Leadership Games: Multiple Followers, Multiple Leaders, and Perfection

Over the last years, algorithmic game theory has received growing interest in AI, as it allows to tackle complex real-world scenarios involving multiple artificial agents engaged in a competitive interaction. These settings call for rational agents endowed with the capability of reasoning strategically, which is achieved by exploiting equilibrium concepts from game theory.

there are more than two players and/or arbitrary, i.e., general-sum, utilities are widely unexplored. In such scenarios, there is no clear definition of solution to a game, as this strongly depends on the specific application that one wish to represent. As a result, many solution concepts other than the Nash equilibrium have been introduced and studied. However, there is still a lot of work to be done on the computational side, as the algorithmic works on multi-player general-sum games are only few.
In this work, we study settings beyond two-player zero-sum games, focusing on a particular game paradigm which leads to the definition of what is known in the literature as the Stackelberg equilibrium.

The Stackelberg Paradigm
The Stackelberg paradigm was originally introduced by von Stackelberg in 1934 to model economic situations where a firm (the leader) moves first and, then, another firm (the follower) moves second by reacting to the first firm's move [50]. Recently, this paradigm was brought to new attention by the work of [7], who study a variant of the original Stackelberg paradigm in which the leader commits to a (possibly randomized, i.e., mixed) strategy beforehand, while the follower decides how to play after observing the leader's strategy. In general settings involving multiple players, a Stackelberg game is characterized by a group of players who act as leaders with the ability to commit to (possibly mixed) strategies beforehand, whereas the other players are followers who observe the commitment and decide how to play thereafter.
Over the last years, Stackelberg games and their corresponding Stackelberg equilibria have received growing attention in the AI literature, where the computational problem of finding such equilibria in often referred to as the problem of computing optimal strategies to commit to [20]. This surge of interest was motivated by the successful applications of Stackelberg games in many interesting real-world settings. In particular, among the others, the security domain is the most explored one, and, in it, different game models have been introduced, usually referred to as security games [2,29,42,47]. In such models, there is a defender that has to protect some valuable targets from an attacker, who can wait while observing the defender's protection strategy before deciding where, when and how to attack. This scenario naturally fits into the Stackelberg model, where the defender is the leader and the attacker is the follower. Other interesting applications are found in toll-setting games, where the leader is a central authority which collects tolls from the users of a network who, acting as followers, decide on how to best travel through the network so as to minimize their cost after observing the pricing strategy chosen by the authority [30,31]. Besides the security domain and toll-setting games, applications of Stackelberg games can be found in, among others, interdiction games [12,39], network routing [1], inspection games [3], and mechanism design [44].
Despite the attention that Stackelberg games received from the AI literature, most of the works related to them focus, with some exceptions (see, e.g., [7,19,26]), on particular game settings that involve only two players (i.e., one leader and one follower) and enjoy specific structures, as it is the case in security games. It is worth pointing out two works that study general Stackelberg games with a single leader and multiple follower; specifically, [7] study the case in which the followers play a Nash equilibrium given the leader's commitment, whereas [19] address the case where they play a correlated equilibrium.
Let us also notice that, while some works (see, e.g., [8,16,32]) address the computation of Stackelberg equilibria in games with a sequential (i.e., tree-form) structure, none of them investigates refinements of such equilibria. This is surprising as refinements have been extensively studied for the Nash equilibrium, since it is well-known that classical (unrefined) solution concepts may lead to a sub-optimal behavior off the equilibrium path in games with a sequential structure (see [24,49] for some references on the topic).

Stackelberg Games with Multiple Followers
We address Stackelberg games with a single leader and multiple followers. Following [7], we study settings in which, after observing the leader's commitment, the followers play a Nash equilibrium in the resulting game. We refer to this solution as Stackelberg-Nash equilibrium. We focus on the case in which the followers are restricted to pure (i.e., non-mixed) strategies, as the general problem with followers playing mixed strategies is already known to be computationally intractable [4]. This restriction leads to interesting computational complexity results. Moreover, this is without loss of generality in games that always admit pure-strategy Nash equilibria, as it is the case for congestion games [43].
We study the problem of computing Stackelberg-Nash equilibria, focusing on two cases: the one in which the followers break ties in favor of the leader (what is usually referred to as a strong equilibrium), and the case where they break ties against the leader (leading to a weak equilibrium). Moreover, we analyze three different classes of games, namely, normal-form games, polymatrix games, and congestion games. Table 10.1 shows our contributions related to Stackelberg games with a single leader, summarizing the computational complexity and the algorithmic aspects of the problems we study, with focus on normal-form, extensive-form, Bayesian, and polymatrix games. The table also shows, for comparison, other state-of-the-art results, including those about single-leader single-follower Stackelberg games (our original contributions are those without a reference). Our contributions on Stackelberg congestion games are instead detailed in Table 10.2.

Norma-Form Stackelberg Games
Since a strong Stackelberg-Nash equilibrium (with followers restricted to pure strategies) can be computed efficiently (in polynomial time) by solving multiple linear programs (LPs), we entirely devote our analysis to the weak case (with, again, followers restricted to pure strategies). In terms of computational complexity, we show that, differently from the strong case, in the weak one the equilibrium-finding prob- Algorithm Multi-LP Spatial branch-and-bound [5] Multi-lex-MILP Algorithm Multi-LP Spatial branch-and-bound [5] Multi-lex-MILP Stackelberg polymatrix games Algorithm Multi-LP Spatial branch-and-bound [5,6] Multi-lex-MILP n ≥ 4 (fixed) Algorithm Multi-LP Spatial branch-and-bound [5,6] Multi-lex-MILP Algorithm Multi-LP Spatial branch-and-bound [5] Multi-lex-MILP lem is NP-hard with two or more followers, while, when the number of followers is three or more, the problem cannot be approximated in polynomial time to within any polynomial multiplicative factor unless P = NP (i.e., in formal terms, it is not in the class Poly-APX unless P = NP). To establish these two results, we introduce two reductions, one from Independent Set and the other one from 3-SAT.
After analyzing the complexity of the problem, we focus on its algorithmic aspects. First, we formulate the problem as a bilevel programming problem. We then show how to recast it as a single-level quadratically constrained quadratic program (QCQP), which we show to be impractical to solve due to admitting a supremum, but not a maximum. We then introduce a restriction based on a mixed-integer linear program (MILP) which, while forsaking optimality, always admits an optimal (restricted) solution. Next, we propose an exact algorithm to compute the value of the supremum of the problem based on an enumeration scheme which, at each iteration, solves a lexicographic MILP (lex-MILP) where the two objective functions are optimized in sequence. Subsequently, we embed the enumerative algorithm within a branch-andbound scheme, obtaining an algorithm which is, in practice, much faster. We also extend the algorithms so that, for cases where the supremum is not a maximum, they return a strategy by which the leader can obtain a utility within an additive loss α with respect to the supremum, for any α > 0. 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].

Stackelberg Polymatrix Games
We identify two classes of Stackelberg polymatrix games that allow to characterize the complexity of computing Stackelberg-Nash equilibria (with followers restricted to pure strategies). The key property of these games is that, once fixed the number of players, computing a strong or weak equilibrium presents the same complexity, namely polynomial (again assuming that the followers play pure strategies). These games are of practical interest in security problems. Moreover, they are equivalent to Bayesian Stackelberg games with one leader and one follower, where the latter may be of different types. Our first class is equivalent to games with interdependent types, while the second one is equivalent to games with independent types (i.e., the leader's utility is independent of the follower's type). Thus, every result that holds for a game class also holds for its equivalent class.
We investigate whether the problem keeps being easy when the number of players is not fixed. We show that it is NP-hard to compute a weak Stackelberg-Nash equilibrium, and we provide an exact (exponential-time) algorithm (conversely, to compute a strong equilibrium, one can adapt the algorithm provided in [20] for Bayesian games, by means of our mapping). We also prove that, in all the instances where the weak Stackelberg-Nash equilibrium is a supremum but not a maximum, an α-approximation of the supremum can be found in polynomial time (also in the number of players) for any given additive loss α > 0. As for approximation complexity, we show that the problem is Poly-APX-complete. This also shows that, in Bayesian Stackelberg games with uncertainty over the follower, computing a weak Stackelberg-Nash equilibrium is as hard as finding a strong one [33].
Next, we investigate whether, in general polymatrix games with followers restricted to play pure strategies, the problem admits polynomial-time approximation algorithms. We provide a negative answer, showing that in the strong case the problem is not in Poly-APX if the number of players is non-fixed, unless P = NP. We also prove that the same inapproximability result holds for the weak case, even with a fixed number of players.
Our detailed results on Stackelberg polymatrix games appeared in [21] (see [22] for an extended version).

Stackelberg Congestion Games
We provide a comprehensive study of the computational complexity of finding Stackelberg-Nash equilibria in congestion games. These are games with a large number of players that compete for the use of some shared resources, where the cost of each resource is a function of the number of players using that resource, i.e., its congestion. Notice that, in such setting, assuming that the followers play a pure-strategy Nash equilibrium is without loss of generality, as congestion games always admit one [43].
First, we focus on games with singleton actions, i.e., where each player selects only one resource at a time. We draw a complete picture of the computational complexity of the problem of finding equilibria in Stackelberg singleton congestion games, with pure or mixed-strategy commitments, and considering the cases of finding either a strong equilibrium or a weak one. Interestingly, we identify two features which allow for thoroughly characterizing hard and easy game instances. The first one concerns the relationship among the action spaces of the players, with two possibilities: the one where the players are symmetric as they have identical action spaces and therefore they share the same set of resources, and the one where their action spaces may differ. The second feature is related to the shape of the players' cost functions. Two cases are possible: the one where these functions are monotonically increasing in the resource congestion and the one in which they may be not.
In particular, we show that, in games where the players' action spaces can be different, computing a (strong or weak) Stackelberg-Nash equilibrium is not in Poly-APX unless P = NP even when the players' cost functions are monotonic, the leader has only one action available, and her costs are equal to the followers'. This result also holds if we restrict the leader to pure-strategy commitments, given that the leader has only one action available. For symmetric games where the players have identical action spaces, we show that the complexity of computing an equilibrium depends on the nature of the players' cost functions. For the case where the players' costs are generic (monotonic or not) functions of the resource congestion, we prove that the problem is not in Poly-APX unless P = NP. On the other hand, we show that, in symmetric games, the problem of computing a strong or weak Stackelberg-Nash equilibrium can be solved in polynomial time when the cost functions are monotonic by proposing an algorithm for it. We also consider the case where the leader is restricted to pure-strategy commitments, providing a polynomial-time algorithm for its solution which applies even to symmetric games with generic cost functions. This algorithm is based on a polynomial-time dynamic programming algorithm available in the literature for computing a socially optimal Nash equilibria in non-Stackelberg singleton congestion games with identical action spaces, which we improve and extend to solve our problem.
Then, we switch the attention to games beyond singleton ones. We show that having actions made of only one resource is necessary to have efficient (polynomial-time) algorithms. Indeed, we prove that finding a strong Stackelberg-Nash equilibrium is NP-hard and not in Poly-APX unless P = NP, even if players' actions contain only two resources, costs are monotonic, and players are symmetric. We also introduce and study singleton congestion games in which the players are partitioned into classes, with followers of the same class sharing the same set of actions. These are a generalization of singleton games with symmetric players, capturing the common case in which users can be split into (usually few) different classes, such as, e.g., users with different priorities. For these games, we provide a dynamic programming algorithm that computes a strong Stackelberg-Nash equilibrium in polynomial time, when the number of classes is fixed and the leader is restricted to play pure strategies. On the other hand, we prove that, if the leader is allowed to play mixed strategies, then the problem becomes NP-hard even with only four classes and monotonic costs.
Finally, for all the settings we study, we design MILP formulations for computing a strong Stackelberg-Nash equilibrium, and we experimentally evaluate them on a testbed containing both randomly generated game instances and worst-case instances based on our hardness reductions.
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).

Stackelberg Games with Multiple Leaders
We study games with multiple leaders, providing a new way to apply the Stackelberg paradigm to any finite (underlying) game. Our approach extends the idea of commitment to correlated strategies in settings involving multiple leaders and followers, generalizing the work of [19]. The crucial component of our framework is that a leader can decide whether to participate in the commitment or to defect from it by becoming a follower. This induces a preliminary agreement stage that takes place before the underlying game is played, where the leaders decide, in turn, whether to opt out from the commitment or not. We model this stage as a sequential game, whose size is factorial in the number of players. Our goal is to identify commitments guaranteeing some desirable properties on the agreement stage. The first one requires that the leaders do not have any incentive to become followers. It comes in two flavors, called stability and perfect stability, which are related to, respectively, Nash and subgame perfect equilibria of the sequential game of the agreement stage. The second property is also defined in two flavors, namely efficiency and perfect efficiency, both enforcing Pareto optimality with respect to the leaders' utility functions, though at different levels of the agreement stage.
We introduce three solution concepts, which we generally call Stackelberg correlated equilibria. They differ depending on the properties they call for. Specifically, (simple) Stackelberg correlated equilibria, Stackelberg correlated equilibria with perfect agreement, and Stackelberg correlated equilibria with perfect agreement and perfect efficiency require, respectively, stability and efficiency, perfect stability and efficiency, and both perfect stability and perfect efficiency.
First, we investigate the game theoretic properties of our solution concepts. We show that Stackelberg correlated equilibria with or without perfect agreement are guaranteed to exist in any game, while Stackelberg correlated equilibria with perfect agreement and perfect efficiency may not. Moreover, we compare the former with other solution concepts, both Stackelberg and non-Stackelberg ones.
Then, we switch the attention to the computational complexity perspective. We show that, provided a suitably defined stability oracle is solvable in polynomial time, a Stackelberg correlated equilibrium optimizing some linear function of leaders' utilities (such as the leaders' social welfare) can be computed in polynomial time, even in the number of players. The same holds for finding a Stackelberg correlated equilibrium with perfect agreement, while we prove that computing an optimal one is an intractable problem. Nevertheless, in the latter case, we provide an (exponential in the game size) upper bound on the necessary number of queries to the oracle.
In conclusion, we study which classes of games admit a polynomial-time stability oracle, focusing on succinct games of polynomial type [41]. The problem solved by our oracle is strictly connected with the weighted deviation-adjusted social welfare problem introduced by [28]. As a result, we get that our oracle is solvable in polynomial time in all game classes where the same holds for finding an optimal correlated equilibrium.
Our results on Stackelberg games with multiple leaders appeared in [13] (see [14] for an extended version).

Trembling-Hand Perfection in Stackelberg Games
We study Stackelberg games with a sequential structure, usually referred to as extensive-form Stackelberg games. In particular, we show that classical Stackelberg equilibria may prescribe the players to play sub-optimally off the equilibrium path, as it is the case for the Nash equilibrium. Thus, in order to amend these weaknesses, we propose a way to refine Stackelberg equilibria thorough trembling-hand perfection, which is based on the idea that each player might play each action with low-but-non-zero probabilities, usually called trembles [45].
We show that for every perturbation scheme (i.e., any possible way of introducing trembles), the set of limit points of Stackelberg equilibria for perturbed games with vanishing perturbations is always a nonempty subset of the Stackelberg equilibria of the non-perturbed game. This does not hold when focusing only on strong (or weak) equilibria: for a given game, the set of strong Stackelberg equilibria (or weak Stackelberg equilibria) in the non-perturbed game may be disjoint from the set of limit points of strong Stackelberg equilibria (or weak Stackelberg equilibria) in the perturbed game. We resort to the perturbation schemes used for quasi-perfect equilibria [48] and extensive-form perfect equilibria [45] to define their Stackelberg counterpart-and their strong and weak versions-as refinements of the Stackelberg equilibrium.
Next, we focus on quasi-perfection. We formally define the quasi-perfect Stackelberg equilibrium refinement game theoretically in the same axiomatic fashion as the quasi-perfect equilibrium was defined for non-Stackelberg games [48]. Thus, our definition is based on a set of properties of the players' strategies, and it cannot be directly used to search for a quasi-perfect Stackelberg equilibrium. Subsequently, we define a class of perturbation schemes for the sequence form such that any limit point of a sequence of Stackelberg equilibria in perturbed games with vanishing perturbation is a quasi-perfect Stackelberg equilibrium. This class of perturbation schemes strictly includes those used to find a quasi-perfect equilibrium by [40]. Then, we extend the algorithm by [16] to the case of quasi-perfect Stackelberg equilibrium computation. We derive the corresponding mathematical program for computing a Stackelberg extensive-form correlated equilibrium when a perturbation scheme is introduced and we discuss how the individual steps of the algorithm change. In particular, the implementation of our algorithm is much more involved, requiring the combination of branch-and-bound techniques with arbitrary-precision arithmetic to deal with small perturbations. This does not allow a direct application of off-the-shelf solvers. Finally, we experimentally evaluate the scalability of our algorithm.
In conclusion, we also study the computational complexity of finding Stackelberg equilibrium refinements, showing that the problem of deciding the existence of a Stackelberg equilibrium-refined or not-that gives the leader expected value at least ν is NP-hard.
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