1. **Problem Statement:** We want to create a math equation to model film distribution waterfall calculations, which allocate revenue in tiers to different parties. 2. **Understanding the Waterfall:** A waterfall splits total revenue $R$ into sequential tiers with thresholds $T_1, T_2, \ldots$, and percentages $p_1, p_2, \ldots$ paid to parties in each tier. 3. **General Formula:** For tier $i$, the payout $P_i$ is: $$ P_i = p_i \times \max(0, \min(R, T_i) - T_{i-1}) $$ where $T_0 = 0$. 4. **Explanation:** This means each tier pays its percentage on the revenue slice between $T_{i-1}$ and $T_i$, capped by total revenue $R$. 5. **Example:** For 3 tiers with thresholds $T_1=1,000,000$, $T_2=5,000,000$, $T_3=\infty$, and percentages $p_1=0.5$, $p_2=0.3$, $p_3=0.2$, the payouts are: $$ P_1 = 0.5 \times \min(R, 1,000,000) $$ $$ P_2 = 0.3 \times \max(0, \min(R, 5,000,000) - 1,000,000) $$ $$ P_3 = 0.2 \times \max(0, R - 5,000,000) $$ 6. **Total payout check:** The sum $\sum_i P_i$ equals total revenue $R$ if $\sum_i p_i = 1$. This formula models film distribution waterfall calculations mathematically.