We study an economy
in which there is a single homogeneous good offered for sale by S
sellers and of interest to B buyers, with 
. Each buyer b
generates purchase orders at random times, at rate 
, and
each seller s reconsiders (and potentially resets) its price 
at random times, at rate 
. The value of the good to buyer b
is 
, and the cost of production for seller s is 
.
A buyer b's utility for a good is a risk-neutral function of its price as follows:
We do not assume that buyers are necessarily utility maximizers.
Instead, we assume that they use one of a set of fixed sample
size search rules in selecting the seller from whom to purchase. A
buyer of type i (where 
) searches for the
lowest price among i sellers chosen at random from the set of S
sellers, and purchases the good if that seller's price is less than
the buyer's valuation . (Price ties are broken randomly.)
A few special cases are worth mentioning. A buyer of type i=0
simply opts out of the market without checking any prices. Buyers of
types i=1, i=2, and i=S have been referred to in previous
work [9, 10] as employing the Any Seller, Compare
Pair and Bargain Hunter strategies, respectively; the latter
corresponds to buyers who take advantage of shopbots. 
The buyer population is assumed to consist of a mixture of buyers
employing one or another of these strategies. Specifically, a fixed,
exogenously determined fraction 
of buyers employs strategy i,
and 
.
The profit function 
for seller s per unit time
is determined by the price vector 
, which describes all
seller's prices:
,
where 
is the rate of demand for the good produced by
seller s. This rate of demand is determined by the overall buyer
rate of demand, the likelihood that the chosen seller's price
will not exceed the buyer's valuation , and the likelihood of
buyers selecting seller s as their potential seller. If 
, and if 
denotes the probability that
seller s is selected, while 
denotes the fraction of buyers
whose valuations satisfy 
, then 
. Note that the function g (p) can be
expressed as 
, where 
is the probability density function describing the likelihood
that a given buyer has valuation x.
Without loss of generality, define the time scale s.t. 
.
Now 
is interpreted as the expected profit for seller
s per unit sold systemwide. Moreover, seller s's profit is such
that 
. We
analyze the functions and g (p) in the next section.