Optimal Stopping for Maximum of Uniforms

You observe $n$ i.i.d. random variables $X_1, X_2, \ldots, X_n$, each drawn from $\text{Unif}(0, 1)$, one at a time. After seeing each $X_t$, you must immediately decide whether to stop and collect payoff $X_t$, or continue to the next observation. If you reach $X_n$ without stopping, you must take $X_n$. Your goal is to maximize $E[X_{\tau}]$, where $\tau$ is your stopping time. 1. Show that the optimal policy is a **threshold rule**: at each step $t$, stop if and only if $X_t \geq c_t$ for some sequence of thresholds $c_1, c_2, \ldots, c_n$. 2. Derive the recursion that determines the thresholds $c_t$, and compute $c_n$ and $c_{n-1}$ explicitly.