Sales-Based Rebate Design

In this paper, we study a family of sales-based rebate mechanisms as an effective tool to implement price discrimination across a population of buyers with correlated heterogeneous valuations on indivisible goods and services. In order to implement such sales-based rebate mechanisms, the seller charges each buyer a fixed price at the time of purchase contingent on a rebate that is a function of the ex post sales volume to be realized at the end of the sales period. The seller declares both a price and a menu of rebates as a function of sales. We show that, when there is a common component of uncertainty in consumers' valuations (to which we refer as the quality of the product), such rebates enable a seller to effectively induce different expected net prices at different valuations. Importantly, this effective price discrimination over valuations is achieved keeping both the base price and the rebate uniform across all buyers. This uniformity of price and rebate across buyers is a key advantage of our proposed rebate mechanism, thereby providing a new mechanism for price discrimination in crowd-based markets. We use tools and techniques from game theory and variational optimization to provide insight into the economics of such mechanisms. In particular, we identify two mechanisms that are monotone functions of the sales volume that are easy to implement in practice and perform well when compared with the much more complex optimal mechanism. We provide a rigorous analysis of the optimal mechanism and discuss practical limitations in implementing the globally optimal design, further demonstrating the efficacy of our proposed monotone mechanisms.