  For a zero-sum stochastic game which does not satisfy the Isaacs condition,
we provide a value function representation for an Isaacs-type equation whose
Hamiltonian lies in between the lower and upper Hamiltonians, as a convex
combination of the two. For the general case (i.e. the convex combination is
time and state dependent) our representation amounts to a random change of the
rules of the game, to allow each player at any moment to see the other player's
action or not, according to a coin toss with probabilities of heads and tails
given by the convex combination appearing in the PDE. If the combination is
state independent, then the rules can be set all in advance, in a deterministic
way. This means that tossing the coin along the game, or tossing it repeatedly
right at the beginning leads to the same value. The representations are
asymptotic, over time discretizations. Space discretization is possible as
well, leading to similar results.
