In this section, we shall demonstrate the
existence of complex analogs of
limit-cycle price wars in systems
with more brokers and categories.
In this case, exact analysis or computation of the
profit landscape 
for a given broker b
becomes very difficult.
For each point y in the
(J+1)B-dimensional state space
, a game-theoretic analysis must
be performed to compute BC
subscription matrix elements 
.
must then be computed from y
and the matrix elements using a generalization
of Eq. 2 (see [Kephart et al.,
1998]).
Consider for example what would be involved in
computing the landscape for a system
with B=3 brokers, J=3 categories, and 10,000 consumers.
The reduced state space is 12-dimensional,
as compared to the full state space 
,
which is 30,012-dimensional.
Suppose that the set of allowed prices is quantized,
such that it runs from 0 to 1 in increments of 0.002.
The optimal interest level in each category is known to
be either 0 or 1, so broker 1 can be in any of
distinct
discrete states, as can brokers 2 and 3. For each of
the resulting several billion discrete states in ,
computation of the landscape 
would require a
game-theoretic computation of B C = 30,000
subscription matrix elements -- an absolutely monstrous task.
A reasonable alternative to computing the entire landscape
is to start the system in some initial configuration, and
simulate its evolution as follows. At any given time step t,
a broker b is randomly selected, and it attempts to maximize
by setting its own parameters to 
,
resulting in a new system state in which the parameters of
all brokers other than b are equal to what they were at time
t-1. For example, in the three-broker system, suppose that
broker 1 is selected at time t. It will try to choose
such that
is optimized. The myoptimal strategy introduced in
the previous section can be implemented as
an exhaustive search over all
4008 possibilities at each time step -- still a lot
of computation, but (just barely) feasible.
Figure 5 shows the resulting price dynamics
for the system just described, with 3 myoptimal
brokers, 3 goods, and 10,000 consumers.
The interest levels 
are generated independently for
each category j and consumer c from a uniform distribution
between 0 and 1. The computational
and transport costs are the same as in the example of the
previous section: 
, V=1. The set of
possible prices is quantized in increments of 0.002, and
each broker performs an exhaustive search among the 4008
possible states. If a consumer perceives two brokers to be
equally attractive, the broker with the lower index is preferred.
Expressing an individual broker's state b using the notation
, we
can now follow the dynamics, starting from an initial configuration
in which each broker is in
the state (0.480,111) (i.e. each has price 0.48 and is interested
in all three categories). In the simulation run depicted in Fig.
5, broker 3 moves first, and chooses
to set its state to (0.560,100). Broker 2 follows, choosing
(0.564,010). Broker 1 is selected next. By choosing (0.586,001),
broker 1 would make a profit 
.
However, the random generation of consumer interests yields
a very slight bias in favor of category 1, and it turns out that
broker 1 can do even better ( 
)
by choosing (0.560,100), undercutting broker 3 and triggering a
price war over the interest vector (100). Meanwhile,
broker 2, in the absence of any other competition for category
2, increases its price to the optimal single-category-monopoly
value: (0.584,010). Note that this is very close to the price
that optimizes 
in a system with an infinite number of
consumers, as computed by maximizing Eq. 7
in the previous section: 
(see also
Fig. 3).
Figure 5: Price war time series for 3 myoptimal brokers, 3 categories,
with , V=1.
See text for other parameters.
Figure 6: Profit for broker 2 as a function of time;
same simulation run as in Fig. 5.
Figure 7: Sum over all consumer's utilities and total number of subscriptions;
same simulation run as in Fig. 5.
Now the high price for category 2 increases its attractiveness, and broker 3 immediately gives up its fight over (100) with broker 1, and now undercuts broker 2 with (0.582,010). With brokers 2 and 3 now specializing in category 2, broker 1 finds it most profitable to offer both categories 1 and 3: (0.564,101). Immediately thereafter, all three brokers join in a price war over the 101 configuration, during which the price is ultimately driven down to 0.540. Now it becomes most profitable to specialize purely in category 2, with price 0.584 (0.584,010). Immediately, a second broker joins into the battle over category 2, causing the remaining broker at (0.540,101) to raise its price, resulting in (0.564,101), instigating yet another price war over the 101 configuration. Although the stochasticity of the order in which brokers make decisions causes some variation in the exact details, the price war cycle continues in this fashion indefinitely.
In summary, after a short initial transient, the system alternates between two price wars: a short-lived one between two brokers vying for the 010 configuration, and a longer-lived one in which all three brokers vie for the 101 configuration. A broker participating in the 010 price war receives its expected profit when it undercuts its competitor, and zero when it is being undercut. During the long 101 price war, a broker will be undercut two thirds of the time, and will thus receive just one third of what it expects. This is illustrated in Fig. 6, which tracks the profit of broker 2 as a function of time.
Price wars are clearly harmful to brokers. In this particular model, they hurt the consumers as well, as illustrated in Fig. 7. During the 010 price war, a single broker is left to offer both categories 1 and 3, which is unsatisfactory to consumers who are highly interested only in one of the two categories. During the long 101 price war, category 2 is completely unavailable to consumers, so the total consumer utility is even lower during this phase than during the 010 price war. Generally, when some or all of the brokers are competing for the same interest vector, a gap is created in the coverage of categories, adversely affecting some consumers.
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.
Consider a second strategy that still assumes full knowledge of
the system state, but requires less
computational power. Instead of performing an exhaustive
search for the optimal state 
, the
random-explorer strategy randomly selects
a few candidate states, computes the
expected profit for each using a
generalization of Eq. 2,
and chooses the candidate that provides the maximal
expected profit. The candidates are biased towards incremental
changes in the prices, but occasional large random jumps
in price and interest vector are permitted.
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:
, 
, 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 interest vector
(101), overlaps partially with broker 3's interest vector (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 near-optimal prices that persist until the next
price war.
Figure 8: Price war time series for 3 random-explorer
brokers and 3 categories. All other parameters are as in
Fig. 5.
Figure 9: Close-up of Fig. 8.
Figure 10: Broker 1's profit for time interval of
Fig. 9.
Figure 11: Total consumer utility and number of subscriptions;
same simulation run as in Fig. 8.
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, the price has dropped to roughly 0.565, at which point brokers 1 and 3 both switch to the interest vector (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 interest vector, yet apparently the partial overlap between their interest vectors 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 coupling would be valuable, and appears to be feasible. After some further switches in broker 1's interest vector, 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 consumers tend to be better off on the whole, as seen by comparing Figs. 7 and 11. For the most part, customers are happiest when the brokers are specialized, because this allows them to receive any articles that they desire without being forced to receive much junk. There are some conditions under which the consumer population as a whole is even better off than during the metastable regime, namely, after a price war, when the brokers are fully specialized but their prices are still rising back towards the equilibrium monopolistic levels. This accounts for the blips in overall utility and subscription rate just prior to the reestablishment of each metastable period, for example at (approximately) times 480 and 610.