Leadership in singleton congestion games: What is hard and what is easy

Castiglioni, Matteo; Marchesi, Alberto; Gatti, Nicola; Coniglio, Stefano

doi:10.1016/j.artint.2019.103177

Cited by 16 publications

(18 citation statements)

References 28 publications

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“…Future developments include establishing the approximability status of the problem with two followers, the generalization to the case with both leader and followers playing mixed strategies, partially addressed in [4,5] (even though we conjecture that this problem could be much harder, probably p 2 -hard), and the study of structured games (e.g., congestion games beyond the special case of singleton games with monotonic costs which are shown to be polynomially solvable in [11,20]).…”

Section: Discussionmentioning

confidence: 99%

“…The complete enumerative algorithm is detailed in Algorithm 1. In the pseudocode, CheckEmptyness(S + , S − ) is a subroutine which looks for a value of 0 which is optimal for Problem (7), while Solve-lex-MILP(S + , S − ) is another subroutine which solves Problem (11). Note that Problem (7) may be infeasible.…”

Section: Outline Of the Explicit Enumeration Algorithmmentioning

confidence: 99%

“…Since max x n ∈X (S + )∩X (S − ) ψ(x n ; S + ) is not well-defined for nodes where S + = ∅, for them we solve, rather than an instance of Problem (11), a restriction of the optimistic problem (see Sect. 3) with constraints imposing that all followers' action profiles in S − are not NEs.…”

Section: Computing Sup X N ∈1 N F (X N )mentioning

confidence: 99%

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

2019

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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...

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Section: Discussionmentioning

confidence: 99%

Section: Outline Of the Explicit Enumeration Algorithmmentioning

confidence: 99%

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

2019

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“…The results related to singleton games appeared in [36] and its extended version [15]. Instead, all the other results are provided by [34] (see [35] for an extended version).…”

Section: Stackelberg Congestion Gamesmentioning

confidence: 99%

Leadership Games: Multiple Followers, Multiple Leaders, and Perfection

Marchesi

2021

Special Topics in Information Technology

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“…This is often the case in the security domain (An et al 2011;Kiekintveld et al 2009), where a defender, aiming to protect a set of valuable targets from the attackers, plays first, while the attackers, acting as followers, make their move only after observing the leader's defensive strategy. Other noteworthy cases are interdiction problems (Caprara et al 2016;Matuschke et al 2017), toll setting problems (Labbé and Violin 2016), network routing problems (Amaldi et al 2013) and (singleton) congestion games (Castiglioni et al 2018;Marchesi et al 2018).…”

Section: Introductionmentioning

confidence: 99%

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

Basilico

Coniglio

Gatti

et al. 2020

EURO Journal on Computational Optimization

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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 N P-hard and not in Poly-APX unless P = N P. 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.

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Leadership in singleton congestion games: What is hard and what is easy

Cited by 16 publications

References 28 publications

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

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

Leadership Games: Multiple Followers, Multiple Leaders, and Perfection

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

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