  Traditional monopoly pricing assumes sellers have full information about
consumer valuations. We consider monopoly pricing under limited information,
where a seller only knows the mean, variance and support of the valuation
distribution. The objective is to maximize expected revenue by selecting the
optimal fixed price. We adopt a distributionally robust framework, where the
seller considers all valuation distributions that comply with the limited
information. We formulate a maximin problem which seeks to maximize expected
revenue for the worst-case valuation distribution. The minimization problem
that identifies the worst-case valuation distribution is solved using
primal-dual methods, and in turn leads to an explicitly solvable maximization
problem. This yields a closed-form optimal pricing policy and a new fundamental
principle prescribing when to use low and high robust prices. We show that the
optimal policy switches from low to high prices when variance becomes
sufficiently large, yielding significant performance gains compared with
existing robust prices that generally decay with market uncertainty. This
presents guidelines for when the seller should switch from targeting mass
markets to niche markets. Similar guidelines are obtained for delay-prone
services with rational utility-maximizing customers, underlining the
universality and wide applicability of the low-high pricing principle.
