If the consumer interest levels 
are distributed uniformly,
the broker profitability per consumer I(J, P) can be calculated
analytically in the limit 
, if we make the
assumption that 
for all j, i.e., all categories are
equally prevalent. This makes all categories equally profitable,
effectively replacing the question of which categories
to offer with the simpler one of how many to offer.
Fig. 2 shows the result in the form of
profit cross-sections I(J, P) vs. P for 
, for a
characteristic choice of extrinsic costs 
, 
and consumer
value V.
Figure 2: I(J, P) vs. P for J=1, 2, 3, 10, when 
.
Focusing for the moment on the curve for J=1, we observe that the
profit per customer has a single peak at 
. As P increases
beyond 
, the broker's profit is reduced as fewer and fewer consumers
purchase articles. On the other hand, as P decreases away from , the
broker's profit per customer per article is less.
More important, however, is the fact that as the number of categories
J increases, the profitability value at the peak changes drastically.
The analytic solution shows that 
is a strictly decreasing
function of J, and 
as
. In other words, the broker can always charge
a higher price if it offers fewer categories. However, the number of
consumers actually buying articles, and hence profit per consumer,
may increase or decrease with the
number of offered categories.
For J = 1, the peak profit per customer is 
,
while for J = 10, the peak value is 
. Evidently,
for optimal profit, the broker should not automatically offer as many
categories as possible. In this case, for example, the best situation is J=2,
because it has the highest peak, even though the overall curve is not
universally higher than other curves for arbitrary P.
Figure 3: 
vs. J for 
, 
and
V=1 for all three curves.
The dependence of the broker's profit on the number of categories is summarized
in Fig. 3. It shows the peak profit per customer
vs. J, for three different settings of and , with V=1.
For 
and 
, the broker's profit is
maximized for a small number
of categories (J=1 and J=2, respectively). For 
, however,
profit increases monotonically with J. We may call this the ``spam'' regime,
in which the extrinsic costs of computation and transportation are so low that
the broker can always make more profit by offering more categories.
As Fig. 3 shows, for each setting of extrinsic costs
costs and , there is an optimal number 
of categories
that maximizes broker profit.
Fig. 4 shows a contour plot of as a function of
and , again for V = 1. Note that, for 
,
. In this region, it is impossible to simultaneously satisfy
the consumers' and broker's conflicting desires as indictated in
Eq. 4. Even for a maximally interested consumer
( 
), 
for all positive prices P.
Figure 4: The optimal number of categories for the broker
to offer as a function of and , for the uniform distribution
of consumer interests. As before, V=1.