Suppose we have three groups: $A$, $B$, and $C$, each with $n$ men and $n$ women. Every person has uniform random preferences over the whole opposite sex, no matter to which group they belong.
We first run the deferred acceptance algorithm and find the men-optimal stable matching for each society. Then we do the same for the large society that contains everybody. The question is:
How many people strictly prefer the partner they got in their group alone over the partner they got when all groups become one?
I have simulated the problem and the answer is always around 25% for any number of groups and of agents. Can anybody unveil the math behind?