We consider an economy
in which there is a single homogeneous good
that is offered for sale by S sellers and of interest to B buyers,
with 
. Each buyer b generates purchase orders at random
times, with rate 
, while each seller s reconsiders (and
potentially resets) its price 
at random times, with rate
. The value of the good to buyer b is 
; the cost of
production for seller s is 
.
A buyer b's utility for the good is a function of price:
which states that a buyer obtains positive utility if and only if the
seller's price is less than the buyer's valuation of the good;
otherwise, the buyer's utility is zero.
We do not assume that buyers are utility maximizers; instead we assume
that they consider the prices offered by sellers using one of the
following strategies:
The buyer population consists of a mixture of buyers employing one of
these strategies, with a fraction 
using the Any Seller
strategy and a fraction 
using the Bargain Hunter strategy,
where 
. Buyers employing these respective strategies
are referred to as type A and type B buyers.
A seller s's expected profit per unit time 
is a function of
the price vector 
, as follows:
,
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 of the buyers selecting seller s as
their potential seller, and the likelihood that seller s's price
does not exceed the buyer's valuation . If 
, and if 
denotes the probability that seller s is selected, while 
denotes the fraction of buyers whose valuations satisfy 
, then
.
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
discuss the functions and g (p) presently.
The probability that buyers select seller s as their
potential seller depends on the buyer distribution 
as follows:
where 
and 
are the
probabilities that seller s is selected by buyers of type A and
B, respectively. The probability that a buyer of type A selects a
seller s is independent of the ordering of sellers' prices: 
. Buyers of type B, however, select a seller s
if and only if s is one of the lowest price sellers. Given that the
buyers' strategies depend on the relative ordering of the sellers'
prices, it is convenient to define the following functions:
is the number of sellers charging a lower price than s,
is the number of sellers charging the same price as s, excluding s itself, and
is the number of sellers charging a higher price than s.
Now a buyer of type b selects a seller s iff s is s.t. \
, in which case a buyer selects a particular
such seller s with probability 
.
Therefore,
where 
is the Kronecker delta function, equal to 1
whenever i = j, and 0 otherwise.
The function g (p) can be expressed as 
, where 
is the probability density function
describing the likelihood that a given buyer has valuation x. For
example, suppose that the buyers' valuations are uniformly distributed
between 0 and v, with v > 0; then the integral yields g (p) =
1 - p / v. This case was studied in Greenwald, et al. [20].
In this paper, we assume 
for all buyers b, in which case
is the Dirac delta function 
, and the
integral yields a step function 
as follows:
The preceding results can be assembled to express the profit function
for seller s in terms of the distribution of strategies and
valuations within the buyer population. Recalling that for
all buyers b, and assuming 
for all sellers s, yields
the following:
where
