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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!

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  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.
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  • $\begingroup$ Thank you so much, that helps a lot! $\endgroup$ – Sandy.Davidson Oct 13 '16 at 17:07
  • $\begingroup$ By the way, could you please explain to me what's the difference between equilibria and equilibrium? Sometimes I was told to find an equilibria and sometimes an equilibrium, I'm pretty confused by the terminology. Thanks. $\endgroup$ – Sandy.Davidson Oct 13 '16 at 17:12
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    $\begingroup$ You are welcome! Equilibria is just the plural of equilibrium. So it is correct to write "find an equilibrium" or "find all equilibria". $\endgroup$ – Jimmy R. Oct 13 '16 at 17:57
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    $\begingroup$ To say "find an equilibria" is certainly false. $\endgroup$ – Jimmy R. Oct 13 '16 at 19:26
  • $\begingroup$ Great, thanks : ) $\endgroup$ – Sandy.Davidson Oct 13 '16 at 19:42

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