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Specialization without competition
As we will see, a crucial first step in understanding specialization in a multi-broker economy is to consider the case of a single broker. In this section we analyze the profitability of a single broker as a function of the number of categories J that it carries, in the limit of a large consumer population ( The broker utility W is given by (cf. ref. [6]):
The product Similarly, the utility for consumer c is given by:
Here,
These can be simplified somewhat. First we note that the broker's utility is maximized by setting
We can summarize a consumer's interest vector by its effective interest level,
If both brokers and consumers
set their subscriptions independently to optimize
their utilities (i.e., they pursue a game-theoretic solution),
it can be shown (cf. ref. [6]) that the subscription
vector element
where
The only remaining free variables are the number of active categories J
and the broker's price P. To determine the values of P and J that
optimize the broker's profitability, it is necessary to specify the consumers'
effective interest levels First let us re-express the broker's profitability in terms of its expected income per consumer I(J,P):
In a sufficiently large consumer population, the first term
dominates for any finite source price
|
). Since B=1, we drop the broker index b: the broker's sale price is denoted by P, its interest vector is 
with elements 
. The subscription matrix S is now a vector with elements 
.
is the probability, per published article, that the broker will buy an article in category j. The term in square braces is the expected net revenue per article in category j. This breaks up into 
, the price the broker pays to the source for the article, and the sum over consumers c, which gives the total revenue from sales of the article to consumers. For each subscribing consumer (i.e., 
), the broker pays 
to transport the article, while the consumer buys it with probability 
, paying P to the broker if it does so.
is the probability, per published article, that the consumer will be offered an article in category j. For each article it is offered, it pays computation cost 
to decide whether to buy it, which it does with probability 
for each category in which its net revenue is positive, and 
for all others. Note, however, that categories j for which 
to preserve the normalization 
.
. This is the probability that a consumer will be interested in a randomly selected article offered by the broker. A little algebra now gives simplified forms of the broker and consumer utilities:
, and is given by
is the step function: 
for
x>0, and 0 otherwise. The first term represents the veto power
of the brokers, who only want subscribers for whom the expected
income 
is greater than the cost 
that exceeds cost 
,
and an ``all or nothing'' distribution in which a consumer is either
completely interested ( 
) in a category with
probability 
, and otherwise completely uninterested
( 
).
