Economic and Game Theory
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Q1) Does Nash Equilibrium condition is same as Pareto Optimality ? A1: No, none of Nash equilibrium and Pareto optimality implies another. For example, in prisoner's dellema, (don't confess, don't confess) is PO and (confess, confess)is NE.
Q2) Does Nash equlibrium or solution holds for all forms of games whether its cooperative or non cooperative. Or whether its zero sum game or non zero sum game or whether its n person game. A2: I suppose that you mean "exist" by "hold" becasue otherwise this question does not make sense. So, if you mean "exist" by "hold". You are asking the famous Theorem of Nash (1950). This theorem basically says that if the number of the players are finite and the strategy is also finite for each players then there exist at least one NE. According to this theorem, the answer to this question is YES.
Q3) If we take for example the classic prisoners dillema here the solution has been proved and executed in a Nash Equilibrium. Therefore if we take the assumption that the payoffs changes then will Nash equilibrium still holds in this case. A3: The NE will be the same if in the changed game playing (don't confess, don't confess) is still dominate strategy for both players.
Q4) If Nash equlibrium (NE) is same as Pareto Optimality (PO) condition then will the reverse be the same i.e., PO = NE A4: No, see A1.
Q5) Will Pareto Optimality condition holds when there are different or mixed strategies and when the payoffs changes when changing strategies or adopting mixed strategies? A5: I have read this question 5 times and could understand it. Can you try again?