For large (N) families, this is approximately deterministic:
This creates negative feedback: If boys exceed girls nationally, (p_n < 0.5), and vice versa. At each step, before having another child, the family estimates current national ratio (\hatR) using:
| (\lambda) | Final national (E[b/g]) | Avg. children per family | Avg. utility per family | |-------------|----------------------------|--------------------------|--------------------------| | 0.05 | 1.023 | 2.91 | 0.955 | | 0.10 | 1.007 | 2.68 | 0.891 | | 0.15 | 0.994 | 2.44 | 0.847 | the hardest interview 2
Additionally, the government secretly measures not the raw gender ratio, but a :
Set (\Delta U = 0) → threshold (p_\textthresh = 2\lambda). For large (N) families, this is approximately deterministic:
[ U = \frac\text# boys\text# girls - \lambda \cdot \text(total births) ]
Given uniform prior (\lambda \sim U[0.05,0.15]), after seeing (m) other families’ early stops, they update via Bayes. The problem becomes a with incomplete information. 6. Key Result (Numerical Simulation Summary) Monte Carlo simulations with (N=10^5) families, 1000 days, yield: yield: [ R_n = \fracB_nG_n
[ R_n = \fracB_nG_n,\quad B_n = B_n-1 + X_n,\ G_n = G_n-1 + (1-X_n) ] where (X_n \sim \textBernoulli(p_n)).