scholarly journals The Frequency of Convergent Games under Best-Response Dynamics

2021 ◽  
Author(s):  
Samuel C. Wiese ◽  
Torsten Heinrich
Keyword(s):  
Nash Equilibrium ◽  
Nash Equilibria ◽  
Pure Strategy ◽  
Response Dynamics ◽  
Novel Approach ◽  
Best Response ◽  
Random Games ◽  
Payoff Matrices ◽  
Best Response Dynamic ◽  
Strategy Nash Equilibrium

AbstractWe calculate the frequency of games with a unique pure strategy Nash equilibrium in the ensemble of n-player, m-strategy normal-form games. To obtain the ensemble, we generate payoff matrices at random. Games with a unique pure strategy Nash equilibrium converge to the Nash equilibrium. We then consider a wider class of games that converge under a best-response dynamic, in which each player chooses their optimal pure strategy successively. We show that the frequency of convergent games with a given number of pure Nash equilibria goes to zero as the number of players or the number of strategies goes to infinity. In the 2-player case, we show that for large games with at least 10 strategies, convergent games with multiple pure strategy Nash equilibria are more likely than games with a unique Nash equilibrium. Our novel approach uses an n-partite graph to describe games.

On the Openness of Unique Pure-Strategy Nash Equilibrium

2018 ◽  
Vol 19 (1) ◽  
Author(s):  
Anna A. Klis
Keyword(s):  
Nash Equilibrium ◽  
Pure Strategy ◽  
Response Curves ◽  
Small Perturbations ◽  
Transversal Intersection ◽  
Interior Equilibrium ◽  
Best Response ◽  
Unique Nash Equilibrium ◽  
Strict Concavity ◽  
Strategy Nash Equilibrium

AbstractThis paper investigates whether small perturbations to a game with continuous strategy spaces and unique Nash equilibrium also yields a game with unique equilibrium. The answer is affirmative for games with smooth payoffs, differentiable strict concavity in own actions, and transversal intersection of best response curves. Though intuitive for games with unique interior equilibrium, this result holds even for equilibria at the boundaries of strategy sets.

scholarly journals Learning Deviation Payoffs in Simulation-Based Games

2019 ◽  
Vol 33 ◽  
pp. 2173-2180 ◽  
Author(s):  
Samuel Sokota ◽  
Caleb Ho ◽  
Bryce Wiedenbeck
Keyword(s):  
Neural Networks ◽  
Nash Equilibrium ◽  
Nash Equilibria ◽  
Mixed Strategy ◽  
State Of The Art ◽  
Pure Strategy ◽  
Normal Form Games ◽  
Novel Approach ◽  
Simulation Based ◽  
Expected Values

We present a novel approach for identifying approximate role-symmetric Nash equilibria in large simulation-based games. Our method uses neural networks to learn a mapping from mixed-strategy profiles to deviation payoffs—the expected values of playing pure-strategy deviations from those profiles. This learning can generalize from data about a tiny fraction of a game’s outcomes, permitting tractable analysis of exponentially large normal-form games. We give a procedure for iteratively refining the learned model with new data produced by sampling in the neighborhood of each candidate Nash equilibrium. Relative to the existing state of the art, deviation payoff learning dramatically simplifies the task of computing equilibria and more effectively addresses player asymmetries. We demonstrate empirically that deviation payoff learning identifies better approximate equilibria than previous methods and can handle more difficult settings, including games with many more players, strategies, and roles.

scholarly journals Pure Nash Equilibria and Best-Response Dynamics in Random Games

2021 ◽  
Author(s):  
Ben Amiet ◽  
Andrea Collevecchio ◽  
Marco Scarsini ◽  
Ziwen Zhong
Keyword(s):  
Nash Equilibria ◽  
Approximation Error ◽  
Pure Nash Equilibrium ◽  
Response Dynamics ◽  
Asymptotic Results ◽  
Finite Games ◽  
Best Response Dynamics ◽  
Best Response ◽  
Random Games ◽  
Pure Equilibria

In finite games, mixed Nash equilibria always exist, but pure equilibria may fail to exist. To assess the relevance of this nonexistence, we consider games where the payoffs are drawn at random. In particular, we focus on games where a large number of players can each choose one of two possible strategies and the payoffs are independent and identically distributed with the possibility of ties. We provide asymptotic results about the random number of pure Nash equilibria, such as fast growth and a central limit theorem, with bounds for the approximation error. Moreover, by using a new link between percolation models and game theory, we describe in detail the geometry of pure Nash equilibria and show that, when the probability of ties is small, a best-response dynamics reaches a pure Nash equilibrium with a probability that quickly approaches one as the number of players grows. We show that various phase transitions depend only on a single parameter of the model, that is, the probability of having ties.

scholarly journals A Mixed 0-1 Linear Programming Approach to the Computation of All Pure-Strategy Nash Equilibria of a Finiten-Person Game in Normal Form

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Zhengtian Wu ◽  
Chuangyin Dang ◽  
Hamid Reza Karimi ◽  
Changan Zhu ◽  
Qing Gao
Keyword(s):  
Linear Programming ◽  
Nash Equilibrium ◽  
Normal Form ◽  
Nash Equilibria ◽  
Selection Process ◽  
Pure Strategy ◽  
Programming Approach ◽  
Main Concern ◽  
Person Game ◽  
Strategy Nash Equilibrium

A main concern in applications of game theory is how to effectively select a Nash equilibrium, especially a pure-strategy Nash equilibrium for a finiten-person game in normal form. This selection process often requires the computation of all Nash equilibria. It is well known that determining whether a finite game has a pure-strategy Nash equilibrium is an NP-hard problem and it is difficult to solve by naive enumeration algorithms. By exploiting the properties of pure strategy and multilinear terms in the payoff functions, this paper formulates a new mixed 0-1 linear program for computing all pure-strategy Nash equilibria. To our knowledge, it is the first method to formulate a mixed 0-1 linear programming for pure-strategy Nash equilibria and it may work well for similar problems. Numerical results show that the approach is effective and this method can be easily distributed in a distributed way.

scholarly journals Non-cooperative queueing games on a network of single server queues

2021 ◽  
Author(s):  
Corine M. Laan ◽  
Judith Timmer ◽  
Richard J. Boucherie
Keyword(s):  
Nash Equilibrium ◽  
Pure Strategy ◽  
Single Server ◽  
Strategy Space ◽  
Single Server Queues ◽  
Best Response ◽  
Multiple Routes ◽  
Service Rates ◽  
Expected Sojourn Time ◽  
Strategy Nash Equilibrium

AbstractThis paper introduces non-cooperative games on a network of single server queues with fixed routes. A player has a set of routes available and has to decide which route(s) to use for its customers. Each player’s goal is to minimize the expected sojourn time of its customers. We consider two cases: a continuous strategy space, where each player is allowed to divide its customers over multiple routes, and a discrete strategy space, where each player selects a single route for all its customers. For the continuous strategy space, we show that a unique pure-strategy Nash equilibrium exists that can be found using a best-response algorithm. For the discrete strategy space, we show that the game has a Nash equilibrium in mixed strategies, but need not have a pure-strategy Nash equilibrium. We show the existence of pure-strategy Nash equilibria for four subclasses: (i) N-player games with equal arrival rates for the players, (ii) 2-player games with identical service rates for all nodes, (iii) 2-player games on a $$2\times 2$$ 2 × 2 -grid, and (iv) 2-player games on an $$A\times B$$ A × B -grid with small differences in the service rates.

ON COURNOT COMPETITION UNDER RANDOM YIELD

2013 ◽  
Vol 30 (04) ◽  
pp. 1350007 ◽  
Author(s):  
XIAOMING YAN ◽  
YONG WANG
Keyword(s):  
Nash Equilibrium ◽  
Production Cost ◽  
Pure Strategy ◽  
Cournot Competition ◽  
Random Yield ◽  
Cournot Model ◽  
Random Capacity ◽  
Strategy Nash Equilibrium

We look at a Cournot model in which each firm may be unreliable with random capacity, so the total quantity brought into market is uncertain. The Cournot model has a unique pure strategy Nash equilibrium (NE), in which the number of active firms is determined by each firm's production cost and reliability. Our results indicate the following effects of unreliability: the number of active firms in the NE is more than that each firm is completely reliable and the expected total quantity brought into market is less than that each firm is completely reliable. Whether a given firm joins in the game is independent of its reliability, but any given firm always hopes that the less-expensive firms' capacities are random and stochastically smaller.

EQUILIBRIUM BALKING STRATEGIES IN THE REPAIRABLE M/M/1 G-RETRIAL QUEUE WITH COMPLETE REMOVALS

2019 ◽  
pp. 1-20
Author(s):  
Shan Gao ◽  
Deran Zhang ◽  
Hua Dong ◽  
Xianchao Wang
Keyword(s):  
Nash Equilibrium ◽  
Queueing Theory ◽  
Fundamental Problem ◽  
Pure Strategy ◽  
Retrial Queue ◽  
Noncooperative Game ◽  
Net Benefit ◽  
Negative Customers ◽  
Stable Strategy ◽  
Best Response

We consider an M/M/1 retrial queue subject to negative customers (called as G-retrial queue). The arrival of a negative customer forces all positive customers to leave the system and causes the server to fail. At a failure instant, the server is sent to be repaired immediately. Based on a natural reward-cost structure, all arriving positive customers decide whether to join the orbit or balk when they find the server is busy. All positive customers are selfish and want to maximize their own net benefit. Therefore, this system can be modeled as a symmetric noncooperative game among positive customers and the fundamental problem is to identify the Nash equilibrium balking strategy, which is a stable strategy in the sense that if all positive customers agree to follow it no one can benefit by deviating from it, that is, it is a strategy that is the best response against itself. In this paper, by using queueing theory and game theory, the Nash equilibrium mixed strategy in unobservable case and the Nash equilibrium pure strategy in observable case are considered. We also present some numerical examples to demonstrate the effect of the information together with some parameters on the equilibrium behaviors.

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