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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), ...



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