When Nash Meets Stackelberg

Motivated by international energy trade between countries with profit-maximizing domestic producers, we analyze Nash games played among leaders of Stackelberg games (\NASP). We prove it is both $\Sigma^p_2$-hard to decide if the game has a pure-strategy (\PNE) or a mixed-strategy Nash equilibrium (\MNE). We then provide a finite algorithm that computes exact \MNEs for \NASPs when there is at least one, or returns a certificate if no \MNE exists. To enhance computational speed, we introduce an inner approximation hierarchy that increasingly grows the description of each Stackelberg leader feasible region. Furthermore, we extend the algorithmic framework to specifically retrieve a \PNE if one exists. Finally, we provide computational tests on a range of \NASPs instances inspired by international energy trades.