[willdearden]

There are a ton of papers on the properties of stable matching under random preferences. The classic paper (Pittel 1989 https://epubs.siam.org/doi/abs/10.1137/0402048?journalCode=s...) calculates the average rank of men and women for their partners under male-proposing Gale Shapley. They find the men's average rank is O(log(n)) and the women's average rank is O(n / log(n)). Depending on how you calculate percent advantage, that roughly matches your simulations.

[likelybear]

The difference based on which side proposes turns out to be a knife-edge result for balanced markets! For imbalanced markets, the short side gets average rank `O(log(n))` in every stable matching and the long side gets `O(n / log(n))`

Edit: reference is Ashlagi et al (2015). "Unbalanced Random Matching Markets: The Stark Effect of Competition" http://web.mit.edu/iashlagi/www/papers/UnbalancedMatchingAKL...