Possible output of a firm using game theory Let there be two firms A and B. Let the price(P) output(X) graph be a linear one $P =a-bX=5-(1)X,c=2$
The parameter 


*

*a is MAX price.

*b is the slope of the curve profit vs output(line) 

*c is the price at perfect competitive market ie large number of firms.


Now firm A wants to put a production unit.Both firms A and B are rational and A knows that B is rational. No firm has negative amount of production. Using the knowledge 
of game theory and Nash equilibrium choose the options which the firm A can produce. $$\text{The option set(in some proper units)}$$
$A\in(0,
0.25, 0.5, 0.75, 1 ,1.25, 1.5, 1.75 ,2)$
$$\text{Attempt}$$
Clearly the firm wont produce 0 units so it isn't one of the choices.Now we know from Nash equilibrium that total output for $N$ firms is $(1-\frac{1}{N+1})(\frac{a-c}{b})$.Here $N=2$ and putting in other values we see the total ouput$=2$. Also at equilibrium the players(firms)mutually take the best option available so each firm produces same quantity. Hence each firm's produce is $=1$.So $1$ is the equilibrium production. Now we know firms are rational but that doesn't mean that A will have the best reply right from the beginning. It may eventually start producing $1$ units. So other options do exist for firm A. We know that the profit vs output curve is quadratic $\text{Profit of A}=k(\frac{a-c}{b}-B-A)A$ where $$\text{A,B are outputs of the respective firms.}$$ I dont know how to proceed to see which other options are available to A.
$$\text{Note:}$$ Assume I have no knowledge of economics so if possible limit your answer to basics of maths and game theory.
$$\text{Edit}$$
In part b I was asked that if B doesn't know that he is rational then which of the above outputs would the firm A never choose.
 A: *

*Write out a payoff matrix, often called the strategic form.  Here it is a $9 \times 9$ matrix/tableau with the quantities along the left side and the top, with two numbers in the $(i,j)$ box, one the payoff for row and the other the payoff for column when row plays $q_i \in A$ and column plays $q_j \in A$:
$$
\pi_{row}(q_i,q_j) = (A-b(q_i+q_j))q_i - cq_i
$$
and
$$
\pi_{col}(q_i,q_j) = (A-b(q_i+q_j))q_j - cq_j.
$$
Include every strategy, including zero: there are often surprising equilibria where agents adopt strategies you might not expect.

*Construct the best reply correspondence for each player. Fix a column, and find the highest payoff for the row player, and underline; do this for all columns. Fix a row, and find the highest payoff for the column player, and underline; do this for all rows. In general, this correspondence exists and is upper hemi-continuous (by extending the pure strategies to random, mixed strategies) by Berge's Theorem of the Maximum.

*Solve for a fixed point of the best reply correspondences, which is a Nash equilibrium. Find all boxes where both payoffs are underlined. In your game there are probably three-ish. In general, a pure-strategy equilibrium might not exist, but if you allow for random strategies, you can use Kakutani's fixed-point theorem along with convexity and upper hemi-continuity of the best-reply correspondences to prove existence of a fixed point (i.e., a Nash eqm).
That is all of game theory in a nutshell: solve for best reply correspondences for the players (use Berge's Theorem), then find a fixed point using Brouwer, Kakutani, Tarski, Glicksberg, Schauder, Banach, Eilenberg-Montgomery, or whatever the appropriate FPT is.
This game is actually solvable by iterated deletion of weakly dominated strategy, which is an epistemically stronger concept than Nash equilibrium: for each player, remove any strategy where there exists an alternative strategy that always gives a weakly higher payoff, and repeat until no further strategies can be deleted for either player. That is one way of picking a unique equilibrium in situations where there are multiple Nash equilibria, and it will be the closest one to the continuous version of the eqm you posted.
I guess the issue is that you solved the continuous version of the model using calculus. The discrete version ($A$ is finite) is going to have multiple eqa, and the question is asking, "What values in $A$ correspond to a Nash eqm for the row player?"  A more sophisticated question would be, ``What values in $A$ are rationalizable: i.e., constitute a best reply to some strategy that the column player might use?"  If you take the iterated deletion route, the unique answer as a result of common knowledge of rationality will be the Nash strategies.
