Calibrated Stackelberg Games: Learning Optimal Commitments Against
  Calibrated Agents
Nika Haghtalab Chara Podimata Kunhe Yang
In this paper, we introduce a generalization of the standard Stackelberg
Games (SGs) framework: Calibrated Stackelberg Games (CSGs). 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 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.