  A growing body of work in game theory extends the traditional Stackelberg
game to settings with one leader and multiple followers who play a Nash
equilibrium. Standard approaches for computing equilibria in these games
reformulate the followers' best response as constraints in the leader's
optimization problem. These reformulation approaches can sometimes be
effective, but often get trapped in low-quality solutions when followers'
objectives are non-linear or non-quadratic. Moreover, these approaches assume a
unique equilibrium or a specific equilibrium concept, e.g., optimistic or
pessimistic, which is a limiting assumption in many situations. To overcome
these limitations, we propose a stochastic gradient descent--based approach,
where the leader's strategy is updated by differentiating through the
followers' best responses. We frame the leader's optimization as a learning
problem against followers' equilibrium, which allows us to decouple the
followers' equilibrium constraints from the leader's problem. This approach
also addresses cases with multiple equilibria and arbitrary equilibrium
selection procedures by back-propagating through a sampled Nash equilibrium. To
this end, this paper introduces a novel concept called equilibrium flow to
formally characterize the set of equilibrium selection processes where the
gradient with respect to a sampled equilibrium is an unbiased estimate of the
true gradient. We evaluate our approach experimentally against existing
baselines in three Stackelberg problems with multiple followers and find that
in each case, our approach is able to achieve higher utility for the leader.
