On the Manipulability of Maximum Vertex-Weighted Bipartite $b$-matching Mechanisms

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.