  Digital advertising constitutes one of the main revenue sources for online
platforms. In recent years, some advertisers tend to adopt auto-bidding tools
to facilitate advertising performance optimization, making the classical
\emph{utility maximizer} model in auction theory not fit well. Some recent
studies proposed a new model, called \emph{value maximizer}, for auto-bidding
advertisers with return-on-investment (ROI) constraints. However, the model of
either utility maximizer or value maximizer could only characterize partial
advertisers in real-world advertising platforms. In a mixed environment where
utility maximizers and value maximizers coexist, the truthful ad auction design
would be challenging since bidders could manipulate both their values and
affiliated classes, leading to a multi-parameter mechanism design problem. In
this work, we address this issue by proposing a payment rule which combines the
corresponding ones in classical VCG and GSP mechanisms in a novel way. Based on
this payment rule, we propose a truthful auction mechanism with an
approximation ratio of $2$ on social welfare, which is close to the lower bound
of at least $\frac{5}{4}$ that we also prove. The designed auction mechanism is
a generalization of VCG for utility maximizers and GSP for value maximizers.
