As noted in the last section, broker b deactivates
category j--so far as it is concerned, at least--by setting the
corresponding interest level 
to 0. In the multi-broker
system, brokers are still free to select any set of categories they wish, so
they may in general have different interest vectors, thereby offering
different selections of categories. Specialization then becomes a
co-evolutionary phenomenon, as brokers move themselves into or
out of direct competition with each other.
When a broker offers more than one category, it may find itself in competition with a different set of brokers in each category. Since each of its competitors is in the same situation, what emerges is a competitive web of brokers linked by partially overlapping category selections. Two brokers may offer disjoint sets of categories, but may still be in indirect competition with each other because of a third broker in partial competition with each.
In this section, we will experimentally study the evolution of a system with multiple brokers. An exact solution of the type discussed in section 3 is not accessible due to the dimensionality of the state space: each broker contributes J+1 dimensions, making calculations prohibitively difficult. Therefore it becomes necessary for brokers to have some sort of dynamical algorithm to help them set prices and select categories.
For the experiments reported here, we choose one broker at random in each time step and, holding all other brokers fixed, allow it to attempt to optimize its profitability by making a fixed number of hypothetical changes to its price and interest vector. Hypotheses are generated in two ways, neither of which uses information about any of the consumers or other brokers: by incremental changes to current parameter values, and (less frequently) by choosing values at random. For each hypothesis, the broker's profitability is accurately determined under the assumption that the other brokers do not change. The broker then sets its parameters to match the hypothesis that gave the best profit. The profitability calculation, described in ref. [7], is feasible because only one broker's parameters are being set. We might think of the evaluation of a few hypotheses as a form of market research. Note, however, that the change actually made by the broker depends crucially on the hypotheses it generated, and that the profitability calculation assumes--falsely--that the other brokers do not change; thus the results of the market research are neither complete nor completely accurate.
We consider an information filtering economy with B=5 brokers,
J=5 categories in all, and C=1000 consumers in the
``all-or-nothing'' distribution with 
,
for which the single-broker category contour plot is shown in Fig.
6. We will take
3 points in that figure corresponding to optimal
number of active categories 
and measure the category coverage of each broker as a function
of time.
The category coverage 
of broker b is the number of
categories it offers at time t:
The category coverage is the quantity that most closely
corresponds to the number of active categories in the single-broker
system ( 
).
Fig. 7 shows for each of the five
brokers b in the system. The costs are 
,
V = 1, corresponding to a point in the 
region of the single-broker system. Each broker starts out offering
all five categories, but all but one quickly specialize to a single
category by time 
. The last broker continues to offer
all five categories until
, when it suddenly specializes as well.
Figure 7: Category coverage vs. time t for brokers
when extrinsic costs are .
A detailed
examination of the timeseries shows that the late-specializing broker,
though offering all articles in all categories, was in fact not
selling any of them in any category, not even the category in
which
it held a monopoly. It could not find any customers that were
sufficiently interested in all five categories. The broker was
behaving as if it were in the spam regime, with disastrous consequences.
Nevertheless, it was eventually able to stumble into the one niche
not filled by any other broker, at which time its profitability rose
from 0 to the same value maintained by the other
brokers. Fig. 8 shows the profit of all five
brokers vs. time. The late-specializing broker is the one whose profit
curve rises abruptly from 0 at . Also note the
initial wild fluctuations in broker profit, with attendant
fluctuations in category coverage. These are caused by a
sorting-out period in which brokers are collectively forming,
breaking, and reforming price-and-category wars of the type
reported in ref. [7].
At this setting of extrinsic costs, a single-broker system prefers to offer only one category. In the multi-broker system, we find that through an initial transient of price-and-category wars, the brokers succeed in spontaneously achieving full specialization, with one broker per category and one category per broker. The same result was found in experiments for which B and J were both set to 8 or to 10.
Figure 8: Broker profit 
vs. time t for brokers
when extrinsic costs are .
Next, we choose 
.
At this setting of extrinsic costs, an isolated broker would
prefer to offer two categories. If that preference persists
in the multi-broker system, then we must expect brokers to find themselves
in direct competition. Fig. 9
shows the category coverage vs. time for this case. (note the
different time scale from Fig. 7). The initial
sorting-out period shows up here as well, but by 
,
the brokers have again fully specialized, with one broker covering
each category. The brokers' profit indicates the dramatic effect of
specialization. Fig. 10 shows broker 1's profit
vs. time; all the other brokers follow similar curves. The periods
during which 
is steadily increasing or maximal correspond to
periods during which broker 1 has a monopoly on one category or other,
and is gradually finding the optimal price. These are abruptly
terminated when another broker discovers that it can increase its
own revenue by moving in on broker 1's category and undercutting its
price. As Fig. 10 demonstrates, this highly
unprofitable state of affairs ends when all brokers specialize.
Instead of wild fluctuations, each broker's profit is pegged at a
maximal value.
Figure 9: Category coverage vs. time t for brokers
when extrinsic costs are .
Finally, let us look at 
, for which
in the single-broker system. The time evolution of
the each broker's category coverage is shown in
Fig. 11. This is well into the spam regime, where
an isolated broker prefers to offer all five categories. The brokers
did not specialize. They remained locked in a competitive struggle
through time t = 10000, when the simulation was stopped. There is no
reason to expect that the system would ever have settled down. As Fig.
11 shows, however, competition did cause the
brokers to give up some of their categories, leading to a
time-averaged coverage of about 2.1 categories per broker.
Figure 10: Broker 1's profit vs. time t
when extrinsic costs are .
In the experiments reported in this section, we found that under the
proper circumstances our simple model system was able to achieve spontaneous
specialization. This is despite the fact that none of the brokers was
``aware'' of even the notion of specialization, let alone of its
potential benefit.
Specialization was brought about solely by market forces as they
made themselves felt through each broker's profitability. For the
first experiment, 
, specialization is attributable to the
preference of offering a single category even without competition. In
this case, the most notable observation is that, despite the
simplicity of their dynamics, the brokers succeeded in finding the
niches at all. In the second experiment, 
, for which 
in the single-broker system, a single-broker system would not have
specialized fully; it was only due to added disvalue of
competition in price wars that brokers found it
preferable to offer one category instead of two. Finally, when
the extrinsic costs were well into the ``spam'' regime,
( 
), the disvalue of competition forced the brokers'
time-averaged category coverages from 
down to about 2; but
the residual competition was still enough to prevent brokers' category
selections from stabilizing.
Figure 11: Category coverage vs. time t for brokers
when extrinsic costs are .