We consider an economy
in which there is a commodity 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 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 a good is a function of price:
This states that a buyer purchases a good from a given seller if and only if the seller's price is less than the buyer's valuation of the good; if price equals valuation, we make the behavioral assumption that a transaction occurs. 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;
. 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 
:
where 
is the rate of demand for the good produced by
seller s. This rate of demand is the product of the overall buyer
rate of demand 
,
the likelihood of a given buyer selecting seller s as their
potential seller, 
, and the fraction of buyers
whose valuations satisfy 
, denoted 
:
Note that 
, where 
is
the probability density function describing the likelihood that a
given buyer has valuation x.
If 
for all buyers b, then is the Dirac delta
function 
, and the integral yields a step function
:
Without loss of generality, we define the time scale such that 
. It follows that 
, and
is seller s's expected profit per unit sold systemwide.
The probability that buyers select seller s as their
potential seller depends on the distribution of the buyer population,
namely 
. In particular,
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 select a
seller s is independent of the ordering of sellers' prices; in
particular, 
. 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, and
is the number of sellers charging the same price as s, excluding s itself.
Now buyers of type b select 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 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. In particular, assuming (as
we do from here forward) that all buyers share the same valuation v,
and all sellers share the same cost c, then
where
