The Stackelberg equilibrium solution concept describes optimal strategies to commit to: Player 1 (termed the leader) publicly commits to a strategy and Player 2 (termed the follower) plays a best response to this strategy (ties are broken in favor of the leader). We study Stackelberg equilibria in finite sequential games (or extensive-form games) and provide new exact algorithms, approximate algorithms, and hardness results for several classes of these sequential games.
The detailed analysis of the impact when the leader can commit to correlated strategies has, however, not been investigated sufficiently in the existing work. We address this extension and study the complexity for multiple subclasses of extensive-form games. Our results show that for many cases the problem of computing Stackelberg equilibria in correlated strategies is polynomial compared to the NP-hardness in behavioral strategies. Finally, these theoretical results have also practical algorithmic implications. An algorithm that computes a Stackelberg equilibrium in correlated strategies can be used to compute a Stackelberg equilibrium in behavioral strategies allowing a significant speed-up in computation time @cite_0 .
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We study the impact of collusion in network games with splittable flow and focus on the well established price of anarchy as a measure of this impact. We first investigate symmetric load balancing games and show that the price of anarchy is at most m, where m denotes the number of coalitions. For general networks, we present an instance showing that the price of anarchy is unbounded, even in the case of two coalitions. If latencies are restricted to polynomials with nonnegative coefficients and bounded degree, we prove upper bounds on the price of anarchy for general networks, which improve upon the current best ones except for affine latencies. In light of the negative results even for two coalitions, we analyze the effectiveness of Stackelberg strategies as a means to improve the quality of Nash equilibria. In this setting, an α fraction of the entire demand is first routed centrally by a Stackelberg leader according to a predefined Stackelberg strategy and the remaining demand is then routed selfishly by the coalitions (followers). For a single coalitional follower and parallel arcs, we develop an efficiently computable Stackelberg strategy that reduces the price of anarchy to one. For general networks and a single coalitional follower, we show that a simple strategy, called SCALE, reduces the price of anarchy to 1+α. Finally, we investigate SCALE for multiple coalitional followers, general networks, and affine latencies. We present the first known upper bound on the price of anarchy in this case. Our bound smoothly varies between 1.5 for α=0 and full efficiency for α=1. Significant progress has been made recently in the following two lines of research in the intersection of AI and game theory: (1) the computation of optimal strategies to commit to (Stackelberg strategies), and (2) the computation of correlated equilibria of stochastic games. In this paper, we unite these two lines of research by studying the computation of Stackelberg strategies in stochastic games. We provide theoretical results on the value of being able to commit and the value of being able to correlate, as well as complexity results about computing Stackelberg strategies in stochastic games. We then modify the QPACE algorithm ( 2011) to compute Stackelberg strategies, and provide experimental results. Algorithms for finding game-theoretic solutions are now used in several real-world security applications. This work has generally assumed a Stackelberg model where the defender commits to a mixed strategy first. In general two-player normal-form games, Stackelberg strategies are easier to compute than Nash equilibria, though it has recently been shown that in many security games, Stackelberg strategies are also Nash strategies for the defender. However, the work on security games so far assumes that the attacker attacks only a single target. In this paper, we generalize to the case where the attacker attacks multiple targets simultaneously. Here, Stackelberg and Nash strategies for the defender can be truly different. We provide a polynomial-time algorithm for finding a Nash equilibrium. The algorithm gradually increases the number of defender resources and maintains an equilibrium throughout this process. Moreover, we prove that Nash equilibria in security games with multiple attackers satisfy the interchange property, which resolves the problem of equilibrium selection in such games. On the other hand, we show that Stackelberg strategies are actually NP-hard to compute in this context. Finally, we provide experimental results. We develop polynomial-time algorithms for finding correlated equilibria—a well-studied notion of rationality that generalizes the Nash equilibrium—in a broad class of succinctly representable multiplayer games, encompassing graphical games, anonymous games, polymatrix games, congestion games, scheduling games, local effect games, as well as several generalizations. Our algorithm is based on a variant of the existence proof due to Hart and Schmeidler, and employs linear programming duality, the ellipsoid algorithm, Markov chain steady state computations, as well as application-specific methods for computing multivariate expectations over product distributions. For anonymous games and graphical games of bounded tree-width, we provide a different polynomial-time algorithm for optimizing an arbitrary linear function over the set of correlated equilibria of the game. In contrast to our sweeping positive results for computing an arbitrary correlated equilibrium, we prove that optimizing over correlated equilibria is NP-hard in all of the other classes of games that we consider.