The set of
BCE is precisely the set of outcomes that can
arise with additional information for a given
basic game and prior information structure. If there are properties that hold for all
BCE, we have identified predictions that are
robust to the exact information structure.
Identifying the best or worst outcome that
can arise under some information structure
according to some objective function as criterion is the same as solving an information
design problem where the designer is maximizing or minimizing that criterion. In this
section, we will review two such economic
applications of information design. We will
highlight the implications of this approach in
the context of third-degree price discrimination.
(Bergemann, Brooks, and Morris 2015) [...]
A classic issue in the economic analysis
of monopoly is the impact of discriminatory
pricing on consumer and producer surplus.
A monopolist engages in third-degree price
discrimination if he uses additional information about consumer characteristics to offer
different prices to different segments of the
aggregate market. Bergemann, Brooks, and
Morris (2015) characterize what could happen to consumer and producer surplus for all
possible segmentations of the market.
One can provide some elementary bounds
on consumer and producer surplus in any
market segmentation. First, consumer surplus must be nonnegative as a consequence
of the participation constraint: a consumer
will not buy the good at a price above his valuation. Second, the producer must get at least
the surplus that he could get if there was no
segmentation and he had no additional information beyond the prior distribution. In this case, an optimal policy is always to offer the
product with probability one at a given price
to all buyers. We therefore refer to it as uniform monopoly price, and correspondingly,
uniform monopoly profit. Third, the sum
of consumer and producer surplus cannot
exceed the total social value that is generated by the good, which is willingness-to-pay
minus unit cost of production. The shaded
right-angled triangle in figure 5 illustrates
these three bounds.
The main result in Bergemann, Brooks,
and Morris (2015) is that every welfare outcome satisfying these constraints is attainable by some market segmentation. This is
the entire shaded triangle in figure 5. If the
monopolist has no information beyond the
prior distribution of valuations, there will
be no segmentation. The producer charges
the optimal monopoly price and gets the
associated monopoly profit, and consumers
receive a positive surplus; this is marked by
point A in figure 5. If the monopolist has
complete information, then he can charge
each buyer his true valuation, i.e., engage in
perfect or first-degree price discrimination;
this is marked by point B. The point marked
C is where consumer surplus is maximized;
the outcome is efficient and the consumer
gets all the surplus gains over the uniform
monopoly profit. At the point marked D,
social surplus is minimized by holding producer surplus down to uniform monopoly
profits and holding consumer surplus down
to zero.
The main result states that we can make
only very weak predictions about producer
and consumer surplus. It can be understood
as the outcome of a set of metaphorical
information design problems. If an information designer wanted to maximize consumer surplus, she would choose point C.
If she wanted minimize consumer surplus,
or producer surplus, or any weighted combination of the two, she could choose point
D. Any other point on the boundary of the
triangle is the solution to some maximization
problem of the information designer defined
by some preferences over producer and consumer surplus.
The information design problem has a
very clear literal interpretation in the case
where the monopolist knows the consumer’s
valuation. She can then achieve perfect price
discrimination at point B. However, giving a
literal information design interpretation of
point C is more subtle. We would need to
identify an information designer who knew
consumers’ valuations and committed to
give partial information to the monopolist
in order to maximize the sum of consumers’ welfare. Importantly, even though the
disclosure rule is optimal for consumers as
a group, individual consumers would not
have an incentive to truthfully report their
valuations to the information designer, given
the designer’s disclosure rule, since they
would want to report themselves to have low
values.
[...] One can show
that any point where the monopolist is
held down to his uniform monopoly profits with no information beyond the prior
distribution—including outcomes A, C,
and D in figure 5—can be achieved with
the same segmentation. In this segmentation, consumer surplus varies because the
monopolist is indifferent between charging
different prices. [They then give a numerical example, which I omit here.]
[...] Roesler and Szentes (2017) consider a
related information design problem in which
a single buyer can design her own information about her value before she is facing a
monopolist seller. While the analysis of the
third-degree price discrimination proceeds
as a one-player application, the arguments
extend to many-player settings. Bergemann,
Brooks, and Morris (2017a) pursue the question of how private information may impact
the pricing behavior in a many-buyer environment. There, we derive results about
equilibrium behavior in the first-price auction that hold across all common-prior information structures. The results that we obtain
can be used for a variety of applications, e.g.,
to partially identify the value distribution in
settings where the information structure is
unknown and to make informationally robust
comparisons of mechanisms.
