  Pricing based on individual customer characteristics is widely used to
maximize sellers' revenues. This work studies offline personalized pricing
under endogeneity using an instrumental variable approach. Standard
instrumental variable methods in causal inference/econometrics either focus on
a discrete treatment space or require the exclusion restriction of instruments
from having a direct effect on the outcome, which limits their applicability in
personalized pricing. In this paper, we propose a new policy learning method
for Personalized pRicing using Invalid iNsTrumental variables (PRINT) for
continuous treatment that allow direct effects on the outcome. Specifically,
relying on the structural models of revenue and price, we establish the
identifiability condition of an optimal pricing strategy under endogeneity with
the help of invalid instrumental variables. Based on this new identification,
which leads to solving conditional moment restrictions with generalized
residual functions, we construct an adversarial min-max estimator and learn an
optimal pricing strategy. Furthermore, we establish an asymptotic regret bound
to find an optimal pricing strategy. Finally, we demonstrate the effectiveness
of the proposed method via extensive simulation studies as well as a real data
application from an US online auto loan company.
