Sales-Based Rebate Design

We study rebate mechanisms according to which a monopolist selling a product introduces rebates as function of the volume of buyers. This enables the firm to induce payoff externalities that ordinarily do not exist. The monopoly firm sells an indivisible good to a mass of consumers with uncertain valuations corresponding to two sources of uncertainty: a {\it systemic uncertainty} corresponding to the realized quality, and an {\it idiosyncratic uncertainty} modeling the diversity of consumers' tastes. Analyzing the equilibria of the induced global game among the consumers, we show that introducing positive externality via an increasing aggregate reward program reduces the profit. Using variational optimization techniques, we identify several key characteristics of the optimal reward program: the optimal solution is a `full-refund or nothing' policy, fully reimbursing the buyers if the realized quality falls in one of the finitely many refund-eligible intervals. The number of intervals, though finite, grow unboundedly as consumers' tastes become less diverse and valuations concentrate around the true quality. While finding the optimal reward program is in general an intractable problem, we fully characterize the optimal solution in two important instances: one with constraints on the reward size and the other with bounds on the rate of change of the reward with sales volume. Despite their simple and intuitive structures, the corresponding optimal reward programs asymptotically recover the optimal solution. Our analysis sheds light on the potential role of novel technology-enabled features of crowd-based markets in developing new revenue management strategies.