  We present Classification of Cluster GAlaxy MEmbers (C$^2$-GaMe), a
classification algorithm based on a suite of machine learning models that
differentiates galaxies into orbiting, infalling, and background (interloper)
populations, using phase space information as input. We train and test
C$^2$-GaMe with the galaxies from UniverseMachine mock catalog based on
Multi-Dark Planck 2 N-body simulations. We show that probabilistic
classification is superior to deterministic classification in estimating the
physical properties of clusters, including density profiles and velocity
dispersion. We propose a set of estimators to get an unbiased estimation of
cluster properties. We demonstrate that C$^2$-GaMe can recover the distribution
of orbiting and infalling galaxies' position and velocity distribution with
$<1\%$ statistical error when using probabilistic predictions in the presence
of interlopers in the projected phase space. Additionally, we demonstrate the
robustness of trained models by applying them to a different simulation.
Finally, adding a specific star formation rate and the ratio of the galaxy's
halo mass to the cluster's halo mass as additional features improves the
classification performance. We discuss potential applications of this technique
to enhance cluster cosmology and galaxy quenching.
