Alternate Models of Replicator Dynamics

2013 ◽  
Author(s):  
Elizabeth N. Wesson ◽  
Richard H. Rand
Keyword(s):  
Evolutionary Dynamics ◽  
Total Population ◽  
Replicator Dynamics ◽  
Standard Form ◽  
Exponential Model ◽  
Payoff Matrix ◽  
Replicator Equation ◽  
Game Theoretic ◽  
Average Fitness ◽  
Strategy I

Models of evolutionary dynamics are often approached via the replicator equation, which in its standard form is given by x.i=xifix-ϕ,i = 1,…, n, where xi is the frequency, or relative abundance, of strategy i, fi is its fitness, and ϕ=∑i=1nxifi is the average fitness. A game-theoretic aspect is introduced to the model via the payoff matrix A, where Ai,j is the expected payoff of i vs. j, by taking fi(x) = (A·x)i. This model is based on the exponential model of population growth, ẋi = xifi, with ϕ introduced in order both to hold the total population constant and to model competition between strategies. We analyze the dynamics of analogous models for the replicator equation of the form x.i=gxifi-ϕ, for selected growth functions g.

scholarly journals Extinction rates in tumor public goods games

2017 ◽  
Author(s):  
Philip Gerlee ◽  
Philipp M. Altrock
Keyword(s):  
Population Growth ◽  
Evolutionary Dynamics ◽  
Replicator Dynamics ◽  
Evolutionary Game ◽  
Cell Types ◽  
Relative Fitness ◽  
Replicator Equation ◽  
Cellular Interactions ◽  
Cancer Evolution ◽  
Public Goods Games

AbstractCancer evolution and progression are shaped by cellular interactions and Darwinian selection. Evolutionary game theory incorporates both of these principles, and has been proposed as a framework to understand tumor cell population dynamics. A cornerstone of evolutionary dynamics is the replicator equation, which describes changes in the relative abundance of different cell types, and is able to predict evolutionary equilibria. Typically, the replicator equation focuses on differences in relative fitness. We here show that this framework might not be sufficient under all circumstances, as it neglects important aspects of population growth. Standard replicator dynamics might miss critical differences in the time it takes to reach an equilibrium, as this time also depends on cellular turnover in growing but bounded populations. As the system reaches a stable manifold, the time to reach equilibrium depends on cellular death and birth rates. These rates shape the timescales, in particular in co-evolutionary dynamics of growth factor producers and free-riders. Replicator dynamics might be an appropriate framework only when birth and death rates are of similar magnitude. Otherwise, population growth effects cannot be neglected when predicting the time to reach an equilibrium, and cell type specific rates have to be accounted for explicitly.

scholarly journals Extinction rates in tumour public goods games

2017 ◽  
Vol 14 (134) ◽  
pp. 20170342 ◽  
Author(s):  
Philip Gerlee ◽  
Philipp M. Altrock
Keyword(s):  
Population Growth ◽  
Evolutionary Dynamics ◽  
Replicator Dynamics ◽  
Evolutionary Game ◽  
Cell Types ◽  
Relative Fitness ◽  
Replicator Equation ◽  
Cellular Interactions ◽  
Cancer Evolution ◽  
Public Goods Games

Cancer evolution and progression are shaped by cellular interactions and Darwinian selection. Evolutionary game theory incorporates both of these principles, and has been proposed as a framework to understand tumour cell population dynamics. A cornerstone of evolutionary dynamics is the replicator equation, which describes changes in the relative abundance of different cell types, and is able to predict evolutionary equilibria. Typically, the replicator equation focuses on differences in relative fitness. We here show that this framework might not be sufficient under all circumstances, as it neglects important aspects of population growth. Standard replicator dynamics might miss critical differences in the time it takes to reach an equilibrium, as this time also depends on cellular turnover in growing but bounded populations. As the system reaches a stable manifold, the time to reach equilibrium depends on cellular death and birth rates. These rates shape the time scales, in particular, in coevolutionary dynamics of growth factor producers and free-riders. Replicator dynamics might be an appropriate framework only when birth and death rates are of similar magnitude. Otherwise, population growth effects cannot be neglected when predicting the time to reach an equilibrium, and cell-type-specific rates have to be accounted for explicitly.

Generalized Replicator Dynamics as a Model of Specialization and Diversity in Societies

1998 ◽  
Vol 01 (04) ◽  
pp. 325-359 ◽  
Author(s):  
Vivek S. Borkar ◽  
Sanjay Jain ◽  
Govindan Rangarajan
Keyword(s):  
Mixed Strategy ◽  
Replicator Dynamics ◽  
Pure Strategy ◽  
Evolutionary Game ◽  
Payoff Matrix ◽  
Theoretic Model ◽  
Final States ◽  
Game Theoretic ◽  
Game Theoretic Model ◽  
Pure Strategies

We consider a generalization of replicator dynamics as a non-cooperative evolutionary game-theoretic model of a community of N agents. All agents update their individual mixed strategy profiles to increase their total payoff from the rest of the community. The properties of attractors in this dynamics are studied. Evidence is presented that under certain conditions the typical attractors of the system are corners of state space where each agent has specialized to a pure strategy, and/or the community exhibits diversity, i.e., all strategies are represented in the final states. The model suggests that new pure strategies whose payoff matrix elements satisfy suitable inequalities with respect to the existing ones can destabilize existing attractors if N is sufficiently large, and be regarded as innovations that enhance the diversity of the community.

Cooperative Homo economicus

2011 ◽  
Author(s):  
Samuel Bowles ◽  
Herbert Gintis
Keyword(s):  
Private Information ◽  
Repeated Games ◽  
Evolutionary Dynamics ◽  
Public Information ◽  
Folk Theorem ◽  
The Self ◽  
Correlated Equilibrium ◽  
Homo Economicus ◽  
Self Interest ◽  
Game Theoretic

This chapter examines whether recent advances in the theory of repeated games, as exemplified by the so-called folk theorem and related models, address the shortcomings of the self-interest based models in explaining human cooperation. It first provides an overview of folk theorems and their account of evolutionary dynamics before discussing the folk theorem with either imperfect public information or private information. It then considers evolutionarily irrelevant equilibrium as well as the link between social norms and the notion of correlated equilibrium. While the insight that repeated interactions provide opportunities for cooperative individuals to discipline defectors is correct, the chapter argues that none of the game-theoretic models mentioned above is successful. Except under implausible conditions, the cooperative outcomes identified by these models are neither accessible nor persistent, and are thus labeled evolutionarily irrelevant Nash equilibria.

Evolutionary Stable Strategies for Supply Chains: Selfishness, Fairness, and Altruism

2018 ◽  
Vol 6 (6) ◽  
pp. 532-551 ◽  
Author(s):  
Caichun Chai ◽  
Hailong Zhu ◽  
Zhangwei Feng
Keyword(s):  
Supply Chains ◽  
Evolutionary Dynamics ◽  
Replicator Dynamics ◽  
Management Strategies ◽  
Individual Preference ◽  
Short Term ◽  
Evolutionary Stable Strategies ◽  
Pure Strategies ◽  
The Stability

Abstract The management strategies of a firm are inevitable affected by individual behavior preferences. The effect of individual preference on the evolutionary dynamics for supply chains is studied by employing replicator dynamics. Each firm has three behavior preferences: selfishness, fairness, and altruism. Firstly, the case that the strategy set of manufacturers and retailers including two pure strategies is considered and the effect of preference parameter on the equilibrium outcome in the short-term interaction is discussed. Secondly, the equilibrium state in the short-term is always disturbed because the change of the environment, firm’s structure, and so forth. Using the replicator dynamics, the evolutionary stable strategies of manufacturers and retailers in the long-term interaction are analyzed. Finally, the extend case that the strategy set of manufacturers and retailers include three pure strategies is investigated. These results are found that the strategy profile in which both manufacturer and retailer choose fairness or altruism, or one player chooses fair or altruistic strategy and the other player chooses selfish strategy may be evolutionary stable, the stability of these equilibria depends on the the preference parameters.

scholarly journals eGAP: An Evolutionary Game Theoretic Approach to Random Forest Pruning

2020 ◽  
Vol 4 (4) ◽  
pp. 37
Author(s):  
Khaled Fawagreh ◽  
Mohamed Medhat Gaber
Keyword(s):  
Random Forest ◽  
Replicator Dynamics ◽  
Evolutionary Game ◽  
Theoretic Approach ◽  
Healthcare Applications ◽  
Healthcare Data ◽  
Smart Healthcare ◽  
Game Theoretic ◽  
Game Theoretic Approach

To make healthcare available and easily accessible, the Internet of Things (IoT), which paved the way to the construction of smart cities, marked the birth of many smart applications in numerous areas, including healthcare. As a result, smart healthcare applications have been and are being developed to provide, using mobile and electronic technology, higher diagnosis quality of the diseases, better treatment of the patients, and improved quality of lives. Since smart healthcare applications that are mainly concerned with the prediction of healthcare data (like diseases for example) rely on predictive healthcare data analytics, it is imperative for such predictive healthcare data analytics to be as accurate as possible. In this paper, we will exploit supervised machine learning methods in classification and regression to improve the performance of the traditional Random Forest on healthcare datasets, both in terms of accuracy and classification/regression speed, in order to produce an effective and efficient smart healthcare application, which we have termed eGAP. eGAP uses the evolutionary game theoretic approach replicator dynamics to evolve a Random Forest ensemble. Trees of high resemblance in an initial Random Forest are clustered, and then clusters grow and shrink by adding and removing trees using replicator dynamics, according to the predictive accuracy of each subforest represented by a cluster of trees. All clusters have an initial number of trees that is equal to the number of trees in the smallest cluster. Cluster growth is performed using trees that are not initially sampled. The speed and accuracy of the proposed method have been demonstrated by an experimental study on 10 classification and 10 regression medical datasets.

scholarly journals Steering eco-evolutionary game dynamics with manifold control

2020 ◽  
Vol 476 (2233) ◽  
pp. 20190643 ◽  
Author(s):  
Xin Wang ◽  
Zhiming Zheng ◽  
Feng Fu
Keyword(s):  
Evolutionary Dynamics ◽  
Replicator Dynamics ◽  
Evolutionary Game ◽  
Feedback Loops ◽  
Mathematical Framework ◽  
Stable Manifolds ◽  
Selection Gradient ◽  
Game Dynamics ◽  
Control Approach ◽  
Attraction Basin

Feedback loops between population dynamics of individuals and their ecological environment are ubiquitously found in nature and have shown profound effects on the resulting eco-evolutionary dynamics. By incorporating linear environmental feedback law into the replicator dynamics of two-player games, recent theoretical studies have shed light on understanding the oscillating dynamics of the social dilemma. However, the detailed effects of more general nonlinear feedback loops in multi-player games, which are more common especially in microbial systems, remain unclear. Here, we focus on ecological public goods games with environmental feedbacks driven by a nonlinear selection gradient. Unlike previous models, multiple segments of stable and unstable equilibrium manifolds can emerge from the population dynamical systems. We find that a larger relative asymmetrical feedback speed for group interactions centred on cooperators not only accelerates the convergence of stable manifolds but also increases the attraction basin of these stable manifolds. Furthermore, our work offers an innovative manifold control approach: by designing appropriate switching control laws, we are able to steer the eco-evolutionary dynamics to any desired population state. Our mathematical framework is an important generalization and complement to coevolutionary game dynamics, and also fills the theoretical gap in guiding the widespread problem of population state control in microbial experiments.

scholarly journals Bounds and dynamics for empirical game theoretic analysis

2019 ◽  
Vol 34 (1) ◽  
Author(s):  
Karl Tuyls ◽  
Julien Perolat ◽  
Marc Lanctot ◽  
Edward Hughes ◽  
Richard Everett ◽  
...  
Keyword(s):  
Nash Equilibrium ◽  
Evolutionary Dynamics ◽  
Learning Algorithm ◽  
Agent Interactions ◽  
Approximate Nash Equilibrium ◽  
Blotto Game ◽  
Multi Agent ◽  
Payoff Structure ◽  
Game Theoretic ◽  
Colonel Blotto

AbstractThis paper provides several theoretical results for empirical game theory. Specifically, we introduce bounds for empirical game theoretical analysis of complex multi-agent interactions. In doing so we provide insights in the empirical meta game showing that a Nash equilibrium of the estimated meta-game is an approximate Nash equilibrium of the true underlying meta-game. We investigate and show how many data samples are required to obtain a close enough approximation of the underlying game. Additionally, we extend the evolutionary dynamics analysis of meta-games using heuristic payoff tables (HPTs) to asymmetric games. The state-of-the-art has only considered evolutionary dynamics of symmetric HPTs in which agents have access to the same strategy sets and the payoff structure is symmetric, implying that agents are interchangeable. Finally, we carry out an empirical illustration of the generalised method in several domains, illustrating the theory and evolutionary dynamics of several versions of the AlphaGo algorithm (symmetric), the dynamics of the Colonel Blotto game played by human players on Facebook (symmetric), the dynamics of several teams of players in the capture the flag game (symmetric), and an example of a meta-game in Leduc Poker (asymmetric), generated by the policy-space response oracle multi-agent learning algorithm.

scholarly journals Parametric Excitation and Evolutionary Dynamics

2013 ◽  
Vol 80 (5) ◽  
Author(s):  
Rocio E. Ruelas ◽  
David G. Rand ◽  
Richard H. Rand
Keyword(s):  
Differential Equation ◽  
Parametric Excitation ◽  
Variable Coefficient ◽  
Evolutionary Dynamics ◽  
Replicator Equation ◽  
Periodic Coefficients ◽  
Forcing Function ◽  
Governing Differential Equation ◽  
Social Applications ◽  
Dynamics Problems

Parametric excitation refers to dynamics problems in which the forcing function enters into the governing differential equation as a variable coefficient. Evolutionary dynamics refers to a mathematical model of natural selection (the “replicator” equation) which involves a combination of game theory and differential equations. In this paper we apply perturbation theory to investigate parametric resonance in a replicator equation having periodic coefficients. In particular, we study evolution in the Rock-Paper-Scissors game, which has biological and social applications. Here periodic coefficients could represent seasonal variation. We show that 2:1 subharmonic resonance can destabilize the usual “Rock-Paper-Scissors” equilibrium for parameters located in a resonant tongue in parameter space. However, we also show that the tongue may be absent or very small if the forcing parameters are chosen appropriately.

scholarly journals Evolutionary game theory for physical and biological scientists. I. Training and validating population dynamics equations

2014 ◽  
Vol 4 (4) ◽  
pp. 20140037 ◽  
Author(s):  
David Liao ◽  
Thea D. Tlsty
Keyword(s):  
Game Theory ◽  
Population Dynamics ◽  
Evolutionary Game Theory ◽  
Evolutionary Dynamics ◽  
Evolutionary Game ◽  
Treatment Strategies ◽  
Analysis Method ◽  
Analysis Techniques ◽  
Game Theoretic ◽  
And Control

Failure to understand evolutionary dynamics has been hypothesized as limiting our ability to control biological systems. An increasing awareness of similarities between macroscopic ecosystems and cellular tissues has inspired optimism that game theory will provide insights into the progression and control of cancer. To realize this potential, the ability to compare game theoretic models and experimental measurements of population dynamics should be broadly disseminated. In this tutorial, we present an analysis method that can be used to train parameters in game theoretic dynamics equations, used to validate the resulting equations, and used to make predictions to challenge these equations and to design treatment strategies. The data analysis techniques in this tutorial are adapted from the analysis of reaction kinetics using the method of initial rates taught in undergraduate general chemistry courses. Reliance on computer programming is avoided to encourage the adoption of these methods as routine bench activities.

Sign in / Sign up

Export Citation Format

Share Document