Repeated Sales with Heterogeneous Buyer Sophistication

This paper considers behavior-based price discrimination in the repeated sale of a non-durable good to a single long-lived buyer, by a seller without commitment power. We assume that there is a mixed population of forward-looking ``sophisticated'' buyers and myopic ``naive'' buyers. We investigate the impact of these dynamics on the seller's ability to learn about the buyer and exploit this learning for revenue. We obtain conclusions that differ dramatically with the time horizon of the interactions. To understand short time horizons, we analyze a two-period model, and find that the strategic demand reduction observed with fully sophisticated buyers is robust to the introduction of naive types. In fact, despite the inability of naive buyers to game the pricing algorithm, their introduction can further harm the seller's revenue, due to more intense demand reduction overall. For long horizons, we consider an infinite-horizon model with time discounting. We find that the extreme demand reduction predicted by previous work does not survive the introduction of naive buyers. Instead, we observe equilibria where the seller learns meaningfully despite the sophisticated buyers' demand reduction. We prove that for a natural family of such equilibria, the seller's revenue is not just high, but approximates the revenue attainable with commitment power, even when the fraction of naive types is vanishingly small.