where (k > 0) is a sensitivity parameter (here, (k=2)).
where (\lambda) is unknown to the families but fixed. Families stop early if they a negative marginal utility from another child, but they have only noisy public information about the global ratio. the hardest interview 2
[ p_n = \frac11 + e^-k \cdot (R_n-1 - 1) ] where (k > 0) is a sensitivity parameter (here, (k=2))
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: [ p_n = \frac11 + e^-k \cdot (R_n-1
If (\Delta U < 0), they stop even if formal stopping rule not met (early stop). [ U_\texttotal = \sum_\textfamilies \left( \fracb_fg_f - \lambda \cdot t_f \right) ]
If (\lambda = 0.1), threshold (p=0.2). If estimated (p < 0.2), they stop early. Families observe historical stops and national ratio changes. Using Bayesian learning, after several days they form a posterior on (\lambda). This influences future stopping.
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: