In this section, we perform a game-theoretic analysis assuming sellers
are profit maximizers. In particular, we first show that there is no
pure strategy Nash equilibrium, and we then compute and describe the
symmetric mixed strategy Nash equilibrium. Recall that 
; in
particular, the number of buyers is assumed to be very large, while
the number of sellers is a great deal smaller. In accordance with
this assumption, it is reasonable to consider the strategic
decision-making of the sellers alone, since their relatively small
number suggests that the behavior of individual sellers indeed
influences market dynamics, while the large number of buyers renders
the effects of individual buyers' actions negligible. A Nash
equilibrium is a vector of prices 
at which sellers
maximize their individual profits and from which they have no
incentive to deviate [Nash1951]. Throughout this exposition, we
adopt the notation 
, which distinguishes the
price offered by seller s from the prices offered by the remaining
sellers.
Traditional economic models consider the case in which all buyers are
bargain hunters: i.e., 
. In this case, prices are driven down
to marginal cost; in particular, 
, for all sellers s
(see, for example, Tirole Tirole88).
In contrast, consider the case in which all buyers are of type A,
meaning that they randomly select a potential seller: i.e., 
.
In this situation, tacit collusion arises, in which all sellers charge
the monopolistic price, in the absence of explicit coordination; in
particular, 
, for all sellers s.
Of particular interest in this study, however, is the dynamics of
interaction among buyers of various types: i.e., 
.
We begin our analysis with the following observation: at
equilibrium, at most one seller s charges 
. Suppose
that two distinct sellers 
set their equilibrium prices to
be 
, while all other sellers set their
equilibrium prices at the buyers' valuation v. In this case, 
,
for small values of 
, whenever 
, which implies that
is not an equilibrium price for seller s. Now suppose that
two distinct sellers set their equilibrium prices to be
, while all other sellers set their equilibrium
prices at v. In this case, seller s prefers price v to ,
since 
, which implies that is
not an equilibrium price for seller s.
Therefore, at most one seller charges .
On the other hand, at equilibrium, at least one seller s
charges . Given that all sellers other than s
set their equilibrium prices at v, seller s maximizes its profits
by charging price 
, since 
, for small values of
, whenever . Thus v is not an
equilibrium price for seller s.
It follows from these two observations that at equilibrium, exactly
one seller s sets its price below the buyers' valuation v, while
all other sellers set their equilibrium prices 
. Note, however, that 
, for all v' > v,
if 
, implying that all other sellers s' maximize their
profits by charging price v. Thus, the unique form of pure strategy
equilibrium which arises in this setting requires that a single seller
s set its price while all other sellers set
their prices 
.
The price vector 
, with 
,
however, is not a Nash equilibrium. While v is in fact an optimal
response to , since the profits of seller are
maximized at v given that there exists low-priced seller s,
is not an optimal response to v. On the contrary, 
.
In particular, the low-priced seller s has incentive to deviate. It
follows that there is no pure strategy Nash equilibrium in the
proposed model of shopbots.
There does, however, exist a symmetric mixed strategy Nash
equilibrium. Let f(p) denote the density function according to
which sellers set their prices, and let F(p) be the corresponding
cumulative distribution function. 
The event that seller s is the low-priced seller occurs
with probability 
. Substituting this into
Eq. 5, we obtain the demand expected by
seller s:
The precise value of F(p) is determined by noting that at
equilibrium expected profits are equal for all sellers, and moreover
the expected profit level is given by the guaranteed minimum achieved
at price v, namely 
. Now, by setting 
equal to this value and solving for F(p), we
obtain:
Notice that F(p) = 0 for 
defined as follows:
and F(p) = 1 for p = v. Thus, Eq. 10 is valid only in
the range 
.
The functions F(p) and f(p) are plotted in Figure 1.
When 
exceeds a critical threshold 
(equal to 0.1071 for S=5), f(p) is bimodal.
In this regime, as either or S increases, the probability
density concentrates either just below v, where sellers expect high
margins but low volume, or just above 
, where they expect low
margins but high volume; moreover, the latter solution becomes
increasingly probable.
Since itself decreases under these conditions (see
Eq. 11), it follows that both the average price paid by
buyers and the average profit earned by sellers decrease. These
relationships have a simple interpretation: buyers' use of shopbots
catalyzes competition among sellers, and moreover, smaller fractions
of shopbot users induce competition among larger numbers of sellers.
