Equilibrium Reward for Liquidity Providers in Automated Market Makers

TL;DR

The paper studies the design problem of an AMM by modeling a leader-follower stochastic game where a venue offers a contract to a strategic LP to maximize order flow. By solving the follower’s exponential-utility optimization and the leader’s contract optimization under risk-neutral and risk-averse settings, it derives approximate closed-form equilibrium policies and interprets the reward structure. A key finding is that higher pool depth only attracts noise trading, enabling more liquidity provision when the external venue fees are favorable, and the equilibrium contract depends on the external price, pool reference price, and reserves. Numerical experiments calibrated to ETH–USDC data illustrate the interplay among fees, depth, and noise trading and demonstrate positive profits for both players under the equilibrium contract. The framework offers actionable guidance for AMM design to stimulate activity while preserving LP profitability.

Abstract

We find the equilibrium contract that an automated market maker (AMM) offers to their strategic liquidity providers (LPs) in order to maximize the order flow that gets processed by the venue. Our model is formulated as a leader-follower stochastic game, where the venue is the leader and a representative LP is the follower. We derive approximate closed-form equilibrium solutions to the stochastic game and analyze the reward structure. Our findings suggest that under the equilibrium contract, LPs have incentives to add liquidity to the pool only when higher liquidity on average attracts more noise trading. The equilibrium contract depends on the external price, the pool reference price, and the pool reserves. Our framework offers insights into AMM design for maximizing order flow while ensuring LP profitability.

Figures (10)

• Figure 1: Histogram of price differences between Binance and Uniswap V2 for ETH-USDC between 1 January 2022 and 30 April 2022. The red shaded area represents the region in which $a_1\pm\,a_3 (S_t-Z_t)$ becomes negative. We take $a_2=0$ for simplicity so that the violation boundary is fixed and does not depend on the number of ETH units in the pool.
• Figure 2: Histogram of simulated price differences between Binance and Uniswap V2 for ETH-USDC using 1,000 simulations and the model parameters at the start of the section. The red shaded area represents the region in which $a_1\pm\,a_3 (S_t-Z_t)$ becomes negative. We take $a_2=0$ for simplicity so that the violation boundary is fixed and does not depend on the number of ETH units in the pool.
• Figure 3: Sample path (with 90% bands across time) for the inventory of ETH in the pool (left panel), and the instantaneous exchange rate in the pool and outside (right panel).
• Figure 4: Sample path (with 90% bands across time) for the speed at which the LP adds/removes liquidity from the pool (left panel), and the cumulative change in liquidity provided (right panels), given by $\int_0^t \nu_s\,\mathrm{d} s$.
• Figure 5: Sample path of the optimal strategy of the LP as model parameters $\mathfrak{a}$ and $\sigma$ change.

Theorems & Definitions (14)

• Theorem 1
• Theorem 2
• Remark 1
• Proposition 1
• Proposition 2
• Remark 2
• Lemma 1
• proof
• Lemma 2
• proof