Zero Sum game how to find NE in pure and mixed strategy

Question

The zero sum game matrix as follows

  p   q   r
T 5   3   y
B 9   5  9y

The question is to find all the NE in pure and mixed strategy by using different case distinctions values of y

Can you please explain how to approach this? For example if y=1

Answer 1

HINT

Assuming the $p,q.r$ chooser is the minimizer, $q$ always dominates $p$. Then since you cannot find any $y$ such that $y>3$ but $9y < 5$, the two by two sub-game involving strategies $q$ and $r$ will always have a saddle point (pure strategy at equilibrium).

For what value(s) of $y$ does that saddle point change to a different element of the game matrix?