We consider the problem of model multiplicity in downstream decision-making, a setting where two predictive models of equivalent accuracy cannot agree on the best-response action for a downstream loss function. We show that even when the two predictive models approximately agree on their individual predictions almost everywhere, it is still possible for their induced best-response actions to differ on a substantial portion of the population. We address this issue by proposing a framework that calibrates the predictive models with regard to both the downstream decision-making problem and the individual probability prediction. Specifically, leveraging tools from multi-calibration, we provide an algorithm that, at each time-step, first reconciles the differences in individual probability prediction, then calibrates the updated models such that they are indistinguishable from the true probability distribution to the decision-maker. We extend our results to the setting where one does not have direct access to the true probability distribution and instead relies on a set of i.i.d data to be the empirical distribution. Finally, we provide a set of experiments to empirically evaluate our methods: compared to existing work, our proposed algorithm creates a pair of predictive models with both improved downstream decision-making losses and agrees on their best-response actions almost everywhere.
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