I have a game with a player. The player's type is unknown (privately known to him) and distributed within a known interval. I play the payoff-maximizing strategy. When the player's type or its distribution is known the game can be easily solved. Then my equilibrium strategy is some sort of a weighted sum of best-responses to each possible player type realization.

The problem is what to do if player's type distribution is completely unknown. The only thing I know is that the type lies within a known interval, say, [a,b].

Any suggestion on how to approach this?

I tried β-distribution, but it does not completely fit the described setup as it requires some initial parameters to set.

Thanks in advance.