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
Matrix Classes That Generate All Matrices with Positive Determinant
SIAM Journal on Matrix Analysis and Applications
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
Nash Equilibria in Random Games
FOCS '05 Proceedings of the 46th 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
Complexity results about Nash equilibria
IJCAI'03 Proceedings of the 18th international joint conference on Artificial intelligence
Rank-1 bimatrix games: a homeomorphism and a polynomial time algorithm
Proceedings of the forty-third annual ACM symposium on Theory of computing
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We prove that an equilibrium of a nondegenerate bimatrix gamehas index + 1 if and only if it can be made the unique equilibriumof an extended game with additional strategies of one player. Themain tool is the "dual construction". A simplicial polytope, dualto the common best-response polytope of one player, has its facetssubdivided into best-response regions, so that equilibria arecompletely labeled points on the surface of that polytope. Thatsurface has dimension m- 1 for an m×ngame, which is much lower than the dimension m+ nofthe polytopes that are classically used.