The Cost of Permissionless Liquidity Provision in Automated Market Makers

Julian Ma and Davide Crapis

Robust Incentives Group, Ethereum Foundation

MARBLE

July 9th, 2024

Introduction: Automated Market Maker (AMM)

• Currently about 9 billion USD provided as liquidity in AMMs.
• 1.5 million USD in Uniswap fees in the last 24 hours.
• Importantly: Fees are distributed pro-rata to liquidity providers.

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AMM

Traders

Deposit capital

Trade

Pay Fees

Introduction: Limit Order Book (LOB)

• Limit Order Book is the most popular trading method in traditional finance.
• Most LOB markets use price-time priority to distribute fees.

Introduction: Comparison

• From a trader perspective, AMM and LOBs are similar.
• From a liquidity provider perspective it is completely different, by design!

Introduction: Comparison

• From a trader perspective, AMM and LOBs are similar.
• From a liquidity provider perspective it is completely different, by design!

• What are the consequences for passive liquidity providers in AMMs?
• What are the consequences for liquidity traders using AMMs?
• How do these consequences affect future AMM designs?

What are the economic consequences of the pro-rata allocation of trading fees amongst liquidity providers?

Related Literature

Endogenous Liquidity Supply

• Multiple papers endogenize the supplied liquidity analyzing various factors that influence trading costs: Capponi and Jia (2021), Hasbrouck et al. (2022) and Hasbrouck et al. (2023).
• Most closely related: Capponi et al. (2023).

Loss-Versus-Rebalancing

• Milionis et al. (2021), and Milionis et al. (2023), decompose liquidity provider costs into 1) arbitrage losses and 2) market risk. We use their arbitrage losses, known as Loss-Versus-Rebalancing (LVR) as expected LP costs.
• New AMM designs aim to minimize LVR. We contribute to this growing literature: Adams et al. (2024), Canidio and Fritsch (2023), Josojo (2022).

Concave pro-rata

• Johnson et al. (2023) introduce a family of games and have similar price of anarchy results.

Contributions

Endogenous Liquidity Supply

• There is a welfare loss amongst liquidity providers that scales linearly with the number of LPs.
• We show how inelastic and elastic liquidity traders complement each other.
• The amount of liquidity is not a good indicator of the welfare of all market participants.

Loss-Versus-Rebalancing

• LVR minimization does not lead to a welfare increase for risk-neutral LPs.
• LVR minimization does not necessarily reduce protocol concerns around MEV extraction.

Loss-Versus-Rebalancing (LVR)

• Prices on AMMs become stale between blocks while prices still move on centralized exchanges. This creates arbitrage opportunities.
• The LVR framework provides a closed-form solution for the amount of arbitrage losses passive liquidity providers face.

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• This work does not focus on adverse selection but on the distribution of trading fees.
• Therefore, we build on the LVR framework.
• Contemporaneously, Adams et al. (2024) have a similar model setup.

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• Building on LVR allows us to connect to the future of AMM design, more on this later!

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Model

• First, N liquidity providers deposit risky assets and numéraire into an AMM.
• Maximize Profit = Liquidity Fraction * Fee Revenue - LVR - Opportunity Costs.

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• Then for all blocks in a time interval, elastic and inelastic liquidity traders arrive.
• Fixed number of inelastic liquidity traders per block.
• Elastic liquidity traders receive private valuations and maximize their payoffs.

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• The solution concept is subgame-perfect Nash equilibrium reached by backward induction.
• The number of LPs, size of inelastic demand, and private valuation shocks are common knowledge.

Model

Results: Liquidity Provision Subgame

Results: Full game

• Elastic traders benefit from the additional liquidity.
• Even without endogenizing inelastic trader utility, we have the following:

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• Inelastic demand subsidizes elastic demand!

LVR Rebating: Motivation

• Reducing Loss-Versus-Rebalancing is a big goal of AMM design.
• Stated goals:
• Increase the profitability of liquidity providers.
• Decrease the centralizing pressures that LVR puts on the Ethereum protocol.

LVR Rebating: Model

• Introduce a “rebate parameter” that resembles what fraction of LVR is rebated to liquidity providers.

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Research Questions:

• How are liquidity provider revenues affected?
• How is the total amount of MEV affected?

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LVR Rebating: Results

• LP profits are not affected by LVR rebates because supplied liquidity increases as LVR decreases.

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• Therefore, there is also more MEV. Not clear that MEV minimization at the application layer decreases the problems MEV causes at the protocol layer.

Summary

• Pro-rata fee allocation means LPs deposit capital inefficiently, leading to a welfare loss that scales linearly in the amount of LPs.

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• LVR minimization does not affect risk-neutral LP profits, nor does it necessarily improve the MEV problems for the Ethereum protocol.

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Thank you!

Julian Ma

Research Scientist, Ethereum Foundation

julian.ma@ethereum.org

@_julianma

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References

• Adams, A., Moallemi, C.C., Reynolds, S., Robisnon, D.: am-amm: An auction-�managed automated market maker (2024), URL: https://arxiv.org/abs/2403.03367.
• Capponi, A., Jia, R.: The adoption of blockchain-based decentralized exchanges (2021). https://doi.org/10.2139/ssrn.3805095, SSRN Scholarly Paper 3805095.
• Capponi, A., Jia, R., Zhu, B.: The paradox of just-in-time liquidity in decentralized exchanges: More providers can lead to less liquidity (February 12 2024), available at SSRN: https://ssrn.com/abstract=4648055.

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References

• Hasbrouck, J., Rivera, T., Saleh, F.: The need for fees at a dex: How increases�in fees can increase dex trading volume (August 17 2022), available at SSRN:�https://ssrn.com/abstract=4192925 or http://dx.doi.org/10.2139/ssrn.4192925.
• Hasbrouck, J., Rivera, T., Saleh, F.: An economic model of a decentralized�exchange with concentrated liquidity (August 2 2023), available at SSRN:�https://ssrn.com/abstract=4529513.
• Johnson, N.A.G., Diamandis, T., Evans, A., de Valence, H., Angeris, G.: Concave�pro-rata games (2023), URL: https://arxiv.org/pdf/2302.02126v1.

References

• Josojo: Mev capturing amm (mcamm) (August 2022), uRL:�https://ethresear.ch/t/mev-capturing-amm-mcamm/13336/4.
• Milionis, J., Moallemi, C.C., Roughgarden, T.: Automated market making and�arbitrage profits in the presence of fees. In: Financial Cryptography and Data Security (FC 2024) (2024).
• Milionis, J., Moallemi, C.C., Roughgarden, T., Zhang, A.L.: Automated market�making and loss-versus-rebalancing. In: Proceedings of the 2022 ACM Computer and Communications Security (CCS) Workshop on Decentralized Finance and Security (ACM CCS DeFi 2022) (2022), extended abstract.