The Hardest Interview | 2

[ 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)).

[ p_n = \frac11 + e^-k \cdot (R_n-1 - 1) ] the hardest interview 2

For large (N) families, this is approximately deterministic: [ R_n = \fracB_nG_n,\quad B_n = B_n-1 +

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

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)):

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.