Figure 1: Vertically differentiated market economy of software agents.
The model, illustrated in Fig. 1, consists of S sellers and B buyers. Each seller offers a single product or service (henceforth referred to generically as a unit). The unit may have a number of different attributes, each with several discrete possible values, or a continuous range of possible values. However, we shall make the simplifying assumption that the preferences of the buyers are correlated in such a way that there exists a universally agreed-upon mapping that transforms a unit's set of attributes and values, however complex this may be, into a simple scalar quality
Through some mechanism (such as a bulletin board), all buyers are informed about the price 
and quality 
of units offered by each seller s. The qualities may be reported by the sellers (assumed to be honest), or alternatively they could be measured or derived and then reported by an independent, trusted third party.
For simplicity, we assume that time is discrete. (This is not an essential aspect of the model.) At any given moment t, each buyer purchases a unit from at most one of the sellers -- either the one it perceives to be the best choice given its understanding of the prices and qualities, or none if no seller's offer is sufficiently attractive. Note that a buyer's understanding of the prices and qualities may be imperfect and delayed, because it may not have the resources or ability to continuously monitor them.
We model a buyer's decision-making process as follows. Each buyer b has a utility function 
, which is a single-valued function of the perceived price 
and perceived quality 
of a prospective supplier. The buyer b will select the single seller s for which is maximized, so long
as that maximal utility is positive, and purchase a unit at the actual price . If the maximal utility is zero or negative, the buyer does not purchase a unit from any seller. In this paper, we will take the utility functions to have the simple form
where 
is the buyer's price ceiling (the maximum
price it is willing to pay), 
is
its quality floor (the minimum quality it is willing to accept),
is a parameter that ranges between 0 and 1, and 
represents the step function: 1 for x>0 and 0 otherwise.
A buyer with 
is at the extreme limit of price sensitivity:
it will choose the seller with the lowest price, just so long as its
quality is no less than the quality floor . A buyer with
is at the extreme limit of quality sensitivity: it will
choose the seller with the highest quality, just so long as its price
is no more than the price ceiling .
For a given set of assumptions about how the estimated prices and
qualities 
and 
are obtained from the actual
values and , a buyer b can be characterized completely
by its set of three parameters 
.
The sellers are somewhat more complex in behavior. At time t,
several events may occur. First, at most one randomly-selected
seller is given the opportunity to
revise and publicize its price and/or quality .
There is no cost for doing so. Then, each buyer is given the
opportunity to receive and make use of this updated information
(if it wishes) to revise its choice of seller. Finally,
each seller receives from each buyer that has selected it,
and pays a cost 
(assumed to increase monotonically with )
to produce this good and deliver it to the buyer.
When a seller has the opportunity to modify its price and/or quality,
its decision is based upon an attempt to maximize its own profit.
Various assumptions may be made about the nature and accuracy of
information available to the seller, as well as its computational
capability, In this paper we will consider five strategies that cover
a broad range of assumptions:
- A game-theoretic strategy can be used in the limit of
perfect knowledge about the entire market, including
all buyers and sellers and their strategies, and unlimited
computational capability.
- A myoptimal, or ``myopically optimal'' strategy requires
virtually unlimited computational capability and full information
about the consumer population's desires and the prices and qualities
of competitors. However, the strategies of the competitors are unknown,
and the myoptimal seller simply assumes that the status quo will be
maintained (i.e. the other sellers will not change their parameters
before the seller has another opportunity to re-set its parameters).
- A computationally-limited myoptimal strategy,
which is like the myoptimal
strategy except that the computational capabilities are weaker:
an exhaustive search of all possible new parameter settings is replaced
with a tighter search in the neighborhood of the current parameter
set, with a few random forays further afield.
- A trial-and-error strategy, in the extreme limit
where sellers have no knowledge of one another and no knowledge
of the consumer population. In the trial-and-error approach,
new prices are generated randomly and tried out for a short
period of time. If profits are found to improve after the adoption
of a new price, that price is retained. Otherwise, the previous
price is reinstated.
- A derivative-following strategy, in the extreme limit
where sellers have no knowledge of one another and no knowledge
of the consumer population. A derivative follower simply experiments
with its parameters, continuing to change them in the same direction
until the observed profitability is reduced, at which point the
direction of change is reversed.
There are numerous questions that one might ask about
the collective behavior that arises in such a model
under various assumptions about the consumer population,
including the accuracy of the buyers' estimates of current
prices and qualities, the joint distribution of consumer parameter
sets 
, the sellers' parameter
update strategies, etc. Can a reasonable, stable equilibrium set
of prices be reached under conditions of information delay
and uncertainty? Are there reasonable price-adjustment or
quality-adjustment strategies for sellers to
pursue, even when they have imperfect, incomplete information
and/or limited computational capabilities? If an equilibrium is
reached, how favorable is it for the various sellers and buyers
in the system?
This work addresses only a subset of these questions.
We restrict ourselves to the study of
two consumer populations generated
by widely different consumer parameter distributions.
First, we will study a population of buyers that are
quality sensitive, i.e. each buyer
seeks the highest quality
seller whose price does not exceed that buyer's budget.
The second population of buyers
is price sensitive, i.e. each buyer seeks the cheapest
seller that meets that buyer's minimum quality requirement.
For each of these populations, we will perform five
experiments. In each experiment, we will
assume that each seller adopts
a given one of the five price adjustment algorithms outlined
above, and we will observe the resulting price dynamics.
