On the complexity of the parity argument and other inefficient proofs of existence
Journal of Computer and System Sciences - Special issue: 31st IEEE conference on foundations of computer science, Oct. 22–24, 1990
Efficient and constructive algorithms for the pathwidth and treewidth of graphs
Journal of Algorithms
Graphical Models for Game Theory
UAI '01 Proceedings of the 17th Conference in Uncertainty in Artificial Intelligence
Reducibility among equilibrium problems
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
The complexity of computing a Nash equilibrium
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
Reducibility among equilibrium problems
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
Computing pure nash equilibria in graphical games via markov random fields
EC '06 Proceedings of the 7th ACM conference on Electronic commerce
Computing good nash equilibria in graphical games
Proceedings of the 8th ACM conference on Electronic commerce
The Local and Global Price of Anarchy of Graphical Games
SAGT '08 Proceedings of the 1st International Symposium on Algorithmic Game Theory
Equilibria of Graphical Games with Symmetries
WINE '08 Proceedings of the 4th International Workshop on Internet and Network Economics
On the complexity of constrained Nash equilibria in graphical games
Theoretical Computer Science
Approximate Pure Nash Equilibria via Lovász Local Lemma
WINE '09 Proceedings of the 5th International Workshop on Internet and Network Economics
Ranking games that have competitiveness-based strategies
Proceedings of the 11th ACM conference on Electronic commerce
Equilibria of graphical games with symmetries
Theoretical Computer Science
Local and global price of anarchy of graphical games
Theoretical Computer Science
Neighbourhood structure in large games
Proceedings of the 13th Conference on Theoretical Aspects of Rationality and Knowledge
Finding pure nash equilibrium of graphical game via constraints satisfaction approach
ESCAPE'07 Proceedings of the First international conference on Combinatorics, Algorithms, Probabilistic and Experimental Methodologies
Ranking games that have competitiveness-based strategies
Theoretical Computer Science
Graph formation effects on social welfare and inequality in a networked resource game
SBP'13 Proceedings of the 6th international conference on Social Computing, Behavioral-Cultural Modeling and Prediction
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Graphical games have been proposed as a game-theoretic model of large-scale distributed networks of non-cooperative agents. When the number of players is large, and the underlying graph has low degree, they provide a concise way to represent the players' payoffs. It has recently been shown that the problem of finding Nash equilibria in a general degree-3 graphical game with two actions per player is complete for the complexity class PPAD, indicating that it is unlikely that there is any polynomial-time algorithm for this problem. In this paper, we study the complexity of graphical games with two actions per player on bounded-degree trees. This setting was first considered by Kearns, Littman and Singh, who proposed a dynamic programming-based algorithm that computes all Nash equilibria of such games. The running time of their algorithm is exponential, though approximate equilibria can be computed efficiently. Later, Littman, Kearns and Singh proposed a modification to this algorithm that can find a single Nash equilibrium in polynomial time. We show that this modified algorithm is incorrect-the output is not always a Nash equilibrium. We then propose a new algorithm that is based on the ideas of Kearns et al. and computes all Nash equilibria in quadratic time if the input graph is a path, and in polynomial time if it is an arbitrary graph of maximum degree 2. Moreover, our algorithm can be used to compute Nash equilibria of graphical games on arbitrary trees, but the running time can be exponential, even when the tree has bounded degree. We show that this is inevitable -- any algorithm of this type will take exponential time, even on bounded-degree trees with pathwidth 2. It is an open question whether our algorithm runs in polynomial time on graphs with pathwidth 1, but we show that finding a Nash equilibrium for a 2-action graphical game in which the underlying graph has maximum degree 3 and constant pathwidth is PPAD-complete (so is unlikely to be tractable).