Pessimistic Leader-Follower Equilibria with Multiple Followers

Coniglio, Stefano; Gatti, Nicola; Marchesi, Alberto

doi:10.24963/ijcai.2017/25

Cited by 22 publications

(24 citation statements)

References 14 publications

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“…As shown in [13], all these LPs can be encoded into a single LP-a slight variation of the LP that is used to compute a correlated equilibrium (the solution concept where all the players can exploit a correlation device to coordinate their strategies). 3 Some works study the equilibrium-finding problem (only in the 2 A preliminary version of this work appeared in [12]. Compared to it, this paper extends the complexity results by studying the inapproximability of the problem (Sect.…”

Section: Previous Workmentioning

confidence: 89%

“…Proof Let x * n ∈ X (S + )∩X (S − ) be the strategy where the supremum is attained according to the formula in Theorem 5, namely, where ψ(x * n , S + ) = max x n ∈X (S + )∩X (S − ) ψ(x n ; S + ) = s. Problem (12) calls for a solution x n of value at least s − α (thus, for an α-approximate strategy) belonging to X (S + )∩ X (S − ; ) with as large as possible, whose existence is guaranteed by Lemma 3. Due to the lexicographic nature of the algo-rithmLet (x n ,ˆ ) be an optimal solution to Problem (12)…”

Section: Finding An˛-approximate Strategymentioning

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: Previous Workmentioning

confidence: 89%

Section: Finding An˛-approximate Strategymentioning

confidence: 99%

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

2019

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“…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.…”

Section: O/p-lfne With Leader In Mixed and Followers In Pure (O/p-lmfp)mentioning

confidence: 91%

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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“…To conclude, we experimentally evaluate the scalability of our methods over a testbed of randomly generated instances. A preliminary version of our results on normal-form Stackelberg games appeared in [17], while a complete version is [18].…”

Section: Norma-form Stackelberg Gamesmentioning

confidence: 99%