In this appendix, it is shown that there is no pure strategy Nash
equilibrium in our model of shopbot economics whenever 
;
if 
, then the unique Nash equilibrium is such that all
sellers charge the monopoly price v, if 
, then the unique
Nash equilibrium is such that all sellers charge price the competitive
price r (see [8, 9]). The proof proceeds by
contradiction: we assume the existence of a pure strategy Nash
equilibrium and we derive the unique form of such an equilibrium, but
we argue that a strategy profile of this form is not in fact an equilibrium.
Assuming the existence of a pure strategy Nash equilibrium, we
rederive Eq. 2 of Sec. 3.1, which
describes the probability 
that buyers choose
seller s. As in Sec. 3.1,
where 
denotes the demand for seller s by buyers
of type i. Since we are now assuming pure rather than mixed
strategies, it is convenient to define the following functions in
deriving , a quantity whose probabilistic analog
was previously expressed in terms of cumulative distribution functions:
is the number of sellers charging a higher price than s;
is the number of sellers charging the same price as s,
excluding s itself;
is the number of sellers charging a lower price than s.
The quantity is is the product of the probability
that s is among i potential sellers selected at random, and
the conditional probability 
that s is either the lowest-priced
seller or the lucky one in a set of lowest-priced sellers, given a set
of i potential sellers that includes s. The probability
that s is one of a set of i potential sellers selected at random
is simply 
.
The conditional probability is a function of 
, and .
For convenience, fix seller s, and abbreviate 
, for 
. Given 
sellers charging the same price as seller s, the probability
that s is one of the lowest-priced sellers among i potential
sellers, and moreover, that s is randomly selected from among
such lowest-priced sellers, is given by:
where 
denotes the probability that s is one of the
t lowest-priced sellers of i potential sellers. The quantity is computed by determining the number of ways in which to
arrange i-1 of the remaining S-1 sellers, exclusive of seller s,
such that t sellers charge the same price as s, but no seller
charges a price less than s, divided by the total number of ways in
which to choose i-1 other potential sellers from the remaining
S-1: i.e.,
Combining the expressions for , , and 
and simplifying,
we arrive at the following:
Finally, as in Sec. 3.1, let 
and
, which yields the sellers' profit function
with
defined by Eqs. 22 and 25.
We now proceed to derive the unique form of a possible pure strategy
Nash equilibrium. Suppose that the sellers are ordered 
such that the indices j < j+1 whenever equilibrium
prices 
. First, note that equilibrium prices
, since 
yields strictly negative profits,
while 
and 
yield zero profits, but 
yields strictly positive profits. The following observation describes
the form of a pure strategy Nash equilibrium whenever :
at equilibrium, no two sellers charge identical prices.
Suppose on the contrary, that two distinct sellers offer equivalent
prices: i.e., 
.
In this case, seller j stands to gain by undercutting seller j+1
by 
, implying that 
is not in fact an equilibrium
price. In particular,
for sufficiently small values of , namely
whenever
Note that 
just in case i > 1. In other words, it
pays for sellers to undercut only so long as the market contains
buyers of type i > 1; otherwise, 
, which implies that
sellers stand to gain by increasing prices.
Further, we also observe that seller S charges price v at equilibrium,
since for all 
, 
. Therefore, the relevant price vector consists of S
distinct prices ordered such that
.
The price vector 
, however, is
not a Nash equilibrium. While 
is in fact an optimal
response to 
, since the profits of seller S are maximized
at v given that there exists lower priced sellers 
,
price is not an optimal response to 
. On the
contrary, for all sellers 
, seller j has incentive to
deviate, since
It follows that there is no pure strategy Nash equilibrium
in the proposed shopbot model, whenever .
