  We compare the profit of the optimal third-degree price discrimination policy
against a uniform pricing policy. A uniform pricing policy offers the same
price to all segments of the market. Our main result establishes that for a
broad class of third-degree price discrimination problems with concave profit
functions (in the price space) and common support, a uniform price is
guaranteed to achieve one half of the optimal monopoly profits. This profit
bound holds for any number of segments and prices that the seller might use
under third-degree price discrimination. We establish that these conditions are
tight and that weakening either common support or concavity can lead to
arbitrarily poor profit comparisons even for regular or monotone hazard rate
distributions.
