I think my question is somewhat related to this one: What happens if the explanatory and response variables are sorted independently before regression?
But my interest is slightly different. Suppose that I have an original dataset $\{(X_i,Y_i)\}_{i=1}^n$, run OLS of $Y$ on $X$, and calculate MSE and $R^2$.
Now I permute (randomly) the observations of the dependent variable $Y$. Denote the new dataset as $\{(X_i,\tilde{Y}_i)\}_{i=1}^n$.
What is the probability that I will end up with better measures of MSE and $R^2$ when I run OLS of $\tilde{Y}$ on $X$?
Is there a way to bound this probability?