This is a quantity-setting oligopoly where both firms supply respectively the quantities $x_1, x_2$ and the resulting price (at which all supply is sold) is $p=1-x_1-x_2$. Production involves consideration of a fixed cost $a>0$ to produce any positive amount of the good, and no variable cost. Hence the game
$$
\begin{aligned}
& x_1=x_2=[0,1] \\
& \mathrm{u}_{\mathrm{i}}\left(\mathrm{x}_1, \mathrm{x}_2\right)=\mathrm{x}_{\mathrm{i}}\left(1-\mathrm{x}_1-\mathrm{x}_2\right)-\mathrm{a} \\
& =0 \\
&
\end{aligned}
$$
We discuss this game with respect to the parameter a.
a) Show that for a small ( $0<a<\alpha$ for some $\alpha$ to be computed) the game is strategically similar to the costless game $(a=0)$. There is a unique Nash equilibrium; it is also the sophisticated equilibrium of the game. b) For a not-too-small and not-too-big ( $\alpha \leq a \leq \beta$ for some $\beta$ to be computed) there are three equilibria (in pure strategies). One of them is the sophisticated equilibrium. c) For a large $(\beta<a \leq 1 / 4)$ there are two asymmetrical equilibria in pure strategies, where only one firm is active. In this case there is also a symmetrical equilibrium in mixed strategies (see Chapter 7 below).