papers/lux-tokenomics/sections/03-staking-reward-mechanism.tex

\section{Staking Reward Mechanism}
\label{sec:staking}

\subsection{Reward Schedule}

The per-epoch reward emission follows an exponentially decaying schedule:
\begin{equation}\label{eq:reward_schedule}
    R_t = R_0 \cdot \alpha^t \cdot f(\sigma_t)
\end{equation}
where $R_0 = 5{,}000{,}000$ LUX is the initial epoch reward, $\alpha = 0.97$ is the annual decay factor, and $f(\sigma_t)$ is a staking ratio adjustment function.

\subsection{Staking Ratio Adjustment}

Let $\sigma_t = \text{Staked}_t / S_t$ denote the staking ratio at epoch $t$, and let $\sigma^* = 0.67$ be the target staking ratio. The adjustment function is:
\begin{equation}\label{eq:staking_adj}
    f(\sigma_t) = \begin{cases}
        1 + \beta \cdot (\sigma^* - \sigma_t) & \text{if } \sigma_t < \sigma^* \\
        1 - \gamma \cdot (\sigma_t - \sigma^*) & \text{if } \sigma_t \geq \sigma^*
    \end{cases}
\end{equation}
where $\beta = 2.0$ and $\gamma = 1.5$ are elasticity parameters. When staking is below target, rewards increase to attract stakers. When above target, rewards decrease to release liquidity.

\begin{proposition}[Staking Ratio Convergence]
Under rational validator assumptions, the staking ratio $\sigma_t$ converges to $\sigma^*$ as $t \to \infty$.
\end{proposition}

\begin{proof}
A rational validator stakes if the expected staking yield $y_t = R_t \cdot f(\sigma_t) / (\sigma_t \cdot S_t)$ exceeds the opportunity cost $r_t$ of capital. If $\sigma_t < \sigma^*$, then $f(\sigma_t) > 1$, increasing $y_t$ and attracting additional stake. If $\sigma_t > \sigma^*$, then $f(\sigma_t) < 1$, decreasing $y_t$ below $r_t$ for marginal validators, causing unstaking. The unique fixed point is $\sigma_t = \sigma^*$ where $f(\sigma^*) = 1$.
\end{proof}

\subsection{Individual Validator Rewards}

For a validator $i$ with stake $s_i$ in epoch $t$, the reward is:
\begin{equation}
    r_{i,t} = R_t \cdot \frac{s_i \cdot u_i}{\sum_{j \in \mathcal{V}_t} s_j \cdot u_j}
\end{equation}
where $u_i \in [0,1]$ is the uptime factor for validator $i$ and $\mathcal{V}_t$ is the active validator set. The uptime factor is computed as $u_i = \min(1, T_{\text{online},i} / T_{\text{epoch}})$, where $T_{\text{online},i}$ is the number of slots in which validator $i$ participated in consensus.

\begin{algorithm}[H]
\caption{Per-Epoch Reward Distribution}
\label{alg:reward}
\begin{algorithmic}[1]
\Require Active validators $\mathcal{V}_t$, stakes $\{s_i\}$, uptimes $\{u_i\}$
\Ensure Reward allocations $\{r_{i,t}\}$
\State $\sigma_t \gets \sum_{i \in \mathcal{V}_t} s_i / S_t$
\State $R_t \gets R_0 \cdot \alpha^t \cdot f(\sigma_t)$
\State $W \gets \sum_{i \in \mathcal{V}_t} s_i \cdot u_i$
\For{each validator $i \in \mathcal{V}_t$}
    \State $r_{i,t} \gets R_t \cdot (s_i \cdot u_i) / W$
    \State $r_{i,\text{self}} \gets r_{i,t} \cdot c_i$ \Comment{Commission portion}
    \State $r_{i,\text{delegators}} \gets r_{i,t} - r_{i,\text{self}}$ \Comment{Delegator portion}
    \State Distribute $r_{i,\text{delegators}}$ pro-rata to delegators of $i$
\EndFor
\end{algorithmic}
\end{algorithm}

\subsection{Delegation Mechanics}

Delegators select a validator and stake tokens through delegation contracts. Each validator sets a commission rate $c_i \in [0.05, 0.25]$ with a minimum of 5\% and maximum of 25\%. Delegators receive rewards proportional to their delegated stake, net of commission:
\begin{equation}
    r_{d,t} = r_{i,t} \cdot (1 - c_i) \cdot \frac{s_d}{\sum_{d' \in \mathcal{D}_i} s_{d'}}
\end{equation}
where $\mathcal{D}_i$ is the set of delegators to validator $i$ and $s_d$ is delegator $d$'s stake.