Solving for mixed strategies makes no sense in bayesian cooperative/coordination games with types
Suppose both Player 1 and Player 2 want to maximize their own payoffs. Wouldn't it make the most sense to simply reason as follows: since there is a 70% probability that a=2, Player 2 should assume that Player 1 is type a=2? And if Player 2 makes that assumption, they should also assume that Player 1 will definitely choose A, and therefore Player 2 should choose A as well.
Edit:
I also solved this game for mixed strategy in the case where a=2 and the mixed equilibrium I found was (1/3,2/3) for player 1 and (1/2,1/2) for player 2.
Though I have a hard time see the reasoning behind it. Since why would any player ever want to do that, if they randomize their choices in a effort to make the other player indifferent between choices then they simply just lower the expected payoff?