# Calibrated Stackelberg Games: Learning Optimal Commitments Against Calibrated Agents

arXiv: https://arxiv.org/abs/2306.02704 · HTML: https://mlsanity.com/paper/2306.02704/

## Key facts

- Cited by 25 papers in the map (data through 14 Oct 2026).
- Builds on 26 of its 71 references that are in the map.
- First posted on arXiv 5 Jun 2023.

## Builds on (most cited first, 26 of 26)

- [Strategic Classification](https://mlsanity.com/paper/1506.06980/) (2015)
- [Strategic Classification from Revealed Preferences](https://mlsanity.com/paper/1710.07887/) (2017)
- [Strongly Adaptive Online Learning](https://mlsanity.com/paper/1502.07073/) (2015)
- [Who Leads and Who Follows in Strategic Classification?](https://mlsanity.com/paper/2106.12529/) (2021)
- [Improved Strongly Adaptive Online Learning using Coin Betting](https://mlsanity.com/paper/1610.04578/) (2016)
- [Strategizing against No-regret Learners](https://mlsanity.com/paper/1909.13861/) (2019)
- [The Strategic Perceptron](https://mlsanity.com/paper/2008.01710/) (2020)
- [Selling to a No-Regret Buyer](https://mlsanity.com/paper/1711.09176/) (2017)
- [Learning Prices for Repeated Auctions with Strategic Buyers](https://mlsanity.com/paper/1311.6838/) (2013)
- [Strategizing against Learners in Bayesian Games](https://mlsanity.com/paper/2205.08562/) (2022)
- [Advancing subgroup fairness via sleeping experts](https://mlsanity.com/paper/1909.08375/) (2019)
- [Learning in Stackelberg Games with Non-myopic Agents](https://mlsanity.com/paper/2208.09407/) (2022)
- [Convergence of Learning Dynamics in Stackelberg Games](https://mlsanity.com/paper/1906.01217/) (2019)
- [Online Minimax Multiobjective Optimization: Multicalibeating and Other Applications](https://mlsanity.com/paper/2108.03837/) (2021)
- [Efficient Convex Optimization with Membership Oracles](https://mlsanity.com/paper/1706.07357/) (2017)
- [Smooth Calibration, Leaky Forecasts, Finite Recall, and Nash Dynamics](https://mlsanity.com/paper/2210.07152/) (2022)
- [Forecast Hedging and Calibration](https://mlsanity.com/paper/2210.07169/) (2022)
- [Is Learning in Games Good for the Learners?](https://mlsanity.com/paper/2305.19496/) (2023)
- [Mechanisms for a No-Regret Agent: Beyond the Common Prior](https://mlsanity.com/paper/2009.05518/) (2020)
- [Calibrated Forecasts: The Minimax Proof](https://mlsanity.com/paper/2209.05863/) (2022)
- [Fundamental Bounds on Online Strategic Classification](https://mlsanity.com/paper/2302.12355/) (2023)
- [Dual-Mandate Patrols: Multi-Armed Bandits for Green Security](https://mlsanity.com/paper/2009.06560/) (2020)
- [Efficient Stackelberg Strategies for Finitely Repeated Games](https://mlsanity.com/paper/2207.04192/) (2022)
- [Robust Stackelberg Equilibria](https://mlsanity.com/paper/2304.14990/) (2023)
- [Robust Stackelberg Equilibria in Extensive-Form Games and Extension to Limited Lookahead](https://mlsanity.com/paper/1711.08080/) (2017)
- [Robust Commitments and Partial Reputation](https://mlsanity.com/paper/1905.11555/) (2019)

## Built on by (most cited first, 25 of 25)

- [High-Dimensional Prediction for Sequential Decision Making](https://mlsanity.com/paper/2310.17651/) (2023)
- [Fast swap regret minimization and applications to approximate correlated equilibria](https://mlsanity.com/paper/2310.19647/) (2023)
- [Forecasting for Swap Regret for All Downstream Agents](https://mlsanity.com/paper/2402.08753/) (2024)
- [Full Swap Regret and Discretized Calibration](https://mlsanity.com/paper/2502.09332/) (2025)
- [Bayesian Strategic Classification](https://mlsanity.com/paper/2402.08758/) (2024)
- [High dimensional online calibration in polynomial time](https://mlsanity.com/paper/2504.09096/) (2025)
- [Group-wise oracle-efficient algorithms for online multi-group learning](https://mlsanity.com/paper/2406.05287/) (2024)
- [The Double-Edged Sword of Behavioral Responses in Strategic Classification: Theory and User Studies](https://mlsanity.com/paper/2410.18066/) (2024)
- [Swap Regret and Correlated Equilibria Beyond Normal-Form Games](https://mlsanity.com/paper/2502.20229/) (2025)
- [Regret Minimization in Stackelberg Games with Side Information](https://mlsanity.com/paper/2402.08576/) (2024)
- [AI-Assisted Decision Making with Human Learning](https://mlsanity.com/paper/2502.13062/) (2025)
- [Impact of Decentralized Learning on Player Utilities in Stackelberg Games](https://mlsanity.com/paper/2403.00188/) (2024)
- [Is Knowledge Power? On the (Im)possibility of Learning from Strategic Interactions](https://mlsanity.com/paper/2408.08272/) (2024)
- [Neural Operators Can Play Dynamic Stackelberg Games](https://mlsanity.com/paper/2411.09644/) (2024)
- [Should Decision-Makers Reveal Classifiers in Online Strategic Classification?](https://mlsanity.com/paper/2506.01936/) (2025)
- [Learning to Play Against Unknown Opponents](https://mlsanity.com/paper/2412.18297/) (2024)
- [Sparsity-Based Interpolation of External, Internal and Swap Regret](https://mlsanity.com/paper/2502.04543/) (2025)
- [Best of Both Worlds: Regret Minimization versus Minimax Play](https://mlsanity.com/paper/2502.11673/) (2025)
- [Learning in Structured Stackelberg Games](https://mlsanity.com/paper/2504.09006/) (2025)
- [Dimension-Free Decision Calibration for Nonlinear Loss Functions](https://mlsanity.com/paper/2504.15615/) (2025)
- [Near-optimal Swap Regret Minimization for Convex Losses](https://mlsanity.com/paper/2602.08862/) (2026)
- [Explanation Design in Strategic Learning: Sufficient Explanations that Induce Non-harmful Responses](https://mlsanity.com/paper/2502.04058/) (2025)
- [Learning Local Stackelberg Equilibria from Repeated Interactions with a Learning Agent](https://mlsanity.com/paper/2510.22471/) (2025)
- [Locally Adaptive Multi-Objective Learning](https://mlsanity.com/paper/2602.14952/) (2026)
- [Robust Strategic Classification under Decision-Dependent Cost Uncertainty](https://mlsanity.com/paper/2606.30136/) (2026)

## Abstract (arXiv)

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.