# The am-AMM Stochastic Game

## Overview
The **Auction-Managed Automated Market Maker (am-AMM)** transitions the relationship between *Liquidity Providers (LPs)* and *arbitrageurs* from a competitive "zero-sum" struggle into a dynamic, multi-stage stochastic game. In traditional Kyle-type frameworks or models of informed trading, a single trader wins a priority auction for one block to capture an arbitrage opportunity based on a private signal $s$.

In this stochastic adaptation, the informed trader is replaced by a **Pool Manager**. This agent secures a temporary monopoly over the pool’s fee-setting and arbitrage rights via an on-chain "Harberger lease," acting as a sophisticated intermediary that optimizes the pool's performance in exchange for rent.

### The Core Conflict
The game shifts the burden of adverse selection from passive LPs to the active Manager. The stochasticity of the game is driven by two primary factors:

1. **Price Dynamics**: The fundamental price of the risky asset follows a geometric Brownian motion.
2. **Temporal Dynamics**: Information and trading opportunities are revealed at the frequency of discrete block generation times.

### Key Game Components
* **The Harberger Lease Auction**: A continuous English auction where potential managers bid a per-block rent $R$. Tenure is maintained as long as the manager’s deposit $D$ remains sufficient.
* **Manager Decision Logic**: The manager solves an optimization problem to set the swap fee $f \in [0, f_{max}]$. The goal is to maximize the sum of noise trader revenue and internalized arbitrage profits, minus the rent $R$ paid to the pool.
* **LP Participation**: LPs provide liquidity $L$ based on the guaranteed rent $R$. Because rent is paid ex-ante, LPs are shielded from the "lumpy" volatility of trade-by-trade fees, effectively transferring market risk to the manager.

## The Trading Game Scenario
In this decentralized market, the game is structured as a sequence of interactions across discrete blockchain slots, governed by a Harberger lease mechanism:

1. **Continuous Bidding Phase**:
* Potential managers submit a per-block rent $R$ and a deposit $D$.
* The deposit must be at least $R \cdot K$, where $K$ is the delay parameter.
* The auction is a continuous English auction; the current highest bidder (Manager) pays rent $R$ out of their deposit $D$ directly to LP token holders.
* The Manager for block $N$ is determined by the top active bid as of block $N-K$.
2. **Liquidity Provision**:
* LPs observe the active rent $R$ and anticipated changes during the $K$-block delay.
* LPs adjust liquidity $\ell$ until the pool reaches a competitive equilibrium.
* The zero-profit condition is met when the rent $R$ compensates for the adverse selection cost (LVR) and the cost of capital.
* LPs pay a small withdrawal fee to the Manager to prevent strategic exits during high volatility.
3. **The Trading Block**:
* At the start of a slot, the Manager observes the mispricing between the AMM and the fundamental price.
* Accessibility Property: Unlike other designs, the pool remains open for trade; if the Manager does not act, other traders can still swap against the pool.
4. **Execution & Fee Setting**:
* *Internalized Arbitrage*: The Manager captures arbitrage by trading against the pool. Because they receive all collected fees, they effectively trade at a zero-fee rate, capturing small price movements within the "no-trade" region.
* *Dynamic Fee Setting*: The Manager sets the swap fee $f$ (subject to a cap $f_{max}$) for the next block to maximize the noise trading revenue.
* *Retail Flow*: Uninformed traders execute against the pool at the fee $f$ set by the Manager.

### Comparison of Frameworks

| Feature | Kyle-Type Game | am-AMM Stochastic Game |
| :--- | :--- | :--- |
| Winning Actor | Informed Trader (Single Block) | Pool Manager (Continuous tenure via Harberger lease) |
| Payment | Priority Fee $\varphi$ | Ongoing Rent $R$ (Continuous per-block payment) |
| LP Revenue | Variable fees $f$ from stochastic trader flow | Guaranteed Rent $R$ paid from the Manager’s deposit |
| Price Discovery | Optimal volume $x^*$ incorporates private signal $s$ | Manager trades with zero fees to drive mispricing to zero each block |
| Risk | LPs bear the cost of adverse selection (LVR/IL) | Manager absorbs LVR risk; LPs earn predictable, "risk-free" rent |

## Single Period Model
This model analyzes the strategic interaction within a single block interval $[0, \Delta t]$.

### Traded Assets
The market consists of two assets:
* Numéraire ($y$): A safe asset with a price normalized to $1$.
* Risky Asset ($x$): An asset whose fundamental price $S_t$ follows a driftless geometric Brownian motion $dS_t = \sigma S_t dW_t$.

### Manager’s Problem
The *pool manager* secures the right to set fees and capture arbitrage by committing a rent $R$ to the LPs. The manager is assumed to be a sophisticated entity with exponential utility $U_M(x) = -\exp(-\gamma_M x)$, where $\gamma_M$ represents risk aversion.

#### Optimization Functional
The manager maximizes their utility by choosing the optimal fee $f$ and bidding the equilibrium rent $R$:

$$
V_M(\ell) := \max_{R,f} \mathbb{E}[U_M(Arb_{int} + F - R)]
$$

where
* *Internalized Arbitrage $Arb_{int}$*: The profit captured by the manager trading at zero fee to drive the AMM price $P$ to the fundamental price $S$.
* *Fee Revenue $F$*: The revenue generated from noise traders, defined as $F = f \cdot H(f, \ell)$, where demand $H$ is decreasing in $f$.

#### Arbitrage Profit
For a constant product pool with liquidity $\ell = \sqrt{xy}$, the manager's ability to trade without fees allows them to monetize price discrepancies that other arbitrageurs cannot. The profit from rebalancing the pool from $P_0$ to $S_0$ is:

$$
Arb_{int} = \ell \frac{(\sqrt{S_0} - \sqrt{P_0})^2}{\sqrt{P_0}}
$$

### LP's Problem
The *Liquidity Provider* supplies liquidity $\ell$ to maximize their utility. In the am-AMM model, the LP's payoff is decoupled from trade-by-trade fees and is instead driven by the guaranteed rent $R$.

#### Optimization Functional
$$
V_{LP}(R) := \max_{\ell} \mathbb{E}[U_{LP}(R - LVR)]
$$

where
* *Utility*: $U_{LP}(x) = -\exp(-\gamma_{LP} x)$, where $\gamma_{LP}$ represents risk aversion.
* *Rent $R$*: The per-block payment received from the manager.
* *LVR (Loss-versus-Rebalancing)*: The adverse selection cost incurred because the pool price is constantly arbed to the market price.

#### Loss-versus-Rebalancing (LVR)
LVR is the expected difference between the AMM value and a market-hedged position. For a single block:

$$
LVR = \ell \frac{(\sqrt{P_1} - \sqrt{P_0})^2}{\sqrt{P_0}}
$$

*Remark.*
* The am-AMM introduces a unique intra-block price path: first, the pool manager (acting as the privileged arbitrageur) executes a transaction at the start of the block to align the pool price $P_0$ with the fundamental price $S_0$. Subsequently, the pool price evolves from $S_0$ to a terminal state $P_1$ driven by the stochastic arrival of noise trader demand.
* This initial "reset" to $S_0$ means that noise traders always face a pool that has been "pre-arbed," effectively reducing the relative mispricing they encounter compared to a standard fixed-fee AMM.
* To formalize this process, one can model the terminal price $P_1$ as a function of the net noise trade volume $H(f, \ell)$. Given the constant product invariant $x \cdot y = L^2$, the price shift $\Delta P$ from noise trading can be modeled as: $$P_1 = P(S_0, \Delta x_{noise})$$ where $\Delta x_{noise}$ is a random variable representing the net order flow, the variance of which may be parameterized by the market volatility $\sigma$, the chosen fee $f$ and the liquidity depth $\ell$.

<font color=red>Q: How could we model the pool price process?</font>

### To Do
* **Derive the Competitive Equilibrium**:
* Solve the system of equations where the Manager’s expected utility $V_M$ and the LP’s expected utility $V_{LP}$ reach an equilibrium.
* Identify the unique equilibrium liquidity level $\ell^*$ and rent $R^*$ as functions of market volatility $\sigma$ and noise trader demand $H$.
* **Extend to a Multi-Period Stochastic Game**:
* Incorporate the *$K$-Block Delay* Parameter, ensuring that the rent $R$ and manager identity for block $N$ are fixed at block $N-K$.
* Integrate a *Withdrawal Fee* to protect managers from strategic liquidity removal during high-volatility events.
* **Formulate the Continuous-Time Setup**:
* Transition discrete-block arbitrage profits into a continuous-time LVR framework, modeled as a running cost per unit of time.
* Analyze the Manager's Value Function under continuous price paths $S_t$, accounting for the exponential utility and risk aversion parameter $\gamma_M$.

## 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.
* 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