Kreutzer

If both P(j) and p(i,j) are constants from the [0; 1] interval, then you have the simplex points as the terms of the product. Just set: a(i) = 1 if p(i,j)>1/2 a(i) = 0 if p(i,j)<1/2 which will maximize your expression.

Sorry, but I'm totally confused now. You say: "sum(j=1..z) [P(Tj | a1, a2......,am) ]" What is "Tj" here? Is there any difference in your notation between P(Tj) and P(j)?? Also, if P is the probability measure, then "a1, a2 ..." must be EVENTS, while as I see here they are random discrete variables, aren't they? Let's make it straight, can you rewrite it like optimization problem in form: Maximize ... subject to constraints ... ?

Your problem formulation is still quite obfuscating, cannot understand what the real problem is. Can you reformulate it more precisely? e.g. Why not? What is your "application"? Is it still math or programming problem?

I don't see any problem here given constraints for P(j). The maximum of the given function is equal to "z" (number of terms in the sum), isn't it? For instance, just set all a(i)=P(i,j) = 1 and you will obtain this supremum.

What are the values of "Pij" here ?

I disagree here though. If they didn't charge for sending scores, then I'm certain almost everyone would want his scores to be sent all over the places. Which surely would be of high cost for them.