text: We consider the classic Moran process modeling the spread of genetic mutations, as extended to structured populations by (Nature, 2005). In this process, individuals are the vertices of a connected graph G. Initially, there is a single mutant vertex, chosen uniformly at random. In each step, a random vertex is selected for reproduction with a probability proportional to its fitness: mutants have fitness r > 1, while non-mutants have fitness 1. The vertex chosen to reproduce places a copy of itself to a uniformly random neighbor in G, replacing the individual that was there. The process ends when the mutation either reaches fixation (i.e., all vertices are mutants), or gets extinct. The principal quantity of interest is the probability with which each of the two outcomes occurs. A problem that has received significant attention recently concerns the existence of families of graphs, called strong amplifiers of selection, for which the fixation probability tends to 1 as the order n of the graph increases, and the existence of strong suppressors of selection, for which this probability tends to 0. For the case of directed graphs, it is known that both strong amplifiers and suppressors exist. For the case of undirected graphs, however, the problem has remained open, and the general belief has been that neither strong amplifiers nor suppressors exist. In this paper we disprove this belief, by providing the first examples of such graphs. The strong amplifier we present has fixation probability 1 − ˜ O(n^ −1 3), and the strong suppressor has fixation probability ˜O(n^−1 4). Both graph constructions are surprisingly simple. We also prove a general upper bound of 1 − ˜Ω(n^−1 3) on the fixation probability of any undirected graph. Hence, our strong amplifier is existentially optimal.
  @cite_27: Evolutionary graph theory (EGT), studies the ability of a mutant gene to overtake a nite structured population. In this review, we describe the original framework for EGT and the major work that has followed it. This review looks at the calculation of the probability" - the probability of a mutant taking over a population and focuses on game-theoretic applications. We look at varying topics such as alternate evolutionary dynamics, time to xation, special topological cases, and game theoretic results. Throughout the review, we examine several interesting open problems that warrant further research.
  @cite_17: Abstract Game theory is one of the key paradigms behind many scientific disciplines from biology to behavioral sciences to economics. In its evolutionary form and especially when the interacting agents are linked in a specific social network the underlying solution concepts and methods are very similar to those applied in non-equilibrium statistical physics. This review gives a tutorial-type overview of the field for physicists. The first four sections introduce the necessary background in classical and evolutionary game theory from the basic definitions to the most important results. The fifth section surveys the topological complications implied by non-mean-field-type social network structures in general. The next three sections discuss in detail the dynamic behavior of three prominent classes of models: the Prisoner's Dilemma, the Rock–Scissors–Paper game, and Competing Associations. The major theme of the review is in what sense and how the graph structure of interactions can modify and enrich the picture of long term behavioral patterns emerging in evolutionary games.
  @cite_20: Every form of behavior is shaped by trial and error. Such stepwise adaptation can occur through individual learning or through natural selection, the basis of evolution. Since the work of Maynard Smith and others, it has been realized how game theory can model this process. Evolutionary game theory replaces the static solutions of classical game theory by a dynamical approach centered not on the concept of rational players but on the population dynamics of behavioral programs. In this book the authors investigate the nonlinear dynamics of the self-regulation of social and economic behavior, and of the closely related interactions among species in ecological communities. Replicator equations describe how successful strategies spread and thereby create new conditions that can alter the basis of their success, i.e., to enable us to understand the strategic and genetic foundations of the endless chronicle of invasions and extinctions that punctuate evolution. In short, evolutionary game theory describes when to escalate a conflict, how to elicit cooperation, why to expect a balance of the sexes, and how to understand natural selection in mathematical terms. Comprehensive treatment of ecological and game theoretic dynamics Invasion dynamics and permanence as key concepts Explanation in terms of games of things like competition between species
  @cite_2: Since its original publication in 2000, Game Theory Evolving has been considered the best textbook on evolutionary game theory. This completely revised and updated second edition of Game Theory Evolving contains new material and shows students how to apply game theory to model human behavior in ways that reflect the special nature of sociality and individuality. The textbook continues its in-depth look at cooperation in teams, agent-based simulations, experimental economics, the evolution and diffusion of preferences, and the connection between biology and economics.Recognizing that students learn by doing, the textbook introduces principles through practice. Herbert Gintis exposes students to the techniques and applications of game theory through a wealth of sophisticated and surprisingly fun-to-solve problems involving human and animal behavior. The second edition includes solutions to the problems presented and information related to agent-based modeling. In addition, the textbook incorporates instruction in using mathematical software to solve complex problems. Game Theory Evolving is perfect for graduate and upper-level undergraduate economics students, and is a terrific introduction for ambitious do-it-yourselfers throughout the behavioral sciences. Revised and updated edition relevant for courses across disciplines Perfect for graduate and upper-level undergraduate economics courses Solutions to problems presented throughout Incorporates instruction in using computational software for complex problem solving Includes in-depth discussions of agent-based modeling
summary: More involved population dynamics have been studied in evolutionary game theory in the context of well-mixed populations @cite_20 @cite_2 , and more recently in populations on graphs @cite_17 @cite_27 . The fitness of an individual is no longer fixed (determined by its type: mutant or non-mutant), but is determined by the expected payoff for the individual when playing a two-player game against a randomly chosen neighbor. In this game mutants play with one strategy and non-mutants with another.