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3 random-explorer brokers
As the number of brokers and consumers in the system grows,
the myoptimal strategy becomes completely impractical
because of the tremendous demands it makes upon on exact knowledge
of the system state, the strategies used by other agents,
and computational power. Therefore the random-explorer strategy
mentioned above becomes the brokers' preferred method.
This strategy still assumes full knowledge of
the system state, but requires vastly less
computational power. Instead of performing an exhaustive
search for the optimal state
Figure 8 shows a simulation run for such
a system. The consumer population and the intial conditions
are identical to those in Fig. 5.
Price wars are still in evidence, but now there are
metastable periods during which configurations and prices hold
steady. During one such metastable period, lasting from
roughly time 283 until time 413,
the brokers have specialized into separate monopolistic niches:
Figure 9 shows a close-up of a somewhat different price war that starts near time 1750. At first, brokers 1 and 3 vie for category 1, leaving category 3 uncovered. At time 1803, their prices have dropped to roughly P = 0.565, at which point brokers 1 and 3 both switch to the niche (101). Prices continue to drop. At time 1829, broker 1 cedes category 1 to broker 3, and the price war continues with broker 1 at (001) and broker 3 at (101). Eventually, at time 1860, broker 2 gets drawn into a price war with broker 1 when broker 1 switches to (011) and broker 2 remains at (010). At this point, the system is in the state ((0.547,011),(0.584,010),(0.547,101)). No two brokers share the same niche, yet apparently the partial overlap between their niches provides sufficient coupling to sustain a price war! At time 1878, broker 3 finds that it can drop its competition with broker 1 by dropping category 3, leaving it a specialist in category 1. While brokers 1 and 2 are fighting it out, broker 3's price drifts back to the single-category-monopoly level. Meanwhile, it is interesting to note that the price war between brokers 1 and 2 does not involve undercutting: broker 1's price is consistently lower than broker 2's price. A quantitative analysis of this apparent coupling would be valuable, and is probably feasible. After some further switches in broker 1's niche, brokers 1 and 2 eventually specialize in categories 2 and 3 respectively, their prices drift up towards the monopolistic optimal point, and the brokers are once again in specialized niches. On average, a universally-adopted random-explorer strategy is more advantageous to the brokers than a universally-adopted myoptimal strategy, as can be seen by comparing Figs. 6 and 10. During the metastable regimes, each broker has a single-category monopoly. The monopoly is inherently unstable because eventually a broker specializing in a slightly less profitable category will undercut a broker that is better off. Since all brokers have this tendency, price wars are still inevitable, but less frequent because it takes random explorers longer to find a good opportunity for undercutting. On the whole, the price wars tend not to be quite as devastating as they are in societies consisting entirely of myoptimals, because they often involve brokers that are only partially in competition with one another, so a broker that is being undercut in price may still retain some customers. Likewise, the consumer population as a whole tends to be better off, as can be seen by comparing Figs. 7 and 11. Broker specialization gives consumers access to pure streams of categorized information from which they can pick and choose. This freedom from junk is counteracted somewhat by the fact that the brokers are niche monopolists, and can charge a somewhat higher price than they could if they were fully or partially competing with another broker. Thus in fact the average consumer utility is highest at the beginning of a metastable period, just after a price war has ended, because then the brokers are specialized but their prices have not yet climbed from competitive to monopolistic levels. This accounts for the upward blips in overall utility and subscription rate just prior to the reestablishment of each metastable period, for example at (approximately) times 480 and 610.
While Fig. 11 is helpful
in describing the average experience of the consumer population,
further insights can be gained by examining the behavior of
individual consumers during the simulation run of Fig. 8.
In order to visualize
the population, we represent each consumer c as a point in the unit cube
corresponding to its interest vector
Figure 12 shows isosurfaces of the field of
consumer utility values (computed for each consumer according
to Eq. 1)
in the space of consumer interests, measured at time 0. At this time,
each broker is offering all three categories. The consumers
with the highest utility, represented by the color red, are
those closest to the corner
The threshold value of
1.875 can be derived analytically. For prices To illustrate other specific features of the consumers' behavior, it is useful to view these same data from another angle. Figs. 13 and 14 display an individualized view of both consumer subscriptions and consumer utilities at time 0 and four other times during the simulation run referred to in Figs. 8 and 11. Note that the orientation of the images is different from Fig. 12: the origin is now the front top right corner. In these two figures, the images on the left consist of tiny cubes, each representing a consumer positioned at its point in the space of consumer interests and colored according to which broker(s) it subscribes to at the given time. Thus the total number of cubes at a given moment in time is given by the ``Total Subscriptions'' curve in Fig. 11. The color scheme is given in the Table 1.
This color scheme represents consumers subscribing to a single
broker as a primary color (red for broker 3, yellow for broker 2, and blue
for broker 1) and consumers subscribing to multiple
brokers as mixtures of the single-broker colors. For example, consumers
that subscribe to brokers 2 and 3 but not broker 1 have
The images on the right represent isosurfaces of consumer utility exactly as in Fig. 12, viewed from the new angle. This view looks through the consumers with zero utility towards the boundary that separates them from consumers with positive utility. Isosurfaces behind the boundary follow its contours closely. For any given frame, the total utility summed over all consumers is shown in the ``Total Utility'' curve in Fig. 11.
|
, the
random-explorer strategy randomly selects
a small number of candidate states, computes the
expected profit for each using a
generalization of Eq. 
, 
, and 
.
The corresponding profits per unit time are
.
At time 414, broker 2 discovers that it can improve its profitability
from 
to 190.88 by switching from (0.585,010) to
(0.580,100), which undercuts broker 1. A brief battle
between brokers 1 and 2 over category 1 ensues, with broker 1
finally giving up and settling for category 2 at time 424.
But just when it looks as though order is going to be restored,
brokers 2 and 3 start to fight over category 1. This quickly
evolves into a price war in which broker 2's niche, (101),
overlaps partially with broker 3's niche, (100).
As the brokers undercut each other in price, their profits
shoot up and down, never going to zero because the sets
of consumers served by the two brokers do not overlap perfectly.
Finally, at time 473, broker 2 cedes category 1 to broker 3, and the three
brokers are once again fully specialized, one to each category.
(Note, however, that the brokers have switched roles since the previous
period of full specialization.) Brokers 2 and 3 now proceed
to raise their prices independently with no interference from one another,
eventually reaching prices that are near-optimal for niche monopolists.
The system holds in this metastable state until the next price war
begins at time 584.
at each moment.
The color codes for the eight possible
niches are shown in the figure.
.
Since the consumers' interest vectors were generated from uniform
distributions of interest values in each category, this leads to a
uniform distribution of consumers in the space of interests.
. They regard
virtually every article as interesting, and are therefore content with
an unfiltered stream of articles. Apart from
statistical fluctuations caused by the random distribution of consumer
interests, the isosurfaces are all normal to the body
diagonal (111), corresponding to equations of the form
.
As one travels along this body diagonal
towards the origin 
,
the consumer utility decreases, represented
by a spectral shift towards blue. Finally, at the
surface defined by 
, consumers'
utilities drop to zero. Any consumer c for which
does not subscribe to any
broker, and thus will have exactly zero utility; the isosurface at which
consumer utility is zero is represented in black (and is thus invisible).
,
one can show that a broker b offering all three categories will
only accept subscribers with
. (For 
, marginal
subscriptions are limited by consumers, not by brokers. The change in
behavior at 
is discussed in Appendix A.)
For broker 3, this boundary value is given by 1.8932, while for brokers 1
and 2, it is 1.8750. Therefore consumers with
subscribe to broker
3, and consumers in the very narrow range
subscribe to one of the others.
and are colored orange.
Consumers that are not subscribed to any broker (and therefore must
have zero utility) are not shown.
