Two firms sell the same good. Their strategy is to establish a price $x_i \geq 0$. If $x_i<x_j$, firm i must satisfy all the demand $D\left(x_i\right)=300-5 x_i$ and firm $j$ sells nothing. If $x_1=x_2$, the firms share equally the demand $D\left(x_1\right)$.
a) Assume first that the cost function is $C(q)=10 q$ for both players. Write the normal form of the game, with strategy sets
$$
x_1=x_2=[0,60]
$$
Show that there is a unique Nash equilibrium: compare the corresponding profits to the minimal guaranteed profits.
b) Assume next that the cost functions differ
$$
C_1\left(q_1\right)=10 q_1 \quad C_2\left(q_2\right)=20 q_2
$$
Prove that, strictly speaking, there is no Nash equilibrium in this game. However, define the $\varepsilon$ Nash equilibrium as an outcome where no player, by a unilateral deviation, can improve his utility by more than $\varepsilon$. Then prove that there are infinitely many $\varepsilon$ Nash equilibria in this game and describe them.
c) In the game in b) perform two rounds of elimination of dominated strategies; show that the reduced game is inessential and its equilibrium corresponds to the $\varepsilon \mathrm{Nash}$ equilibrium most favorable to Player 1 .