Residual Prophet Inequalities

We introduce a variant of the classic prophet inequality, called \emph{residual prophet inequality} (RPI). In the RPI problem, we consider a finite sequence of $n$ nonnegative independent random values with known distributions, and a known integer $0\leq k\leq n-1$. Before the gambler observes the sequence, the top $k$ values are removed, whereas the remaining $n-k$ values are streamed sequentially to the gambler. For example, one can assume that the top $k$ values have already been allocated to a higher-priority agent. Upon observing a value, the gambler must decide irrevocably whether to accept or reject it, without the possibility of revisiting past values. We study two variants of RPI, according to whether the gambler learns online of the identity of the variable that he sees (FI model) or not (NI model). Our main result is a randomized algorithm in the FI model with \emph{competitive ratio} of at least $1/(k+2)$, which we show is tight. Our algorithm is data-driven and requires access only to the $k+1$ largest values of a single sample from the $n$ input distributions. In the NI model, we provide a similar algorithm that guarantees a competitive ratio of $1/(2k+2)$. We further analyze independent and identically distributed instances when $k=1$. We build a single-threshold algorithm with a competitive ratio of at least 0.4901, and show that no single-threshold strategy can get a competitive ratio greater than 0.5464.