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Consider symmetric and finite game. By Nash (1950), the game must have at least one symmetric equilibrium (proof). Also, it must have at least one trembling hand perfect equilibrium (proof).

Question: must it have at least one symmetric trembling hand perfect equilibrium?

My thoughts on this:

  • We demonstrate the existence of a trembling hand perfect equilibrium by looking at a sequence of perturbed games which converge to the original game.
  • By Nash (1950), each perturbed game has a symmetric equilibrium.
  • Therefore (?), the original game has a symmetric trembling hand perfect equilibrium.
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