Game Theoretic Liquidity Provisioning in Concentrated Liquidity Market Makers

TL;DR

A game theoretic model is formulated and analyzed to study the incentives of LPs in CLMMs and shows that the Nash equilibrium of this simplified game follows a waterfilling strategy, in which low-budget LPs use up their full budget, but rich LPs do not.

Abstract

Automated marker makers (AMMs) are a class of decentralized exchanges that enable the automated trading of digital assets. They accept deposits of digital tokens from liquidity providers (LPs); tokens can be used by traders to execute trades, which generate fees for the investing LPs. The distinguishing feature of AMMs is that trade prices are determined algorithmically, unlike classical limit order books. Concentrated liquidity market makers (CLMMs) are a major class of AMMs that offer liquidity providers flexibility to decide not only \emph{how much} liquidity to provide, but \emph{in what ranges of prices} they want the liquidity to be used. This flexibility can complicate strategic planning, since fee rewards are shared among LPs. We formulate and analyze a game theoretic model to study the incentives of LPs in CLMMs. Our main results show that while our original formulation admits multiple Nash equilibria and has complexity quadratic in the number of price ticks in the contract, it can be reduced to a game with a unique Nash equilibrium whose complexity is only linear. We further show that the Nash equilibrium of this simplified game follows a waterfilling strategy, in which low-budget LPs use up their full budget, but rich LPs do not. Finally, by fitting our game model to real-world CLMMs, we observe that in liquidity pools with risky assets, LPs adopt investment strategies far from the Nash equilibrium. Under price uncertainty, they generally invest in fewer and wider price ranges than our analysis suggests, with lower-frequency liquidity updates. We show that across several pools, by updating their strategy to more closely match the Nash equilibrium of our game, LPs can improve their median daily returns by \$116, which corresponds to an increase of 0.009\% in median daily return on investment.

Figures (15)

• Figure 1: Example AMM bonding curve and trade.
• Figure 2: Comparison between Legacy AMM and CLMM in their bonding curves and liquidity distributions.
• Figure 3: Trades in CLMM.
• Figure 4: $N=2$ LPs sharing fee across $M=2$ atomic intervals. Interval $m \in \{1,2\}$ has a fee reward of $f_m$. Both LPs invested in 3 liquidity positions, one for each of the general price ranges $(a,b)$, $(b,c)$, and $(a,c)$. Recall that $L_{n, (a, b)}$ denotes LP $n$'s liquidity in interval $(a,b)$ (which can span multiple ticks), whereas $K_{n, m}$ denotes all of $n$'s active liquidity in the atomic range $(t_{m-1},t_m)$.
• Figure 5: The waterfilling pattern of LPs' strategies at Nash equilibrium among $N=4$ LPs across $M=2$ atomic ranges. Each LP's budget is represented by the height of their corresponding bar.

Theorems & Definitions (23)

• Definition 2.1
• Definition 3.1: Nash equilibrium
• Proposition 3.2
• Theorem 3.3: Uniqueness
• Definition 3.4: Twin games
• Theorem 3.5: Original game and atomic game are twin games
• Proposition 3.6: Waterfilling
• Proposition 3.7
• Proposition 3.8: Positive liquidity at equilibrium
• Proposition 3.9