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Solving $N$-player general sum matrix games is known to be PPAD-complete, and even for two players algorithms such as Lemke-Howson can take exponential time in the worst case. On the other hand, verifying that a solution is indeed a Nash equilibrium can be done in polynomial time.

Are there standard examples/benchmarks/libraries of bimatrix or $N$-player games that

  1. are relevant for applications or concrete problems (i.e., are not just synthetic numerical examples), and
  2. ideally have not been solved yet, or can not be solved without significant computational effort with current methods?
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