When Individually Calibrated Models Become Collectively Miscalibrated

Probabilistic prediction systems often aggregate probability estimates from multiple models into a single decision. A natural assumption is that if each model is individually calibrated, the aggregate prediction will also be well calibrated. We show that this assumption fails in multi-agent settings: individually calibrated predictors can become collectively miscalibrated when their predictions interact strategically. This phenomenon arises in settings where predictions originate from multiple independent agents, including federated healthcare, multi-vendor intrusion detection, and crowdsourced forecasting, where agents optimize their own objectives. Specifically, we prove that under Brier-score-based aggregation with correlated beliefs, each agent’s individually optimal report systematically underestimates the positive-class probability, producing a Price of Anarchy (PoA) of 7.25x (mean aggregate bias -0.375). In contrast, VCG-based aggregation, which rewards each agent’s marginal contribution to aggregate accuracy, achieves the lowest PoA among all mechanisms studied (PoA $\approx$ 1.0x under dominant-strategy equilibrium). On three real-world datasets (NSL-KDD, UNSW-NB15, Credit Card Fraud) with feature-partitioned agents, VCG provides the strongest robustness guarantees among the aggregation methods we evaluate, while maintaining comparable accuracy. In data-sparse regimes ($n \leq 500$), VCG consistently outperforms stacking and majority voting; under adversarial agents, VCG maintains substantially lower false-negative rates than robust aggregation baselines. Adaptive weight updates further reduce false negatives by 20–22% under distribution shift, with $O(\sqrt{T})$ online regret guarantees. These results establish that how probabilistic predictions are aggregated matters as much as how well individual models are calibrated.