Algorithms, games, and the internet
STOC '01 Proceedings of the thirty-third annual ACM symposium on Theory of computing
On the NP-completeness of finding an optimal strategy in games with common payoffs
International Journal of Game Theory
Evolution towards the Maximum Clique
Journal of Global Optimization
On Standard Quadratic Optimization Problems
Journal of Global Optimization
On the Complexity of Two-PlayerWin-Lose Games
FOCS '05 Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science
The complexity of uniform Nash equilibria and related regular subgraph problems
Theoretical Computer Science
Complexity results about Nash equilibria
IJCAI'03 Proceedings of the 18th international joint conference on Artificial intelligence
On the computational complexity of Nash equilibria for (0,1) bimatrix games
Information Processing Letters
Algorithms for playing games with limited randomness
ESA'07 Proceedings of the 15th annual European conference on Algorithms
Parameterized Complexity
Single parameter FPT-algorithms for non-trivial games
IWOCA'10 Proceedings of the 21st international conference on Combinatorial algorithms
Parameterized two-player nash equilibrium
WG'11 Proceedings of the 37th international conference on Graph-Theoretic Concepts in Computer Science
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In this paper we show that some decision problems regarding the computation of Nash equilibria are to be considered particularly hard. Most decision problems regarding Nash equilibria have been shown to be NP-complete. While some NP-complete problems can find an alternative to tractability with the tools of Parameterized Complexity Theory, it is also the case that some classes of problems do not seem to have fixed-parameter tractable algorithms. We show that k-Uniform Nash and k-Minimal Nash support are W[2]-hard. Given a game G=(A, B) and a nonnegative integer k, the k-Uniform Nash problem asks whether G has a uniform Nash equilibrium of size k. The k-Minimal Nash support asks whether G has Nash equilibrium such that the support of each player's Nash strategy has size equal to or less than k. First, we show that k-Uniform Nash (with k as the parameter) is W[2]-hard even when we have 2 players, or fewer than 4 different integer values in the matrices. Second, we illustrate that even in zerosum games k-Minimal Nash support is W[2]-hard (a sample Nash equilibrium in a zerosum 2-player game can be found in polynomial time (von Stengel 2002)). Thus, it must be the case that other more general decision problems are also W[2]-hard. Therefore, the possible parameters for fixed parameter tractability in those decision problems regarding Nash equilibria seem elusive.