How to deal with unknown distribution?



I have a game with two players. Player 1's type is publicly known, while player 2's type is unknown (privately known to him) and distributed, say, from 0 to 1. Both players play their payoff maximizing strategies. Need to find players' equilibrium strategies. I'm able to solve the game when player 2's type distribution is known. Then player 1's equilibrium strategy is some sort of a weighted sum of best-responses to each possible player 2 type realization. And player 2's equilibrium strategy is a best-response to player 1's equilibrium strategy. The problem is what to do if player 2's type distribution is unknown. The only thing I know is that player 2's type lies within a known interval [a,b].

Any suggestion on how to approach this?

I tried β-distribution, but need more help. For example, how to choose the initial parameters for β?

Thanks in advance.