Let us perform the same analysis for the all-or-nothing distribution,
in which a consumer's interest in a category is either 
or
, with probability 
. Again
assuming that articles are published uniformly in all active categories
( 
for all j), we again found an analytic solution
for 
as a function of 
and 
.
Figs. 5 and 6
show contour plots of the optimal number of categories
, for the two bias values 
and 
, repectively.
One thing to note in Fig.5 is the small
``jump'' in the contours at 
, 
. This
does not appear to be an artifact, though we do not at this time have
an interpretation of it. In general, though most of the contours are
nearly parallel, they are not exactly so.
In Fig. 6, the bias has caused
the spam regime to be much reduced in area, with a similar shift in
all the other contours. This is due to the overall reduction in the
number of articles a typical consumer would buy, caused by the change
in bias. This bias value, which corresponds to the average consumer being
very interested in 10% of the categories and not
at all interested in the remaining 90%, is perhaps not an
unreasonable distribution for an information filtering context.
The shape of the contours differs only in detail from the uniform
distribution of Fig. 4. In all cases, there
is a large region of 
space in which a broker prefers
to specialize to a small number of categories, even though it has a
monopoly over any category it chooses to offer. The fact that such
dissimilar distributions of consumer interests give rise to
qualitatively similar results suggests that contours like
Figs. 4, 5 and
6 would be found for a wide class of
consumer interest profiles. And as we will see in the next section,
the optimal single-broker plays a central role in the behavior
of multi-broker systems.
Figure 5: The optimal number of categories for the broker
to offer as a function of and , for the all-or-nothing
distribution of consumer interests with . As before, V=1.
How can we understand these contours? In the region where 
, as we
have already noted, the extrinsic costs are too high to permit any value
of price P that simultaneously gives profit to the broker and
positive net value to the consumers. But why should there be large
regions for which is nonzero but small?
Consider for the moment the
interest profile distribution that gives rise to
Fig. 6. As the number of categories offered
by the broker grows, the likelihood of any given consumer being
interested in all categories decreases as 
. In general,
for the typical consumer, additional categories means additional junk.
This means there is an overall decrease in the percentage of articles
each consumer pays for, regardless of the price P. Since the
broker has to pay for each article it sends, there comes a point
at which the transportation costs outweigh the expected revenue from
sales to consumers. Similarly, each consumer has to pay
to evaluate whether it is interested in an article. At some point, the
percentage of positive evaluations becomes so low that it fails to
overcome the cost of evaluating. Reducing and reduces the
disvalue of sending (and receiving) junk, pushing out the
threshold at which the cost of
junk outweighs the value of good information, and thereby increasing
. As the extrinsic costs are further reduced, they become so low
that the broker is assured of a profit on every additional category it
offers, and the system finds itself in the spam regime where
.
Figure 6: The optimal number of categories for the broker
to offer as a function of and , for the all-or-nothing
distribution of consumer interests with . As before, V=1.