Section 22.2

Thresholds for Continuous Draws

Stopping with uniform draws: costly sampling, take-or-roll thresholds, secretary-type problems, continuous blackjack.

Toolkit

Idea: with i.i.d. continuous draws $U_1,\dots,U_n\sim\mathrm{Unif}(0,1)$ (or exchangeable relative ranks), optimal stopping reduces to a threshold rule found by equating the value of stopping now with the expected value of continuing one more step, then working backward.

Backward induction: if $V_k$ is the value of having $k$ draws left, stop with current value $x$ iff $x\ge \mathbb{E}[V_{k-1}]$ , and $\mathbb{E}[V_k]=\int_0^1\max(x,\mathbb{E}[V_{k-1}])\,dx$ .
Costly sampling with fee $c$ : stop once current best $x$ satisfies $\int_x^1(u-x)\,du=\frac{(1-x)^2}{2}\le c$ , giving threshold $T=1-\sqrt{2c}$ .
Rank-based (secretary-type) rules: only relative order of draws seen so far is informative; optimal rule skips an initial fraction then stops at the first “record” value.
Continuous blackjack: with current total $x$ and one more uniform draw $Y$ , bust probability after hitting is $x$ , so expected value of hitting is $\int_0^{1-x} e^x(1-x-y)\,dy \cdot(\text{appropriate scaling})$ ; threshold $x^*$ solves standing value $=$ hitting value, e.g. $(1-x)e^x =$ hit payoff equality.

Problems