The PDFs and CDFs for the equilibrium prices appear
in Figure 2. The lower price support is defined by Firm 1’s
minimum price because Firm 2 never needs to discount
below this to compete for the switchers. Firm 3 is content to
be a niche player, pricing high to serve only its relatively
small loyal segment. Given that n1 > n3, the P2 condition
that φ3 ≤ φ1 essentially represents the case in which s123 is
small compared with s12 (s123 = 0 will always meet the condition
of P2). Firm 3 recognizes that most of the switchers
compare prices only between Firms 1 and 2, so Firm 3
becomes a high-priced retailer.