Smoothed analysis to compare algorithms
As far as I am aware, a comparison of approximation algorithms has not been done so far with respect to smoothed analysis. There are only a few papers on smoothed analysis of approximation ratios in the first place.

Maybe only loosely related: bijective analysis has been used to directly compare the competitive ratio of online algorithms (Angelopoulos, Schweitzer, "Paging and List Update under Bijective Analysis", J. ACM, 2013).

See also here for more about smoothed analysis of approximation algorithms: "Smoothed analysis of approximation algorithms".