  Stackelberg games originate where there are market leaders and followers, and
the actions of leaders influence the behavior of the followers. Mathematical
modelling of such games results in what's called a Bilevel Optimization
problem. There is an entire area of research dedicated to analyzing and solving
Bilevel Optimization problems which are often complex, and finding solutions
for such problems is known to be NP-Hard. A generalization of Stackelberg games
is a Multilevel Stackelberg game where we may have nested leaders and
followers, such that a follower is, in turn, a leader for all lower-level
players. These problems are much more difficult to solve, and existing solution
approaches typically require extensive cooperation between the players (which
generally can't be assumed) or make restrictive assumptions about the structure
of the problem. In this paper, we present a stochastic algorithm to approximate
the local equilibrium solutions for these Multilevel games. We then construct a
few examples of such Multilevel problems, including: a) a nested toll-setting
problem; and b) an adversarial initial condition determination problem for
Robust Trajectory Optimization. We test our algorithm on our constructed
problems as well as some trilevel problems from the literature, and show that
it is able to approximate the optimum solutions for these problems within a
reasonable error margin. We also provide an asymptotic proof for the
convergence of the algorithm and empirically analyze its accuracy and
convergence speed for different parameters. Lastly, we compare it with existing
solution strategies from the literature and demonstrate that it outperforms
them.
