In this paper, we study the Maximum Vertex-weighted $b$-Matching (MVbM) problem on bipartite graphs in a new game-theoretical environment. In contrast to other game-theoretical settings, we consider the case in which the value of the tasks is public and common to every agent so that the private information of every agent consists of edges connecting them to the set of tasks. In this framework, we study three mechanisms. Two of these mechanisms, namely $\MB$ and $\MD$, are optimal but not truthful, while the third one, $\MG$, is truthful but sub-optimal. Albeit these mechanisms are induced by known algorithms, we show $\MB$ and $\MD$ are the best possible mechanisms in terms of Price of Anarchy and Price of Stability, while $\MG$ is the best truthful mechanism in terms of approximated ratio. Furthermore, we characterize the Nash Equilibria of $\MB$ and $\MD$ and retrieve sets of conditions under which $\MB$ acts as a truthful mechanism, which highlights the differences between $\MB$ and $\MD$. Finally, we extend our results to the case in which agents' capacity is part of their private information.

翻译：本文研究了二分图上最大顶点加权$b$-匹配（MVbM）问题在一种新的博弈论环境中的表现。与其他博弈论设置不同，我们考虑任务价值对所有智能体公开且相同的情形，因此每个智能体的私有信息由连接它们与任务集的边组成。在此框架下，我们研究了三种机制。其中两种机制，即$\MB$和$\MD$，是最优的但不真实，而第三种机制$\MG$是真实的但非最优。尽管这些机制由已知算法推导得出，我们证明$\MB$和$\MD$在价格无政府性和价格稳定性方面是最优的可能机制，而$\MG$在近似比方面是最优的真实机制。此外，我们刻画了$\MB$和$\MD$的纳什均衡，并找出了$\MB$作为真实机制成立的条件集，从而凸显了$\MB$与$\MD$的差异。最后，我们将结果扩展到智能体容量属于其私有信息的情形。