Game Theory with Applications to Finance and Marketing

The first-order condition, upon replacing p 1 into the objective function,
is
[a − c − (n − 1)q1 ∗ − 1
− 2q] + δρn
n + 1 E[ψ∗ (p 1 , q)] = 0,
where
E[ψ ∗ (p 1 , q)] = a − c
n + 1 − ρ n − 1
2 n + 1 (q∗ 1 − q).
Solving, and then letting q = q1 ∗ , we have
and
q ∗ 1 = (a − c)[(n + 1)2 + δρ(n − 1)]
(n + 1) 3 ,
σ2(p ∗ 1 ) = ρ(p 1 + nq1 ∗ ) + (1 − ρ)a − c
.
n + 1
The bottom line here is that σ2 ∗(p 1) coincides with the static Cournot
symmetric output when a 1 = p 1 + nq1 ∗ ; that is, no firms are fooled
by their rivals’ output expansion activities. Still, q1 ∗ is greater than
the static Cournot output level: the firms cannot resist trying to fool
their rivals at date 1. In this sense, we conclude that with imperfect
information about demand and the rivals’ outputs, the date-1 efficiency
is lower for the firms, but the date-2 efficiency is unchanged. Of course,
if we discuss efficiency from the perspective of social benefit, then the
date-1 efficiency is actually higher!
32. Example 13. (Fudenberg and Tirole, 1986, Rand Journal of Economics.)
Suppose that two firms are located at the two ends of the
Hotelling main street [0, 1], facing consumers uniformly located on the
street with a total population of one. Each consumer is willing to pay
2 dollars for one unit of the product produced by either firm 1 (located
at the left endpoint of the Hotelling main street) or firm 2 (located at
the right endpoint of the Hotelling main street). Firm 1 has no production
costs, but firm 2 is faced with a random fixed cost ˜F , which is
uniformly distributed on [0, 1 ]. A consumer located at t ∈ [0, 1] must
2
pay t and 1 − t respectivley if he wants to visit firm 1 and firm 2. The
firms compete for two periods (t = 1, 2), and they seek to maximize
the sum of profits over the two periods.
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