Selling Tomorrow's Bargains Today

Consider a good (such as a hotel room) which, if not sold on time, is worth nothing to the seller. For a customer who is considering a choice of such goods, their prices may change dramatically by the time the customer needs to use the good; thus a customer who is aware of this fact might choose to gamble, delaying buying until the last moment in the hopes of better prices. While this gamble can yield large savings, it also carries much risk. However, a coordinator can offer customers a compromise between these extremes and benefits in aggregate. Here we explore how a coordinator might profit from forecasts of such future price fluctuations. Our results can be used in a general setting where customers buy products or services in advance and where market prices may significantly change in the future. We model this as a two-stage optimization problem, where the coordinator first agrees to serve some buyers, and then later executes all agreements once the final values have been revealed. Agreements with buyers consist of a set of acceptable options and a price where the details of agreements are proposed by the buyer. We investigate both the profit maximization and loss minimization problems in this setting. For the profit maximization problem, we show that the profit objective function is a non-negative submodular function, and thus we can approximate its optimal solution within an approximation factor of 0:5 in polynomial time. For the loss minimization problem, we first leverage a sampling technique to formulate our problem as an integer program. We show that there is no polynomial algorithm to solve this problem optimally, unless P = NP. In addition, we show that the corresponding integer program has a high integrality gap and it cannot lead us to an approximation algorithm via a linear-programming relaxation. Nevertheless, we propose a bicriteria-style approximation that gives a constant-factor approximation to the minimal loss by allowing a fraction of our options to overlap. Importantly, however, we show that our algorithm provides a strong, uniform bound on the amount the overlap per options. We propose our algorithm by rounding the optimal solution of the relaxed linear program via a novel dependent-rounding method.