On the complexity of deciding degeneracy in a bimatrix game with sparse payoff matrix (original) (raw)
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On the computational complexity of Nash equilibria for bimatrix games
Information Processing Letters, 2005
The computational complexity of finding a Nash equilibrium in a nonzero sum bimatrix game is an important open question. We put forward the notion of (0, 1)-bimatrix games, and show that some associated computational problems are as hard as in the general case.
On the complexity of deciding bimatrix games similarity
Theoretical Computer Science, 2008
In this paper, we show that it is NP-complete to decide whether two bimatrix games share a common Nash equilibrium. Furthermore, it is co-NP-hard to decide whether two bimatrix games have exactly the same set of Nash equilibria.
A note on strategy elimination in bimatrix games
1987
In bimatrix games we study the process of successively eliminating strategies which are dominated by others. ~We show how to perform this simplification in, O(n) time, where n is the number of strategies. We also prove that the problem is P-complete, which suggests that it is inherently sequential.
Lecture Notes in Computer Science
In Shapley observed that a matrix has a saddle point whenever every ×2 submatrix of it has one. In contrast, a bimatrix game may have no Nash equilibrium (NE) even when every 2 × 2 subgame of it has one. Nevertheless, Shapley's claim can be generalized for bimatrix games in many ways as follows. We partition all 2 × 2 bimatrix games into fifteen classes S = {c 1 ,. .. , c 15 } depending on the preference pre-orders of the two players. A subset t ∈ S is called a NE-theorem if a bimatrix game has a NE whenever it contains no subgame from t. We suggest a general method for getting all minimal (that is, strongest) NE-theorems based on the procedure of joint generation of transversal hypergraphs given by a special oracle. By this method we obtain all (six) minimal NE-theorems.
A note on bimatrix games with an unknown payoff matrix
Economics Letters, 1985
We show that, using a simple decision rule, two players repeatedly playing the same zero-sum game without the direct knowledge of the payoff matrix will ultimately achieve the Nash Equilibrium if the game possesses a unique pure strategy Nash Equilibrium.
On mutual concavity and strategically-zero-sum bimatrix games
Theoretical Computer Science, 2012
We study the fundamental problem 2NASH of computing a Nash equilibrium (NE) point in bimatrix games. We start by proposing a novel characterization of the NE set, via a bijective map to the solution set of a parameterized quadratic program (NEQP), whose feasible space is the highly structured set of correlated equilibria (CE). This is, to our knowledge, the first characterization of the subset of CE points that are in ''1-1'' correspondence with the NE set of the game, and contributes to the quite lively discussion on the relation between the spaces of CE and NE points in a bimatrix game (e.g., [15,26,33]). We proceed with studying a property of bimatrix games, which we call mutually concavity (MC), that assures polynomial-time tractability of 2NASH, due to the convexity of a proper parameterized quadratic program (either NEQP, or a parameterized variant of the Mangasarian & Stone formulation [23]) for a particular value of the parameter. We prove various characterizations of the MC-games, which eventually lead us to the conclusion that this class is equivalent to the class of strategically zero-sum (SZS) games of Moulin & Vial [25]. This gives an alternative explanation of the polynomial-time tractability of 2NASH for these games, not depending on the solvability of zero-sum games. Moreover, the recognition of the MC-property for an arbitrary game is much faster than the recognition SZS-property. This, along with the comparable time-complexity of linear programs and convex quadratic programs, leads us to a much faster algorithm for 2NASH in MC-games. We conclude our discussion with a comparison of MC-games (or, SZS-games) to k-rank games, which are known to admit for 2NASH a FPTAS when k is fixed [18], and a polynomialtime algorithm for k = 1 [2]. We finally explore some closeness properties under wellknown NE set preserving transformations of bimatrix games.
Proceedings of the 43rd annual ACM symposium on Theory of computing - STOC '11, 2011
Given a rank-1 bimatrix game (A, B), i.e., where rank(A+ B) = 1, we construct a suitable linear subspace of the rank-1 game space and show that this subspace is homeomorphic to its Nash equilibrium correspondence. Using this homeomorphism, we give the first polynomial time algorithm for computing an exact Nash equilibrium of a rank-1 bimatrix game. This settles an open question posed in . In addition, we give a novel algorithm to enumerate all the Nash equilibria of a rank-1 game and show that a similar technique may also be applied for finding a Nash equilibrium of any bimatrix game. This technique also proves the existence, oddness and the index theorem of Nash equilibria in a bimatrix game. Further, we extend the rank-1 homeomorphism result to a fixed rank game space, and give a fixed point formulation on [0, 1] k for solving a rank-k game. The homeomorphism and the fixed point formulation are piece-wise linear and considerably simpler than the classical constructions. be a NESP of a non-degenerate non-negative bimatrix game (A, B). Let I = {i ∈ S 2 | x i > 0} and J = {j ∈ S 1 | y j > 0} with the corresponding submatrices A J I and B J
Exploiting Concavity in Bimatrix Games: New Polynomially Tractable Subclasses
Lecture Notes in Computer Science, 2010
We study the fundamental problem of computing an arbitrary Nash equilibrium in bimatrix games. We start by proposing a novel characterization of the set of Nash equilibria, via a bijective map to the solution set of a (parameterized) quadratic program, whose feasible space is the (highly structured) set of correlated equilibria. We then proceed by proposing new subclasses of bimatrix games for which either an exact polynomial-time construction, or at least a FPTAS, is possible. In particular, we introduce the notion of mutual (quasi-) concavity of a bimatrix game, which assures (quasi-) convexity of our quadratic program, for at least one value of the parameter. For mutually concave bimatrix games, we provide a polynomial-time computation of a Nash equilibrium, based on the polynomial tractability of convex quadratic programming. For the mutually quasi-concave games, we provide (to our knowledge) the first FPTAS for the construction of a Nash equilibrium. Of course, for these new polynomially tractable subclasses of bimatrix games to be useful, polynomial-time certificates are also necessary that will allow us to efficiently identify them. Towards this direction, we provide various characterizations of mutual concavity, which allow us to construct such a certificate. Interestingly, these characterizations also shed light to some structural properties of the bimatrix games satisfying mutual concavity. This subclass entirely contains the most popular subclass of polynomial-time solvable bimatrix games, namely, all the constantsum games (rank−0 games). It is though incomparable to the subclass of games with fixed rank [16]: Even rank−1 games may not be mutually concave (eg, Prisoner's dilemma), but on the other hand, there exist mutually concave games of arbitrary (even full) rank. Finally, we prove closeness of mutual concavity under (Nash equilibrium preserving) positive affine transformations of bimatrix games having the same scaling factor for both payoff matrices. For different scaling factors the property is not necessarily preserved.