Mechanism design on social networks is a hot research direction recently, and we have seen many interesting results in auctions and matching. Compared to the traditional settings, the new goal of the network settings is that we need to design incentives to incentivize the participants of the game to invite their neighbors on the network to join the game. This is challenging because they are competing for something (e.g., resources or matches) in the game. In one-sided matching, especially house exchange, the well-known unique truthful, stable and optimal solution called Top Trading Cycle (TTC) cannot achieve the new goal. Existing works have tried to add constraints on TTC to obtain the incentive, but it only works in trees and it does not guarantee any stability. In this paper, we move this forward and propose the first mechanism called Leave and Share (LS) which not only achieves the goal in all networks but also gives the most stable solution in the new settings. In terms of optimality, as it is impossible to achieve it in any network, we conduct simulations to compare it with the extensions of TTC.

翻译：社交网络上的机制设计是近年来热门的研究方向，我们在拍卖和匹配领域已看到许多有趣成果。与传统场景相比，网络场景的新目标是需要设计激励措施，促使博弈参与者邀请其网络中的邻居加入博弈。由于他们在博弈中争夺资源（如资源或匹配对象），这充满挑战性。在单向匹配（特别是房屋交换）中，著名的唯一诚实、稳定且最优的解——顶轮交易循环（TTC）无法实现这一新目标。现有研究尝试在TTC上增加约束以获得激励，但仅适用于树形结构且无法保证任何稳定性。本文推进了该研究，提出首个名为"离开与分享（LS）"的机制，该机制不仅能在所有网络中实现目标，还能在新场景下提供最稳定的解。关于最优性，由于在任何网络中均无法实现，我们通过模拟将其与TTC的扩展方案进行了比较。