  In this paper, we explore the Mechanism Design aspects of the Maximum
Vertex-weighted $b$-Matching (MVbM) problem on bipartite graphs $(A\cup T, E)$.
The set $A$ comprises agents, while $T$ represents tasks. The set $E$ is the
private information of either agents or tasks. In this framework, we
investigate three mechanisms - $\MB$, $\MD$, and $\MG$ - that, given an MVbM
problem as input, return a $b$-matching. We examine scenarios in which either
agents or tasks are strategic and report their adjacent edges to one of the
three mechanisms. In both cases, we assume that the strategic entities are
bounded by their statements: they can hide edges, but they cannot report edges
that do not exist. First, we consider the case in which agents can manipulate.
In this framework, $\MB$ and $\MD$ are optimal but not truthful. By
characterizing the Nash Equilibria induced by $\MB$ and $\MD$, we reveal that
both mechanisms have a Price of Anarchy ($PoA$) and Price of Stability ($PoS$)
of $2$. These efficiency guarantees are tight; no deterministic mechanism can
achieve a lower $PoA$ or $PoS$. In contrast, the third mechanism, $\MG$, is not
optimal, but truthful and its approximation ratio is $2$. We demonstrate that
this ratio is optimal; no deterministic and truthful mechanism can outperform
it. We then shift our focus to scenarios where tasks can exhibit strategic
behaviour. In this case, $\MB$, $\MD$, and $\MG$ all maintain truthfulness,
making $\MB$ and $\MD$ truthful and optimal mechanisms. In conclusion, we
investigate the manipulability of $\MB$ and $\MD$ through experiments on
randomly generated graphs. We observe that (1) $\MB$ is less prone to be
manipulated by the first agent than $\MD$ (2) $\MB$ is more manipulable on
instances in which the total capacity of the agents is equal to the number of
tasks (3) randomizing the agents' order reduces the agents' ability to
manipulate $\MB$.
