Initially, we focus on the strategic decision-making of the sellers,
by assuming the distribution of the buyer population is fixed and
exogenously determined.
In this case, there are no pure strategy Nash equilibria whenever
; a proof of this
claim is provided in Appendix A. There does, however,
exist a symmetric Nash equilibrium in mixed strategies, which we
derive presently.
Let f(p) denote the probability density function according to which
sellers set their equilibrium prices, and let F(p) denote the
corresponding cumulative distribution function. Following
Varian [19], we note that in the range for which it is
defined, F(p) has no mass points, since otherwise a seller could
decrease its price by an arbitrarily small amount and experience a
discontinuous increase in profits. Moreover, there are no gaps in the
distribution, since otherwise prices would not be optimal -- a seller
charging a price at the low end of the gap could increase its price to
fill the gap while retaining its market share, thereby increasing its
profits. The cumulative distribution function F(p) is computed in
terms of the quantity 
.
Recall that represents the probability that
buyers select seller s as their potential seller. This function is
expressed in terms of the probabilistic demand for seller s by
buyers of type i, namely 
, for 
.
The first component 
. Consider the next
component 
. Buyers of type 1 select sellers at
random; thus, the probability that seller s is selected by such
buyers is simply 
. Now consider buyers of
type 2. In order for seller s to be selected by a buyer of type
2, s must be included within the pair of sellers being sampled --
which occurs with probability 
-- and
s must be lower in price than the other seller in the pair. Since,
by the assumption of symmetry, the other seller's price is drawn from
the same distribution, this occurs with probability 1 - F(p).
Therefore 
. In
general, seller s is selected by a buyer of type i with
probability 
, and seller
s is the lowest-priced among the i sellers selected with
probability 
, since these are i - 1
independent events. Thus, 
, and
A Nash equilibrium in mixed strategies requires that all prices that
receive positive probability yield equal payoffs, otherwise it would
not be optimal to randomize. Thus, assuming 
for all sellers
s, the equilibrium payoff 
, for all prices p. The precise value of 
can
be derived by considering the maximum price that sellers are willing
to charge, say 
. At this price, 
, which by
Eq. 2 implies that 
.
Identifying the expression (p - r) g (p) as the profit function of a
monopolist, this function attains its maximal value (the monopolist's
profit) at say 
. Therefore, for all sellers s, and for all
prices p,
,
and hence,
implicitly defines p and F (p) in terms of one another and g (p),
for all p such that 
.
The function g (p) can be expressed as 
, where 
is the probability density function
describing the likelihood that a given buyer has valuation x. For
example, suppose that the buyers' valuations are uniformly distributed
between 0 and v, with v > 0; then the integral yields g (p) =
1 - p / v, for 
. This case was studied in Greenwald,
et al. [10]. In this paper, we assume 
for all
buyers b, in which case is the Dirac delta function
, and the integral yields a step function 
as follows:
For the assumed distribution of buyer valuations, the monopolist's
profit function is simply p - r, for , which is maximized
at price 
. At this price, the monopolist's profits 
. Inserting these values into Eq. 3 and solving
for p in terms of F yields:
Eq. 5 has several important implications. First of
all, in a population in which there are no buyers of type 1 (i.e.,
) the sellers charge the production cost c and earn zero
profits; this is the traditional Bertrand equilibrium. On the other
hand, if the population consists of just two buyer types, 1 and some
, then it is possible to invert p(F) to obtain:
The case in which i = S was studied previously by Varian [19]; in this model, buyers either choose a single seller at random (type 1) or search all sellers and choose the lowest-priced among all sellers (type S).
Since F(p) is a cumulative probability distribution, it is only
valid in the domain for which its valuation is between 0 and 1. The
upper boundary is 
, since prices above this threshold lead to
Des-creases in market share that exceed the benefits of increased
profits per unit. The lower boundary 
can be computed by setting
in Eq. 3, which yields:
In general, Eq. 5 cannot be inverted to obtain an analytic
expression for F(p). It is possible, however, to plot F(p)
without resorting to numerical root finding techniques. We use
Eq. 5 to evaluate p at equally spaced intervals in 
; this produces unequally spaced values of p ranging from
to v.
We now consider the probability density function f (p). For the
given choice of g (p), the sellers' profits are as follows:
Differentiating both sides of Eq. 8 with respect to p and substituting Eq. 2, we obtain an expression for f(p) in terms of F(p) and p that is conducive to numerical evaluation:
The values of f(p) at the boundaries and v are as follows:
Fig. 2(a) and 2(b) depict the PDFs in the
prescribed model under varying distributions of buyer strategies --
in particular, 
and 
-- when S = 5 and
S = 20, respectively. In both figures, f(p) is bimodal when 
, as is derived in Eq. 10. Most of the probability
density is concentrated either just above , where sellers expect low
margins but high volume, or just below v, where they expect high
margins but low volume. In addition, moving from S = 5 to S = 20,
the boundary decreases, and the area of the no-man's land
between these extremes diminishes.
In contrast, when 
, a peak appears in the distribution.
If a seller does not charge the absolute lowest price when ,
then it fails to obtain sales from any buyers of type S. In the
presence of buyers of type 2, however, sellers can obtain increased
sales even when they are priced moderately. Thus,
there is an incentive to price in this manner, as is depicted by the
peak in the distribution. The case in which 
: i.e., 
is explored in more detail in the next section.
Recall that the profit earned by each seller is 
; this quantity is strictly positive so long as 
.
It is as though only buyers of type 1 are contributing to sellers'
profits, although the actual distribution of contributions from buyers
of type 1 vs. buyers of type i > 1 is not as one-sided as it
appears. In reality, buyers of type 1 are charged less than v on
average, and buyers of type i > 1 are charged more than r on
average, although total profits are equivalent to what they would be
if the sellers practiced perfect price discrimination. In effect,
buyers of type 1 exert negative externalities on buyers of type
i > 1, by creating surplus profits for sellers.
.