  As predictive models are deployed into the real world, they must increasingly
contend with strategic behavior. A growing body of work on strategic
classification treats this problem as a Stackelberg game: the decision-maker
"leads" in the game by deploying a model, and the strategic agents "follow" by
playing their best response to the deployed model. Importantly, in this
framing, the burden of learning is placed solely on the decision-maker, while
the agents' best responses are implicitly treated as instantaneous. In this
work, we argue that the order of play in strategic classification is
fundamentally determined by the relative frequencies at which the
decision-maker and the agents adapt to each other's actions. In particular, by
generalizing the standard model to allow both players to learn over time, we
show that a decision-maker that makes updates faster than the agents can
reverse the order of play, meaning that the agents lead and the decision-maker
follows. We observe in standard learning settings that such a role reversal can
be desirable for both the decision-maker and the strategic agents. Finally, we
show that a decision-maker with the freedom to choose their update frequency
can induce learning dynamics that converge to Stackelberg equilibria with
either order of play.
