Non Collusive Cournot Duopoly model with two firms, zero costs and linear demand curve I am reading Modern Microeconomics by Koutsoyiannis. In a Non Collusive Cournot Duopoly model with two firms, zero costs and linear demand curve. 

Firm A produces half the total market demand to maximise revenue.



Further, Firm B takes A's output as given and operates on the left over demand curve eD' and produces 1/4th of output (AB).

Now Firm A in period 3 should respond by taking the leftover demand curve e'D' and produce $\frac{1}{2}$ of the leftover market that is $ (1 - \frac{1}{2} - \frac{1}{4})\frac{1}{2} = \frac{1}{8}th $ of total market output. 

But it is mentioned
The temporal constraints (i.e. what happens in a period) are not very clear in this question. It is likely that the good sold is not a durable good and hence there is no "leftover demand" between periods, demand is simply 'reset'.

In period 2 leftover demand appears because firm B assumes firm A will not change its production from period 1.

Then in period 3 firm A will best respond to the unchanging production of firm B from period 2.