In this appendix, we compute the game-theoretic equilibrium
price vector in the case where the buyers are all purely
price sensitive.
Let the S sellers be ordered in the following
fashion: 
, where 
. We assume that the cost to
produce a unit of quality Q is given by 
, which is a
monotonically increasing function of quality. In addition, we assume
that there is a monotonic
functional relationship 
between a buyer's b's price
ceiling 
and its quality floor 
(i.e. these
parameters are perfectly correlated).
First, note that, if all sellers act selfishly in their best interests,
then the components of the equilibrium price vector, if it exists,
will be ordered in just the
same way as the qualities: 
.
This can be seen from the following argument. Suppose the
prices are not so ordered. Then there exist at least two
sellers i and j for whom
but 
, i.e. a higher quality seller
is undercutting a lower quality seller in price. If this
were to happen, the lower quality seller would sell nothing
because any consumer would gladly pay a lower price to obtain
a higher quality item. In such a situation, the lower quality
seller could increase its profit from zero to some positive
value by undercutting its higher-quality
competitor. Thus the supposed equilibrium would not be an equilibrium at all!
By contradiction, we can conclude that
any equilibrium must satisfy the above price ordering.
Due to the correlated orderings of seller quality and price,
the market undergoes a natural segmentation, such that
all buyers within a given segment j will either be served
by seller j (if j's price is acceptable), or will
opt out of the market entirely. As shown in Fig. 14,
market segment j consists of all buyers b for whom
. This can be understood as follows.
For a buyer with a quality floor in this range, seller j
is the lowest quality seller that meets the quality requirements,
and since the sellers' prices and qualities have the same rank
ordering, seller j is also the lowest-priced seller
that meets the buyer's requirements. The boundaries of market
segment j can also be expressed on a price scale by using
the assumed functional relationship between a buyer's price
ceiling and quality floor. As indicated on the righthand side
of Fig. 14, the lower price boundary of the
segment j is defined as 
,
and the upper price boundary is 
.
Figure 14: Seller market segments. Quality and price boundaries are
indicated by solid lines, and labeled on the left and right sides
of the figure, respectively. Seller prices are indicated by dashed
lines, and are labeled inside the rectangle representing each
segment.
Now we are prepared to compute seller j's optimal price 
.
First, note that the optimal price must lie in the range
.
By setting its price at the lower
boundary, seller j will
capture all buyers within its segment, so
there is no benefit to setting the price lower than this.
If it sets its price at or above the upper boundary, it will capture
none of the buyers in its segment.
Note that if the production cost 
exceeds the upper
price boundary 
, then seller j cannot
make a positive profit under any circumstances, and it will
opt out of the market. In what follows, we shall assume that
the production costs are not so high as to prevent any of
the sellers from entering the market.
In order to compute an exact value for ,
we make the simplifying assumption that the quality floor
and price ceiling parameters of the buyer population are
distributed uniformly between ranges 
and 
and 
and 
respectively. Then seller j's
profit as a function of its price 
is simply proportional to:
This quadratic function of is maximized at the value
. If this price is between
the lower and upper boundaries, then
it will be the optimal price for seller j. However, if
the maximum of the quadratic fails to occur within these
boundaries, then the true maximum within the segment
must occur at one of the price boundaries.
In fact, the lower boundary 
is the only option, because at
the upper boundary the number of buyers served (and therefore
the profit) is zero.
A little thought shows that these two
cases can be expressed as a single equation:
which holds for 
, provided that we
introduce a fictitious seller S+1 with quality 
.
In section 4, we set 
and 
.
In this case, the function f that associates buyers' price ceilings
and quality floors is just the identity, and 
.
Thus we obtain:
