Computing a Pessimistic Stackelberg Equilibrium with Multiple Followers: The Mixed-Pure Case

Abstract

The search problem of computing a Stackelberg (or leader-follower) equilibrium (also referred to as an optimal strategy to commit to) has been widely investigated in the scientific literature in, almost exclusively, the single-follower setting. Although the optimistic and pessimistic versions 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, polynomialtime 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 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 a pessimistic bilevel programming problem, we show that, with two followers, the problem is NP-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 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 All the authors contributed equally to this manuscript. Extended author information available on the last page of the article 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-theart methods, we propose an exact ad hoc algorithm (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 α-approximate strategy where α > 0 is an arbitrary additive loss (at most as large as the supremum). We conclude the paper by evaluating the scalability of our algorithms via computat...

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

“…It is not difficult to see that the previous algorithm (which, overall, runs in polynomial time) is not correct in the pessimistic case. This is not surprising since, as shown in Coniglio et al (2017Coniglio et al ( , 2018, the optimization problem corresponding to the equilibrium-finding problem is N P-hard in the pessimistic case even with followers restricted to pure strategies. For its solution, we can resort to the same methods proposed in this paper for the LMFM case, simply requiring ρ 1 and ρ 2 to be binary.…”

Coordinating Followers to Reach Better Equilibria: End-to-End Gradient Descent for Stackelberg Games

“…Stackelberg models with multiple followers Multi-follower Stackelberg problems have received a lot of attention in domains with a hierarchical leader-follower structure [29,35,42,44,46]. Although single-follower normal-form Stackelberg games can be solved in polynomial time [8,22,39], the problem becomes NP-hard when multiple followers are present, even when the equilibrium is assumed to be either optimistic or pessimistic [6,11]. Existing approaches [3,6] primarily leverage the leaderfollower structure in a bilevel optimization formulation [10], which can be solved by reformulating the followers' best response into non-convex stationary and complementarity constraints in the leader's optimization problem [43].…”

“…Moreover, many realistic settings involve a single leader with multiple self-interested followers such as wildlife conservation efforts with a central coordinator and a team of defenders [15,16]; resource management in energy [3] with suppliers, aggregators, and end users; or security problems with a central insurer and a set of vulnerable agents [21,34]. Solving Stackelberg games with multiple followers is challenging in general [5,11]. Previous work often reformulates the followers' best response as stationary and complementarity constraints in the leader's optimization [5,6,9,11,40], casting the entire Stackelberg problem as a single optimization problem.…”

“…Solving Stackelberg games with multiple followers is challenging in general [5,11]. Previous work often reformulates the followers' best response as stationary and complementarity constraints in the leader's optimization [5,6,9,11,40], casting the entire Stackelberg problem as a single optimization problem. This reformulation approach has achieved significant success in problems with linear or quadratic objectives, assuming a unique equilibrium or a specific equilibrium concept, e.g., followers' optimistic or pessimistic choice of equilibrium [5,6,18].…”

A Unified Framework for Multistage Mixed Integer Linear Optimization

“…Two main variants of the Stackelberg paradigm are typically considered: one in which the followers can observe the action that the leader draws from its commitment and, therefore, the commitment is in pure strategies Stackelberg [2010], and one in which the followers cannot do that directly and, hence, the leader's commitment can be in mixed strategies Conitzer and Sandholm [2006], von Stengel and Zamir [2010]. While most of the works focus on the case with a single leader and a single follower (which leads to a proper bilevel optimization problem), some work has been done on the case with more than two players: see Conitzer and Korzhyk [2011], Basilico et al [2016Basilico et al [ , 2017, , Basilico et al [2020], Marchesi et al [2018], Castiglioni et al [2019b], Coniglio et al [2020a] for the single-leader multi -follower case, Smith et al [2014], Lou and Vorobeychik [2015], Laszka et al [2016], Lou et al [2017], Gan et al [2018] for the multi -leader single-follower case, or Castiglioni et al [2019a], Pang and Fukushima [2005], Leyffer and Munson [2010], Kulkarni and Shanbhag [2014] for the multi -leader multi -follower case. Practical applications are often found in security games, which correspond to competitive situations where a defender (leader) has to allocate scarce resources to protect valuable targets from an attacker (follower) Paruchuri et al [2008], Kiekintveld et al [2009], , Tamble [2011].…”