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The Cost of Permissionless Liquidity Provision in Automated Market Makers

Julian Ma, Davide Crapis

TL;DR

This paper analyzes how pro-rata fee allocation in permissionless AMMs induces competition among passive LPs, generating excess liquidity and higher adverse selection costs. It introduces a $N$-player symmetric game with two liquidity-trader types (elastic and inelastic) and arbitrageurs, derives equilibrium liquidity, and characterizes the resulting price of anarchy metrics. The main findings show the LP-profit PoA scales as $O(N)$, but the full game including elastic-trader welfare yields a constant PoA $O(1)$ due to complementarity between trader types. It also examines Loss-Versus-Rebalancing rebates, finding deposits and elastic-trader utility rise with rebates while LP profits can be unchanged, and that reducing arbitrage intensity per unit of liquidity may not proportionally cut total arbitrage, with MEV-centralization concerns potentially persisting at the protocol layer.

Abstract

Automated market makers (AMMs) allocate fee revenue \textit{proportional} to the amount of liquidity investors deposit. In this paper, we study the economic consequences of the competition between passive liquidity providers (LPs) caused by this allocation rule. We employ a game-theoretic model in which $N$ strategic agents optimally provide liquidity and two types of liquidity traders trade. In this setting, we find that competition drives LPs to provide excess liquidity. Excess liquidity is costly as more capital is exposed to adverse selection costs. One of our main results is that the price of anarchy, defined over the liquidity provider performance, is $O(N)$, implying that the welfare loss scales linearly with the number of liquidity providers. This inefficient capital allocation is masked when considering the welfare of elastic liquidity traders as the total price of anarchy is $O(1)$. Since this result is driven by elastic liquidity traders benefiting from the liquidity provided because of inelastic liquidity traders, we show that different types of liquidity traders complement each other. Finally, we show that AMM designs that reduce the arbitrage intensity per unit of liquidity do increase utility for liquidity traders but importantly not for LPs nor do they necessarily decrease total arbitrage volume.

The Cost of Permissionless Liquidity Provision in Automated Market Makers

TL;DR

This paper analyzes how pro-rata fee allocation in permissionless AMMs induces competition among passive LPs, generating excess liquidity and higher adverse selection costs. It introduces a -player symmetric game with two liquidity-trader types (elastic and inelastic) and arbitrageurs, derives equilibrium liquidity, and characterizes the resulting price of anarchy metrics. The main findings show the LP-profit PoA scales as , but the full game including elastic-trader welfare yields a constant PoA due to complementarity between trader types. It also examines Loss-Versus-Rebalancing rebates, finding deposits and elastic-trader utility rise with rebates while LP profits can be unchanged, and that reducing arbitrage intensity per unit of liquidity may not proportionally cut total arbitrage, with MEV-centralization concerns potentially persisting at the protocol layer.

Abstract

Automated market makers (AMMs) allocate fee revenue \textit{proportional} to the amount of liquidity investors deposit. In this paper, we study the economic consequences of the competition between passive liquidity providers (LPs) caused by this allocation rule. We employ a game-theoretic model in which strategic agents optimally provide liquidity and two types of liquidity traders trade. In this setting, we find that competition drives LPs to provide excess liquidity. Excess liquidity is costly as more capital is exposed to adverse selection costs. One of our main results is that the price of anarchy, defined over the liquidity provider performance, is , implying that the welfare loss scales linearly with the number of liquidity providers. This inefficient capital allocation is masked when considering the welfare of elastic liquidity traders as the total price of anarchy is . Since this result is driven by elastic liquidity traders benefiting from the liquidity provided because of inelastic liquidity traders, we show that different types of liquidity traders complement each other. Finally, we show that AMM designs that reduce the arbitrage intensity per unit of liquidity do increase utility for liquidity traders but importantly not for LPs nor do they necessarily decrease total arbitrage volume.
Paper Structure (18 sections, 7 theorems, 31 equations, 1 figure)

This paper contains 18 sections, 7 theorems, 31 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Literature Review
  3. Model Setup
  4. Sequential Game
  5. LP Deposits and Withdrawals
  6. Arbitrageurs
  7. Liquidity Traders
  8. Representative Elastic Liquidity Trader (ET)
  9. Representative Inelastic Liquidity Trader (IT)
  10. Reverse Trade Arbitrage Opportunity
  11. Main Results
  12. Loss-Versus-Rebalancing Rebating
  13. Appendix
  14. Proof of Lemma \ref{['demand_function_lemma']}
  15. Proof of Lemma \ref{['total_deposits_lemma']}
  16. ...and 3 more sections

Key Result

lemma thmcounterlemma

The expected aggregate demand of liquidity traders over $T$ periods in terms of the numéraire is where

Figures (1)

  • Figure 1: Example of the $T$-period game in which liquidity providers deposit in the first period and withdraw in the last and in which arbitrageurs, representative elastic and inelastic liquidity traders trade in every period.

Theorems & Definitions (7)

  • lemma thmcounterlemma
  • lemma thmcounterlemma
  • theorem thmcountertheorem
  • theorem thmcountertheorem
  • corollary thmcountercorollary
  • corollary thmcountercorollary
  • lemma thmcounterlemma