Existence of pure strategy Nash equilibrium I understand the reason why mixed strategy Nash equilibrium exists. But what are the conditions for the existence of pure strategy Nash equilibrium?
The most simple example is the example of unlimited periods and Bertrand pricing, in any given period the "pure strategy" (with 2 firms) is to produce 1/2 of monopolist quantity and sell at monopolist price and recieve profits = π(1-d)^t 
Where d is discount value 
t is period
Any deviation will result in 0 profits every period following the deviation, thus in an infinite period model is not worth it.