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Complex Price and Niche Wars
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
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
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
The following subsections explore the dynamics
of economies in which the brokers employ either
the myoptimal strategy or the random-explorer strategy.
Except for the optimization strategies employed by
the brokers, the parameters are identical in the
two cases studied. In each, there are 3 brokers,
3 goods, and 10,000 consumers.
The interest levels
|
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. 
,
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 
would require a
game-theoretic computation of B C = 30,000
subscription matrix elements -- an absolutely monstrous task.
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.
Since the amount of computation involved in performing
the exhaustive search is just barely feasible, it
is also worthwhile to study a second strategy that is
less optimal (in the myopic sense) but more computationally
feasible. In this random-explorer strategy, only
a small, randomly selected subset of the full range
of possible states is sampled, and the best one
is then selected from that sample.
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.
In what follows, we express an individual broker b's
state 
as a combination of its price
and niche, i.e. 
.