The Hardest Interview 2 |link| -

[ \hatR = R_n-2 + \epsilon,\quad \epsilon \sim \mathcalN(0, \sigma^2),\ \sigma=0.03 ]

[ U = \frac\text# boys\text# girls - \lambda \cdot \text(total births) ]

[ 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)). the hardest interview 2

Set (\Delta U = 0) → threshold (p_\textthresh = 2\lambda).

The fixed point (R^ ) satisfies (p(R^ ) = 0.5) → (R^* = 1). So long-term ratio tends to 1 even with feedback. Families compute (\Delta U) using their noisy (\hatR). For a family with ((b,g)): [ \hatR = R_n-2 + \epsilon,\quad \epsilon \sim

[ p_n = \frac11 + e^-k \cdot (R_n-1 - 1) ]

[ R_n \approx R_n-1 \cdot \frac1 + \fracp_nR_n-1 \cdot (1-p_n) \cdot G_n-1/B_n-11 + \frac1-p_nG_n-1 ] So long-term ratio tends to 1 even with feedback

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: