Today's shopbots are used by only a small fraction of
shoppers. This is due at least in part to the fact that many
potential users are unaware of the existence of shopbots, and others
do not know where to find them or how to use them. One way of
modeling buyers who do not use shopbots is to assume that such
uninformed
users are buyers of type 1, for which they incur only fixed cost
. This establishes a lower limit on the fraction 
, which we
denote 
. In particular, represents the fraction of uninformed
buyers who guarantee the sellers a strictly positive profit surplus.
In order to explore the outcome of some proportion
of buyers failing to adopt low-cost search methods (perhaps
due to ignorance about shopbots' existence or about how to
use them), we now impose a lower limit on , denoted
.
Fig. 6(a) depicts the result
of imposing 
,
with linear search costs 
.
Allowing the system to evolve from initial strategy vector
,
the system reaches an equilibrium in which
only types 1 and 4 co-exist, with 
and
, rather than types 1 and 2 as was the case
in the traditional economic setting that was analyzed in
Sec. 3.2.
In numerous experiments with
bounded below and
linear search costs,
we have observed that the final
equilibrium always consists of a mixture of buyer types
1 and i, where i is not necessarily 2, as it
must be when is determined in an entirely
endogenous fashion. The strategy i depends on
the values of
and 
. Table 1 illustrates the
dependence of the strategy i that mixes with
strategy 1 upon
and the incremental cost . Higher
values of
lead to higher equilibrium strategies i
(more extensive search) while
higher incremental costs lead to lower
equilibrium strategies i (less extensive search).
For the table entries
and 
,
multiple equilibria are obtained. In these cases,
the initial setting of
the strategy vector determines which equilibrium obtains.
Table 1: Search strategy or strategies that co-exist with
type 1 search strategy, as a function of
and incremental cost .
Figure 6: (a) Evolution of indicated components of
buyer strategy vector 
for 5 sellers, with
linear search costs 
and .
Starting from 
indicated
in the text, evolves towards
equilibrium with only types 1 and 4
present. (b) Two-dimensional cross-section of basin of attraction
for .
The effect of initial conditions on equilibrium selection
in the case
is illustrated in Fig. 6(b). Four equilibria
are possible, all of the form 
, for i=2,3,4,5.
The set of initial conditions leading to equilibrium i -- its
``basin of attraction'' -- forms a contiguous, smoothly bounded region,
a two-dimensional cross-section of which
is depicted in Fig. 6(b).