Calibrated Stackelberg Games: Learning Optimal Commitments Against Calibrated Agents

We introduce Calibrated Stackelberg Games (CSGs), a generalization of the standard Stackelberg Games (SGs) framework. In CSGs, a principal repeatedly interacts with an agent who (contrary to standard SGs) does not have direct access to the principal's action but instead best-responds to calibrated forecasts about it. This framework provides a powerful and realistic modeling tool that goes beyond assuming that agents use ad hoc and highly specified algorithms for interacting in strategic settings and instead builds on statistical foundations of forecasts and calibration. We show that in CSGs, despite both the principal and the agent having less information than in standard SGs, the principal's optimal utility remains upper and lower bounded by the Stackelberg value of the one-shot game, in both finite and continuous settings. Alongside CSGs, we develop stronger notions of calibration and corresponding algorithms that address two central challenges for calibration in game-theoretic environments. First, achieving point-wise calibration typically incurs an error that scales exponentially with the dimension of the strategy space. Second, the principal's convergence rate in CSGs depends critically on the adaptivity of the agent's calibration algorithm. To address these challenges, we establish a meaningful, efficiently achievable relaxation of calibration based on conditioning on best-response regions. This yields the first notion of calibration in games with a statistical rate that only depends on the number of agents' actions rather than the dimension of the principal's strategy space and that leads to no-swap regret for the agent. We further develop adaptive calibration algorithms for the agents that provide fine-grained, any-time calibration guarantees against adversarial sequences, enabling the principal to achieve faster convergence in CSGs.