The tension of selling to switchers and loyals results in a
lack of pure strategies typical of such models. Therefore,
we solve for a mixed-strategy equilibrium that depends on
the relative sizes of the loyal and switcher segments. A
mixed strategy can be interpreted as arising from a retailer’s
uncertainty about the pricing decisions of competing retailers
(Gibbons 1992). Equilibrium prices are defined by a
probability density function (PDF) that indicates the range
of prices a retailer may charge (the retailer’s price support).
We find that equilibrium price promotion strategies fundamentally
depend on the retailers’ relative SLRs. Thus, we
present the mixed-strategy equilibrium for the case that is
more typical of prior models, in which the large Firm 1 has
the lowest φi, as follows: