Consider an industry in which two firms compete by simultaneously and
independently selecting prices for their goods. Firm 1's product and firm 2's
product are imperfect substitutes, so that an increase in one firm's price
will lead to an increase in the quantity demanded of the other firm's product.
In particular, each firm \(i\) faces the following demand curve:
$$
q_{i}=\max \left\\{0,24-2 p_{i}+p_{j}\right\\}
$$
Consider an industry in which two firms compete by simultaneously and
independently selecting prices for their goods. Firm 1's product and firm 2's
product are imperfect substitutes, so that an increase in one firm's price
will lead to an increase in the quantity demanded of the other firm's product.
In particular, each firm \(i\) faces the following demand curve:
$$
q_{i}=\max \left\\{0,24-2 p_{i}+p_{j}\right\\}
$$
where \(q_{i}\) is the quantity that firm \(i\) sells, \(p_{i}\) is firm \(i\) 's
price, and \(p_{j}\) is the other firm's price. (The "maximum" operator is
needed to ensure that quantity demanded is never negative.)
Suppose that the maximum possible price is 20 and the minimum price is zero,
so the strategy space for each firm \(i\) is \(S_{i}=[0,20]\). Both firms produce
with no costs.
(a) What is firm \(i\) 's profit as a function of the strategy profile
\(\left(p_{1}, p_{2}\right)\) ? Let each firm's payoff be defined as its profit.
(b) Show that for each firm \(i\), prices less than 6 and prices greater than 11
are dominated strategies, whereas prices from 6 to 11 are not dominated, so
\(B_{i}=[6,11]\).
(c) Consider a reduced game in which firms select prices between \(x\) and \(y\),
where \(x
