A typical shopbot such as the one residing at www.DealPilot.com permits users to choose the number of sellers among whom to search. Since the service is free for buyers at present, and moreover, since the search is very fast -- DealPilot searches prices at a few dozen book retailers within about 20 seconds -- there is only a mild disincentive not to request a large number of price quotations. Thus, the effective search cost is only weakly dependent on the number of searches. One way to model weak dependence on the number of searches is via a nonlinear search cost schedule:
where the exponent 
is in the range 
.
Note that 
yields a linear search cost model, while
yields a search cost that is independent of the number of
searches for j > 1.
Suppose that buyers periodically (but at random
times) re-evaluate their search strategies and choose
the strategy j that minimizes 
,
where 
is their estimate of the average
price they are likely to pay when using search strategy
j. In determining ,
one possibility is that the buyers (or an agents
acting on buyers' behalf) use historical
data on sellers' prices to compute their estimates.
We assume here, however, that the buyers are
perfectly knowledgeable about the
sellers' marginal production cost r and the
current state of the strategy vector 
, and
thus they integrate Eq. 11 numerically to compute
. As the buyers modify their
strategies in this manner, we assume further that the
sellers monitor , and instantaneously re-compute
the symmetric price distribution f(p)
according to which they randomly choose their prices.
We can approximate this evolutionary process by a discrete time
process in which, at each time step, a fraction 
of the buyer
population is given the opportunity to switch to the
optimal strategy. Then the strategy vector evolves
according to:
,
where j is the strategy that minimizes
and 
represents the
Kronecker delta function, equal to 1 when i=j and
0 otherwise.
Fig. 5(a) illustrates the evolution
of the components of in a 5-seller system
when 
is completely endogenous ( 
),
the search costs are linear ( ,
, and 
), and the value of
is 0.002. Recall that according to
Burdett and Judd [2],
must evolve toward an equilibrium consisting of
a finite number of type 1 and type 2 buyers. Indeed, this
does occur, but what is most interesting is the trajectory
of the on its route toward equilibrium.
Figure 5: (a) Evolution of indicated components of
buyer strategy vector for 5 sellers, with
linear search costs 
.
Final equilibrium oscillates with small amplitude around
theoretical solution involving a mixture of strategy
types 1 and 2. (b) Evolution of indicated components of
buyer strategy vector for 5 sellers, with
nonlinear search costs 
.
Final equilibrium oscillates chaotically around a
mixture of strategy types 1, 2, and 3.
Initially, 
.
In this situation, the favored strategy is type 3, and
so 
begins to grow at the expense of , 
and 
.
However, as diminishes, the total amount of search
in the system diminishes, and f(p) flattens and shifts
in such a way that eventually the favored
strategy shifts from 3 to 2. Thereafter, grows
at the expense of and the other components.
In this simulation, near but imperfect equilibrium
is achieved: due to the finite size of
(equal to 0.002),
there are small oscillations in around an average
value that is close to the theoretical value of
0.9641721. This value can
be derived by identifying the value of
corresponding to in
Fig. 4(b).
In Fig. 4(b), there
is a second value of satisfying , near
. However, this is the unstable
equilibrium, and as discussed in the previous section
it marks the boundary between two
basins of attraction, one in which the final equilibrium
is 
, and the other
in which 
.
The derivation of an equilibrium in which
only type 1 and type 2 strategies could co-exist was founded on
the assumption that search costs are linear in the amount of
search. In order to investigate the effect of nonlinear
search costs that grow only weakly with the amount of search,
we run the same experiment, in which all parameters are
identical except for the exponent , which is reduced
from 1.0 to 0.25. Fig. 5(b) depicts the
result. Interestingly, in this case the system evolves to an equilibrium
in which types 1, 2 and 3 co-exist: oscillates
around the value (0.0217,0.5357,0.4426,0.0000,0.0000)
in a way that appears to be chaotic; it remains to
conduct further tests of this phenomenon.
While these oscillations are an artifact of the finite size
of , and would likely disappear in the limit 
,
they hint at the fact that the system might still undergo
large-scale nonlinear and possibly chaotic oscillations if the
buyers were to revise their strategies synchronously
rather than asynchronously.
