Game Theory with Applications to Finance and Marketing

22. Now we summarize the equilibria. First suppose that for one firm i,
∆F i (v) > 0. Then ∆F j (v) = 0, implying that ∆F j (V ) > 0 and hence
Π j = αV . It follows that Π i = αV also. Since v is a best response for
firm i, we must have
∆F j (V )v(1−α)+[1−∆F j (V )]vα = αV, ⇒ ∆F j (V ) =
Alternatively, we can obtain the same result from
α(V − v)
v(1 − 2α) . (5)
∆F j (V ) = 1 − F j (v−). (6)
In this case, we have
⎧
⎪⎨
F i (p i ) =
⎪⎩
0, p i ≤ αV
1−α ;
1−α− αV
p i
1−2α , p i ∈ [ αV
1−α , v);
p ∗ , p i ∈ [v, V );
1, p i ≥ V,
and
⎧
⎪⎨
F j (p j ) =
⎪⎩
0, p j ≤ αV 1−α
1−α− αV
p j
1−2α p j ∈ [ αV 1−α
1−α− αV v
1−2α , p j ∈ (v, V );
1, p j ≥ V,
(7)
where p ∗ ∈ ( 1−α− αV v
1−2α , 1].
Next consider the case where ∆F A (v) = ∆F B (v) = 0. In this case, the
equilibrium is symmetric, and we have both F A (·) and F B (·) characterized
by the F j (·) above. 11
23. Let S A and S B be the supports of equilibrium prices p A and p B . Then
for i = A, B, we can interpret sup S i as the regular price of firm i,
and any price strictly lower than sup S i as a dealing price. Now we
can compute that the dealing frequency for a firm, which is the firm’s
probability of selecting a dealing price. In the symmetric equilibrium,
11 It can be verified that in the former equilibrium v is a not a pure-strategy best response
for firm j, but it is in the latter equilibrium.
21