Introduction to Game Theory

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.

International Journal of Game Theory, 1971

A bi-matrix threat game is defined as a triple (A,B,S) where A and B are m x n payoff matrices, and S is a closed convex subset of the plane, with (aij, blj ) e S for each i,j. Given (threat) mixed strategies x and y, NASH'S model suggests that the eventual outcome will be that point (u,v) ~ S which maximizes the product (u -xAy') (v -xBy), subject to u >_ xAy ~, v >_ xBy ~. Optimality of the threat strategies is then defined in the obvious way.

We assemble in this note some basic results about the existence of equilibrium for bi-matrix games, along with their proofs.

Annals of Operations Research, 1998

In this paper, we propose a new way to analyze bimatrix games. This new approach consists of considering the game as a bicriteria matrix game. The solution concepts behind this game are based on getting the probability to achieve some prespecified goals. We consider as a part of the solution, not only the payoff values, but also the probability to get them. In addition, to avoid the choice of only one goal, two different approaches are used. Firstly, sensitivity analysis of the solution set is carried out on the range of goals, secondly a partition of the goal space in a finite number of regions is presented. Some examples are included to illustrate the results in the paper.

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.

In this paper we prove the existence of constrained equilibrium points for bi-matrix 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

Mathematics of Operations Research, 2016

We show that in zero-sum polymatrix games, a multiplayer generalization of two-person zero-sum games, Nash equilibria can be found efficiently with linear programming. We also show that the set of coarse correlated equilibria collapses to the set of Nash equilibria. In contrast, other important properties of two-person zero-sum games are not preserved: Nash equilibrium payoffs need not be unique, and Nash equilibrium strategies need not be exchangeable or max-min.

International Journal for Research in Applied Science and Engineering Technology, 2019

In classical game theory, a conflict of two opponents can be modelled as an equilibrium-based matrix game. We assume a conflict of two non-cooperative antagonistic opponents with a finite number of strategies with zero-sum or constant sum pay-offs. At the same time, we suppose that the elements of the payoff matrix describing the game are not fixed and are allowed to change within a specified interval. Supposing that some of the elements of the payoff matrix are uncertain, it is evident that this would influence the utilities of both players at the same time and moreover, such entropy of the model would eventually influence the position of equilibria or its very existence. We propose a modelling approach that allows one to find a solution of the game with either pure or mixed strategies of opponents with the guaranteed payoffs under the assumption that a specified number of unspecified entries would attain different values than expected. The chosen robust approach is presented briefly as well as the necessary circumstances of matrix game solutions. Our novelty approach follows and is accompanied by an explanatory example in the end of the paper.

Journal of Mathematical Analysis and Applications, 1976