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 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.

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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