Bilevel programming methods for computing single-leader-multi-follower equilibria in normal-form and polymatrix games

The concept of leader-follower (or Stackelberg) equilibrium plays a central role in a number of real-world applications bordering on mathematical optimization and game theory. While the single-follower case has been investigated since the inception of bilevel programming with the seminal work of von Stackelberg, results for the case with multiple followers are only sporadic and not many computationally affordable methods are available. In this work, we consider Stackelberg games with two or more followers who play a (pure or mixed) Nash equilibrium once the leader has committed to a (pure or mixed) strategy, focusing on normal-form and polymatrix games. As customary in bilevel programming, we address the two extreme cases where, if the leader’s commitment originates more Nash equilibria in the followers’ game, one which either maximizes (optimistic case) or minimizes (pessimistic case) the leader’s utility is selected. First, we show that, in both cases and when assuming mixed strategies, the optimization problem associated with the search problem of finding a Stackelberg equilibrium is $$\mathcal {NP}$$NP-hard and not in Poly-$$\mathcal {APX}$$APX unless $$\mathcal {P} = \mathcal {NP}$$P=NP. We then consider different situations based on whether the leader or the followers can play mixed strategies or are restricted to pure strategies only, proposing exact nonconvex mathematical programming formulations for the optimistic case for normal-form and polymatrix games. For the pessimistic problem, which cannot be tackled with a (single-level) mathematical programming formulation, we propose a heuristic black-box algorithm. All the methods and formulations that we propose are thoroughly evaluated computationally.