House assignment with identical preferences:-

Find all the assignments in the core of the n-player game in which every player ranks the houses in the same way. I now argue that for any (strict preferences), the core of a house exchange game contains the assignment induced by the top trading cycle procedure. The following argument establishes this result.

Every player assigned a house in the first round receives her favorite house, so that no coalition containing such a player can make all its members better off than they are in aN. Now consider a coalition that contains players assigned houses in the second round, but no players assigned houses in the first round.

Such a coalition does not own any of the houses assigned on the first round, so that its members who were assigned in the second round obtain their favorite houses among the houses it owns. Thus such a coalition has no action that makes all its members better off than they are in aN. A similar argument applies to coalitions containing players assigned in later rounds.