papers/lux-validator-economics/sections/03-delegation-mechanism.tex

\section{Delegation Mechanism}
\label{sec:delegation}

\subsection{Delegation Lifecycle}

\begin{algorithm}[H]
\caption{Delegation Protocol}
\label{alg:delegation}
\begin{algorithmic}[1]
\Procedure{Delegate}{delegator $d$, validator $v_i$, amount $s_d$}
    \State \textbf{require} $s_d \geq s^{\min}_{\text{del}}$
    \State \textbf{require} $s_i + s_d \leq \omega \cdot s_i^{\text{self}}$
    \State \textbf{require} $v_i$ is active and not jailed
    \State Lock $s_d$ in delegation contract
    \State $\mathcal{D}_i \gets \mathcal{D}_i \cup \{d\}$
    \State Effective at epoch $e_{\text{current}} + 1$
\EndProcedure
\Procedure{Undelegate}{delegator $d$}
    \State Begin unbonding period $\tau_{\text{unbond}} = 14$ days
    \State $d$ earns no rewards during unbonding
    \State After $\tau_{\text{unbond}}$: release $s_d$ to $d$
    \State During unbonding: $d$'s stake still subject to slashing
\EndProcedure
\end{algorithmic}
\end{algorithm}

The 14-day unbonding period serves two purposes: (i) it prevents rapid stake movement in response to anticipated slashing events, and (ii) it provides economic finality---actions taken during the bonded period can still be penalized during unbonding.

\subsection{Delegation Cap and Decentralization}

The delegation multiplier $\omega = 15$ limits concentration:

\begin{proposition}[Decentralization Lower Bound]
With total stake $S$ and minimum self-stake $s^{\min}_{\text{val}}$, the minimum number of validators needed to secure $2S/3$ is:
\begin{equation}
    n^{\min} = \left\lceil \frac{2S}{3 \cdot \omega \cdot s^{\min}_{\text{val}}} \right\rceil
\end{equation}
\end{proposition}

\begin{proof}
Each validator's effective stake is at most $\omega \cdot s^{\min}_{\text{val}} = 30{,}000$ LUX. To accumulate $2S/3$ stake requires at least $\lceil 2S / (3 \cdot 30{,}000) \rceil$ validators.
\end{proof}

For $S = 300{,}000{,}000$ LUX (60\% of 500M staked), $n^{\min} = 6{,}667$. In practice, the equilibrium is more concentrated because most validators attract more self-stake, but the cap ensures a lower bound on decentralization.

\subsection{Commission Market}

Validators compete for delegators by setting commission rates. We model this as a Bertrand competition:

\begin{theorem}[Commission Equilibrium]\label{thm:commission}
In a market with $n \geq 2$ validators with identical costs, the unique Nash equilibrium commission rate is $c^* = c^{\min} + \epsilon$ where $\epsilon \to 0$ as $n \to \infty$.
\end{theorem}

\begin{proof}
Delegators prefer lower commissions (holding uptime constant). If validator $i$ sets $c_i > c^{\min}$, any competitor $j$ can attract $i$'s delegators by setting $c_j = c_i - \epsilon$. The only equilibrium under Bertrand dynamics is $c_i = c^{\min}$ for all $i$. With the floor $c^{\min} = 0.05$, the equilibrium approaches $c^* = 0.05$ as competition increases.
\end{proof}

In practice, validators differentiate on uptime, appchain coverage, and reputation, allowing premium commission rates:

\begin{definition}[Quality-Adjusted Commission]
The effective cost to delegators of validator $v_i$ is:
\begin{equation}
    \text{EffCost}_i = \frac{c_i}{u_i \cdot (1 + \sum_{k \in \mathcal{K}_i} \rho_k)}
\end{equation}
where $\rho_k$ is the reward premium for appchain $k$ participation. Delegators minimize this effective cost.
\end{definition}