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31
Evolutionary games on graphs
, 2007
"... 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 ..."
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Cited by 54 (0 self)
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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 nonequilibrium statistical physics. This review gives a tutorialtype 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 nonmeanfieldtype 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.
The replicator equation on graphs
, 2006
"... We study evolutionary games on graphs. Each player is represented by a vertex of the graph. The edges denote who meets whom. A player can use any one of n strategies. Players obtain a payoff from interaction with all their immediate neighbors. We consider three different update rules, called ‘birth– ..."
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Cited by 20 (6 self)
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We study evolutionary games on graphs. Each player is represented by a vertex of the graph. The edges denote who meets whom. A player can use any one of n strategies. Players obtain a payoff from interaction with all their immediate neighbors. We consider three different update rules, called ‘birth–death’, ‘death–birth ’ and ‘imitation’. A fourth update rule, ‘pairwise comparison’, is shown to be equivalent to birth–death updating in our model. We use pair approximation to describe the evolutionary game dynamics on regular graphs of degree k. In the limit of weak selection, we can derive a differential equation which describes how the average frequency of each strategy on the graph changes over time. Remarkably, this equation is a replicator equation with a transformed payoff matrix. Therefore, moving a game from a wellmixed population (the complete graph) onto a regular graph simply results in a transformation of the payoff matrix. The new payoff matrix is the sum of the original payoff matrix plus another matrix, which describes the local competition of strategies. We discuss the application of our theory to four particular examples, the Prisoner’s Dilemma, the SnowDrift game, a coordination game and the Rock–Scissors–Paper game.
Active linking in evolutionary games
, 2006
"... In the traditional approach to evolutionary game theory, the individuals of a population meet each other at random, and they have no control over the frequency or duration of interactions. Here we remove these simplifying assumptions. We introduce a new model, where individuals differ in the rate at ..."
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Cited by 13 (6 self)
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In the traditional approach to evolutionary game theory, the individuals of a population meet each other at random, and they have no control over the frequency or duration of interactions. Here we remove these simplifying assumptions. We introduce a new model, where individuals differ in the rate at which they seek new interactions. Once a link between two individuals has formed, the productivity of this link is evaluated. Links can be broken off at different rates. In a limiting case, the linking dynamics introduces a simple transformation of the payoff matrix. We outline conditions for evolutionary stability. As a specific example, we study the interaction between cooperators and defectors. We find a simple relationship that characterizes those linking dynamics which allow natural selection to favour cooperation over defection.
Intention recognition promotes the emergence of cooperation. Adaptive Behavior
, 2011
"... Few problems have created the combined interest of so many unrelated areas as the evolution of cooperation. As a result, several mechanisms have been identified to work as catalyzers of cooperative behavior. Yet, these studies, mostly grounded on evolutionary dynamics and game theory, have neglected ..."
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Cited by 12 (11 self)
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Few problems have created the combined interest of so many unrelated areas as the evolution of cooperation. As a result, several mechanisms have been identified to work as catalyzers of cooperative behavior. Yet, these studies, mostly grounded on evolutionary dynamics and game theory, have neglected the important role played by intention recognition in behavioral evolution. Here we address explicitly this issue, characterizing the dynamics emerging from a population of intention recognizers. We derive a Bayesian Network model for intention recognition in the context of repeated social dilemmas and evolutionary game theory, by assessing the internal dynamics of trust between intention recognizers and their opponents. Intention recognizers are then able to predict the next move of their opponents based on past direct interactions, which, in turn, enables them to prevail over the most famous strategies of repeated dilemmas of cooperation, even in presence of noise. Overall, our framework offers new insights on the complexity and beauty of behavioral evolution driven by elementary forms of cognition.
Analytical Results for Individual and Group Selection of Any Intensity
 BULLETIN OF MATHEMATICAL BIOLOGY (2008)
, 2007
"... The idea of evolutionary game theory is to relate the payoff of a game to reproductive success ( = fitness). An underlying assumption in most models is that fitness is a linear function of the payoff. For stochastic evolutionary dynamics in finite populations, this leads to analytical results in th ..."
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Cited by 12 (4 self)
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The idea of evolutionary game theory is to relate the payoff of a game to reproductive success ( = fitness). An underlying assumption in most models is that fitness is a linear function of the payoff. For stochastic evolutionary dynamics in finite populations, this leads to analytical results in the limit of weak selection, where the game has a small effect on overall fitness. But this linear function makes the analysis of strong selection difficult. Here, we show that analytical results can be obtained for any intensity of selection, if fitness is defined as an exponential function of payoff. This approach also works for group selection ( = multilevel selection). We discuss the difference between our approach and that of inclusive fitness theory.
The role of intention recognition in the evolution of cooperative behavior
 In IJCAI’2011
, 2011
"... Given its ubiquity, scale and complexity, few problems have created the combined interest of so many unrelated areas as the evolution of cooperation. Using the tools of evolutionary game theory, here we address, for the first time, the role played by intention recognition in the final outcome of coo ..."
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Cited by 10 (9 self)
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Given its ubiquity, scale and complexity, few problems have created the combined interest of so many unrelated areas as the evolution of cooperation. Using the tools of evolutionary game theory, here we address, for the first time, the role played by intention recognition in the final outcome of cooperation in large populations of selfregarding individuals. By equipping individuals with the capacity of assessing intentions of others in the course of repeated Prisoner’s Dilemma interactions, we show how intention recognition opens a window of opportunity for cooperation to thrive, as it precludes the invasion of pure cooperators by random drift while remaining robust against defective strategies. Intention recognizers are able to assign an intention to the action of their opponents based on an acquired corpus of possible intentions. We show how intention recognizers can prevail against most famous strategies of repeated dilemmas of cooperation, even in the presence of errors. Our approach invites the adoption of other classification and pattern recognition mechanisms common among Humans, to unveil the evolution of complex cognitive processes in the context of social dilemmas. 1
Direct reciprocity on graphs
, 2007
"... Direct reciprocity is a mechanism for the evolution of cooperation based on the idea of repeated encounters between the same two individuals. Here we examine direct reciprocity in structured populations, where individuals occupy the vertices of a graph. The edges denote who interacts with whom. The ..."
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Cited by 7 (2 self)
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Direct reciprocity is a mechanism for the evolution of cooperation based on the idea of repeated encounters between the same two individuals. Here we examine direct reciprocity in structured populations, where individuals occupy the vertices of a graph. The edges denote who interacts with whom. The graph represents spatial structure or a social network. For birth–death or pairwise comparison updating, we find that evolutionary stability of direct reciprocity is more restrictive on a graph than in a wellmixed population, but the condition for reciprocators to be advantageous is less restrictive on a graph. For death–birth and imitation updating, in contrast, both conditions are easier to fulfill on a graph. Moreover, for all four update mechanisms, reciprocators can dominate defectors on a graph, which is never possible in a wellmixed population. We also study the effect of an error rate, which increases with the number of links per individual; interacting with more people simultaneously enhances the probability of making mistakes. We provide analytic derivations for all results.
Spatial invasion of cooperation
, 2008
"... The evolutionary puzzle of cooperation describes situations where cooperators provide a fitness benefit to other individuals at some cost to themselves. Under Darwinian selection, the evolution of cooperation is a conundrum, whereas noncooperation (or defection) is not. In the absence of supporting ..."
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Cited by 6 (0 self)
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The evolutionary puzzle of cooperation describes situations where cooperators provide a fitness benefit to other individuals at some cost to themselves. Under Darwinian selection, the evolution of cooperation is a conundrum, whereas noncooperation (or defection) is not. In the absence of supporting mechanisms, cooperators perform poorly and decrease in abundance. Evolutionary game theory provides a powerful mathematical framework to address the problem of cooperation using the prisoner’s dilemma. One wellstudied possibility to maintain cooperation is to consider structured populations, where each individual interacts only with a limited subset of the population. This enables cooperators to form clusters such that they are more likely to interact with other cooperators instead of being exploited by defectors. Here we present a detailed analysis of how a few cooperators invade and expand in a world of defectors. If the invasion succeeds, the expansion process takes place in two stages: first, cooperators and defectors quickly establish a local equilibrium and then they uniformly expand in space. The second stage provides good estimates for the global equilibrium frequencies of cooperators and defectors. Under hospitable conditions, cooperators typically form a single, ever growing cluster interspersed with specks of defectors, whereas under more hostile conditions, cooperators form isolated, compact clusters that minimize exploitation by defectors. We provide the first quantitative assessment of the way cooperators arrange in space during invasion and find that the macroscopic