scholarly journals Structural symmetry in evolutionary games

2015 ◽  
Vol 12 (111) ◽  
pp. 20150420 ◽  
Author(s):  
Alex McAvoy ◽  
Christoph Hauert
Keyword(s):  
Population Structure ◽  
Evolutionary Game ◽  
Evolutionary Process ◽  
Evolutionary Games ◽  
Structured Populations ◽  
Fixation Probability ◽  
Matrix Games ◽  
Single Mutant ◽  
Symmetric Games ◽  
Monomorphic Population

In evolutionary game theory, an important measure of a mutant trait (strategy) is its ability to invade and take over an otherwise-monomorphic population. Typically, one quantifies the success of a mutant strategy via the probability that a randomly occurring mutant will fixate in the population. However, in a structured population, this fixation probability may depend on where the mutant arises. Moreover, the fixation probability is just one quantity by which one can measure the success of a mutant; fixation time , for instance, is another. We define a notion of homogeneity for evolutionary games that captures what it means for two single-mutant states, i.e. two configurations of a single mutant in an otherwise-monomorphic population, to be ‘evolutionarily equivalent’ in the sense that all measures of evolutionary success are the same for both configurations. Using asymmetric games, we argue that the term ‘homogeneous’ should apply to the evolutionary process as a whole rather than to just the population structure. For evolutionary matrix games in graph-structured populations, we give precise conditions under which the resulting process is homogeneous. Finally, we show that asymmetric matrix games can be reduced to symmetric games if the population structure possesses a sufficient degree of symmetry.

scholarly journals The effect of population structure on the rate of evolution

2013 ◽  
Vol 280 (1762) ◽  
pp. 20130211 ◽  
Author(s):  
Marcus Frean ◽  
Paul B. Rainey ◽  
Arne Traulsen
Keyword(s):  
Population Structure ◽  
Evolutionary Process ◽  
Ecological Factors ◽  
Structured Populations ◽  
Fixation Time ◽  
Fixation Probability ◽  
Star Graphs ◽  
Rate Of Evolution ◽  
Population Structures ◽  
Evolution Proceeds

Ecological factors exert a range of effects on the dynamics of the evolutionary process. A particularly marked effect comes from population structure, which can affect the probability that new mutations reach fixation. Our interest is in population structures, such as those depicted by ‘star graphs’, that amplify the effects of selection by further increasing the fixation probability of advantageous mutants and decreasing the fixation probability of disadvantageous mutants. The fact that star graphs increase the fixation probability of beneficial mutations has lead to the conclusion that evolution proceeds more rapidly in star-structured populations, compared with mixed (unstructured) populations. Here, we show that the effects of population structure on the rate of evolution are more complex and subtle than previously recognized and draw attention to the importance of fixation time. By comparing population structures that amplify selection with other population structures, both analytically and numerically, we show that evolution can slow down substantially even in populations where selection is amplified.

scholarly journals Evolutionary games on graphs and the speed of the evolutionary process

2009 ◽  
Vol 466 (2117) ◽  
pp. 1327-1346 ◽  
Author(s):  
M. Broom ◽  
C. Hadjichrysanthou ◽  
J. Rychtář
Keyword(s):  
Analytical Approach ◽  
Directed Graphs ◽  
Evolutionary Game ◽  
Evolutionary Process ◽  
Evolutionary Games ◽  
Fixation Probability ◽  
Numerical Examples ◽  
Games On Graphs ◽  
The Mean ◽  
Mean Time

In this paper, we investigate evolutionary games with the invasion process updating rules on three simple non-directed graphs: the star, the circle and the complete graph. Here, we present an analytical approach and derive the exact solutions of the stochastic evolutionary game dynamics. We present formulae for the fixation probability and also for the speed of the evolutionary process, namely for the mean time to absorption (either mutant fixation or extinction) and then the mean time to mutant fixation. Through numerical examples, we compare the different impact of the population size and the fitness of each type of individual on the above quantities on the three different structures. We do this comparison in two specific cases. Firstly, we consider the case where mutants have fixed fitness r and resident individuals have fitness 1. Then, we consider the case where the fitness is not constant but depends on games played among the individuals, and we introduce a ‘hawk–dove’ game as an example.

scholarly journals Aspiration dynamics of multi-player games in finite populations

2014 ◽  
Vol 11 (94) ◽  
pp. 20140077 ◽  
Author(s):  
Jinming Du ◽  
Bin Wu ◽  
Philipp M. Altrock ◽  
Long Wang
Keyword(s):  
Evolutionary Game ◽  
Pairwise Comparison ◽  
Evolutionary Games ◽  
Structured Populations ◽  
Weak Selection ◽  
Additional Information ◽  
Average Abundance ◽  
Imperfect Rationality ◽  
Update Rules ◽  
Limiting Case

On studying strategy update rules in the framework of evolutionary game theory, one can differentiate between imitation processes and aspiration-driven dynamics. In the former case, individuals imitate the strategy of a more successful peer. In the latter case, individuals adjust their strategies based on a comparison of their pay-offs from the evolutionary game to a value they aspire, called the level of aspiration. Unlike imitation processes of pairwise comparison, aspiration-driven updates do not require additional information about the strategic environment and can thus be interpreted as being more spontaneous. Recent work has mainly focused on understanding how aspiration dynamics alter the evolutionary outcome in structured populations. However, the baseline case for understanding strategy selection is the well-mixed population case, which is still lacking sufficient understanding. We explore how aspiration-driven strategy-update dynamics under imperfect rationality influence the average abundance of a strategy in multi-player evolutionary games with two strategies. We analytically derive a condition under which a strategy is more abundant than the other in the weak selection limiting case. This approach has a long-standing history in evolutionary games and is mostly applied for its mathematical approachability. Hence, we also explore strong selection numerically, which shows that our weak selection condition is a robust predictor of the average abundance of a strategy. The condition turns out to differ from that of a wide class of imitation dynamics, as long as the game is not dyadic. Therefore, a strategy favoured under imitation dynamics can be disfavoured under aspiration dynamics. This does not require any population structure, and thus highlights the intrinsic difference between imitation and aspiration dynamics.

scholarly journals Two conceptions of evolutionary games: reductive vs effective

2017 ◽  
Author(s):  
Artem Kaznatcheev
Keyword(s):  
Game Theory ◽  
Evolutionary Game ◽  
Evolutionary Games ◽  
Theoretical Work ◽  
Structured Populations ◽  
Agent Based ◽  
Specific Interpretation ◽  
The Difference ◽  
Games Theory ◽  
Future Work

Evolutionary game theory (EGT) was born from economic game theory through a series of analogies. Given this heuristic genealogy, a number of central objects of the theory (like strategies, players, and games) have not been carefully defined or interpreted. A specific interpretation of these terms becomes important as EGT sees more applications to understanding experiments in microscopic systems typical of oncology and microbiology. In this essay, I provide two interpretations of the central objects of games theory: one that leads to reductive games and the other to effective games. These interpretation are based on the difference between views of fitness as a property of individuals versus fitness as a summary statistic of (sub)populations. Reductive games are typical of theoretical work like agent-based models. But effective games usually correspond more closely to experimental work. However, confusing reductive games for effective games or vice-versa can lead to divergent results, especially in spatially structured populations. As such, I propose that we treat this distinction carefully in future work at the interface of EGT and experiment.

scholarly journals Dynamical patterns of coexisting strategies in a hybrid discrete-continuum spatial evolutionary game model

2016 ◽  
Author(s):  
A.E.F. Burgess ◽  
P.G. Schofield ◽  
S.F. Hubbard ◽  
M.A.J. Chaplain ◽  
T. Lorenzi
Keyword(s):  
Diffusion Equation ◽  
Evolutionary Game ◽  
Evolutionary Games ◽  
Reaction Diffusion ◽  
Social Contexts ◽  
Random Motion ◽  
Structured Populations ◽  
Reaction Diffusion Equation ◽  
Modelling Framework ◽  
Dynamical Patterns

AbstractWe present a novel hybrid modelling framework that takes into account two aspects which have been largely neglected in previous models of spatial evolutionary games: random motion and chemotaxis. A stochastic individual-based model is used to describe the player dynamics, whereas the evolution of the chemoattractant is governed by a reaction-diffusion equation. The two models are coupled by deriving individual movement rules via the discretisation of a taxis-diffusion equation which describes the evolution of the local number of players. In this framework, individuals occupying the same position can engage in a two-player game, and are awarded a payoff, in terms of reproductive fitness, according to their strategy. As an example, we let individuals play the Hawk-Dove game. Numerical simulations illustrate how random motion and chemotactic response can bring about self-generated dynamical patterns that create favourable conditions for the coexistence of hawks and doves in situations in which the two strategies cannot coexist otherwise. In this sense, our work offers a new perspective of research on spatial evolutionary games, and provides a general formalism to study the dynamics of spatially-structured populations in biological and social contexts where individual motion is likely to affect natural selection of behavioural traits.

scholarly journals IsoMaTrix: a framework to visualize the isoclines of matrix games and quantify uncertainty in structured populations

2020 ◽  
Author(s):  
Jeffrey West ◽  
Yongqian Ma ◽  
Artem Kaznatcheev ◽  
Alexander R. A. Anderson
Keyword(s):  
Spatial Configuration ◽  
Replicator Dynamics ◽  
Evolutionary Game ◽  
Payoff Matrix ◽  
Structured Populations ◽  
Three Dimensions ◽  
Continuous Treatment ◽  
Hybrid Automata ◽  
Matrix Games ◽  
Structured Matrix

AbstractSummaryEvolutionary game theory describes frequency-dependent selection for fixed, heritable strategies in a population of competing individuals using a payoff matrix, typically described using well-mixed assumptions (replicator dynamics). IsoMaTrix is an open-source package which computes the isoclines (lines of zero growth) of matrix games, and facilitates direct comparison of well-mixed dynamics to structured populations in two or three dimensions. IsoMaTrix is coupled with a Hybrid Automata Library module to simulate structured matrix games on-lattice. IsoMaTrix can also compute fixed points, phase flow, trajectories, velocities (and subvelocities), delineated “region plots” of positive/negative strategy velocity, and uncertainty quantification for stochastic effects in structured matrix games. We describe a result obtained via IsoMaTrix’s spatial games functionality, which shows that the timing of competitive release in a cancer model (under continuous treatment) critically depends on the initial spatial configuration of the tumor.Availability and implementationThe code is available at: https://github.com/mathonco/isomatrix.

scholarly journals IsoMaTrix: a framework to visualize the isoclines of matrix games and quantify uncertainty in structured populations

2020 ◽  
Author(s):  
Jeffrey West ◽  
Yongqian Ma ◽  
Artem Kaznatcheev ◽  
Alexander R A Anderson
Keyword(s):  
Spatial Configuration ◽  
Evolutionary Game ◽  
Payoff Matrix ◽  
Structured Populations ◽  
Continuous Treatment ◽  
Supplementary Information ◽  
Matrix Games ◽  
Spatial Games ◽  
Competitive Release ◽  
Zero Growth

Abstract   Evolutionary game theory describes frequency-dependent selection for fixed, heritable strategies in a population of competing individuals using a payoff matrix. We present a software package to aid in the construction, analysis, and visualization of three-strategy matrix games. The IsoMaTrix package computes the isoclines (lines of zero growth) of matrix games, and facilitates direct comparison of well-mixed dynamics to structured populations on a lattice grid. IsoMaTrix computes fixed points, phase flow, trajectories, (sub)velocities, and uncertainty quantification for stochastic effects in spatial matrix games. We describe a result obtained via IsoMaTrix’s spatial games functionality, which shows that the timing of competitive release in a cancer model (under continuous treatment) critically depends on the initial spatial configuration of the tumor. Availability The code is available at: https://github.com/mathonco/isomatrix. Supplementary information Supplementary data are available at Bioinformatics online.

scholarly journals Evolutionary dynamics of group interactions on structured populations: a review

2013 ◽  
Vol 10 (80) ◽  
pp. 20120997 ◽  
Author(s):  
Matjaž Perc ◽  
Jesús Gómez-Gardeñes ◽  
Attila Szolnoki ◽  
Luis M. Floría ◽  
Yamir Moreno
Keyword(s):  
Statistical Physics ◽  
Evolutionary Dynamics ◽  
Evolutionary Game ◽  
Evolutionary Games ◽  
Structured Populations ◽  
Public Goods Game ◽  
Group Interactions ◽  
The Public ◽  
Living Organisms ◽  
Class Representative

Interactions among living organisms, from bacteria colonies to human societies, are inherently more complex than interactions among particles and non-living matter. Group interactions are a particularly important and widespread class, representative of which is the public goods game. In addition, methods of statistical physics have proved valuable for studying pattern formation, equilibrium selection and self-organization in evolutionary games. Here, we review recent advances in the study of evolutionary dynamics of group interactions on top of structured populations, including lattices, complex networks and coevolutionary models. We also compare these results with those obtained on well-mixed populations. The review particularly highlights that the study of the dynamics of group interactions, like several other important equilibrium and non-equilibrium dynamical processes in biological, economical and social sciences, benefits from the synergy between statistical physics, network science and evolutionary game theory.

scholarly journals Evolutionary graph theory revisited: when is an evolutionary process equivalent to the Moran process?

2015 ◽  
Vol 471 (2182) ◽  
pp. 20150334 ◽  
Author(s):  
Karan Pattni ◽  
Mark Broom ◽  
Jan Rychtář ◽  
Lara J. Silvers
Keyword(s):  
Graph Theory ◽  
Evolutionary Dynamics ◽  
Evolutionary Process ◽  
Structured Populations ◽  
Fixation Probability ◽  
Finite Populations ◽  
Moran Model ◽  
Moran Process ◽  
Evolutionary Graph Theory ◽  
Associated Matrix

Evolution in finite populations is often modelled using the classical Moran process. Over the last 10 years, this methodology has been extended to structured populations using evolutionary graph theory. An important question in any such population is whether a rare mutant has a higher or lower chance of fixating (the fixation probability) than the Moran probability, i.e. that from the original Moran model, which represents an unstructured population. As evolutionary graph theory has developed, different ways of considering the interactions between individuals through a graph and an associated matrix of weights have been considered, as have a number of important dynamics. In this paper, we revisit the original paper on evolutionary graph theory in light of these extensions to consider these developments in an integrated way. In particular, we find general criteria for when an evolutionary graph with general weights satisfies the Moran probability for the set of six common evolutionary dynamics.

scholarly journals Evolutionary dynamics in structured populations

2010 ◽  
Vol 365 (1537) ◽  
pp. 19-30 ◽  
Author(s):  
Martin A. Nowak ◽  
Corina E. Tarnita ◽  
Tibor Antal
Keyword(s):  
Game Theory ◽  
Evolutionary Game Theory ◽  
Evolutionary Dynamics ◽  
Evolutionary Game ◽  
Evolutionary Process ◽  
Structured Populations ◽  
Evolution Of Cooperation ◽  
Evolutionary Graph Theory ◽  
Phenotype Space ◽  
Multicellular Organisms

Evolutionary dynamics shape the living world around us. At the centre of every evolutionary process is a population of reproducing individuals. The structure of that population affects evolutionary dynamics. The individuals can be molecules, cells, viruses, multicellular organisms or humans. Whenever the fitness of individuals depends on the relative abundance of phenotypes in the population, we are in the realm of evolutionary game theory. Evolutionary game theory is a general approach that can describe the competition of species in an ecosystem, the interaction between hosts and parasites, between viruses and cells, and also the spread of ideas and behaviours in the human population. In this perspective, we review the recent advances in evolutionary game dynamics with a particular emphasis on stochastic approaches in finite sized and structured populations. We give simple, fundamental laws that determine how natural selection chooses between competing strategies. We study the well-mixed population, evolutionary graph theory, games in phenotype space and evolutionary set theory. We apply these results to the evolution of cooperation. The mechanism that leads to the evolution of cooperation in these settings could be called ‘spatial selection’: cooperators prevail against defectors by clustering in physical or other spaces.

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