Stability and venture structures in multilateral matching

We consider a multilateral matching market, where two or more agents can engage in a joint venture via multilateral contracts. Possible joint ventures are exogenously given. We study four stability concepts: stability, weak setwise stability, strong group stability, and setwise stability. We characterize the structure of possible joint ventures that guarantee the efficiency and existence of outcomes satisfying these stability concepts under general preference profiles. Specifically, we show that any stable outcome, weakly setwise stable outcome, and setwise stable outcome are efficient for any preference profile if and only if the structure of possible joint ventures satisfies a condition called the acyclicity. We also show that the acyclicity is a necessary and sufficient condition for the existence of a stable outcome and strongly group stable outcome for any preference profile. Moreover, we show that a weaker condition called the no-crossing property is a necessary and sufficient condition for the existence of a weakly setwise stable outcome and setwise stable outcome for any preference profile.