  When several two-sided matching markets merge into one, it is inevitable that
some agents will become worse off if the matching mechanism used is stable. I
formalize this observation by defining the property of integration
monotonicity, which requires that every agent becomes better off after any
number of matching markets merge. Integration monotonicity is also incompatible
with the weaker efficiency property of Pareto optimality.
  Nevertheless, I obtain two possibility results. First, stable matching
mechanisms never hurt more than one-half of the society after the integration
of several matching markets occurs. Second, in random matching markets there
are positive expected gains from integration for both sides of the market,
which I quantify.
