Delegated Online Search

In a delegation problem, a principal P with commitment power tries to pick one out of $n$ options. Each option is drawn independently from a known distribution. Instead of inspecting the options herself, P delegates the information acquisition to a rational and self-interested agent A. After inspection, A proposes one of the options, and P can accept or reject. Delegation is a classic setting in economic information design with many prominent applications, but the computational problems are only poorly understood. In this paper, we study a natural online variant of delegation, in which the agent searches through the options in an online fashion. For each option, he has to irrevocably decide if he wants to propose the current option or discard it, before seeing information on the next option(s). How can we design algorithms for P that approximate the utility of her best option in hindsight? We show that in general P can obtain a $\Theta(1/n)$-approximation and extend this result to ratios of $\Theta(k/n)$ in case (1) A has a lookahead of $k$ rounds, or (2) A can propose up to $k$ different options. We provide fine-grained bounds independent of $n$ based on two parameters. If the ratio of maximum and minimum utility for A is bounded by a factor $\alpha$, we obtain an $\Omega(\log \log \alpha / \log \alpha)$-approximation algorithm, and we show that this is best possible. Additionally, if P cannot distinguish options with the same value for herself, we show that ratios polynomial in $1/\alpha$ cannot be avoided. If the utilities of P and A for each option are related by a factor $\beta$, we obtain an $\Omega(1/ \log \beta)$-approximation, where $O(\log \log \beta/ \log \beta)$ is best possible.