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.
For one-shot games, the problem of computing a Stackelberg equilibrium is polynomial @cite_4 in contrast to the PPAD-completeness of a Nash equilibrium @cite_1 @cite_16 . The situation changes in extensive-form games where Letchford and Conitzer showed @cite_15 that for many cases the problem is NP-hard, while it still remains PPAD-complete for a Nash equilibrium @cite_23 . More specifically, computing Stackelberg equilibria is polynomial only for: games with perfect information with no chance on DAGs where the leader commits to a pure strategy, games with perfect information with no chance on trees. Introducing chance or imperfect information leads to NP-hardness. However, several cases were unexplored by the existing work, namely extensive-form games with perfect information and concurrent moves. We address this subclass in this work.
@cite_4 @cite_1 @cite_23 @cite_15 @cite_16
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A recent sequence of results established that computing Nash equilibria in normal form games is a PPAD-complete problem even in the case of two players [11,6,4]. By extending these techniques we prove a general theorem, showing that, for a far more general class of families of succinctly representable multiplayer games, the Nash equilibrium problem can also be reduced to the two-player case. In view of empirically successful algorithms available for this problem, this is in essence a positive result — even though, due to the complexity of the reductions, it is of no immediate practical significance. We further extend this conclusion to extensive form games and network congestion games, two classes which do not fall into the same succinct representation framework, and for which no positive algorithmic result had been known. 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. In 1951, John F. Nash proved that every game has a Nash equilibrium [Ann. of Math. (2), 54 (1951), pp. 286-295]. His proof is nonconstructive, relying on Brouwer's fixed point theorem, thus leaving open the questions, Is there a polynomial-time algorithm for computing Nash equilibria? And is this reliance on Brouwer inherent? Many algorithms have since been proposed for finding Nash equilibria, but none known to run in polynomial time. In 1991 the complexity class PPAD (polynomial parity arguments on directed graphs), for which Brouwer's problem is complete, was introduced [C. Papadimitriou, J. Comput. System Sci., 48 (1994), pp. 489-532], motivated largely by the classification problem for Nash equilibria; but whether the Nash problem is complete for this class remained open. In this paper we resolve these questions: We show that finding a Nash equilibrium in three-player games is indeed PPAD-complete; and we do so by a reduction from Brouwer's problem, thus establishing that the two problems are computationally equivalent. Our reduction simulates a (stylized) Brouwer function by a graphical game [M. Kearns, M. Littman, and S. Singh, Graphical model for game theory, in 17th Conference in Uncertainty in Artificial Intelligence (UAI), 2001], relying on “gadgets,” graphical games performing various arithmetic and logical operations. We then show how to simulate this graphical game by a three-player game, where each of the three players is essentially a color class in a coloring of the underlying graph. Subsequent work [X. Chen and X. Deng, Setting the complexity of 2-player Nash-equilibrium, in 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS), 2006] established, by improving our construction, that even two-player games are PPAD-complete; here we show that this result follows easily from our proof. Strong Stackelberg Equilibrium (SSE) is a fundamental solution concept in game theory in which one player commits to a strategy, while the other player observes this commitment and plays a best response. We present a new algorithm for computing SSE for two-player extensive-form general-sum games with imperfect information (EFGs) where computing SSE is an NP-hard problem. Our algorithm is based on a correlated version of SSE, known as Stackelberg Extensive-Form Correlated Equilibrium (SEFCE). Our contribution is therefore twofold: (1) we give the first linear program for computing SEFCE in EFGs without chance, (2) we repeatedly solve and modify this linear program in a systematic search until we arrive to SSE. Our new algorithm outperforms the best previous algorithms by several orders of magnitude.