Abstract
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}-hard and not in Poly-mathcal {APX} unless mathcal {P} = mathcal {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.
Highlights
Leader-follower games model the interaction between rational agents when a hierarchical decision-making structure is in place
We focus on the fundamental case of single-leader multi-follower games with a finite number of actions per player where the overall game can be represented as a normal-form or polymatrix game—the latter is of interest as it plays an important role in a number of applications such as in the security domain, where the defender may need to optimize against multiple uncoordinated attackers solely interested in damaging the leader
For the problem of computing an O/P-leader-follower Nash equilibrium (LFNE), we show the following result: Proposition 3 The optimization problem associated with the search problem of computing an O/P-LFNE in the leader in mixed and followers in mixed (LMFM) and leader in pure and followers in mixed (LPFM) cases is N P-hard and it is not in Poly-APX unless P = N P, even when the game is polymatrix
Summary
Leader-follower (or Stackelberg) games model the interaction between rational agents (or players) when a hierarchical decision-making structure is in place. The number of real-world problems where a leader-follower (or Stackelberg) structure can be identified is extremely large 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. We refer to an equilibrium in such games as leader-follower Nash equilibrium (LFNE) As it is typical in bilevel programming, we study two extreme cases: the optimistic one where, if the leader’s commitment originates more NE in the followers’ game, one which maximizes the leader’s utility is selected, and the pessimistic one where an equilibrium which minimizes the leader’s utility is chosen. We conclude by providing a thorough experimental evaluation of our techniques on a (normal-form and polymatrix) test bed generated with GAMUT (Nudelman et al 2004), encompassing some structured games, employing different solvers: BARON, SCIP, CPLEX, SNOPT and RBFOpt. (The latter is used for black-box optimization)
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