papers/lux-economics/sections/09-network-effects-and-ecosystem-dynamics.tex

\section{Network Effects and Ecosystem Dynamics}

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\subsection{Security Externalities}

Each new appchain joining the Lux ecosystem creates \emph{positive security externalities}
for existing appchains through two channels:

\begin{enumerate}[leftmargin=1.5em]
  \item \textbf{Validator supply expansion}: New appchains incentivize new validators to
    join the Primary Network, increasing total stake $W$ and improving security for all.
  \item \textbf{SSF contribution}: SSF payments from new appchains increase Primary Network
    yield, attracting more stake.
\end{enumerate}

\begin{proposition}[Superadditive Security]
Under the SSF mechanism, the security of any appchain $s$ is strictly increasing in the
number of other appchains:
\[
  \frac{\partial \Pi(s)}{\partial |\mathcal{S}|} > 0.
\]
\end{proposition}

\begin{proof}
More appchains $\Rightarrow$ higher total SSF $\Rightarrow$ higher primary staking yield
$\Rightarrow$ more stake enters primary network ($dW/dt > 0$ from Eq.~\ref{eq:stakeflow})
$\Rightarrow$ larger validator pool $\Rightarrow$ more validators available for appchain $s$
$\Rightarrow$ $\Pi(s)$ increases.
\end{proof}

\subsection{Competition Among Appchains}

While appchain growth creates positive externalities for security, appchains also compete
for scarce validator attention ($K$ capacity per validator). We model this as a market
where appchains post reward schedules and validators allocate their capacity optimally.

The equilibrium is a Nash equilibrium of the validator capacity allocation game, which
we analyze as follows. Let $R_s$ be the total reward rate offered by appchain $s$ (USD per
year per unit stake), and let $\hat{w}_s$ be the total stake allocated to appchain $s$.
The validator equilibrium satisfies:
\[
  R_s(\hat{w}_s) = R_{s'}(\hat{w}_{s'}) \quad \forall s, s' \text{ in the active set},
\]
where $R_s(\hat{w}_s) = r_s^{\text{base}} + \beta_s \cdot \text{FeeRevenue}(s) / \hat{w}_s$.
This equalizes marginal returns across all opted-into appchains, which is the competitive
equilibrium condition.

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