Game Theory with Applications to Finance and Marketing

show that the unique equilibrium in mixed strategy is such that 13
⎧
0, x ≤ p ≡ αV ; 1−β
1− ⎪⎨
p x
, x ∈ [p, v);
1−
F A (x) =
β
1−α
1− p v
, x ∈ [v, V );
1−
⎪⎩
β
1−α
1, x ≥ V,
(9)
13 Let us demonstrate in detail how to get this equilibrium.
• At first, the supports of F A and F B must share the same greatest lower bound
p j+p
p, because given p i
, each p j < p i
is strictly dominated by, say, i
2
, for i ≠ j,
i, j ∈ {A, B}. Note that p > 0 because both firms can ensure a strictly positive
profit by serving their own loyals only.
• Second, in equilibrium at least one firm i must price at V with a strictly positive
probability. To see this, define p j ≡ sup{p j : p j ≤ v} and suppose instead that
F A (v) = F B (v) = 1, and we shall demonstrate a contradiction. We first claim that
p A = p B = p: any p j ∈ (p i , p j ) would otherwise be dominated by V . Next, we claim
that neither F A nor F B can have a jump at p: if ∆F i (p) > 0 then firm j would rather
price at p − ɛ than at p, for some sufficiently small ɛ > 0, and hence ∆F j (p) = 0, but
then pricing at p is dominated by pricing at V from firm i’s perspective! Now, by
the fact that F A and F B are both continuous at p, again, p must be a best response
because for some δ > 0, every price contained in (p − δ, p) is a best response (cf.
Lemmas 4 and 6), but it is clear that pricing at p is still dominated by pricing at
V from each firm’s perspective! We conclude that a contradiction would always
arise unless at least one firm will price at V with a strictly positive probability in
equilibrium.
• Third, for i = A, B, we must have p i = v. Again, it must be that p A = p B = p:
any p j ∈ (p i , p j ) would otherwise be dominated by V . Recall also that neither F A
nor F B can have a jump at p: if ∆F i (p) > 0 then firm j would rather price at
p − ɛ than at p, for some sufficiently small ɛ > 0, and hence ∆F j (p) = 0, but then
pricing at p is dominated by pricing at v+p
2
from firm i’s perspective! By definition
of the least upper bound, there exists an increasing sequence of pure-strategy best
responses {p n i ; n = 1, 2, · · ·} converging to p, which, by the fact that F A and F B
are both continuous at p, implies that the limit p is also a best response from both
firms’ perspective. However, pricing at p is apparently dominated by pricing at v
from each firm’s perspective, which is a contradiction.
• Fourth, firm i’s equilibrium payoff (expected profit) is Π i = α i V if with a strictly
positive probability it may price at V . This pins down p as follows. Note that for
all x ∈ [p, v),
Π i = α i V = x{α i + (1 − α i − α j )[1 − F j (x)]} ⇒ F j (x) = 1 −
− α i
,
1 − α i − α j
which, by the fact that F j (p) = 0, implies that
p 23 =
α iV
.
1 − α j
The question is which firm will price at V with a strictly positive probability. Recall
that the firm with a larger loyal base cannot compete as aggressively as its rival
α iV
x