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 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.

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.

HADAMARD AND TYKHONOV WELL-POSEDNESS IN TWO PLAYER GAMES

2003 ◽  
Vol 05 (04) ◽  
pp. 375-384 ◽  
Author(s):  
GRAZIANO PIERI ◽  
ANNA TORRE
Keyword(s):  
Nash Equilibrium ◽  
Equilibrium Point ◽  
Nash Equilibria ◽  
Well Posedness ◽  
Nash Equilibrium Point ◽  
Suitable Definition ◽  
Payoff Functions ◽  
The Stability ◽  
Definition Of ◽  
Zero Sum

We give a suitable definition of Hadamard well-posedness for Nash equilibria of a game, that is, the stability of Nash equilibrium point with respect to perturbations of payoff functions. Our definition generalizes the analogous notion for minimum problems. For a game with continuous payoff functions, we restrict ourselves to Hadamard well-posedness with respect to uniform convergence and compare this notion with Tykhonov well-posedness of the same game. The main results are: Hadamard implies Tykhonov well-posedness and the converse is true if the payoff functions are bounded. For a zero-sum game the two notions are equivalent.

scholarly journals From Nash Equilibria to Chain Recurrent Sets: An Algorithmic Solution Concept for Game Theory

Entropy ◽  
2018 ◽  
Vol 20 (10) ◽  
pp. 782 ◽  
Author(s):  
Christos Papadimitriou ◽  
Georgios Piliouras
Keyword(s):  
Game Theory ◽  
Nash Equilibrium ◽  
Dynamical Systems ◽  
Nash Equilibria ◽  
Equilibrium Solution ◽  
Solution Concept ◽  
Constructive Proof ◽  
Specific Information ◽  
Potential Game ◽  
Chain Recurrence

In 1950, Nash proposed a natural equilibrium solution concept for games hence called Nash equilibrium, and proved that all finite games have at least one. The proof is through a simple yet ingenious application of Brouwer’s (or, in another version Kakutani’s) fixed point theorem, the most sophisticated result in his era’s topology—in fact, recent algorithmic work has established that Nash equilibria are computationally equivalent to fixed points. In this paper, we propose a new class of universal non-equilibrium solution concepts arising from an important theorem in the topology of dynamical systems that was unavailable to Nash. This approach starts with both a game and a learning dynamics, defined over mixed strategies. The Nash equilibria are fixpoints of the dynamics, but the system behavior is captured by an object far more general than the Nash equilibrium that is known in dynamical systems theory as chain recurrent set. Informally, once we focus on this solution concept—this notion of “the outcome of the game”—every game behaves like a potential game with the dynamics converging to these states. In other words, unlike Nash equilibria, this solution concept is algorithmic in the sense that it has a constructive proof of existence. We characterize this solution for simple benchmark games under replicator dynamics, arguably the best known evolutionary dynamics in game theory. For (weighted) potential games, the new concept coincides with the fixpoints/equilibria of the dynamics. However, in (variants of) zero-sum games with fully mixed (i.e., interior) Nash equilibria, it covers the whole state space, as the dynamics satisfy specific information theoretic constants of motion. We discuss numerous novel computational, as well as structural, combinatorial questions raised by this chain recurrence conception of games.

ON A CLASS OF NASH EQUILIBRIA WITH MEMORY STRATEGIES FOR NONZERO-SUM DIFFERENTIAL GAMES

2000 ◽  
Vol 02 (02n03) ◽  
pp. 173-192 ◽  
Author(s):  
JEAN MICHEL COULOMB ◽  
VLADIMIR GAITSGORY
Keyword(s):  
Nash Equilibrium ◽  
Feedback Control ◽  
Differential Games ◽  
Differential Game ◽  
Nash Equilibria ◽  
Payoff Function ◽  
Feedback Controls ◽  
Memory Strategies ◽  
Payoff Functions ◽  
With Memory

A two-player nonzero-sum differential game is considered. Given a pair of threat payoff functions, we characterise a set of pairs of acceptable feedback controls. Any such pair induces a history-dependent Nash δ-equilibrium as follows: the players agree to use the acceptable controls unless one of them deviates. If this happens, a feedback control punishment is implemented. The problem of finding a pair of "acceptable" controls is significantly simpler than the problem of finding a feedback control Nash equilibrium. Moreover, the former may have a solution in case the latter does not. In addition, if there is a feedback control Nash equilibrium, then our technique gives a subgame perfect Nash δ-equilibrium that might improve the payoff function for at least one player.

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 Synthesis of Strategic Games with Multiple Nash Equilibria - A Global Optimization Approach

2019 ◽  
Author(s):  
Hime Oliveira
Keyword(s):  
Global Optimization ◽  
Nash Equilibrium ◽  
Nash Equilibria ◽  
Optimization Method ◽  
Optimization Techniques ◽  
Optimization Approach ◽  
Strategic Games ◽  
Finite State ◽  
Payoff Functions ◽  
Multiple Nash Equilibria

This paper presents an extension of the resuts obtained in previous work by the author concerning the application of global optimization techniques to the design of finite strategic games with mixed strategies. In that publication the Fuzzy ASA global optimization method was applied to many examples of synthesis of strategic games with one previously specified Nash equilibrium, evidencing its ability in finding payoff functions whose respective games present those equilibria, possibly among others. That is to say, it was shown it is possible to establish in advance a Nash equilibrium for a generic finite state strategic game and to compute payoff functions that will make it feasible to reach the chosen equilibrium, allowing players to converge to the desired profile, considering that it is an equilibrium of the game as well. Going beyond this state of affairs, the present article shows that it is possible to "impose" multiple Nash equilibria to finite strategic games by following the same reasoning as before, but with a slight change: using the same fundamental theorem of Richard D. McKelvey, modifying the original prescribed objective function and globally minimizing it. The proposed method, in principle, is able to find payoff functions that result in games featuring an arbitrary number of Nash equiibria, paving the way to a substantial number of potential applications.

scholarly journals A new evolutionary approach for computing Nash equilibria in bimatrix games with known support

2012 ◽  
Vol 2 (2) ◽  
Author(s):  
Urszula Boryczka ◽  
Przemyslaw Juszczuk
Keyword(s):  
Nash Equilibrium ◽  
Nash Equilibria ◽  
Probability Distributions ◽  
Pure Strategy ◽  
Global Optimum ◽  
Adaptive Mutation ◽  
De Algorithm ◽  
Approximate Nash Equilibria ◽  
Zero Sum ◽  
Continuous Problem

AbstractIn this paper, we present the application of the Differential Evolution (DE) algorithm to the problem of finding approximate Nash equilibria in matrix, non-zero sum games for two players with finite number of strategies. Nash equilibrium is one of the main concepts in game theory. It may be classified as continuous problem, where two probability distributions over the set of strategies of both players should be found. Every deviation from the global optimum is interpreted as Nash approximation and called ε-Nash equilibrium. The main advantage of the proposed algorithm is self-adaptive mutation operator, which direct the search process. The approach used in this article is based on the probability of chosing single pure strategy. In optimal mixed strategy, every strategy has some probability of being chosen. Our goal is to determine this probability and maximize payoff for a single player.

scholarly journals Characterizations of Pareto-Nash Equilibria for Multiobjective Potential Population Games

Mathematics ◽  
2021 ◽  
Vol 9 (1) ◽  
pp. 99
Author(s):  
Guanghui Yang ◽  
Chanchan Li ◽  
Jinxiu Pi ◽  
Chun Wang ◽  
Wenjun Wu ◽  
...  
Keyword(s):  
Nash Equilibrium ◽  
Potential Function ◽  
Nash Equilibria ◽  
Jacobi Matrix ◽  
Practical Applications ◽  
Population Games ◽  
Population Potential ◽  
Potential Population ◽  
Payoff Functions ◽  
Vector Valued

This paper studies the characterizations of (weakly) Pareto-Nash equilibria for multiobjective population games with a vector-valued potential function called multiobjective potential population games, where agents synchronously maximize multiobjective functions with finite strategies via a partial order on the criteria-function set. In such games, multiobjective payoff functions are equal to the transpose of the Jacobi matrix of its potential function. For multiobjective potential population games, based on Kuhn-Tucker conditions of multiobjective optimization, a strongly (weakly) Kuhn-Tucker state is introduced for its vector-valued potential function and it is proven that each strongly (weakly) Kuhn-Tucker state is one (weakly) Pareto-Nash equilibrium. The converse is obtained for multiobjective potential population games with two strategies by utilizing Tucker’s Theorem of the alternative and Motzkin’s one of linear systems. Precisely, each (weakly) Pareto-Nash equilibrium is equivalent to a strongly (weakly) Kuhn-Tucker state for multiobjective potential population games with two strategies. These characterizations by a vector-valued approach are more comprehensive than an additive weighted method. Multiobjective potential population games are the extension of population potential games from a single objective to multiobjective cases. These novel results provide a theoretical basis for further computing (weakly) Pareto-Nash equilibria of multiobjective potential population games and their practical applications.

A Theory of Patronized Goods: Inefficient and Efficient Equilibria

2011 ◽  
pp. 65-87 ◽  
Author(s):  
A. Rubinstein
Keyword(s):  
Nash Equilibrium ◽  
Social Policy ◽  
Nash Equilibria ◽  
Institutional Environment ◽  
Social Motivation ◽  
Pareto Optimal ◽  
Market Failures ◽  
Equilibrium Conditions ◽  
Policy Issues

The article considers some aspects of the patronized goods theory with respect to efficient and inefficient equilibria. The author analyzes specific features of patronized goods as well as their connection with market failures, and conjectures that they are related to the emergence of Pareto-inefficient Nash equilibria. The key problem is the analysis of the opportunities for transforming inefficient Nash equilibrium into Pareto-optimal Nash equilibrium for patronized goods by modifying the institutional environment. The paper analyzes social motivation for institutional modernization and equilibrium conditions in the generalized Wicksell-Lindahl model for patronized goods. The author also considers some applications of patronized goods theory to social policy issues.

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