# AMM Liquidity Provision with Adverse Selection

This model adapts the classical Kyle framework to *decentralized exchanges (DEXs)*. Unlike traditional continuous markets where orders execute instantly if liquidity is available, price discovery in blockchains is a **"step process"** where information is revealed periodically—specifically, at the frequency of block creation.

## The Trading Game Scenario
In this decentralized market, trading occurs in discrete intervals (slots) defined by the block time $\Delta t$.

* **Stage Zero**: Multiple traders decide whether to pay a fixed cost $C$ to gather private information.
* **Stage One**: LPs establish the liquidity profile $\mathcal{L}$ (e.g., bonding curve) by balancing fee revenue from uninformed demand against expected losses to informed traders.
* **Stage Two**: A new blockchain slot begins. Informed traders obtain private signals regarding the future liquidation value $s$. They then determine their optimal volumes $x$ and bid priority fees $\varphi$ in a *Priority Auction* to secure queue position in the block.

## The Informed Trader’s Problem
The informed trader seeks to maximize expected profits by optimizing two interdependent variables: trade volume $x$ and priority fee $\varphi$.

### Optimization Functional
A risk-neutral trader $i$ solves:

$$
\sup_{x_i, \varphi_i} \mathbb{E}_i [J_i^{\mathcal{L}}(s, x_i, \varphi_i)]
$$

Where the profit $J_i^{\mathcal{L}}$ is defined as:

$$
J_i^{\mathcal{L}}(s, x) =
\begin{cases}
s \cdot x - y(x) - \varphi_i - C & \text{if win the auction} \\
-C & \text{if lose the auction}
\end{cases}
$$

#### Optimal Quote for Informed Trader
Given exactly one informed trader wins the auction, we analyze his optimal quote strategy by taking the derivative of $J_i^{\mathcal{L}}$ with respect to volume $x$:

$$
0 = \frac{dJ_i^{\mathcal{L}}(s,x)}{dx} = s - p^+(x^+) \pmb{1}_{x>0} - p^-(x^-) \pmb{1}_{x<0}
$$

Here, $p^+$ and $p^-$ are the marginal prices reported by the AMM determined by the profile $\mathcal{L}$. The optimal quote is given by:

$$
x^* = \begin{cases}
(p^+)^{-1}(s) &\text{if } p^+(0) < s \\
0 &\text{if } p^+(0) \ge s \ge p^-(0) \\
-(p^-)^{-1}(s) &\text{if } p^-(0) > s
\end{cases}
$$

Beyond the no-arbitrage regime, the informed trader trades against the AMM until the marginal price incorporates their private "alpha" $s$.

Remark. In reality, common choice for $s$ is the CEX price or the aggreated price (ChainLink Oracle) .

## The LP’s Problem
The LP chooses the liquidity profile $\mathcal{L}$ (e.g., Uniswap v3 parameters) to maximize a utility function $J_{LP}$:

$$
J_{LP}(\mathcal{L}) = \mathbb{E}\left[U(p_0, p_1, f) \right]
$$

where $U$ is the LP's utility and $f$ is the fee earned from uninformed/noise traders.

A common choice for $U$ is the impermanent loss plus fee $IL(\mathcal{L}, p) + f$ or its risk-adjusted counterpart.

### AMM Dynamics
The AMM's marginal price at the end of the block follows the rule:

$$
p = p(x^*) + \sigma \sqrt{\Delta t} \epsilon
$$

* $p(x^*)$: The price movement caused by the winner of the informed trading auction.
* $\sigma \sqrt{\Delta t} \epsilon$: Models price movement caused by noise/uninformed trading, where $\epsilon \sim \mathcal{N}(0,1)$.

and the expected earned fee $F$ is a function of $\sigma$ and $\mathcal{L}$:

$$
\mathbb{E}[f] = F(\mathcal{L}, \sigma)
$$

## Uniswap v4
Uniswap v4 introduces "hooks," allowing LPs to implement custom pricing rules and internalized priority auctions for informed traders.

### Uniswap v3 Core Pricing Mechanism (Base for v4)
The profile $\mathcal{L}$ depends on $(\ell, \gamma, p_u, p_l)$:

* $\ell$: Liquidity depth (affects price impact).
* $\gamma$: Fee tier.
* $p_u$: Upper price bound for the liquidity provision.
* $p_u, p_l$: Price boundaries.

#### Inventory & Impermanent Loss
Under the Uniswap framework, the final inventory updates and resulting impermanent loss ($IL$) are given by:

$$
\Delta x = x(p_0) - x(p_1), \quad \Delta y = y(p_0) - y(p_1)
$$

$$
IL(\mathcal{L}, p_0, p_1) = p_1 \Delta x + \Delta y
$$

with

$$
x(p) = \ell (\frac{1}{\sqrt{p \wedge p_l}} - \frac{1}{\sqrt{p_u}}), \quad y(p) = \ell (\sqrt{p \wedge p_u} - \sqrt{p_l})
$$

### LP's Response
To solve the LP's optimization problem, the following modeling challenges must be addressed:

* **Estimating Terminal Price $p_1$**: Since the LP is agnostic to the private signal $s$, they must treat the informed trade volume $x^*$ as a random variable. The distribution of $p_1$ is modeled by combining the expected price impact of the informed winner—$p(x^*)$—with the volatility of noise trading over the block time, $\sigma \sqrt{\Delta t} \epsilon$.
* **Modeling the Fee Function $F(\ell, \gamma, \sigma)$**: Following standard microstructure models, fee revenue is driven by uninformed liquidity demand that is elastic to the price of liquidity.
* *$\ell$ (Liquidity)*: $F$ is increasing but sublinear in $\ell$. Increased depth $\ell$ reduces price impact, attracting more uninformed volume, but with diminishing returns as the marginal benefit of additional depth decreases.
* *$\gamma$ (Fee Tier)*: $F$ is decreasing in $\gamma$. Higher fee rates increase revenue per trade but lower the aggregate demand due to price elasticity.
* *$\sigma$ (Volatility)*: $F$ is increasing in $\sigma$. Higher volatility typically correlates with higher trading activity from noise traders rebalancing or responding to external price signals.

#### Optimization Constraint
The LP must ensure that $F(\ell, \gamma, \sigma) \ge \mathbb{E}[IL(\mathcal{L}, p_0, p_1)]$. If the variance of $s$ increases (due to longer $\Delta t$) faster than fee revenue accumulates, the LP faces a "liquidity freeze" where the optimal $\ell$ drops to zero.

### Main Research Question
* **LP Best Response in Uniswap v4**: How can an LP utilize v4 hooks to dynamically adjust the liquidity profile $\mathcal{L}$? Specifically, can a hook adjust $\ell$ or $\gamma$ in real-time as a response to changes in block time $\Delta t$ or market volatility $\sigma$ to maintain the break-even condition?
* **Auction Integration and Bidding**: How does the equilibrium bidding strategy change if the priority auction is internalized within a v4 hook? If the LP captures a portion of the priority fee $\varphi$, it may offset adverse selection losses and allow for deeper liquidity provision than a public memory pool PA.
* **Optimal Bonding Curve Design**: Beyond the constant product model, can we design "strategic bonding curves" using complex price impact models (e.g., non-linear $p(x)$) to better protect LPs from informed takers while remaining attractive to uninformed flow?

## References
* Adams, A., Moallemi, C. C., Reynolds, S., & Robinson, D. (2025). am-AMM: An auction-managed automated market maker. In Proceedings of the 29th International Conference on Financial Cryptography and Data Security (FC 25). ACM.
* Bachu, B., Hasbrouck, J., Saleh, F., & Wan, X. (2025). An overview of Uniswap v4 for researchers. http://dx.doi.org/10.2139/ssrn.5152215
* Biais, B., Martimort, D., & Rochet, J.-C. (2000). Competing mechanisms in a common value environment. *Econometrica*, *68*(4), 799–837. https://doi.org/10.1111/1468-0262.00138
* Capponi, A., Cartea, Á., & Drissi, F. (2025). Do longer block times impair market efficiency in decentralized markets? http://dx.doi.org/10.2139/ssrn.5290232
* Herdegen, M., Muhle-Karbe, J., & Stebegg, F. (2023). Liquidity provision with adverse selection and inventory costs. *Mathematics of Operations Research*, *48*(3), 1286–1315.