A related realm in which economically-motivated software agents are likely to play a significant role is information bundling. To a much greater extent than is possible in print, an electronic publisher can unbundle an issue and sell individual articles, or re-bundle articles together into personalized sets. Negotiation over the price and composition of bundles is likely to become a natural application for software agents.
Just as in the information filtering domain, agents will need to set or
interpret a number of price and product parameters simultaneously. For
example, suppose that N items are available for inclusion in a
bundle. 
Then each seller could offer 
different products.
Rather than requiring the seller to choose one of them (as we did
in the information filtering model), we permit the seller to offer
all of these choices. This requires the seller to maintain prices
for all possible products.
The seller can reduce the complexity of managing so many separate
prices by introducing a pricing structure. One simple pricing scheme
has just a single parameter: all items are priced identically at
, with no volume discounts. Thus the consumer pays 
to purchase n of the N possible items. Another one-parameter
pricing scheme amounts to what is traditionally meant by ``bundling'':
all N items are included in the bundle at a fixed bundle price of
F -- regardless of how many the consumer actually wants. An example
of a two-parameter price structure is the ``two-part tariff'' scheme,
in which the charge for n items is 
, i.e. there
is a per-item charge of , plus a fixed fee F that is
assessed if any items are purchased at all. If prices are based solely
on the number of items in the bundle, then the most general
price structure is a nonlinear scheme with N parameters --
potentially an arbitrary monotonically nondecreasing function of the
number of items purchased.
Suppose that there is just a single seller. Then its objective is to choose a price structure and the optimal price parameters for that structure. If the buyers' valuations of that seller's wares are known by the buyers and by the seller, this becomes a standard optimization problem. In general, the seller can extract greater profits from more complex price structures [1].
However, if the seller does not know the buyers' valuations a priori, it must use an adaptive procedure to adjust its price parameters. We have adapted the ``amoeba'' optimization method to this problem [23]. Starting from an arbitrary setting of price parameters, amoeba selects new parameters, measures the profit obtained at those parameters for a while, and uses these measurements to guide the choice of the next set of parameters. It is typically very successful at finding optimal or near-optimal price parameters, but while it is exploring the parameter space it may visit unprofitable regions. Thus it is important to minimize the number of evaluations required to attain near optimality.
Figure 9: Average cumulative profit vs. time for five different pricing
structures using the amoeba algorithm.
Figure 9 shows the time-averaged profit extracted by a monopolist seller that uses amoeba to learn the optimal setting of its parameters. The five curves represent various pricing structures ranging in complexity from 1 to 10 parameters. Although the nonlinear pricing structure with 10 parameters yields the highest profit asymptotically, it takes much longer to learn than the simpler pricing structures. If the time scale on which the market changes is shorter than the amount of time it takes the amoeba to conduct 1000 or more evaluations, the two-part tariff scheme may be preferable [2].
If the buyers themselves do not know their own valuations until
they have a chance to sample the seller's wares, and if they
allow for the possibility that the valuations shift in time,
then the problem becomes much more complex. Suppose that the
seller has adopted a two-part tariff scheme, and that buyers
must pay the subscription fee F prior to examining the items
and deciding how many to purchase at per item.
Suppose as well that the consumers use a simple form of
maximum likelihood estimation to estimate the likely value to
be obtained by subscribing.
In this scenario, we have observed an interesting ``leakage'' effect.
If the seller sets F and to the values that are optimal for
perfectly informed buyers, profits decrease over time. Profit leakage
occurs because, as buyers estimate the average value of the seller's
wares, statistical errors sometimes lead a buyer to the false
conclusion that the
subscription fee F is not worthwhile. Once that buyer stops
subscribing, it will stop receiving the information that it would need
to revise its estimate, and therefore it will become disenfranchised
permanently unless the seller lowers its prices.
One solution is for the buyers to periodically resample what the seller has to offer. However, even if buyers do not do this, the seller can still prevent profit erosion by lowering its prices. Once disenfranchised buyers re-enter the market, they will soon discover that their previous valuations were overly pessimistic, and they will (temporarily) stay in the market even if prices are raised back to previous levels. Unfortunately, as illustrated in Fig. 10, the amoeba algorithm as it is typically described is unable to discover how to manipulate prices dynamically so as to maintain profits [16]. Because the standard amoeba algorithm assumes that the optimization problem is not changing with time, it fails to notice that prices that were once profitable may no longer be after a while. This causes it to get stuck at a fixed setting of parameters, leading to profit erosion.
Figure 10: (a)Profit 
(solid line; normalized to ``ideal'' value of
0.41367) and proportion of subscribed consumers m (dashed line)
vs. time (in subscription periods) and (b) f (solid) and
(dashed) vs. time when the producer uses amoeba for online learning.
Horizontal dashed lines indicate the optimal f and values
for fully informed consumers. The market consists of M=10000
consumers and one seller offering N=10 articles per subscription
period.
However, we have implemented a simple modification of the amoeba that recognizes the dynamic nature of the optimization problem. This version adjusts prices dynamically. As illustrated in Fig. 11, the price fluctuations are large and rapid, but they are generated in such a way that steady long-term profits are maintained. Ironically, it is precisely the fact that the modified amoeba recognizes the dynamic nature of the market that causes it to interact with buyers in such a way as to stabilize the market in the long term.
Figure 11: (a)Profit (solid line; normalized to ``ideal'' value of
0.41367) and proportion of subscribed
consumers m (dashed line) vs. time (in subscription periods) and
(b) f (solid) and (dashed) vs. time
when the producer uses the modified amoeba algorithm for online learning.
The parameter settings remain unchanged from Fig. 10.