Calibrated Stackelberg Games: AI agents learn optimal commitments from forecasts

A new paper from CMU and MIT researchers introduces Calibrated Stackelberg Games (CSGs), a generalization of the classic Stackelberg game framework. In standard Stackelberg games, a leader (principal) commits to a strategy, and a follower (agent) observes it directly before best-responding. CSGs remove that direct observation: the agent instead receives only calibrated forecasts about the principal's actions. This models real-world scenarios where agents (e.g., users, competitors) infer strategies from signals rather than seeing the exact move. The paper proves that despite this added uncertainty, the principal's optimal utility remains tightly bounded by the Stackelberg value of the underlying one-shot game, in both finite and continuous action spaces.

To make CSGs practical, the authors develop two key algorithmic contributions. First, they introduce a relaxation of calibration—conditioning on best-response regions—that achieves a statistical error rate depending only on the number of agent actions, not the dimension of the principal's strategy space. This is the first notion of calibration in games with such efficient guarantees and leads to no-swap regret for the agent. Second, they design adaptive calibration algorithms that provide anytime guarantees against adversarial sequences, enabling the principal to converge faster to optimal commitments. The work bridges game theory and statistical learning, offering a principled way to model strategic interactions where agents rely on predictions rather than perfect information.