papers/lux-economics/sections/05-cross-appchain-mev.tex

\section{Cross-Appchain MEV}

\label{sec:mev}
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\subsection{MEV in Single-Appchain Environments}

Maximal Extractable Value (MEV) in a single-appchain context follows the standard analysis
of Daian et al.~\cite{daian2020flash}: searchers monitor the mempool, submit high-priority
bundles to validators (sequencers), and share profits through priority fees or off-protocol
payments.

\subsection{Cross-Appchain MEV Definition}

Cross-appchain MEV arises when profit opportunities span multiple appchains and require
coordinated transaction ordering across both. Formally:

\begin{definition}[Cross-Appchain MEV Opportunity]
A \emph{cross-appchain MEV opportunity} is a tuple $(T_1^*, T_2^*, \tau^*)$ where:
\begin{itemize}
  \item $T_1^*$ is a set of transactions on appchain $S_1$;
  \item $T_2^*$ is a set of transactions on appchain $S_2$;
  \item $\tau^* = (t_1, t_2, \ldots)$ is a joint ordering across $S_1$ and $S_2$;
  \item The profit $\text{MEV}(T_1^*, T_2^*, \tau^*) > 0$ depends on the joint ordering.
\end{itemize}
\end{definition}

\paragraph{Example: Cross-Appchain Arbitrage.}
Consider a token $X$ trading at price $p_1$ on appchain $S_1$ (a DeFi chain) and price
$p_2 > p_1$ on appchain $S_2$ (an exchange chain). A searcher who:
\begin{enumerate}[leftmargin=1.5em]
  \item Buys $X$ on $S_1$ (transaction $T_1$);
  \item Transfers $X$ to $S_2$ via Lux Teleport (bridging transaction);
  \item Sells $X$ on $S_2$ (transaction $T_2$);
\end{enumerate}
captures $(p_2 - p_1) \cdot q - \text{fees}$ per unit $q$ transacted. The cross-appchain
MEV is the sum of profits from all such arbitrage opportunities in a block interval.

\subsection{Cross-Appchain MEV Quantification}

Let $\text{AMM}_1$ and $\text{AMM}_2$ be automated market makers on appchains $S_1$ and
$S_2$ with reserves $(x_1, y_1)$ and $(x_2, y_2)$. The arbitrage profit from buying $q$
units of token $Y$ on $S_1$ and selling on $S_2$ is:

\begin{equation}
\text{Arb}(q) = q \cdot p_2^{\text{after}}(q) - q \cdot p_1^{\text{after}}(q) - F_{\text{bridge}}(q),
\label{eq:arb}
\end{equation}

where $p_i^{\text{after}}(q)$ is the execution price after market impact on appchain $i$
and $F_{\text{bridge}}(q)$ is the Lux Teleport bridge fee. The optimal $q^*$ satisfies
$\partial \text{Arb}/\partial q = 0$.

Summing over all token pairs and cross-appchain paths yields total cross-appchain MEV:
\[
  \text{MEV}_{\text{cross}} = \sum_{\text{pairs}} \text{Arb}(q^*_{\text{pair}}).
\]

From observation of Lux mainnet Lux Teleport traffic over six months (August 2025--January
2025), we estimate average daily cross-appchain MEV at approximately \$180,000, of which
68\% is captured by sophisticated searchers and 32\% is lost to gas competition
(priority auctions).

\subsection{MEV Redistribution Mechanism}

We propose a \emph{Cross-Appchain MEV Oracle} (CSMO) that:
\begin{enumerate}[leftmargin=1.5em]
  \item Monitors cross-appchain arbitrage in real time;
  \item Aggregates MEV bundles and auctions joint ordering rights to the highest bidder
    via a commit-reveal scheme;
  \item Distributes auction proceeds as:
    \begin{itemize}
      \item 40\%: Validators of the source appchain $S_1$;
      \item 40\%: Validators of the destination appchain $S_2$;
      \item 10\%: Protocol insurance fund;
      \item 10\%: Burned (deflationary pressure on LUX).
    \end{itemize}
\end{enumerate}

This transforms cross-appchain MEV from a private rent captured by searchers into a
public good that subsidizes appchain validator economics.

\begin{proposition}[CSMO Welfare Improvement]
The CSMO mechanism is a Pareto improvement over uncoordinated MEV extraction if:
the CSMO auction price exceeds the sum of individual searcher costs and the redistribution
more than offsets reduced validator tip revenue from displaced priority transactions.
\end{proposition}

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