  We introduce balancedness a fairness axiom in house allocation problems. It
requires a mechanism to assign the top choice, the second top choice, and so
on, on the same number of profiles for each agent. This axiom guarantees equal
treatment of all agents at the stage in which the mechanism is announced when
all preference profiles are equally likely. We show that, with an interesting
exception for the three-agent case, Top Trading Cycles from individual
endowments is the only mechanism that is balanced, efficient, and group
strategy-proof.
