Is there a constructive parallel repetition theorem for nice MIP protocols?
In his simplified proof of the parallel repetition theorem, Holenstein gives the following explicit bound on the value of the $n$-fold parallel repetition of a game with value $v$:

$$
\left(1-\frac{(1-v)^3}{6000}\right)^{n/\log(|\Sigma_1|\cdot |\Sigma_2|)},
$$
where $\Sigma_1, \Sigma_2$ are the alphabets of the two players. 

It's not clear to me that Raz's bound is not constructive in your sense; I am wondering what step in his proof you think is not computable. He does not make his bound fully explicit, but I thought this was done more for exposition.