Figure 1: Shopbot Model. Buyers' numerical labels indicate the
number of price quotes compared before selecting a seller
from which to potentially make a purchase.
We consider an economy in which there is a single commodity that is
offered for sale by S sellers and of interest to B buyers (see
Figure 1). Periodically, at a rate 
, a buyer
b attempts to purchase a unit of the commodity. Each attempted
purchase proceeds as follows. First, buyer b conducts a search of
fixed sample size i, which entails requesting 
price quotes.
A search mechanism (which could be manual or
shopbot-assisted) instantly provides price quotes for i randomly
chosen sellers. Buyer b then selects a seller s whose quoted
price 
is lowest among the i (ties are broken randomly), and
purchases the commodity from seller s if and only if 
, where 
is buyer b's valuation of the commodity.
In addition to the purchase price, buyers incur search costs. The
cost 
of using search strategy i, however, does not enter into
the purchasing decision of the buyers, because buyers must commit to
conducting a search before the results of the search become available.
In other words, search payments are sunk costs. Instead, search costs
affect the choice ( ) of search strategy utilized by
buyers. A buyer b is assumed to periodically re-evaluate its
strategy at a rate 
, where typically,
. Upon re-evaluation, the rational buyer
estimates a price 
that it would expect to pay for the
commodity if it were to abide by strategy i, and then selects the
strategy j that minimizes 
, provided that
. If this condition is not satisfied, then
j = 0: i.e., the rational buyer does not search or participate in the
market at that time.
The buyer population at any given moment is characterized by the
strategy vector 
, in which component 
represents the
fraction of buyers employing strategy i and 
. A seller s's expected profit per unit time 
depends on
the strategy vector , the price vector 
describing
all sellers' prices, and the cost of production 
for seller s.
In particular, 
. where 
is the rate of demand for
the good produced by seller s, in terms of the current price and
search strategy vectors. The demand is the
product of three terms: (i) the overall buyer rate of demand, namely
, (ii) the likelihood that seller s is
selected as a potential seller, denoted 
, and
(iii) the fraction of buyers whose valuations satisfy 
,
denoted 
. Specifically, 
. Without loss of generality, we define
the time scale such that 
, and we then interpret
as seller s's expected profit per unit sold systemwide.
Now, seller s's profits are given by:
