  This paper studies one-sided matching with initial endowments and the social
connections between participants are specifically considered (their social
network). Different from the traditional setting, the matching starts with a
group of initial participants, and the others need their invitation to join the
matching game. Thus, the incentive compatibility (IC) notion here considers
both reporting preferences and inviting others via their social connections.
This is challenging because inviters and invitees might compete in the game.
Besides IC, stability and optimality are two properties extensively studied in
matching, but they both are incompatible with the new IC. In the new setting,
compatible stability has been discussed, but no discussion about compatible
optimality yet. We complete this and prove the theoretical boundaries regarding
both stability and optimality in the new setting. We then propose a mechanism
called Connected Trading Cycles to satisfy all the desirable properties for the
first time. Unlike the previous solutions that add restrictions on
participants' matching choices to achieve IC, we allow participants to choose
anyone in the game, which in principle improves everyone's satisfiability in
the matching. Finally, we give the first characterization of IC in the network
setting to facilitate IC mechanism design.
