How to prove the existence of a pure Nash equilibrium?
If p=q=s=t=−1, then it is easy to see that a pure NE can be reached by local search starting from (a3, a3).  So in the rest of the proof, assume that at least one of p, q, s, and t is equal to 1.

If p=q=1, then (a1, a2) is a pure NE.  Similarly, if s=t=1, then (a2, a1) is a pure NE.

In the remaining cases, at least one of the pairs (p, q), (q, p), (s, t), and (t, s) is equal to (1, −1).  By symmetry, we may assume without loss of generality that p=1 and q=−1.  By assumption, r=1.  Then (a1, a3) is a pure NE.