Constructing bimatrix games with unique equilibrium points

We show that to prove the existence of an equilibrium point of a bimatrix game, it is enough to start out with any equilibrium point-not necessarily symmetric-of the symmetrization for the original bimatrix game that is invoked in the proof of Theorem 9 of Chandrasekaran (undated).

International Journal of Game Theory, 2015

In 1964 Shapley observed that a matrix has a saddle point in pure strategies whenever every its 2 × 2 submatrix has one. In contrast, a bimatrix game may have no pure strategy Nash equilibrium (NE) even when every 2 × 2 subgame has one. Nevertheless, Shapley's claim can be extended to bimatrix games as follows. We partition all 2×2 bimatrix games into fifteen classes C = {c 1 ,. .. , c 15 } depending on the preferences of two players. A subset t ⊆ C 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) strongest NE-theorems.

Economic Theory, 2007

Informationally robust equilibria (IRE) are introduced in Robson (Games Econ Behav 7: 233–245, 1994) as a refinement of Nash equilibria for strategic games. Such equilibria are limits of a sequence of (subgame perfect) Nash equilibria in perturbed games where with small probability information about the strategic behavior is revealed to other players (information leakage). Focusing on bimatrix games, we consider a type of informationally robust equilibria and derive a number of properties they form a non-empty and closed subset of the Nash equilibria. Moreover, IRE is a strict concept in the sense that the IRE are independent of the exact sequence of probabilities with which information is leaked. The set of IRE, like the set of Nash equilibria, is the finite union of polytopes. In potential games, there is an IRE in pure strategies. In zero-sum games, the set of IRE has a product structure and its elements can be computed efficiently by using linear programming. We also discuss extensions to games with infinite strategy spaces and more than two players.

Linear Algebra and its Applications, 1989

Dedicated to Alan J. Hoffman on the occasion of his 65th birthday.

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.

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

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.

2005

In this paper it is shown how to compute stable sets, defined by , in the context of bimatrix games only using linear optimization techniques.

In this paper we introduce the class of quadratically optimal (bi-matrix) games, which are bi-matrix games whose set of equilibrium points contain all pairs of probability vectors which maximize the expected pay-off of some pay-off matrix. We call the equilibrium points obtained in this way, quadratically optimal equilibrium points. We prove the existence of quadratically optimal equilibrium points of identical bi-matrix games, i.e. bi-matrix games for which the two pay-off matrices are equal, from which it easily follows that weakly potential bi-matrix games (a generalization of potential bi-matrix games) are quadratically optimal. We also show that those weakly potential square bi-matrix games which have potential matrices that are two-way matrices are quadratically optimal symmetric games, i.e. there exists a square pay-off matrix whose expected pay-off maximizing probability vectors subject to the two probability vectors (row probability vector and column probability vector) being equal, are equilibrium points of the bi-matrix game. None of our results require using a fixed-point theorem argument in the proofs. We show by means of an example of a 22 identical symmetric potential bi-matrix game that for every potential matrix of the game, the set of pairs of probability distributions that maximizes the expected pay-off of the potential matrix is a strict subset of the set of equilibrium points of the potential game.

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.