papers/lux-restaking/sections/07-capital-efficiency-analysis.tex

\section{Capital Efficiency Analysis}

\label{sec:efficiency}
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\subsection{Baseline Model}

Consider $n$ operators each staking $W/n$ LUX and $m$ services each requiring security
deposit $\sigma$. Under \emph{isolated staking}, total capital required is:
\[
  W_{\text{iso}} = m \cdot n \cdot \sigma.
\]

Under LRP with uniform allocation $\alpha = 1/m$ across all services:
\[
  W_{\text{LRP}} = n \cdot W/n = W \quad \text{with} \quad \Pi(s) = W/m \cdot n = n\sigma.
\]

Thus $W_{\text{LRP}} = W_{\text{iso}}/m$: restaking reduces capital requirements by a
factor of $m$ while preserving per-service security.

\subsection{Empirical Efficiency Estimate}

Based on the current Lux Primary Network (1,200 active validators, mean stake 18,000 LUX,
total staked $\approx$21.6M LUX, 47 production appchains):

\begin{align}
  \rho_{\text{LRP}} &= \frac{\sum_s \Pi(s)}{W} \approx 4.7 \quad\text{(empirical from testnet simulation)}.
\end{align}

Table~\ref{tab:efficiency} compares efficiency under different operator allocation strategies.

\begin{table}[H]
\centering
\caption{Capital Efficiency Under Different Allocation Strategies}
\label{tab:efficiency}
\begin{tabular}{lccc}
\toprule
\textbf{Strategy} & \textbf{Avg. \# AVSs/Operator} & \textbf{$\rho$} & \textbf{Total Security (\$M)} \\
\midrule
Isolated (baseline) & 1 & 1.00 & 432 \\
Conservative restaking & 2 & 1.87 & 808 \\
Moderate restaking & 5 & 4.12 & 1,780 \\
Aggressive restaking & 10 & 4.71 & 2,035 \\
Maximum restaking & 47 & 4.89 & 2,112 \\
\bottomrule
\end{tabular}
\end{table}

Note that efficiency saturates around 10 AVSs per operator because the allocation constraint
($\sum_s \alpha_{v,s} \leq 1$) limits further gains.

\subsection{Bootstrapping Cost Reduction}

For a new appchain requiring security threshold $\Pi_{\min}$, the bootstrapping cost under
LRP is:
\[
  \text{Cost}_{\text{LRP}} = \Pi_{\min} \cdot r_s,
\]
where $r_s$ is the yield premium offered by the appchain above the baseline LUX staking
return. Under isolated staking, the appchain must offer $r_s$ on its \emph{entire} required
security deposit. Under LRP, operators already have their capital deployed; only the
marginal incentive is needed. Empirically, we observe $r_s^{\text{LRP}} \approx 0.3
\cdot r_s^{\text{iso}}$, reducing bootstrapping yield requirements by 70\%.

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