Why you randomize your opponents payoff in a mixed nash equilibrium?

Question

I wanted to understand the justification more intuitively -- if that is possible. For example, I'm in a abstract game with another opponent and there is no pure strategy equilibrium: why do I randomize in order to make my opponent's payoff such that he is indifferent between his strategies? Why wouldn't I randomize myself mine strategies?

Answer 1

Under expected utility theory, a player has never a strict incentive to randomize and can randomize only between optimal choices when she is indifferent among them.

Mixed strategy are needed to guarantee equilibrium existence, they are not needed for individual decision theory.

Answer 2

Also recognize that Nash Equilibrium has not to do with optimal strategies but a set of strategies in which the players are stuck in a sense (they can't do better by choosing other strategy) thus to find a nash equilibrium if you make other player indifferent he literally can't do better since any mixed/pure strategy will give same expected utility. Thus though inherently in a game you may not be looking at your opponents strategies to find nash equilibrium we must look in this perspective