Stability Concepts: Beyond Nash Equilibria

2020 ◽  
pp. 63-90
Author(s):  
John M. McNamara ◽  
Olof Leimar
Keyword(s):  
Nash Equilibrium ◽  
Stability Condition ◽  
Replicator Dynamics ◽  
Adaptive Dynamics ◽  
Evolutionarily Stable Strategy ◽  
Stability Criteria ◽  
Evolutionary Dynamic ◽  
Stability Concepts ◽  
Evolutionarily Stable ◽  
Convergence Stability

The concept of an Evolutionarily Stable Strategy (ESS), which is a stronger stability condition than that of a Nash equilibrium, is introduced. A simple evolutionary dynamic, adaptive dynamics, is also introduced. This leads to the concept of convergence stability under adaptive dynamics. It is shown that these two stability criteria are independent for general games: a strategy can be an ESS but not be reachable under adaptive dynamics and a strategy may be an attractor under adaptive dynamics but a fitness minimum and so not an ESS. The latter situation leads to the possibility of evolutionary branching, a phenomenon in which the population splits into a mixture of two or more distinct morphs. Replicator dynamics provide another evolutionary dynamic, although it is argued that it is of limited relevance to biology. In some games, individuals interact with relatives. The effects of kin assortment, and the direct fitness and gene-centred approaches to games between kin are described and illustrated.

Three characterizations of population strategy stability

1980 ◽  
Vol 17 (2) ◽  
pp. 333-340 ◽  
Author(s):  
W. G. S. Hines
Keyword(s):  
Nash Equilibrium ◽  
Dynamical Systems ◽  
Evolutionarily Stable Strategy ◽  
Evolutionarily Stable Strategies ◽  
Stable Strategy ◽  
Standard Population ◽  
Population Strategy ◽  
Operative Strategies ◽  
Nash Equilibrium Strategies ◽  
Evolutionarily Stable

In addition to the concept of the evolutionarily stable strategy (ESS), developed specifically for considering intraspecific conflicts, concepts such as the Nash equilibrium from game theory and the attractor or sink from dynamical systems theory appear relevant to the problem of characterizing populations of stable composition. The three concepts mentioned are discussed for one simple standard population model. It is found that evolutionarily stable strategies of one type are necessarily Nash equilibrium strategies, although the converse is not true. The dynamical systems characterization is found to provide a model for populations susceptible to invasion by ‘co-operative' strategies, but capable of evolving back in average to the original equilibrium.

Three characterizations of population strategy stability

1980 ◽  
Vol 17 (02) ◽  
pp. 333-340 ◽  
Author(s):  
W. G. S. Hines
Keyword(s):  
Nash Equilibrium ◽  
Dynamical Systems ◽  
Evolutionarily Stable Strategy ◽  
Evolutionarily Stable Strategies ◽  
Stable Strategy ◽  
Standard Population ◽  
Population Strategy ◽  
Operative Strategies ◽  
Nash Equilibrium Strategies ◽  
Evolutionarily Stable

In addition to the concept of the evolutionarily stable strategy (ESS), developed specifically for considering intraspecific conflicts, concepts such as the Nash equilibrium from game theory and the attractor or sink from dynamical systems theory appear relevant to the problem of characterizing populations of stable composition. The three concepts mentioned are discussed for one simple standard population model. It is found that evolutionarily stable strategies of one type are necessarily Nash equilibrium strategies, although the converse is not true. The dynamical systems characterization is found to provide a model for populations susceptible to invasion by ‘co-operative' strategies, but capable of evolving back in average to the original equilibrium.

scholarly journals The Stability of Two-Community Replicator Dynamics with Discrete Multi-Delays

Mathematics ◽  
2020 ◽  
Vol 8 (12) ◽  
pp. 2120
Author(s):  
Jinxiu Pi ◽  
Hui Yang ◽  
Yadong Shu ◽  
Chongyi Zhong ◽  
Guanghui Yang
Keyword(s):  
Time Delays ◽  
Replicator Dynamics ◽  
Evolutionarily Stable Strategy ◽  
Sufficient Conditions ◽  
Linear Matrix ◽  
Stable Strategy ◽  
Two Delays ◽  
The Stability ◽  
Delay Dependent ◽  
Evolutionarily Stable

This article investigates the stability of evolutionarily stable strategy in replicator dynamics of two-community with multi-delays. In the real environment, players interact simultaneously while the return of their choices may not be observed immediately, which implies one or more time-delays exists. In addition to using the method of classic characteristic equations, we also apply linear matrix inequality (i.e., LMI) to discuss the stability of the mixed evolutionarily stable strategy in replicator dynamics of two-community with multi-delays. We derive a delay-dependent stability and a delay-independent stability sufficient conditions of the evolutionarily stable strategy in the two-community replicator dynamics with two delays, and manage to extend the sufficient condition to n time delays. Lastly, numerical trials of the Hawk–Dove game are given to verify the effectiveness of the theoretical consequences.

scholarly journals Stability of Replicator Dynamics with Bounded Continuously Distributed Time Delay

Mathematics ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 431
Author(s):  
Chongyi Zhong ◽  
Hui Yang ◽  
Zixin Liu ◽  
Juanyong Wu
Keyword(s):  
Time Delay ◽  
Replicator Dynamics ◽  
Evolutionarily Stable Strategy ◽  
Evolutionary Games ◽  
Stability Conditions ◽  
Stable Strategy ◽  
Distributed Time Delay ◽  
Player Game ◽  
The Stability ◽  
Evolutionarily Stable

In this paper, we consider evolutionary games and construct a model of replicator dynamics with bounded continuously distributed time delay. In many circumstances, players interact simultaneously while impacts of their choices take place after some time, which implies a time delay exists. We consider the time delay as bounded continuously distributed other than some given constant. Then, we investigate the stability of the evolutionarily stable strategy in the replicator dynamics with bounded continuously distributed time delay in two-player game contexts. Some stability conditions of the unique interior Nash equilibrium are obtained. Finally, the simple but important Hawk–Dove game is used to verify our results.

scholarly journals Research on Optimization of Array Honeypot Defense Strategies Based on Evolutionary Game Theory

Mathematics ◽  
2021 ◽  
Vol 9 (8) ◽  
pp. 805
Author(s):  
Leyi Shi ◽  
Xiran Wang ◽  
Huiwen Hou
Keyword(s):  
Replicator Dynamics ◽  
Evolutionarily Stable Strategy ◽  
Evolutionary Game ◽  
Game Model ◽  
Evolutionarily Stable Strategies ◽  
Defense Strategies ◽  
Active Defense ◽  
Defense Technology ◽  
The Stability ◽  
Evolutionarily Stable

Honeypot has been regarded as an active defense technology that can deceive attackers by simulating real systems. However, honeypot is actually a static network trap with fixed disposition, which is easily identified by anti-honeypot technology. Thus, honeypot is a “passive” active defense technology. Dynamic honeypot makes up for the shortcomings of honeypot, which dynamically adjusts defense strategies with the attack of hackers. Therefore, the confrontation between defenders and attackers is a strategic game. This paper focuses on the non-cooperative evolutionary game mechanism of bounded rationality, aiming to improve the security of the array honeypot system through the evolutionarily stable strategies derived from the evolutionary game model. First, we construct a three-party evolutionary game model of array honeypot, which is composed of defenders, attackers and legitimate users. Secondly, we formally describe the strategies and revenues of players in the game, and build the three-party game payoff matrices. Then the evolutionarily stable strategy is obtained by analyzing the Replicator Dynamics of various parties. In addition, we discuss the equilibrium condition to get the influence of the number of servers N on the stability of strategy evolution. MATLAB and Gambit simulation experiment results show that deduced evolutionarily stable strategies are valid in resisting attackers.

scholarly journals SATO–CRUTCHFIELD FORMULATION FOR SOME EVOLUTIONARY GAMES

2003 ◽  
Vol 14 (07) ◽  
pp. 963-971 ◽  
Author(s):  
E. AHMED ◽  
A. S. HEGAZI ◽  
A. S. ELGAZZAR
Keyword(s):  
Prisoner's Dilemma ◽  
Initial Conditions ◽  
Evolutionarily Stable Strategy ◽  
Prisoner’S Dilemma ◽  
Evolutionary Games ◽  
Theoretical Explanation ◽  
Battle Of The Sexes ◽  
Different Types ◽  
Evolutionarily Stable ◽  
Rules Of The Game

The Sato–Crutchfield equations are analytically and numerically studied. The Sato–Crutchfield formulation corresponds to losing memory. Then the Sato–Crutchfield formulation is applied for some different types of games including hawk–dove, prisoner's dilemma and the battle of the sexes games. The Sato–Crutchfield formulation is found not to affect the evolutionarily stable strategy of the ordinary games. But choosing a strategy becomes purely random, independent of the previous experiences, initial conditions, and the rules of the game itself. The Sato–Crutchfield formulation for the prisoner's dilemma game can be considered as a theoretical explanation for the existence of cooperation in a population of defectors.

A dynamic programming approach to evolutionarily stable strategy theory

1980 ◽  
Vol 12 (1) ◽  
pp. 3-5 ◽  
Author(s):  
C. Cannings ◽  
D. Gardiner
Keyword(s):  
Dynamic Programming ◽  
Density Function ◽  
Evolutionarily Stable Strategy ◽  
Programming Approach ◽  
Stable Strategy ◽  
War Of Attrition ◽  
Dynamic Programming Approach ◽  
The Individual ◽  
Image Position ◽  
Evolutionarily Stable

In the war of attrition (wa), introduced by Maynard Smith (1974), two contestants play values from [0, ∞), the individual playing the longer value winning a fixed prize V, and both incurring a loss equal to the lesser of the two values. Thus the payoff, E(x, y) to an animal playing x against one playing y, is A more general form (Bishop and Cannings (1978)) has and it was demonstrated that with and there exists a unique evolutionarily stable strategy (ess), which is to choose a random value from a specified density function on [0, ∞). Results were also obtained for strategy spaces [0, s] and [0, s).

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