Robust Online Selection with Uncertain Offer Acceptance

TL;DR

This work introduces SP-UA, a secretary-style online selection problem where candidate offers are accepted with probability $p$, and the goal is to maximize a robust ratio that protects against unknown top-$k$ candidates who will accept. Using Markov decision processes, the authors derive an exact policy-encoding LP $(LP)_{n,p}$ and its infinite-horizon relaxation $(CLP)_p$, enabling bounds and closed-form policies. They prove the optimal threshold-like policy for $p\ge p^*\approx 0.594$, with a concrete rule to observe a fraction $p^{1/(1-p)}$ before offering to the best observed candidate; they also provide upper and lower bounds on $\gamma_\infty^*(p)$ and show the limit exists and is encapsulated by the continuous LP. Computational results compare robust policies to benchmarks under top-$k$ and power-law utilities, highlighting robustness advantages when $p$ is small and near-term practicality for ad-display and other online selection tasks. Overall, the framework links MDPs and LPs in secretary-type problems and offers a tractable path to performance guarantees in online selection with uncertain acceptance.

Abstract

Online advertising has motivated interest in online selection problems. Displaying ads to the right users benefits both the platform (e.g., via pay-per-click) and the advertisers (by increasing their reach). In practice, not all users click on displayed ads, while the platform's algorithm may miss the users most disposed to do so. This mismatch decreases the platform's revenue and the advertiser's chances to reach the right customers. With this motivation, we propose a secretary problem where a candidate may or may not accept an offer according to a known probability $p$. Because we do not know the top candidate willing to accept an offer, the goal is to maximize a robust objective defined as the minimum over integers $k$ of the probability of choosing one of the top $k$ candidates, given that one of these candidates will accept an offer. Using Markov decision process theory, we derive a linear program for this max-min objective whose solution encodes an optimal policy. The derivation may be of independent interest, as it is generalizable and can be used to obtain linear programs for many online selection models. We further relax this linear program into an infinite counterpart, which we use to provide bounds for the objective and closed-form policies. For $p \geq p^* \approx 0.6$, an optimal policy is a simple threshold rule that observes the first $p^{1/(1-p)}$ fraction of candidates and subsequently makes offers to the best candidate observed so far.

Figures (4)

• Figure 1: Bounds for $\gamma_\infty^*$ as a function of $p$. The solid line represents the theoretical upper bound given in Theorem \ref{['thm:main_upper_bound']}. The dashed-dotted line corresponds to the theoretical lower bound given in Theorem \ref{['thm:main_lower_bound']}; for $p$ close to $0$, the guarantee rises to $0.51$. In dashed line we present numerical results by solving $(LP)_{n,p}$ for $n=200$ candidates.
• Figure 2: Approximation factors for the top $k$ utility function, for $k=1,2,3,4$.
• Figure 3: Approximation factor for the power law utility function. The function has the form $U_i=i^{-(1+\delta)}$. Experiments are run for $\delta \in \{ 10^{-2}, 10^{-1}, 2\cdot 10^{-1} \}$.
• Figure 4: Linear program that finds value function $v^*$ for $\textbf{SP-UA}$ and its dual.

Theorems & Definitions (66)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Proposition 1
• Theorem 5: buchbinder2014secretary
• Lemma 1
• Lemma 2
• Proposition 2
• proof