Computing a Pessimistic Stackelberg Equilibrium with Multiple Followers: The Mixed-Pure Case
Authors: Coniglio Stefano , Gatti Nicola , Marchesi Alberto 
Published: November 2019

The search problem of computing aStackelberg(orleader-follower)equilibrium(also referred to as anoptimal strategy to commit to) has been widely investigated in the scientific literature in, almost exclusively, the single-follower setting. Although theoptimisticandpessimisticversions of the problem, i.e., those where the single follower breaks any ties among multiple equilibria either in favour or against the leader, are solved with different methodologies, both cases allow for efficient, polynomial-time algorithms based on linear programming. The situation is different with multiple followers, where results are only sporadic and depend strictly on the nature of the followers’ game. In this paper, we investigate the setting of a normal-form game with a single leader and multiple followers who, after observing the leader’s commitment, play a Nash equilibrium. When both leader and followers are allowed to play mixed strategies, the corresponding search problem, both in the optimistic and pessimistic versions, is known to be inapproximable in polynomial time to within any multiplicative polynomial factor unless$$\textsf {P}=\textsf {NP}$$P=NP. Exact algorithms are known only for the optimistic case. We focus on the case where the followers play pure strategies—a restriction that applies to a number of real-world scenarios and which, in principle, makes the problem easier—under the assumption of pessimism (the optimistic version of the problem can be straightforwardly solved in polynomial time). After casting this search problem (with followers playing pure strategies) as apessimistic bilevel programming problem, we show that, with two followers, the problem is -hard and, with three or more followers, it cannot be approximated in polynomial time to within any multiplicative factor which is polynomial in the size of the normal-form game, nor, assuming utilities in [0, 1], to within any constant additive loss stricly smaller than 1 unless$$\textsf {P}=\textsf {NP}$$P=NP. This shows that, differently from what happens in the optimistic version, hardness and inapproximability in the pessimistic problem are not due to the adoption of mixed strategies. We then show that the problem admits, in the general case, a supremum but not a maximum, and we propose a single-level mathematical programming reformulation which asks for the maximization of a nonconcave quadratic function over an unbounded nonconvex feasible region defined by linear and quadratic constraints. Since, due to admitting a supremum but not a maximum, only a restricted version of this formulation can be solved to optimality with state-of-the-art methods, we propose an exactad hocalgorithm (which we also embed within a branch-and-bound scheme) capable of computing the supremum of the problem and, for cases where there is no leader’s strategy where such value is attained, also an$$\alpha $$α-approximate strategy where$$\alpha > 0$$α>0is an arbitrary additive loss (at most as large as the supremum). We conclude the paper by evaluating the scalability of our algorithms via computational experiments on a well-established testbed of game instances.