Data-based price discrimination: information theoretic limitations and a minimax optimal strategy

This paper studies the gap between the classical pricing theory and the data-based pricing theory. We focus on the problem of price discrimination with a continuum of buyer types based on a finite sample of observations. Our first set of results provides sharp lower bounds in the worst-case scenario for the discrepancy between any data-based pricing strategies and the theoretical optimal third-degree price discrimination (3PD) strategy (respectively, uniform pricing strategy) derived from the distribution (where the sample is drawn) ranging over a large class of distributions. Consequently, there is an inevitable gap between revenues based on any data-based pricing strategy and the revenue based on the theoretical optimal 3PD (respectively, uniform pricing) strategy. We then propose easy-to-implement data-based 3PD and uniform pricing strategies and show each strategy is minimax optimal in the sense that the gap between their respective revenue and the revenue based on the theoretical optimal 3PD (respectively, uniform pricing) strategy matches our worst-case lower bounds up to constant factors (that are independent of the sample size $n$). We show that 3PD strategies are revenue superior to uniform pricing strategies if and only if the sample size $n$ is large enough. In other words, if $n$ is below a threshold, uniform pricing strategies are revenue superior to 3PD strategies. We further provide upper bounds for the gaps between the welfare generated by our minimax optimal 3PD (respectively, uniform pricing) strategy and the welfare based on the theoretical optimal 3PD (respectively, uniform pricing) strategy.