The PDF and the cumulative distribution function
(CDF) for the equilibrium prices appear in Figure 1. Firm 2
has greater motivation to compete for s123 than Firm 1, and
because Firm 3 can only capture s123, the competition
between Firms 2 and 3 is fairly intense. The low prices Firm
2 quotes while trying to capture s123 make it easier for Firm
2 to sell to s12. Thus, there is no guarantee that Firm 1 will
capture any switchers, even at its minimum price, because
the other two firms already compete for s123 at lower prices.
Therefore, Firm 1 is less inclined to discount. The more
intense competition between Firms 2 and 3 makes the lower
bound of Firm 1’s price range move up to the point in equilibrium
at which Firm 1 no longer directly competes for s123. Thus, Firm 1’s and Firm 3’s equilibrium price ranges
do not overlap. Moreover, when Firm 2 prices below Firm 1
to compete with Firm 3 for segment s123, it also always
serves segment s12. That is, in the price region in which
only Firms 2 and 3 compete, s12 effectively becomes a component
of Firm 2’s loyal segment. Because an effectively
larger loyal segment means that Firm 2 now has more to
lose from price cuts, the lowest price that Firm 2 can profitably
quote increases, which lessens the severity of the
price competition with Firm 3. Thus, the lowest price support
for Firms 2 and 3 rises above the minimum prices
of both firms.