Two questions about Nash Equilibrium

Question

I'm reading an introductory book on game theory, but I'm confused by Nash Equilibrium, can someone explain to me the following questions?

1. Will the iterated elimination of strictly dominated strategies eliminate a strategy that is part of a Nash Equilibrium, why?

2. For each player, will the utility of an equilibrium outcome always be better than the security level? If so, how to prove it?

Any idea will be appreciated, thanks a lot!

Answer 1

1. No, never. A strictly dominated strategy is never part of an equilibrium, because one can profitably deviate to the strictly dominant strategy.
2. Assume that the utility of a player at a strategy profile is less than his security-level. Can you find then a profitable deviation for this player? Of course you can, since that player can deviate to the strategy that guarantees him his sequrity level no-matter what and improve his payoff. Hence, such a profile violates the definition of a Nash equilibrium.