Stopping on the last success with unknown odds: Impossibility barriers and quantitative oracle bounds

Abstract

We consider the classical last-success problem for sequential Bernoulli trials in the homogeneous setting where $X_1,\ldots,X_n$ are i.i.d. $\mathrm{Bernoulli}(p)$ but the success probability $p\in(0,1)$ is unknown to the decision maker. When $p$ is known, Bruss' sum-the-odds theorem yields an optimal threshold rule with value $V_n(p)$. We study a natural oracle-free plug-in rule that replaces $p$ by the online empirical estimate $\hat p_t$ and we denote its win probability by $W_n(p)$. First, we derive an exact expression for $W_n(p)$ via a recursion for the state probabilities, enabling explicit comparisons with $V_n(p)$ and revealing a finite-horizon separation between plug-in and oracle performance. Next, we formalize a first decision-theoretic obstruction inherent to the unknown-$p$ formulation: for every fixed $n\ge2$, the dominance partial order on $p$-blind (possibly randomized) rules has no greatest element. We then identify regimes where oracle-freeness is achievable with sharp bounds. For any $p_0\in(0,1)$, we establish finite-horizon oracle bounds on $[p_0,1)$ and we prove a matching minimax lower bound of order $1/\sqrt{n}$ for $p_0\in(0,\tfrac 12)$ (larger values of $p_0$ do not allow for non-trivial lower bounds). We also show that the rate is exponential for any fixed $p$. In sparse regimes where $p=p_n\to0$ with $np_n\to\infty$, we prove asymptotic oracle-optimality of the plug-in rule, in the sense that $V_n(p_n)-W_n(p_n)\to0$. Together with our non-sparse bounds, this yields a broad uniform convergence guarantee, which we show cannot be extended to the critical regime $p\asymp 1/n$. Finally, we establish another impossibility barrier: even allowing randomization, no oracle-free sequence of rules can converge uniformly to the oracle value over $p\in(0,1)$.