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We provide a unifying framework for the design and analysis of multi-calibrated and moment- multi-calibrated predictors. Placing the multi-calibration problem in the general setting of multiobjective learning—where learning guarantees must hold simultaneously over a set of distribu- tions and loss functions—we exploit connections to game dynamics to obtain state-of-the-art guarantees for a diverse set of multi-calibration learning problems. In addition to shedding light on existing multi-calibration guarantees, and greatly simplifying their analysis, our ap- proach yields a 1/ε2 improvement in the number of oracle calls compared to the state-of-the-art algorithm of Jung et al. [19] for learning deterministic moment-calibrated predictors and an exponential improvement in k compared to the state-of-the-art algorithm of Gopalan et al. [14] for learning a k-class multi-calibrated predictor. Beyond multi-calibration, we use these game dynamics to address existing and emerging considerations in the study of group fairness and multi-distribution learning.