  In this paper, we develop an approximation scheme for solving bilevel
programs with equilibrium constraints, which are generally difficult to solve.
Among other things, calculating the first-order derivative in such a problem
requires differentiation across the hierarchy, which is computationally
intensive, if not prohibitive. To bypass the hierarchy, we propose to bound
such bilevel programs, equivalent to multiple-followers Stackelberg games, with
two new hierarchy-free problems: a $T$-step Cournot game and a $T$-step
monopoly model. Since they are standard equilibrium or optimization problems,
both can be efficiently solved via first-order methods. Importantly, we show
that the bounds provided by these problems -- the upper bound by the $T$-step
Cournot game and the lower bound by the $T$-step monopoly model -- can be made
arbitrarily tight by increasing the step parameter $T$ for a wide range of
problems. We prove that a small $T$ usually suffices under appropriate
conditions to reach an approximation acceptable for most practical purposes.
Eventually, the analytical insights are highlighted through numerical examples.
