# Concave Pro-rata Arbitrage Games

## Overview
This research note proposes a pro-rata framework for modelling trading games within **batched decentralized exchanges (DEXs)**. The model employs a Stackelberg dynamics where the *Automated Market Maker (AMM)* acts as the leader by setting its trading function and fee structure, and *market participants (traders)* act as followers, responding with optimal control strategies. This framework addresses the "abstraction mismatch" between sequential economic mechanisms and block-based consensus, which serves as a primary driver of *Miner Extractable Value (MEV)*.

## Automated Market Maker (AMM)
In this framework, the AMM operates via an exchange function $f$, which maps the input $\Delta$ (the amount of asset $X$ provided to the pool) to the output payout $-\Lambda$ (the amount of asset $Y$ received by the trader):

$$
f(\Delta) = -\Lambda
$$

In accordance with standard *Constant Function Market Maker (CFMM)* theory, we assume $f$ is **concave, increasing, and differentiable** for all $\Delta \neq 0$. The AMM’s *marginal price* $P$ (the price of asset $X$ in terms of the numeraire $Y$) is the derivative of this function:

$$
P(\Delta) = f'(\Delta)
$$

### Examples
* *Linear Impact*: For a model $P(\Delta) = P - \lambda \Delta$, the associated exchange function is $f(\Delta) = (P - \frac{\lambda}{2} \Delta) \Delta$.
* *Uniswap V2*: Defined by the invariant $\phi(x, y) = xy = k$. The resulting exchange function is $f(\Delta) = \frac{y \Delta}{x + \Delta} = P \frac{x}{x+\Delta} \Delta$ with marginal price $P(\Delta) = \frac{k}{(x + \Delta)^2} = P \frac{x^2}{(x + \Delta)^2}$.

### Relation to the Order Book
In traditional order book modelling, the execution price is often expressed as:

$$
P_t = S_t + (\text{temporary impact}) + (\text{permanent impact})
$$

In this setup, the AMM uses $f$ to predetermine *permanent impact* over a fixed horizon based on order flow. *Temporary impact* can be integrated via time-dependency $t$, and directional asymmetry via a pair $(f_+, f_-)$.

## Single-Period Pro-rata Game

### Mechanism and Payoff
In a batched DEX, transactions are processed "all at once," aligning the economic mechanism with the discrete time steps of blockchain consensus.

* *The Process*: Players place assets $x_i$ into a pool; the pool executes an aggregate trade and pays out a concave function $g$ of the total assets.
* *Payoff Function*: For $n$ players, player $i$ receives a pro-rata share: $$U_{i}(x)=\frac{x_{i}}{\mathbf{1}^{T}x} g(\mathbf{1}^{T}x)$$where $x \in \mathbb{R}_{+}^{n}$ and $g: \mathbb{R} \rightarrow \mathbb{R}$ is concave with $g(0)=0$.

### Competitive Arbitrage
Arbitrageurs compete to capture the spread between the AMM's forward exchange function $f(t)$ and the external market price $s$.

* *Net Profit Function*: The pool's total arbitrage profit is $g(t) = f(t) - st$.
* *Strategic Interaction*: Arbitrageurs maximize $U_i$, considering that their contribution $x_i$ impacts the aggregate pool size $\mathbf{1}^T x$ and the resulting marginal payout.

### Summary of Results (Johnson et al., 2023)
* **Existence and Uniqueness**: The game possesses a unique, symmetric, pure strategy equilibrium $x = (q/n)\mathbf{1}$.
* **Equilibrium Characterization**: The total equilibrium contribution $q$ is found by solving the unimodal problem: $\text{maximize } q^{n-1}g(q)$.
* **Overpayment**: In competitive settings ($n > 1$), $q$ exceeds the socially optimal $q^*$ that maximizes $g$, leading to rent dissipation.
* **Price of Anarchy (PoA)**: The PoA—the ratio of optimal fair payoff to equilibrium payoff—is $\Omega(n)$, indicating efficiency decreases linearly as the number of players increases.

## Discrete-Time Pro-rata Arbitrage Game
We consider $n \geq 2$ agents engaging in arbitrage on a CFMM over a discrete time grid $\mathbb{T}_N = \{ \frac{kT}{N} \mid k=0, 1, \dots N\}$.

### Agent Strategies and Flow
Each agent $i$ chooses a strategy $\pmb{\xi}_i = (\xi_{i,0}, \dots, \xi_{i,N})^\top$ consisting of random variables where each $\xi_{i,k}$ is $\mathcal{F}_k$-measurable and bounded.

* **Directionality**: $\xi_{i,k} > 0$ represents a sell to the AMM (inputting asset $X$), while $\xi_{i,k} < 0$ represents a buy from the AMM (extracting asset $X$).
* **Aggregate Flow**: The total batch size at time $t_k$ is the sum of all individual strategies: $\Xi_k = \sum_{i=1}^n \xi_{i,k}$.

### Inventory and Payout
Collecting agents' strategies in the matrix $\pmb{\Xi} = [\pmb{\xi}_1, \dots, \pmb{\xi}_n]$, the AMM's inventory process is:

$$
x^\mathbf{\Xi}_k = x_0 + \Delta_k, \quad y^\mathbf{\Xi}_k = y_0 + \Lambda(\Delta_k)
$$

where $\Delta_k = \sum_{i=1}^k \Xi_k$.

The total arbitrage profit available at each step $j$ is the difference between the value of the assets in the external market (at price $S_j$) and the cost/payout from the AMM. The total profit for agent $i$ over the horizon is:

$$
U_{\mathbb{T}_N}(\pmb{\xi}_i \mid \pmb{\xi}_{-i}) = \sum_{j=1}^N \left[ \frac{\xi_{i,j}}{\Xi_j} \left\{ f(\Delta_j) - f(\Delta_{j-1}) - S_{j-1} \Xi_j \right\} \right]
$$

### Nash Equilibrium
A Nash equilibrium is a profile $(\pmb{\xi}^*_1, \dots, \pmb{\xi}^*_n)$ such that no agent can increase their expected utility by unilaterally changing their strategy:

$$
\mathbb{E}\left[ U_{\mathbb{T}_N}(\pmb{\xi}^*_i \mid \pmb{\xi}^*_{-i}) \right] = \max_{\pmb{\xi}_i} \mathbb{E}\left[ U_{\mathbb{T}_N}(\pmb{\xi}_i \mid \pmb{\xi}^*_{-i}) \right]
$$

for every $i = 1, \dots, n$.

## Continuous-Time Pro-rata Arbitrage Game
We consider $n$ traders with trading processes $X^i = (X^i_t)_{t \in [0,T]}$. An admissible strategy $X^i$ is a càdlàg, predictable process of bounded total variation with $X^i_{0-}=0$. The reference price $S=(S_t)_{t \ge 0}$ is a càdlàg local martingale.

As in the discrete-time model, the aggregate arbitrage flow $X_t = \sum_{i=1}^n X^i_t$ generates the AMM inventory processes:

$$
x^{\mathbf{X}}_t = x_0 + X_t, \quad y^{\mathbf{X}}_t = y_0 - \Lambda(X_t)
$$

where $\Lambda(X_t) = -f(X_t)$ represents the cumulative payout from the AMM. The instantaneous AMM marginal price is determined by the derivative of the forward exchange function:

$$
P^{\mathbf{X}}_t = f'(X_t)
$$

The payoff for agent $i$ given opponents' strategies $\mathbf{X}^{-i}$ is defined by the sum of pro-rata shares of the arbitrage profit across all discrete jumps (batches) in the horizon:

$$
U(X^i \mid \mathbf{X}^{-i}) = \int_0^T \frac{dX^i_t}{dX_t} \left\{ df(X_t) - S_t dX_t \right\}
$$

When $X_t$ is differentiable, the term in the braces represents the instantaneous arbitrage opportunity relative to the marginal price:

$$
d\Lambda(X_t) - S_t dX_t = (P(X_t) - S_t) dX_t
$$

### Nash Equilibrium
A profile $\mathbf{X}^* = (X^{*,1},\dots,X^{*,n})$ is a Nash equilibrium if each $X^{*,i}$ is admissible and satisfies:

$$
\mathbb{E}[U(Z \mid \mathbf{X}^{*,-i})] \le \mathbb{E}[U(X^{*,i} \mid \mathbf{X}^{*,-i})]
$$

for all admissible strategies $Z$.

## Strategic Questions: LP’s Best Response
What is the LP's best response to arbitrage regarding optimal trading function and bonding curve design?

To be more precise, the LP aims to maximize their net profit by choosing an optimal bonding curve $f$:

$$
U(f) = \mathbb{E}[X^*_t S_t - f(X^*_t)]
$$

where $X^*$ represents the arbitrageur's equilibrium strategy.

## Linear Price Impact
We conduct further analysis using a linear price impact model, where the marginal price is defined as $P(\Delta) = P - \lambda \Delta$. This specific structure simplifies the strategic interactions into an LQ control framework.

### Discrete-Time Model
Utilizing the associated exchange function $f(\Delta) = (P - \frac{\lambda}{2} \Delta) \Delta$, the arbitrage profit for agent $i$ is:

$$
U_{\mathbb{T}_N}(\pmb{\xi}_i \mid \pmb{\xi}_{-i}) = \sum_{j=1}^N \left[ \xi_{i,j} \left( P - \lambda \left\{ \sum_{k=1}^{j-1} \Xi_k + \frac12 \Xi_j \right\} - S_{j-1} \right) \right]
$$

#### Explicit Objective Function
The expected utility can be expressed in matrix form, highlighting the quadratic costs and cross-agent impact:

$$
\mathbb{E}[U_{\mathbb{T}_N}(\pmb{\xi}_i \mid \pmb{\xi}_{-i})] = \mathbb{E} \left[ \pmb{\xi}_i^T (P - \mathbf{S}) - \lambda \left\{ \pmb{\xi}^T_i \Gamma \pmb{\xi}_i + \pmb{\xi}^T_i \tilde{\Gamma} \left(\sum_{j \neq i} \pmb{\xi}_j\right) \right\} \right]
$$

In this formulation:
* *Self-Impact Matrix*: $\Gamma_{ij} = \frac12$. This captures the impact of agent $i$'s own trades on their total profit.
* *Cross-Impact Matrix*: $\tilde{\Gamma}_{ij} = \begin{cases} 0, &i<j \\ \frac12, &i=j \\ 1, &i>j \end{cases}$. This models the temporal dependence of profit on opponents' previous and simultaneous trades.

#### Dynamic Programming Principle (DPP)
To solve for the equilibrium, we apply the DPP for each $j=1, \dots, N$. Define the value function as:

$$
V_j(z, \xi_{i,j} | \pmb{\xi}^{-i}_j) = \max_{\xi_{i,j}} \mathbb{E} \left[ \xi_{i,j} (z - \frac{\lambda}{2} \Xi_j) + V^*_{j+1}(z' - \lambda \Xi_j) \right]
$$

Here
* $z = P-S$ is the mispricing.
* $\Xi_j = \sum_i \xi_{i,j}$ is the aggregated flow.
* $z' = z + \sigma \frac{T}{N} \varepsilon$ with $\varepsilon \sim N(0,1)$.
* $V^*_{j+1}$ represents the Nash equilibrium for the remaining periods $\{ \frac{kT}{N} \mid k=j+1 \dots N\}$.

#### First-Order Optimality Condition
The optimal strategy is characterized by the FOC:

$$
z - \frac{\lambda}{2} \sum_{k \neq i} \xi_{k,j} - \lambda \xi_{i,j} - \lambda \frac{d}{dz} \mathbb{E}[V^*_{j+1}](z' - \lambda \Xi_j) = 0
$$

Utilizing symmetry ($\xi^*_{i,j} = \frac{1}{n} \Xi^*_j$ for all $k$), the equilibrium condition becomes:

$$
z - \frac{\lambda (n+1)}{2n} \Xi^*_j - \lambda \frac{d}{dz} \mathbb{E}[V^*_{j+1}](z' - \lambda \Xi^*_j) = 0
$$

#### Closed-Form Solution
Using the linear-quadratic ansatz $\Xi^*_j = A_j z$ and $V^*_j(z) = B_j z^2 + C_j$, we derive the following recurrence relations:

\begin{align}
A_j &= \frac{1 - 2\lambda B_{j+1}}{\frac{\lambda(n+1)}{2n} - 2\lambda^2 B_{j+1}} = \frac{1}{\lambda} \left[ 1 + \frac{n-1}{n+1 - 4 \lambda n B_{j+1}} \right], \\
B_j &= \frac{A_j}{n} (1 - \frac{\lambda}{2}A_j) + B_{j+1} (1 - \lambda A_j)^2 \\
C_j &= C_{j+1} + B_{j+1} \frac{\sigma^2 T}{N}
\end{align}

For the terminal period $N$:

$$
A_N = \frac{2}{\lambda (n+1)}, \quad B_N = \frac{2}{\lambda (n+1)^2}, \quad C_N=0
$$

#### Numerical Analysis
Numerical simulations based on the recurrences above reveal several key behaviors of the pro-rata game:

1. **Scaling with Player Count $n$ for Fixed $N$**
* *Aggregate Flow*: Aggregated order flows are approximately double those of the monopoly case ($n=1$), except near the terminal horizon.
* *Initial Mispricing*: The impact of initial mispricing ($n B_0$) on total arbitrage profits vanishes as $n \to \infty$.
* *Volatility Capture*: Total arbitrage profits derived from volatility ($n C_0$) converge to zero as the number of arbitrageurs grows.
2. **Scaling with Blocktime ($N$) for Fixed $n$**
* *Flow Convergence*: Aggregated order flows converge toward twice the monopoly level as the grid refines.
* *Profit Dynamics*: As blocktime decreases (increasing $N$), the total profits from volatility ($n C_0$) increase and converge to a finite limit.
* *Mispricing Sensitivity*: The effect of initial mispricing on total profits increases but eventually stabilizes as the discretization becomes finer.
3. **Global Parameter Sensitivity**:

### Continuous-Time Model
Numerical experiments in the discrete-time model indicate that a well-defined high-frequency limit ($N \to \infty$) exists. In this continuous-time limit, the trader's payoff accounts for both continuous flows and discrete jumps (batches):

\begin{align}
U(X^i \mid \mathbf{X}^{-i}) &= \int_0^T (P_{t-} - S_t) dX^i_t - \sum_{t \in [0,T]} \frac{\lambda}{2} \Delta X_t \Delta X^i_t
\end{align}

*Remark.* It is expected that "block trades" happens at the beginning and the end. It may be necessary to introduce an additional quadratic cost term to the objective function so that the Nash equilibrium exists.

### LP's Problem: Optimal Mechanism Design
The Liquidity Provider aims to maximize net profit by strategically selecting the linear impact parameter $\lambda$. In this LQ framework, the LP’s objective function is:

$$
U(\lambda) = \mathbb{E}[U_{LP}(X^*_t S_t - f(X^*_t))]
$$

where $U_{LP}$ is the LP's utility function and $X^*_t$ represents the equilibrium strategy of the arbitrageurs.

#### Strategic Implementations of $\lambda$
Experimental results suggest that the LP should set the initial price $P_0 = S_0$ to eliminate immediate arbitrage loss. The remaining challenge is optimizing the bonding curve shape (via $\lambda$) to capture volatility drag.

* *Path Independence*: Current path-independent bonding curves (dependent only on final inventory) are not fully optimized for the LP. They fail to extract the maximum benefit from the competitive rent dissipation between arbitrageurs during the trading horizon.
* *Dynamic Adjustments*: Revenue capture can be improved by implementing time-dependent parameters or imposing a "bid-ask spread" by separating buying and selling flow functions.

#### Adjusted Problem: Asymmetric Price Impact
To address the limitations of path independence, we propose an adjusted problem in the [sequel note](https://hackmd.io/@e41406/r1FVdQEBZe).

## References
* Campbell, S., & Nutz, M. (2025). *Optimal execution among N traders with transient price impact*. arXiv preprint arXiv:2501.09638.
* Johnson, N. A. G., Diamandis, T., Evans, A., de Valence, H., & Angeris, G. (2023). *Concave pro-rata games*. In Financial Cryptography and Data Security (pp. 266–285). Springer. https://doi.org/10.1007/978-3-031-48806-1_18
* Nutz, M., & Prosperi, A. (2025). *High-frequency analysis of a trading game with transient price impact*. arXiv preprint arXiv:2512.11765.
* Schied, A., Strehle, E., & Zhang, T. (2017). *High-frequency limit of Nash equilibria in a market impact game with transient price impact*. SIAM Journal on Financial Mathematics, 8(1), 589–634. https://doi.org/10.1137/16M107030X