Let a* be an evolutionarily stable action

Subject:EconomicsPrice: Bought3

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1. Let a* be an evolutionarily stable action. Does a* necessarily weakly dominate every other action? Is it possible that some action weakly dominates a*?
2. Suppose that α* is a mixed ESS, and let a be an action to which α* assigns positive probability. Show that if α* assigns positive probability to an action different from a, then (a, a) is not a pure strategy Nash equilibrium, and hence a is not an ESS.