On the complexity of approximating a KKT point of quadratic programming
Mathematical Programming: Series A and B
Playing large games using simple strategies
Proceedings of the 4th ACM conference on Electronic commerce
Exponentially Many Steps for Finding a Nash Equilibrium in a Bimatrix Game
FOCS '04 Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science
Settling the Complexity of Two-Player Nash Equilibrium
FOCS '06 Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science
Computing Nash Equilibria: Approximation and Smoothed Complexity
FOCS '06 Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science
Progress in approximate nash equilibria
Proceedings of the 8th ACM conference on Electronic commerce
Efficient computation of nash equilibria for very sparse win-lose bimatrix games
ESA'06 Proceedings of the 14th conference on Annual European Symposium - Volume 14
On the copositive representation of binary and continuous nonconvex quadratic programs
Mathematical Programming: Series A and B
Complexity results about Nash equilibria
IJCAI'03 Proceedings of the 18th international joint conference on Artificial intelligence
The Complexity of Computing a Nash Equilibrium
SIAM Journal on Computing
New algorithms for approximate Nash equilibria in bimatrix games
WINE'07 Proceedings of the 3rd international conference on Internet and network economics
An optimization approach for approximate Nash equilibria
WINE'07 Proceedings of the 3rd international conference on Internet and network economics
Exploiting concavity in bimatrix games: new polynomially tractable subclasses
APPROX/RANDOM'10 Proceedings of the 13th international conference on Approximation, and 14 the International conference on Randomization, and combinatorial optimization: algorithms and techniques
Rank-1 bimatrix games: a homeomorphism and a polynomial time algorithm
Proceedings of the forty-third annual ACM symposium on Theory of computing
Polynomial algorithms for approximating nash equilibria of bimatrix games
WINE'06 Proceedings of the Second international conference on Internet and Network Economics
A note on approximate nash equilibria
WINE'06 Proceedings of the Second international conference on Internet and Network Economics
Hi-index | 0.00 |
In this work we present a simple quadratic formulation for the problem of computing Nash equilibria in symmetric bimatrix games, inspired by the well-known formulation of Mangasarian and Stone [26]. We exploit our formulation to shed light to the approximability of NE points. First we observe that any KKT point of this formulation (and indeed, any quadratic program) is also a stationary point, and vice versa. We then prove that any KKT point of the proposed formulation (is not necessarily itself, but) indicates a ( 0, in time polynomial in the size of the game and quasi-linear in 1/δ, exploiting Ye's algorithm for approximating KKT points of QPs [34]. This is (to our knowledge) the first polynomial time algorithm that constructs ε-NE points for symmetric bimatrix games for any e close to 1/3. We extend our main result to (asymmetric) win lose games, as well as to games with maximum aggregate payoff either at most 1, or at least 5/3. To achieve this, we use a generalization of the Brown & von Neumann symmetrization technique [6] to the case of non-zero-sum games, which we prove that is approximation preserving. Finally, we present our experimental analysis of the proposed approximation and other quite interesting approximations for NE points in symmetric bimatrix games.