  Metaverse has recently attracted much attention from both academia and
industry. Virtual services, ranging from virtual driver training to online
route optimization for smart goods delivery, are emerging in the Metaverse. To
make the human experience of virtual life more real, digital twins (DTs),
namely digital replicas of physical objects, are key enablers. However, DT
status may not always accurately reflect that of its real-world twin because
the latter may be subject to changes with time. As such, it is necessary to
synchronize a DT with its physical counterpart to ensure that its status is
accurate for virtual businesses in the Metaverse. In this paper, we propose a
dynamic hierarchical framework in which a group of IoT devices is incentivized
to sense and collect physical objects' status information collectively so as to
assists virtual service providers (VSPs) in synchronizing DTs. Based on the
collected sensing data and the value decay rate of the DTs, the VSPs can
determine synchronization intensities to maximize their payoffs. In our
proposed dynamic hierarchical framework, the lower-level evolutionary game
captures the VSPs selection by the IoT device population, and the upper-level
differential game captures the VSPs payoffs, which are affected by the
synchronization strategy, IoT devices selections, and the DTs value status,
given VSPs are simultaneous decision makers. We further consider the case in
which some VSPs are first movers and extend it as a Stackelberg differential
game. We theoretically and experimentally show that the equilibrium to the
lower-level game exists and is evolutionarily robust, and provide a sensitivity
analysis with respect to various system parameters. Experiments show that the
proposed dynamic hierarchical game outperform the baseline.
