in continuous rather than discrete time. Because of this, LVR does not exist in LOBs in the same form as in AMMs, due to market makers constantly updating their quotes based on new information. In the case of LOBs “LVR” happens when an arbitrageur can process information from multiple venues a few parts of a second quicker than liquidity providers.

In AMMs, the major selling point is being able to act as a liquidity provider without having to actively manage one’s liquidity providing position. This advantage would vanish if each liquidity provider had to constantly manage their quotes based on new information. Because of this, the potential next evolution for AMMs could be an AMM that passively updates its quotes based on new market information. The AMM could have access to pool state, chain state, or even external state through oracles to bring in values that it then uses to determine what prices it quotes, similar to how LOB market makers adjust their quotes, except in case of an AMM doing it by changing fees instead of quoting bid and ask with a spread. Based on previous research on LOB market making, such as the Avellaneda-Stoikov model, we believe that at least market volatility should play a large role in determining fees AMMs should be charging users. Additional variables to consider could, for example, include trading volume.

### 2.5 Current Solutions
Ambient Finance combines Uniswap V2 (ambient liquidity), Uniswap V3 (concentrated liquidity), and limit orders (knockout liquidity) into a single liquidity curve, stating their goal as bridging the gap between trading and LPing. They also differ from Uniswap's initial implementations by having all pools in a single smart contract rather than a single pool per asset, which provides gas savings to the users. The design that combines different ways of providing liquidity decrease fragmentation of liquidity by allowing users with different needs to all provide liquidity in the same place. However, it should be noted that Uniswap V2-style LPing is possible even in Uniswap V3 by opening a position for the full price range, knockout liquidity is not possible as it would require automatically having your LP position be closed as price moves across it. Ambient Finance uses a dynamic fee so their pools can adjust to different market conditions and match the market demand over longer term. The fee is provided through an oracle which allows both the fee and the fee logic to change over time without smart contract updates. Currently, the oracle looks at which Uniswap V3 pool of an asset has accumulated the highest return over the last 60 minutes and uses that as the fee tier for the pool. The oracle-based approach is very flexible and allows fee to be based on external data but is more costly and won't allow reacting to market changes based on shorter period of data, for example changing fees per trade. The current heuristic is a simple one but it does not directly aim to take the LVR problem and it likely favours those who provide concentrated liquidity providers over ambient liquidity providers as in low-volatility market the fee would go down and concentrated LPs would narrow their positions, taking in a larger share of the pool fees decreasing the share, and percentage fee, that ambient LPs earn significantly. This shows the difficulty of treating ambient and concentrated liquidity providers equally. Currently, Ambient Finance only has \$130M TVL and $10-$30M daily volume.

Trader Joe V2 uses what they call the Liquidity Book (LB), which resembles Uniswap V3 with some subtle differences. First, instead of ticks, LB uses bins where there is zero slippage, meaning that everyone trading inside that bin will use the same price, the liquidity of the bin affecting how much can be traded at that price. The difference between Uniswap V3 and this depends on the bin width but the narrower it becomes, the designs should differ less in pricing. If the bins are too wide, it might also lead to discontinuation points in pricing. Second major difference is that the liquidity positions are fungible. This allows better composability and allows them to be used for example as collateral increasing capital efficiency. Finally, they also use a dynamic fee that depends on what they call a “volatility accumulator”, which uses time since the last transaction and bins crossed to determine the rolling volatility. More frequent trades that move price more increase this fee, while less frequent and smaller trades that do not cross bins decrease this. The dynamic fee should increase pool fees especially when the price moves in a single direction over a short period of time. This type of dynamic fee makes sense as liquidity providers that are hedged at the current price incur more and more losses as the price moves in either direction from the price they were perfectly hedged and this fee can help them with the costs of rehedging their position. This should also decrease LVR as price is more likely stale when the volatility is high. The dynamic fee is combined with a fixed base fee that sets up a baseline for LP fees and can be configured separately. Trader Joe's liquidity pools currently sit at \$87M TVL and do around \$35M daily volume.

CoW AMM implements a function-maximizing (FM) AMM. Their design is based on Canidio and Fritsch [2023] and the idea that batching trades and competition between arbitrageurs eliminates arbitrage profit and sandwich attacks. In their simulation, they find that the FM AMM would outperform most of the tested Uniswap V3 pools although the absolute differences are small. Their design would also eliminate most sandwich attack and arbitrage MEV. That is also currently implemented at CoW AMM, it batches orders together, execution everything at same price. Solvers then bid for the right to execute these orders and the one who can provide the best pricing gets the right to execute them. The bidding for first-access-auction should mean that the price approaches the market price, which in turn should decrease the CEX-DEX arbitrage. In a vacuum this removes the need for liquidity pools, trades are batched on an interval and then solvers compete who can provide the best price for rest of the volume. Here solvers are active market makers and all trading is done with them. CoW AMM however integates this first-access-auction with liquidity pools as well. Solvers will still bid for the best pricing but rather than just settling the open orders, they also will bid for the right to arbitrage the pool. There are some gaps in public documentation of CoW AMM on how this is achieved but the general idea is that the solver who offers most surplus will win the right to rebalance the pool. This auction-driven mechanism should provide an optimal solution for redistributing LVR back to LPs. The design does come with some drawbacks. Competing for the best pricing will ultimately favour those with large balance sheets and inventory that they can distribute across CEXes to gain deeper liquidity, this can have negative effect on how competitive the market is and lead to worse pricing. Matching orders and bids also requires another layer on top of the blockchain, which will need it's own consensus protocol to be considered decentralized. This increases complexity of the system significantly. Final problem is that there might be even fewer solvers who work with the long tail of assets, decreasing the benefits of first-access-auction there.

## 3 Data and methdology
This section discusses data and methodology. The first subsection discusses the data used in the study. The second through fourth subsections discuss how we have calculated LVR, the different fee functions we analyze, and how we analyze first-access auctions.

### 3.1 Data
To calculate LVR, we acquire the AMM price and a price from the venue where price discovery happens. For AMMs, we focus on Uniswap V2 and V3 on the Ethereum blockchain. Being the most liquid venue, we follow the Binance price as our off-chain venue price and assume price discovery process for assets analyzed in this study happen on Binance, or if it happens on another off-chain venue it is reflected on Binance price before analyzed on-chain venues. Using the event data from Uniswap pools, we recreate the pool state at each block and match it with the respective Binance price using the block timestamp.

We collect data for the following Uniswap pools on top of the Ethereum mainnet: Uniswap V2 USDC - WETH pool, Uniswap V3 USDC - WETH pool with a 0.05\% fee, Uniswap V3 WETH - WBTC pool with a 0.3\% fee, and Uniswap V3 UNI - WETH pool with a 0.3\% fee. We collect a dataset of historical transactions for these pools using a geth node. The first block of our dataset is 17 162 286 produced on 11:59 PM UTC April 30th, 2023, and the last block of our dataset is 18 473 542 produced on 11:59 PM UTC October 31st, 2023. The data consists of swap transactions and liquidity transactions.

We collect historical 1-second candle data from Binance for the same period. For WETH - WBTC and UNI - WETH we calculate the implied Binance prices based on 1-second candle data of WBTC - USDT and UNI - USDT combined with WETH - USDT price.

To analyze the efficiency of having a first access auction integrated in an AMM we collect and analyze a dataset from the USDC - WETH pool on CoW AMM. The first block of our dataset is 19 675 204 produced on 12:30 PM UTC April 17th, 2024 and the last block of our data is 20 375 662 produced on 09:59 AM UTC July 24th, 2024. We recreate the pool states and merge them with Binance data as described above.

### 3.2 Calculating LVR and basic characteristics of LVR
To quantify available arbitrage, we look at the LP pools' price at the end of block and compare it with the Binance price at the next block’s timestamp. Rational arbitrageur wants to move the price until the point where their next unit traded would clear the LP and Binance at the same price. We make a simplification that the opposite trade on Binance can be made with no slippage and can be executed at the same price as the previous trade was executed. With LP having fees, this leads to the target price $P_t$:
$$
P_t =
\begin{cases}
\frac{P_b}{1+f} & \text{if } P_{lp} < P_b \\
P_b (1 + f) & \text{if } P_{lp} \geq P_b
\end{cases}
$$

And there is an arbitrage opportunity available when the target price is between the Binance price and the LP price $(P_{lp} < P_t < P_b) \lor (P_b < P_t < P_{lp})$

After determining the target price, we determine the quantity of LVR by calculating what is the average price the arbitrageur pays when moving the LP price into the target price and how big the quantity of the asset moves. This number is derived from the LP liquidity as follows.

Uniswap V2 follows equation $k=xy$ and $P=\frac{y}{x}$. When moving price from $P_0$ to $P_t$, $\delta = \frac{P_t}{P_0}$. The average price is dependent on the quantities that changed hands: $P_a = |\frac{\Delta_y}{\Delta_x}|$, $\Delta y = y_1 - y_0$, $\Delta x = x_1 - x_0$. Using the Uniswaps constant product equation, we can derive

\begin{equation}
\label{eq:deltay}
\Delta_y = \sqrt{kP_0\delta} - \sqrt{kP_0} = y_0(\sqrt{\delta - 1})
\end{equation}

And similarly $\Delta_x = x_0(\sqrt{\frac{1}{\delta}}-1)$. Plugging these into average price, it simplifies to

\begin{equation}
\label{eq:p_avg}
P_a = \sqrt{\delta}P_0 = \sqrt{P_0 \cdot P_t}
\end{equation}

Now the value of arbitrage in asset y is:

\begin{equation}
\label{eq:arb}
\text{arbitrage} = |P_b - \sqrt{P_0 \cdot P_t}|\Delta_x
\end{equation}

and LVR is calculated by subtracting fees from the arbitrage.

For Uniswap V3, one has to consider ticks that are being crossed. Steps to achieve this are as follows:
1. Turn the price from swap events into the same format as the Binance price, now a target price can be calculated similarly as with V2.
2. Use the target price to calculate the respective target tick range and target price in the Uniswap V3 format.
3. The start tick and that tick’s virtual liquidity are available from swap events. If the target tick and start tick are in the same range, we can directly use this to calculate the $\Delta x$ and $\Delta y$ required to move the price to the target price and also calculate the average price.
4. For crossing tick ranges, we have to first fetch liquidities for crossed ticks. That is available by taking the cumulative sum of all mint and burn events that add or remove liquidity to that the specific tick is in range. Then, we move from $P_0$ to the end of the tick range and get the quantities needed to achieve that. We repeat the process until we get to the target tick and there we move from the end of the range to the target price.

When we have access to all the quantity changes needed to achieve the swap from $P_{lp}$ to $P_t$ the average price can be calculated. Using the quantity changes and the Binance price we can derive the value of LVR. We then repeat this process for all our tested assets. The process must also be repeated for trying out different fee functions and function parameters, as changing the fee function changes the target prices.

We calculate LVR statistics for pools using their fee functions. The statistics are listed in [Table 1](#table-1).

### Table 1: Pool Statistics

| **Pool** | **% blocks with LVR** | **Total LVR / Total Fees** |
|----------------|-----------------------|----------------------------|
| USDC-WETH V2 | 6.6% | 22.3% |
| USDC-WETH V3 | 14.7% | 129.5% |
| WBTC-WETH V3 | 1.8% | 1603.7% |
| UNI-WETH V3 | 19.3% | 117.0% |

To provide the reader with some basic characteristics of LVR we explore the base-case fixed-fee data. For this, we have selected the WETH-USDC V3 5bps pool as it is the most liquid pool in Uniswap. Data used for determining these characteristics is from March 2023 to April 2024. First, we find that most of the LVR is generated by very few transactions as shown in [Figure 1](#figure-1). The bottom 90% of transactions that generate LVR are responsible for only 20% of the total LVR, or that top 10% of LVR-generating transactions result in 80% of LVR. This means being able to capture those few transactions is extremely important and implies that the dynamic fee would need to be very quick to respond to changing market conditions.

### <a name="figure-1"></a>Figure 1: LVR Distribution


We also look into how the LVR varies over time, as shown in [Figure 2](#figure-2). We find that LVR is far from constant and there is quite a lot of variation, even when calculating rolling averages as long as 30 days. The peak LVR at the start is almost four times more than the lows of LVR in the middle of August 2023. It seems that LVR spikes at the same times as the standard deviation of LVR, which is likely due to most of LVR being comprised of a few transactions, leading to a significant increase in standard deviation when those happen. This again highlights the importance of dynamic fees for capturing LVR, as a higher base fee could disincentivize other trading as well when LVR is low.

### <a name="figure-2"></a>Figure 2: LVR Volatility


### 3.3 Fee functions
We analyze which functions have the best trade-off between the decrease in the LVR and an increase in the collected fees as well as trade-off between arbitrageurs' profits and mispricing. Different parameters we used as input for dynamic fees are listed in [Table 2](#table-2).

### <a name="table-2"></a>Table 2: Different Inputs

| **Input** | **Source** |
|--------------------|-----------------------------|
| Momentum | Liquidity pool / CEX oracle |
| Volatility | Liquidity pool / CEX oracle |
| Volume | Liquidity pool / CEX oracle |
| Last interaction | Liquidity pool |
| Gas price | Chain |

### 3.4 First-access Auction
We analyze the viability of implementing first-access auctions on AMMs by analyzing how the value from a CEX DEX arbitrage is split between LPs and arbitrageurs in CoW AMM. We do this by examining how big the absolute price difference between CoW AMM and Binance is before a swap and what fee LPs get from each swap. By knowing the absolute price difference between CoW AMM and Binance we can estimate the amount of available CEX DEX arbitrage in the pool. By knowing CoW AMM’s states before and after a swap we can calculate how much fees LPs get from a swap. Based on this information we can interpret how the value in CEX DEX arbitrage is split between LPs and arbitrageurs in first access auctions.

To calculate LP fees of a swap we consider the following.
1. We know that the pool price before a swap is $P_0 = \frac{y}{x}$ and after a swap $P_1 = \frac{y + \Delta y}{x - \Delta x}$.
2. Based on this, we can determine the mid-price without fees as $P_{\text{mid}} = \sqrt{P_0 \cdot P_t}$.
3. On the other hand, we can also define the execution price to be $P_{\text{execution}} = \frac{\Delta y}{\Delta x}$.
4. To determine the fee percentage LPs collect from a swap, we calculate
$$ \text{LP fee} = \left|\frac{P_{\text{mid}}}{P_{\text{execution}}} - 1\right| \cdot \Delta x $$
5. To calculate the profit the arbitrageur collects, we calculate
$$ \text{Arbitrage} = \frac{P_{\text{mid}} - P_t}{P_t} \cdot \Delta x $$

## 4 Results
This section discusses the results of our analysis. The first subsection discusses the analysis of the Uniswap V2 USDC - ETH pool. The second through fourth subsections discuss Uniswap V3 USDC - ETH pool with a 5 bps fee, Uniswap V3 WBTC - ETH pool with a 30 bps fee, and Uniswap V3 UNI - ETH pool with a 30 bps fee on Uniswap V3. Finally, the last subsection discusses our analysis of first-access auctions.

### 4.1 Uniswap V2 USDC - WETH
For Uniswap V2 we study USDC - WETH as it is one of the most liquid pools. [Figure 3](#figure-3) visualizes the results of analyzing the trade-off between the decrease in LVR and an increase in average fees. As a benchmark, the blue line represents just increasing the constant fee and how that would affect LVR and average fee percent. Other lines are created by using a fixed base fee and adding a dynamic fee on top of it by multiplying the respective variable with a parametrized multiplier. Changes in LVR are measured against the baseline of the pool, so a fixed fee of 30bps.

We’ve split the plot into 4 separate subplots to highlight functions based on specific metrics and values: momentum-based metrics, volatility-based metrics, metrics with lower base fees, and other metrics. Overall, looking at the metrics we see that LVR can be significantly decreased by increasing the average fee, for a constant fee, increasing the fee from 30bps to 45bps would decrease the LVR by roughly 60\%. However, better trade-offs can also be made, both momentum and volatility metrics can achieve a decrease of over 80\% in LVR for the same average fee increase. On the other hand, if it is desired to keep the average fee fixed at 30bps, an over 20\% decrease in LVR could be achieved by changing the fee function to 20 bps + volatility or 20 bps + momentum. Overall, the best performance is achieved for volatility and momentum-based metrics. For other metrics, the gas fee-based fee function outperforms the fixed fee while the time since the last interaction does not.

Comparing dynamic fees combined with different base fees momentum based fee with 30 bps base fee is a better fee function than momentum based fee function with 20 bps base fee. The same also applies to volatility based functions with 5-minute volatility based function with 30 bps base fee being better fee function than 5-minute volatility based function with 20 bps base fee. Out of the volatility and momentum based functions the volatility based functions outperform the momentum based functions.

On top of the shown results, we tried multiple different lookback periods and different ways of turning the metric into a fee, for example making it have a negative effect on the total fee while the value is below some rolling mean. We didn’t find any significant improvements from changing how the fee is applied and just kept it simple. Shortening the lookback period, however, significantly improved the performance and this applied to both momentum-based and volatility-based fee functions. This can be seen for example by looking at the difference between 5 and 15-minute volatility. We believe this could be due to shorter lookback periods reacting to changing market conditions more quickly and therefore being able to increase the fee faster when market conditions change and LVR increase.

We also analyze how different fees affect the relation between the standard deviation of mispricing and total LVR. Milionis et al. [2023] state that increasing fees should decrease LVR with the cost of increased mispricing. In this context, mispricing means how far away AMM’s price is from the venue’s price where the price discovery happens after a block. Following their formula we find similar results, as visualized in [Figure 4](#figure-4). It should be noted that the change in arbitrage value is actually by definition the same as the change in LVR in [Figure 3](#figure-3).

Out of the studied fee functions all functions outperform a constant fee function except a function based on time since last interaction that is slightly worse than the constant fee function. From the studied functions momentum based function with 20 bps base fee has the best trade-off and 5-minute volatility based function with 20 bps base fee is the second best function with only slightly worse trade-off than the momentum based function.

Comparing momentum based functions with different base fees the momentum based function with 20 bps base fee outperforms the momentum based function with 30 bps base. The same is also applies to volatility based functions with 5-minutes volatility based function with 20 bps base fee outperforming 5-minutes volatility based function with 30 bps base fee. We also find that function based on 5-minute volatility outperforms function based on 15-minute volatility. Considering both trade-offs we find volatility based function to have better trade-off between increase in average fees and decrease in LVR while momentum based function has a better trade-off between mispricing and change in arbitrage value.

### 4.2 Uniswap V3 USDC - WETH
For Uniswap V3 we start our analysis by studying USDC - WETH pool with 5 bps fees. We test similar functions as with Uniswap V2 using the same visuals as above for results. As visualized in [Figure 5](#figure-5) we included tests with both a 5 bps base fee and a 2.5 bps base fee to highlight the possibilities of dynamic fees depending on the goal. For example, with a 2-minute momentum and 2.5 bps base fee, one could achieve a 20\% reduction in LVR without changing the average transaction fee in the pool. If increasing the pool fee is deemed as a good trade-off, a 60\% reduction in LVR could be achieved by increasing the average fee from 5 bps to 12.5 bps when using 2-minute momentum and a base fee of 5 bps. Comparably this with simply increasing the constant fee from 5 bps to 12.5 bps, a simple increase would only decrease LVR by 40\%. We again also tried out metrics based on gas fee, time since last interaction, and trading volume, but those either underperformed compared to constant fee or performance difference was negligible. For example, function based on average gas fee over past minute has very similar trade-off with constant fee function. Time since last interaction has slightly better trade-off than constant fee when the average fee is below 7.5 bps and slightly worse if the average fee is above 7.5 bps.

Comparing momentum based fee functions with different base fees we find momentum based function with 5 bps based fee to have a better trade-off than momentum based function with 2.5 bps base fee when the average fee is below roughly 7 bps. If the average fee of the function is above 7 bps the momentum based function with 2.5 bps base fee outperform the function with base fee of 5 bps. Comparing volatility based functions we find that volatility based function with 2.5 bps base fee is slightly better than volatility based function with 6 bps base fee if average fee is below 6 bps and the volatility based functions to perform similarly if the average fee is above 6 bps.

As visualized in [Figure 6](#figure-6) mispricing results for Uniswap V3 follow a similar pattern as previously where momentum and volatility-based functions are the best. Now, however, decreasing the base fee does not provide such a significant increase as it does on Uniswap V2. This could be due to the decrease being so small in absolute terms compared to the decrease in absolute terms on Uniswap V2. Constant fee function and fee functions based on time since last interaction and gas fee have the worst trade-off.

### 4.3 Uniswap V3 WBTC - WETH
We continue our analysis for Uniswap V3 with WBTC - WETH pool with 30 bps fees. [Figure 7](#figure-7) visualizes the results of analyzing the trade-off between the decrease in LVR and an increase in average fees. These numbers are relative to the constant 30 bps fees collected by the pool.

Examining the results, we see that gas-based fee functions have the best trade-off if the average fee is not increased above 45 bps after which the volatility-based function with a 25 bps base fee is the best. If the fees are below 45 bps 1-minute average gas fee-based function offers a significantly better trade-off than the constant fee and if the fees are between 45 bps and 60 bps 10-minute volatility-based fee functions offer a better trade-off than the constant fee. We also find that time since last interaction has always a worse trade-off than constant fee.

We find volatility-based fee functions to have the best results when using 10-minutes of rolling volatility and momentum-based fee functions to have the best results when using 5-minutes of rolling momentum. If the volatility was is decreased further or increased the trade-offs are worse. We also find that 10-minutes volatility based function have a better trade-off with 25 bps base fee than volatility based function with 30 bps base fee. On the other hand, momentum based function with 30 bps base fee has a better trade-off than a momentum based function with 25 bps base fee.

The results of analyzing how different fee functions affect the relationship between mispricing and total LVR are visualized in [Figure 8](#figure-8). All fee functions perform very similarly to the constant fee function or slightly worse, except for the volatility-based fee functions which are better in certain cases. Out of the volatility-based functions the function with 25 bps base fee has a better trade-off than 30 bps base fee. On the other hand, from the momentum based functions the function with 25 bps base fee has a worse trade-off than 30 bps base fee. Functions based on momentum, gas price and time since last interaction had worse trade-off at almost all points compared to constant fee function.

### 4.4 Uniswap V3 UNI - WETH
We continue our analysis for Uniswap V3 with UNI - WETH pool with 30 bps fees. [Figure 9](#figure-9) visualizes the results of analyzing the trade-off between the decrease in LVR and an increase in average fees. These numbers are relative to the constant 30 bps fees collected by the pool.

Examining the results, we see that the constant fee function has the best trade-off in all cases. Volatility and momentum-based functions were the best with rolling variables of 10 minutes and 5 minutes, respectively, yet they had worse trade-offs than the constant function. Momentum based function with 25 bps base fee is worse than momentum based function with 30 bps base fee. Similarly, volatility based function with 25 bps base fee is worse than volatility based function with 30 bps base fee. Gas fee and time since last block based functions are the best out of the dynamic fee functions as they are only slightly worse than constant fee function. Comparing volatility and momentum based functions momentum based functions are better than volatility based functions.

The results of analyzing how different fee functions affect the relationship between mispricing and total LVR are visualized in [Figure 10](#figure-10). All fee functions perform very similarly. This time constant fee is the best out of the functions all the time, except in certain cases where the volatility-based function with a 25 bps base fee is marginally better. Between momentum based functions the momentum based function with 30 bps base fee performs better than the momentum based function with 25 bps base fee. Comparing volatility and momentum based functions volatility based functions are better than momentum based functions.

### 4.5 First-Access Auctions
We analyze price differences between CoW AMM and Binance as well as the fees earned LPs get from each swap. The results of the analysis are visualized in [Figure 11](#figure-11).

There is no clear proportionality between LP fees and price differences of swaps. This suggests that in practice an increase in price difference does not directly mean an increase in LP fees, although, in theory, this should be the case. However, if the price difference between CoW AMM and Binance is above 1\% before the swap there is a clear proportionality between the increase in LP fees and the price difference. This indicates that first-access auction mechanism becomes more efficient once price discrepancies reach a significant level. The emergence of this relationship above 1\% suggests that auction participants may have implicit cost considerations that make smaller arbitrage opportunities less attractive, leading to lower auction efficiency for small price differences.

This indicates that first-access auctions work, to an extend, if the price difference is big enough but if the price difference is too small they are not efficient in redistributing value from arbitrages back to LPs. The properties of trendline suggest that for an increase of 100 bps in price difference the LPs will get, on average, 26 bps more fees. However, it is important to note that CoW AMM pools has failed to gain significant TVL which means potential profits for auction participants are limited. The thin liquidity likely increase relative transaction costs and reduce the absolute size of profitable opportunities, potentially discouraging sophisticated market participants to engage in the auction.

## 5 Findings and discussion
We analyze a variety of different dynamic fee functions for different AMM pools. We find that volatility and momentum-based dynamic fee functions have the best results for both Uniswap V2 and Uniswap V3 USDC - WETH AMM pools with significantly better trade-offs between the increase in fees and decrease in LVR when compared to the constant fee function. In addition, we find either gas-based function or a volatility-based function with a decreased base fee to have a significantly better trade-off for the Uniswap V3 WBTC - WETH pool. On the other hand, when analyzing functions for the UNI - WETH Uniswap V3 pool we do not find functions that perform better than the constant fee function.

We also find that optimal base fee for dynamic fee based functions differ on case-by-case basis. For Uniswap V2 USDC - WETH pool functions with base fee of 30 bps outperform functions with base fee of 20 bps. However, for Uniswap WETH - WBTC pool we find a volatility based function with 25 bps base fee to outperform a volatility based function with 30 bps base fee. For Uniswap V3 USDC - WETH pool we find mixed results with 2.5 bps base fee being better than 5 bps at times.

In terms of trade-off between mispricing and change in LVR we find volatility and momentum-based dynamic fee functions to have the best results for both Uniswap V2 and Uniswap V3 USDC - WETH pools with significantly better trade-offs than constant fee function. For Uniswap V3 WBTC - WETH and Uniswap V3 WETH - UNI pools we find either constant fee or volatility based fee to have the best trade-off. This indicates that while for certain pools a single function can optimize both trade-offs between increase in average fees and decrease in LVR as well as mispricing and LVR, like in the case of Uniswap V2 and Uniswap V3 USDC - WETH pools, other pools do not seem to have a function that optimizes all aspects of the pool.

We also analyze CoW AMM and the efficiency of first-access auctions on the USDC - WETH pool. We do not find first-access auctions to be particularly efficient ways of redistributing LVR to LPs on a high level. However, if the price difference between CoW AMM and Binance is above 1\% before the swap there is a clear proportionality between the increase in LP fees and price difference. The emergence of this relationship above 1\% suggests that auction participants may have implicit cost considerations that make smaller arbitrage opportunities less attractive, leading to lower auction efficiency for small price differences. This indicates that first-access auctions work if the price difference is big enough but if the price difference is small first-access auctions are not efficient in redistributing LVR back to LPs.

### 5.1 Implications of the results
Our analysis reveals that dynamic fee functions can offer significantly better trade-offs between the increase in fees and the decrease in LVR compared to constant fee functions to certain pools. The difference in performance between Uniswap V2 and Uniswap V3 USDC - WETH pools and other pools could be attributed to a few things, the most meaningful being liquidity, Uniswap V3 USDC - WETH processes currently about \$1B in volume daily while Uniswap V3 WBTC - WETH only does \$100M. A large portion of this difference can be attributed to the higher base fee in most liquid WBTC pools, but the more liquid Uniswap V3 USDC - WETH pool pricing could be considered more efficient. With more efficient pricing, the market reacts to changing information quickly, which could mean that dynamic fees are more efficient. Another less likely reason is that centralized venues mostly quote prices in USD-denominated stablecoins, which could mean that arbitrage, when considering the impact of market friction, is still happening less in non-USD pairs, but this is not as likely. Based on these results we recommend that AMMs trying to minimize LVR with fee functions would first approach the problem by analyzing whether the pool gets significant benefits from implementing a dynamic fee, after which they either implement an optimal constant fee level or dynamic fee for the pool. More research should also be done to better understand what is the reason for such variations in pool performance, as with a better understanding a generic fee function could be developed.

Our results suggest that LVR cannot be fully eliminated while keeping the average fees at sensible levels. This means that for LVR to be fully eliminated changes in dynamic AMM need to be combined with other solutions to the problem or another stand-alone solution should be implemented. For example, dynamic AMM fees could be combined with MEV-Burn to partially redistribute the value arising from CEX-DEX arbitrage to both AMM LPs and the Ethereum network. Alternatively, there could be fundamental changes either in the Ethereum mainnet or the way AMMs function. Decreasing the block time in the mainnet could significantly reduce CEX DEX arbitrage without other changes as discussed by y Milionis et al. [2023] and Fritsch and Canidio [2024]. Alternatively, AMMs could move to L2s or other sidechain solutions supporting more continuous trade settlements.

There is not an universal optimal dynamic fee function. Optimal fee functions and their parameters differ pool by pool. The optimal fee functions and their parameters also differ for the same pool depending on which characteristics of the pool are optimized for. This implies that dynamic AMM trying to find the most optimal function has to analyze multiple different variables, however, functions based on short-term volatility and momentum seem to outperform the constant fee function the most when there is a chance for significant outperformance. However, even when only trying to optimize volatility and momentum based fee functions there are countless variable combinations that need to be consider based on optimal base fee and lookback window for volatility or momentum.

Empirical analysis on first-access auctions does not show first-access auctions to be efficient when analyzing the whole dataset. However, if only the subset of blocks where mispricing is above 1\% is considered, first-access auctions seem to increase the fees LPs get. We believe the poor performance when the mispricing is below 1\% can be, at least partially, attributed to there being costs associated with arbitraging the pool making smaller arbitrage opportunities less attractive leading to lower auction efficiency for small price differences. Based on this we believe that first access auctions could be a highly efficient way of redistributing LVR to LPs if adopted by big AMMs with some kind of minimum reserve fee, despite poor results when considering the whole dataset.

## 6 Summary
The purpose of this study is to analyze the potential redistribution of Loss-Versus-Rebalancing (LVR) to Automated Market Maker (AMM) Liquidity Providers (LPs) with dynamic fees and first-access auctions. This study analyzes a dataset between blocks 17162286 produced on April 30th, 2023, and 18473542 produced on October 31st, 2023 for following Ethereum mainnet Uniswap pools: Uniswap V2 USDC - WETH pool, Uniswap V3 USDC - WETH pool with a 0.05\% fee, Uniswap V3 WBTC - WETH pool with a 0.3\% fee, and Uniswap V3 UNI - WETH pool with a 0.3\% fee. We also analyze a dataset between blocks 19675204 produced on April 17th, 2024, and 20167248 produced on June 25th, 2024 for the USDC - WETH pool on CoW AMM. In addition to the AMM data, we acquire 1-second candle data from Binance for the same period and match it with the AMM data.

We analyze different dynamic fee alternatives for different pools and find short-term volatility and momentum-based fee functions to have significantly better trade-offs between the decrease in LVR and an increase in average fees compared to constant fees for Uniswap V2 and V3 USDC - WETH pools. In addition, we find Uniswap V3 UNI - WETH pool to have significantly better trade-off in certain cases with gas-based fees and in certain cases with volatility-based fees. On the other hand, we do not find a dynamic fee that would have a better trade-off than the constant fee function for the Uniswap V3 UNI - WETH pool. We also find dynamic fees to have only a slightly better trade-off between LVR and mispricing compared to constant fees. The optimal parameters for fee functions differ pool by pool thus optimal dynamic fee functions would need to be crafted on a pool-by-pool basis.

We also analyze the effectiveness of first-access auctions as a mechanism for redistributing CEX DEX arbitrage to LPs. Our findings suggest that first access auctions are efficient in value redistribution if the price difference exceeds 1\%. However, if a price difference is smaller, the auction mechanisms is significantly less efficient in redistributing value from arbitrages to LPs. We believe this to be, at least partially, due to there being costs assocaited with arbitraging making smaller arbitrage opportunities less attractive. This asymmetric effectiveness indicates that effectiveness of first-access auctions is dependent on the magnitude of mispricing. Based on these observations, we believe first-access auctions could be an efficient way to redistribute LVR to LPs if they were implemented with some kind of reserve fee. This would ensure minimum compensation for LPs while having auction's effectiveness in larger price deviations.

In conclusion, this study analyzes the potential of using dynamic fees and first-access auctinos in redistribution of LVR to AMM LPs. The findings suggest that dynamic fees are significantly better than static fees for only certain pools and that first-access auctions show promising results only when mispricing of the pool is big enough.

Future research could aim to find a generalized dynamic fee that could be applied to all pools with limited configuration, as that would allow it to be used in the long tail of assets. Hybrid solutions that merge dynamic fees and first-access auction could also be an interesting approach as it would allow first-access auction to capture large mispricings while dynamic fees help with smaller discrepancies. In general, first-access auctions should be researched further when there is more data available, as we believe it to be a great solution for more liquid pairs. More research should also be done on what would be the optimal block time to optimize for LVR and the downsides that it would bring.

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## Appendix

### <a name="figure-3"></a>Figure 3: WETH - USDC V2 LVR and average fee change for different fee functions


### <a name="figure-4"></a>Figure 4: WETH - USDC V2 change in arbitrage and mispricing for different fee functions


### <a name="figure-5"></a>Figure 5: WETH - USDC V3 LVR and average fee change for different fee functions


### <a name="figure-6"></a>Figure 6: WETH - USDC V3 change in arbitrage and mispricing for different fee functions


### <a name="figure-7"></a>Figure 7: WETH - WBTC V3 LVR and average fee change for different fee functions


### <a name="figure-8"></a>Figure 8: WETH - WBTC V3 change in arbitrage and mispricing for different fee functions


### <a name="figure-9"></a>Figure 9: WETH - UNI V3 LVR and average fee change for different fee functions


### <a name="figure-10"></a>Figure 10: WETH - UNI V3 change in arbitrage and mispricing for different fee functions


### <a name="figure-11"></a>Figure 11: WETH - USDC CoW AMM first access auctions