Prophet Inequalities with Cancellation Costs (2404.00527v2)

Abstract: Most of the literature on online algorithms in revenue management focuses on settings with irrevocable decisions, where once a decision is made upon the arrival of a new input, it cannot be canceled later. Motivated by modern applications -- such as cloud spot markets, selling banner ads, or online hotel booking -- we introduce and study "prophet inequalities with cancellations" under linear cancellation costs (known as the buyback model). In the classic prophet inequality problem, a sequence of independent random variables $X_1, X_2, \ldots$ with known distributions is revealed one by one, and a decision maker must decide when to stop and accept the current variable in order to maximize the expected value of their choice. In our model, after accepting $X_j$, one may later discard $X_j$ and accept another $X_i$ at a cost of $f \times X_j$, where $f\geq 0$ is a parameter. The goal is to maximize the expected net reward: the value of the final accepted variable minus the total cancellation cost. We aim to design online policies that are competitive against the optimal offline benchmark. Our first main result is an optimal prophet inequality for all parameters $f \ge 0$. We fully characterize the worst-case competitive ratio of the optimal online policy against the optimal offline via the solution to a certain differential equation (for which we provide a constructive solution). Our second main result is to design and analyze a simple and polynomial-time randomized adaptive policy that achieves this optimal competitive ratio. Importantly, our policy is order-agnostic (`a la [Samuel-Cahn, 1984]), as it only needs the set of distributions and not their arrival order. These results are obtained by novel techniques related to factor-revealing LPs and generalized flow, reductions to a differential equation, and embedding of problem instances into specific Poisson point processes.