We define the state of the system at time t,
,
as the collection of broker prices 
,
broker interest vectors 
,
and subscription matrix elements 
at time t. Our goal is to understand the
evolution of , given
,
, the costs 
,
, and 
, the consumer value V, and the consumer interest
vectors 
, and
Even in systems of modest size (e.g., J=10 categories, B=10 brokers,
C=1000 consumers), the state space can be quite large: its dimension
is (J+1+C)B, or more than 
for the numbers just quoted. This is
mainly due to the 
elements of the subscription matrix S.
However, it is possible to reduce the dimensionality by factoring out the
degrees of freedom associated with S. This
is done by assuming that each broker and consumer instantly adapts to
changes in its environment by selecting its optimal
set of subscribers or subscriptions. Recall, however, that both
broker and consumer must consent to a subscription.
The conflicting opinions on what constitutes ``optimal'' may
be resolved via a game-theoretic analysis. Thus
the subscription matrix becomes a function
of the remaining system variables
and . Note that, by factoring out the subscription matrix,
we have in effect factored out the consumer population, so that the
``reduced'' system's state is expressed entirely in terms of the brokers'
states. We will denote the reduced state space by 
, and the
subspace associated with broker b by 
. Thus
,
where is the space of possible values of
the (J+1) broker variables 
.
In the reduced system, the broker utility function 
defines broker b's profit landscape, and the system
dynamics is a co-evolutionary process in which each broker b
attempts to maximize its profit by setting the values of
, given the values of 
for all the other brokers
i.
The remainder of this section is devoted to an analysis of the
simplest possible multi-broker system, in which two brokers compete
in a large consumer market in which there is only one type of
information good. Thus B=2, J=1, and 
.
For the moment, assume that
broker 1 charges less than broker 2 for the good ( 
).
In this case, we can apply the definition of the system given in the
last section to get an expected utility per article for consumer c of
The expected profit per article for the brokers is
From equation 2, it is readily seen that, independent
of any other choices it may make, broker i can maximize its expected profit
by setting its interest
level 
equal to 1 if the quantity in parentheses is positive, and
0 otherwise. In other words, a self-interested broker will never have a negative
expected profit, because it always has the option of pulling out of the market
by setting its interest level to 0. In all that follows, we shall assume that
, and if the resulting expected profit per article for
broker i is negative we shall override this, setting 
.
This reduces the dimensionality of the landscape, nominally 4, to 2.
Having established the optimal setting of interest levels by the brokers,
now consider the subscription matrix elements
and 
.
First, note that each term in the expression for
broker 1's expected utility in Eq. 2
can be maximized independently by setting
,
where 
represents the step function: 
for
x>0, and 0 otherwise.
In other words, it is only worthwhile to send articles to consumer
c if c's interest level is sufficiently high that
c's expected payment for interesting articles exceeds the cost
of sending articles to c. Broker 2 is in a different situation.
If c is already subscribed to broker 1, it will never
purchase articles from broker 2 because it charges a higher
price for the same good. Under such circumstances, broker 2
should not attempt to send articles to c because it will
be paying for article transport with no hope of reimbursement.
However, if c is not subscribed to broker 1, then broker 2
should set 
in order to maximize each term of the sum over consumers in
the expression for 
. Putting all of this together, we have
Now consider the situation from the consumer's perspective,
using Eq. 1.
The consumer c will choose the optimal setting of
from among the
four possible choices: (0,0), (0,1), (1,0) and (1,1).
First, suppose that 
. If c
chooses to set 
, then 
and therefore 
, so that 
.
Alternatively, if c chooses to set 
,
then 
, and so
.
Which is the better choice? If 
,
then this quantity always exceeds
because . In this case, 
should
be set to 1, and the value of 
is immaterial
because substituted into Eq. 3 shows
that 
.
However, if 
,
then 
as well, and both
and should be set to zero.
Now consider the other alternative: 
. In this
case the value of is immaterial, ,
and . The value
of matters only if 
,
in which case the optimal value for is 1 if
and 0 otherwise.
Assembling all of the above analysis, we obtain:
In the expression for 
, the step function on
the left represents the veto power of the brokers, and the
one on the right represents the veto power of the consumers.
The expression for 
is similar, except that it is automatically
zero if the consumer already subscribes to the lower-priced broker.
Having established the subscription matrix elements, and the
brokers' interest levels, the only remaining decisions to be
considered are the optimal brokers' prices 
and 
.
Since the number of consumers ,
we may replace the sums in Eq. 2 with integrals
over a consumer population with a distribution of
interest levels given by 
,
with the result:
where
Suppose that is the uniform distribution, i.e.
for all values of 
. Then
substitution of Eq. 4 into
Eq. 6,
and some integration by parts and other
algebra lead to analytic solutions
for the integrals 
and 
.
In the interval 
,
and outside this interval 
. (Despite the
step function, 
is not discontinuous at
.) The solution for is:
in the region satisfied by the constraints
,
,
and 
. Beyond
this region, 
. Again, 
contains no real discontinuity, despite the step function.
The restriction can be removed by
exploiting the symmetry arising from the fact that
there is no inherent difference between the two brokers.
We obtain the profit landscapes for brokers 1
and 2 as a function of the prices and :
where is given by:
Each broker's profit landscape describes the dependence of its expected profitability as a function of the price vector (all of the brokers' prices, including its own). For any given price vector, the myriad self-interested decisions of the consumers and brokers about subscriptions and interest levels are taken into account.
The profit landscape 
is illustrated for the case
, V = 1 in Fig. 2.
Note that there are two distinct humps, the one on
the right
corresponding to 
in
Eq. 10 and the one on the left
corresponding to 
.
The ``cheap'' hump on the right corresponds to a situation in which broker 1 is
cheaper ( ). The ``expensive'' hump on the left
corresponds to the case in which broker 1 is more expensive than
broker 2, but is still able to find customers. This comes about
when broker 2 charges so little that it cannot afford to
keep marginal customers (those with low interest levels
) as subscribers. (Recall that a broker pays for
each article it sends to each of its subscribers, but receives payment only
for those articles a subscriber is interested in.)
Broker 1 can make money by serving the marginal customers
that were rejected by the lower-priced broker 2.
Suppose that brokers use the profit landscape itself to periodically update their parameters so as to maximize their profitability. Assume further that the updates are asynchronous, and that the entire agent population adjusts its subscription matrix elements in a selfishly optimal way. Such an update strategy is guaranteed to produce the optimal profit in the very short term -- up until the moment when the next broker updates its parameters. Thus we call such a strategy ``myopically optimal'', or ``myoptimal'' for short.
If broker 1 is myoptimal, it could derive from its profit
landscape a function 
that gives the
value of that maximizes 
for
each possible 
.
Figure 3 shows a contour
plot of on which is overlaid as a heavy solid line.
(As before, , V = 1.) For 
,
is given by the solution to a cubic equation
involving cube roots of square roots of ; in this region it looks
fairly linear. The ``vertical'' segment
at 
is a discontinuity as the optimal price jumps
from the peak to the peak.
In the region 
, 
,
where 
is a price quantum -- the minimal amount by which
one price can exceed another. For 
, 
.
This is the value of that maximizes , i.e.
it is the price that would be established by a monopolist.
Any further increase in would cut consumer
demand by too much.
If broker 2 also uses a myoptimal strategy,
then by symmetry its price-setting
function is identical to that of broker 1 with
and interchanged. The profit landscape 
and optimal price 
are likewise identical under an
interchange of and .
Figure 2: Double-peaked rofit landscape 
for broker 1 when
, V = 1.
Figure 3: Contour map of profit landscape, with overlaid optimal price function
for 
.
Figure 4: Iterative graphical construction of price-war time series,
using functions and . See text.
Now the evolution of both and can be obtained
simply by alternate application of the two price optimization
functions. I.e., first broker 1 sets its price 
,
then broker 2 sets its price 
, and so forth.
The time series may be traced graphically on a plot of both
and together, as shown in
Fig. 4. Assume any initial price vector
, and suppose broker 1 is the first to move.
Then the graphical construction starts by
holding constant while moving horizontally to the
curve for . Then, is held constant while moving
vertically to the curve .
Alternate horizontal moves to and
vertical moves to always lead to a price
war during which the brokers
successively undercut each other, corresponding to
zig-zagging between the diagonal segments of the curves. The
horizontal or vertical offset
between the diagonals, equal to the amount by which
a broker's price drops on every other iteration, is
. Eventually,
the price gets driven down to 0.388709, at which point
the other broker (say broker 1) opts out of the price war,
switching to the high-priced peak in its profit
landscape. Raising the price to 
breaks the price war, but unfortunately, as Fig. 4
shows, it triggers the immediate start of another one. The brokers are
caught in a never-ending (and, as we shall see, disastrous) limit cycle
of price wars punctuated by abrupt resets.
Classic models of price wars, including those introduced by Cournot and Bertrand [Tirole, 1988], typically have the feature that prices are driven down to a stable value (e.g. the marginal cost in Bertrand's model). However, limit-cycle price wars have been observed previously in a simple model introduced by Edgeworth, in which it assumed that no single firm is able to satisfy the entire aggregate consumer demand [Shubik, 1980]. On constructing the profit landscape for Edgeworth's model, we find that it has two peaks that are qualitatively similar to those of Figure 2. In our case, the ``expensive'' hump arises because the low-priced broker may reject some consumers; this can be regarded as a sort of self-induced capacity constraint.
