For the little it might be worth, here are the results of some simulations:

[![][1]][1]  
<sub>(source: [landsburg.com](http://www.landsburg.com/ariel.gif))</sub>  

The columns correspond to values of $|A|$ and the rows to values of $|B|$.  The four-tuple in each cell is (Probability winner is in $A$, Probability winner is in $B$, Probability a given member of $A$ is the winner, Probability a given member of $B$ is the winner).

I simulated each of these 10,000 times, rounded results to the nearest percent, and retyped them (which has a small chance of having introduced additional errors).  

I was struck by the non-monotonicity in the third entry as you go down the third column, so I repeated these trials and got the same result.


 [1]: https://i.sstatic.net/kDi0R.gif