A single producer periodically (at discrete times 
) generates sets of N articles. It sets a subscription
fee F and a price schedule 
,
where P(k) represents the price it charges for a subset of 
of the N articles.
At a given time t, each of M consumers are informed about
, and they decide whether to subscribe. Then, each
subscribing consumer receives abstracts of all N articles, and
uses them to assess its value 
from reading each article, for
. We assume that these values are generated
randomly according to a distribution 
, where
is the set of parameters that define the distribution
for consumer i. The parameters that represent
consumer i's valuation distribution are themselves generated
randomly prior to time 0 from a distribution 
.
Once the parameters are generated for consumer i,
they remain fixed for the rest of time (though the agents may not
know their true values for a while, if ever).
After assessing the value of each article, a subscriber decides
which articles to purchase. It does this by choosing a set of
articles K to maximize surplus 
. Henceforth we assume that the articles have been sorted
by consumer j so that the are ordered highest to lowest,
and thus the set K consists of the first k=|K| articles.
The subscription decision depends on consumer expectations.
If a consumer believes correctly that its valuations are drawn
from a distribution with parameters , then its expected
surplus from purchasing k articles would be
The consumer can then derive the vector of values 
, the
probabilities that any k is the expected surplus maximizing
number of articles. Then the consumer's optimized expected surplus
is
The consumer should
subscribe 
if and only if the
expectation 
exceeds the subscription fee
F.
If consumer i does not know its own for the articles
offered by this broker, then values 
are the consumer's
beliefs drawn from a distribution 
after reviewing the abstracts, where the 
are the
agent's current best estimate of the valuation parameters
. When the agent purchases and reads articles, it
learns their true values and can then use this sample
information to update its beliefs about the
distribution of article values. Therefore, a good consumer
strategy should take into account the value of learning. For
example, when uncertainty about is high, the consumer
might deliberately subscribe even when its estimated surplus is
less than F, simply to experience more articles to improve its
valuation estimates. Or, having subscribed the consumer might
purchase more articles than would maximize expected surplus from
current reading.
Turning to the producer's problem, it can choose a subscription
fee F and a price schedule 
in each period. These
should be chosen to maximize some function of expected current and
future profits, where current profit can be expressed as:
with 
if consumer i chooses to
subscribe, and 0 otherwise. The cost of delivering 
articles
to consumer i is denoted as 
.
In performing its maximization, the producer must take into
account the effect of and F on the consumer's
subscription decisions, and the effect of on the
distribution of across the set of subscribers. Higher prices
will decrease the number of subscribers and the expected number of
articles purchased, of course. To compute the optimal subscription
fee and price schedule the broker wants to know the distribution
from which the consumers' parameters were
generated, and the consumer strategies for subscribing and
purchasing. Based on its current beliefs about 
and
consumer strategies, the broker can simulate a consumer population
and its responses to various schedules, and then
pick to maximize a value function. In practice, the
broker may not know the consumer type parameters , nor
the consumer strategies. Thus, the broker may choose
to balance current expected profit according to
equation (3), against the increase in expected
future profit from learning about consumer preferences by
observing their behavior when confronted with varying
combinations.
