scholarly journals Semidefinite Programming and Nash Equilibria in Bimatrix Games

2020 ◽  
Author(s):  
Amir Ali Ahmadi ◽  
Jeffrey Zhang
Keyword(s):  
Nash Equilibrium ◽  
Semidefinite Programming ◽  
Nash Equilibria ◽  
Valid Inequalities ◽  
Bimatrix Games ◽  
Competitive Game ◽  
Approximate Nash Equilibria ◽  
Hard Problems ◽  
Rank 2

We explore the power of semidefinite programming (SDP) for finding additive ɛ-approximate Nash equilibria in bimatrix games. We introduce an SDP relaxation for a quadratic programming formulation of the Nash equilibrium problem and provide a number of valid inequalities to improve the quality of the relaxation. If a rank-1 solution to this SDP is found, then an exact Nash equilibrium can be recovered. We show that, for a strictly competitive game, our SDP is guaranteed to return a rank-1 solution. We propose two algorithms based on the iterative linearization of smooth nonconvex objective functions whose global minima by design coincide with rank-1 solutions. Empirically, we demonstrate that these algorithms often recover solutions of rank at most 2 and ɛ close to zero. Furthermore, we prove that if a rank-2 solution to our SDP is found, then a [Formula: see text]-Nash equilibrium can be recovered for any game, or a [Formula: see text]-Nash equilibrium for a symmetric game. We then show how our SDP approach can address two (NP-hard) problems of economic interest: finding the maximum welfare achievable under any Nash equilibrium, and testing whether there exists a Nash equilibrium where a particular set of strategies is not played. Finally, we show the connection between our SDP and the first level of the Lasserre/sum of squares hierarchy.

An Improved Quasi-Polynomial Algorithm for Approximate Well-Supported Nash Equilibria

2019 ◽  
Vol 33 ◽  
pp. 1926-1932
Author(s):  
Michail Fasoulakis ◽  
Evangelos Markakis
Keyword(s):  
Nash Equilibrium ◽  
Polynomial Time ◽  
Nash Equilibria ◽  
Polynomial Algorithm ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Bimatrix Games ◽  
Approximate Nash Equilibria ◽  
Pure Strategies ◽  
Approximation Parameter

We focus on the problem of computing approximate Nash equilibria in bimatrix games. In particular, we consider the notion of approximate well-supported equilibria, which is one of the standard approaches for approximating equilibria. It is already known that one can compute an ε-well-supported Nash equilibrium in time nO (log n/ε2), for any ε > 0, in games with n pure strategies per player. Such a running time is referred to as quasi-polynomial. Regarding faster algorithms, it has remained an open problem for many years if we can have better running times for small values of the approximation parameter, and it is only known that we can compute in polynomial-time a 0.6528-well-supported Nash equilibrium. In this paper, we investigate further this question and propose a much better quasi-polynomial time algorithm that computes a (1/2 + ε)-well-supported Nash equilibrium in time nO(log logn1/ε/ε2), for any ε > 0. Our algorithm is based on appropriately combining sampling arguments, support enumeration, and solutions to systems of linear inequalities.

scholarly journals On Sparse Discretization for Graphical Games

2020 ◽  
Vol 69 ◽  
pp. 67-84
Author(s):  
Luis Ortiz
Keyword(s):  
Nash Equilibrium ◽  
Nash Equilibria ◽  
Sufficient Conditions ◽  
Research Note ◽  
Graphical Games ◽  
Approximate Nash Equilibrium ◽  
Approximate Nash Equilibria ◽  
Polymatrix Games ◽  
The Impact ◽  
Game Representation

Graphical games are one of the earliest examples of the impact that the general field of graphical models have had in other areas, and in this particular case, in classical mathematical models in game theory. Graphical multi-hypermatrix games, a concept formally introduced in this research note, generalize graphical games while allowing the possibility of further space savings in model representation to that of standard graphical games. The main focus of this research note is discretization schemes for computing approximate Nash equilibria, with emphasis on graphical games, but also briefly touching on normal-form and polymatrix games. The main technical contribution is a theorem that establishes sufficient conditions for a discretization of the players’ space of mixed strategies to contain an approximate Nash equilibrium. The result is actually stronger because every exact Nash equilibrium has a nearby approximate Nash equilibrium on the grid induced by the discretization. The sufficient conditions are weaker than those of previous results. In particular, a uniform discretization of size linear in the inverse of the approximation error and in the natural game-representation parameters suffices. The theorem holds for a generalization of graphical games, introduced here. The result has already been useful in the design and analysis of tractable algorithms for graphical games with parametric payoff functions and certain game-graph structures. For standard graphical games, under natural conditions, the discretization is logarithmic in the game-representation size, a substantial improvement over the linear dependency previously required. Combining the improved discretization result with old results on constraint networks in AI simplifies the derivation and analysis of algorithms for computing approximate Nash equilibria in graphical games.

scholarly journals New algorithms for approximate Nash equilibria in bimatrix games

2010 ◽  
Vol 411 (1) ◽  
pp. 164-173 ◽  
Author(s):  
Hartwig Bosse ◽  
Jaroslaw Byrka ◽  
Evangelos Markakis
Keyword(s):  
Nash Equilibria ◽  
Bimatrix Games ◽  
Approximate Nash Equilibria ◽  
New Algorithms

scholarly journals Constructive Proof of the Existence of Nash Equilibrium in a Finite Strategic Game with Sequentially Locally Nonconstant Payoff Functions

2012 ◽  
Vol 2012 ◽  
pp. 1-8 ◽  
Author(s):  
Yasuhito Tanaka
Keyword(s):  
Nash Equilibrium ◽  
Nash Equilibria ◽  
Constructive Proof ◽  
Constructive Mathematics ◽  
Strategic Game ◽  
Sperner’S Lemma ◽  
Approximate Nash Equilibria ◽  
Payoff Functions

We will constructively prove the existence of a Nash equilibrium in a finite strategic game with sequentially locally nonconstant payoff functions. The proof is based on the existence of approximate Nash equilibria which is proved by Sperner's lemma. We follow the Bishop-style constructive mathematics.

scholarly journals Bounding Regret in Empirical Games

2020 ◽  
Vol 34 (04) ◽  
pp. 4280-4287
Author(s):  
Steven Jecmen ◽  
Arunesh Sinha ◽  
Zun Li ◽  
Long Tran-Thanh
Keyword(s):  
Large Scale ◽  
Nash Equilibria ◽  
Sample Complexity ◽  
Bandit Problem ◽  
Theoretic Analysis ◽  
Present Sample ◽  
Game Theoretic Analysis ◽  
Approximate Nash Equilibria ◽  
Game Theoretic

Empirical game-theoretic analysis refers to a set of models and techniques for solving large-scale games. However, there is a lack of a quantitative guarantee about the quality of output approximate Nash equilibria (NE). A natural quantitative guarantee for such an approximate NE is the regret in the game (i.e. the best deviation gain). We formulate this deviation gain computation as a multi-armed bandit problem, with a new optimization goal unlike those studied in prior work. We propose an efficient algorithm Super-Arm UCB (SAUCB) for the problem and a number of variants. We present sample complexity results as well as extensive experiments that show the better performance of SAUCB compared to several baselines.

Computation of Nash Equilibria for Bimatrix Games

2017 ◽  
Author(s):  
João P. Hespanha
Keyword(s):  
Nash Equilibrium ◽  
Numerical Computation ◽  
Nash Equilibria ◽  
Bimatrix Games ◽  
Football Game ◽  
Battle Of The Sexes

This chapter discusses the computation of the Nash equilibrium for bimatrix games. It begins by considering a different version of the battle of the sexes game introduced in Chapter 9, in which action 1 corresponds to going to the baby shower and action 2 to the football game. In this new version no one really wants to go to the football game alone (cost of 3), but going to the baby shower alone is a little better (cost of 0). After finding the mixed Nash equilibrium for this case, the chapter describes the computation of a completely mixed Nash equilibrium and the numerical computation of a mixed Nash equilibrium. It concludes with a practice exercise and the corresponding solution, along with an additional exercise.

Sign in / Sign up

Export Citation Format

Share Document