We consider zero-sum absorbing games where the payoff depends on two fixed parameters drawn randomly at the outset of the game, one that is announced to Player 1, the other one that is announced to Player 2. We solve two open problems, that correspond to the Mertens conjectures (1986) for this model, that is: -the limit value exists, -when Player 1 knows both parameters (incomplete information on one side), Player 1 can guarantee uniformly the limit value. The proof builds on a new approximation technique of the belief martingale based on an appropriate simplex triangulation, that is of self-interest and extends beyond the zero-sum case.