  A seller is pricing identical copies of a good to a stream of unit-demand
buyers. Each buyer has a value on the good as his private information. The
seller only knows the empirical value distribution of the buyer population and
chooses the revenue-optimal price. We consider a widely studied third-degree
price discrimination model where an information intermediary with perfect
knowledge of the arriving buyer's value sends a signal to the seller, hence
changing the seller's posterior and inducing the seller to set a personalized
posted price. Prior work of Bergemann, Brooks, and Morris (American Economic
Review, 2015) has shown the existence of a signaling scheme that preserves
seller revenue, while always selling the item, hence maximizing consumer
surplus. In a departure from prior work, we ask whether the consumer surplus
generated is fairly distributed among buyers with different values. To this
end, we aim to maximize welfare functions that reward more balanced surplus
allocations.
  Our main result is the surprising existence of a novel signaling scheme that
simultaneously $8$-approximates all welfare functions that are non-negative,
monotonically increasing, symmetric, and concave, compared with any other
signaling scheme. Classical examples of such welfare functions include the
utilitarian social welfare, the Nash welfare, and the max-min welfare. Such a
guarantee cannot be given by any consumer-surplus-maximizing scheme -- which
are the ones typically studied in the literature. In addition, our scheme is
socially efficient, and has the fairness property that buyers with higher
values enjoy higher expected surplus, which is not always the case for existing
schemes.
