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In this paper, we introduce a generalization of the standard Stackelberg Games (SGs) framework: Calibrated Stackelberg Games. 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. CSG is a powerful modeling tool that goes beyond assuming that agents use ad hoc and highly specified algorithms for interacting in strategic settings to infer the principal's actions and thus more robustly addresses real-life applications that SGs were originally intended to capture. Along with CSGs, we also introduce a stronger notion of calibration, termed adaptive calibration, that provides fine-grained any-time calibration guarantees against adversarial sequences. We give a general approach for obtaining adaptive calibration algorithms and specialize them for finite CSGs. In our main technical result, we show that in CSGs, the principal can achieve utility that converges to the optimum Stackelberg value of the game both in finite and continuous settings and that no higher utility is achievable. Two prominent and immediate applications of our results are the settings of learning in Stackelberg Security Games and strategic classification, both against calibrated agents. 

We study the problem of online binary classification where strategic agents can manipulate their observable features in predefined ways, modeled by a manipulation graph, in order to receive a positive classification. We show this setting differs in fundamental ways from classic (non-strategic) online classification. For instance, whereas in the non-strategic case, a mistake bound of ln |H| is achievable via the halving algorithm when the target function belongs to a known class H, we show that no deterministic algorithm can achieve a mistake bound o(Δ) in the strategic setting, where Δ is the maximum degree of the manipulation graph (even when |H| = O(Δ)). We complement this with a general algorithm achieving mistake bound O(Δ ln |H|). We also extend this to the agnostic setting, and show that this algorithm achieves a Δ multiplicative regret (mistake bound of O(Δ · OPT + Δ · ln |H|)), and that no deterministic algorithm can achieve o(Δ) multiplicative regret. Next, we study two randomized models based on whether the random choices are made before or after agents respond, and show they exhibit fundamental differences. In the first, fractional model, at each round the learner deterministically chooses a probability distribution over classifiers inducing expected values on each vertex (probabilities of being classified as positive), which the strategic agents respond to. We show that any learner in this model has to suffer linear regret. On the other hand, in the second randomized algorithms model, while the adversary who selects the next agent must respond to the learner's probability distribution over classifiers, the agent then responds to the actual hypothesis classifier drawn from this distribution. Surprisingly, we show this model is more advantageous to the learner, and we design randomized algorithms that achieve sublinear regret bounds against both oblivious and adaptive adversaries.