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Multi-label image classification aims to predict all possible labels in an image. It is usually formulated as a partial-label learning problem, since it could be expensive in practice to annotate all the labels in every training image. Existing works on partial-label learning focus on the case where each training image is labeled with only a subset of its positive/negative labels. To effectively address partial-label classification, this paper proposes an end-to-end Generic Game-theoretic Network (G2NetPL) for partial-label learning, which can be applied to most partial-label settings, including a very challenging, but annotation-efficient case where only a subset of the training images are labeled, each with only one positive label, while the rest of the training images remain unlabeled. In G2NetPL, each unobserved label is associated with a soft pseudo label, which, together with the network, formulates a two-player non-zero-sum non-cooperative game. The objective of the network is to minimize the loss function with given pseudo labels, while the pseudo labels will seek convergence to 1 (positive) or 0 (negative) with a penalty of deviating from the predicted labels determined by the network. In addition, we introduce a confidence-aware scheduler into the loss of the network to adaptively perform easy-to-hard learning for different labels. Extensive experiments demonstrate that our proposed G2NetPL outperforms many state-of-the-art multi-label classification methods under various partial-label settings on three different datasets.