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A Review of Evolutionary Graph Theory With Applications to Game Theory
"... Evolutionary graph theory (EGT), studies the ability of a mutant gene to overtake a finite 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 “fixation probability ” the probability o ..."
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Evolutionary graph theory (EGT), studies the ability of a mutant gene to overtake a finite 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 “fixation probability ” the probability of a mutant taking over a population and focuses on gametheoretic applications. We look at varying topics such as alternate evolutionary dynamics, time to fixation, special topological cases, and game theoretic results. Throughout the review, we examine several interesting open problems that warrant further research.
Mutation–selection equilibrium in games with multiple strategies.
 J. Theor. Biol.
, 2009
"... a b s t r a c t We develop a new method for studying stochastic evolutionary game dynamics of mixed strategies. We consider the general situation: there are n pure strategies whose interactions are described by an n Â n payoff matrix. Players can use mixed strategies, which are given by the vector ..."
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a b s t r a c t We develop a new method for studying stochastic evolutionary game dynamics of mixed strategies. We consider the general situation: there are n pure strategies whose interactions are described by an n Â n payoff matrix. Players can use mixed strategies, which are given by the vector ðp 1 ; . . . ; p n Þ. Each entry specifies the probability to use the corresponding pure strategy. The sum over all entries is one. Therefore, a mixed strategy is a point in the simplex S n . We study evolutionary dynamics in a wellmixed population of finite size. Individuals reproduce proportional to payoff. We consider the case of weak selection, which means the payoff from the game is only a small contribution to overall fitness. Reproduction can be subject to mutation; a mutant adopts a randomly chosen mixed strategy. We calculate the average abundance of every mixed strategy in the stationary distribution of the mutationselection process. We find the crucial conditions that specify if a strategy is favored or opposed by selection. One condition holds for low mutation rate, another for high mutation rate. The result for any mutation rate is a linear combination of those two. As a specific example we study the HawkDove game. We prove general statements about the relationship between games with pure and with mixed strategies.
Invasion and expansion of cooperators in lattice populations: Prisoner’s dilemma vs. snowdrift games
 JOURNAL OF THEORETICAL BIOLOGY
, 2010
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Games on graphs
 EMS Surveys in Mathematical Sciences. 2014; 1(1):113–151. doi: 10.4171/EMSS/3
"... Abstract. Evolution occurs in populations of reproducing individuals. The trajectories and outcomes of evolutionary processes depend on the structure of the population. Evolutionary graph theory is a powerful approach to studying the consequences of spatial or social population structure. The verti ..."
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Abstract. Evolution occurs in populations of reproducing individuals. The trajectories and outcomes of evolutionary processes depend on the structure of the population. Evolutionary graph theory is a powerful approach to studying the consequences of spatial or social population structure. The vertices of the graph represent individuals. The edges determine who interacts with whom for game payoff and who competes with whom for reproduction. Interaction and competition can be governed by the same graph or by two different graphs. In this paper, we review the basic approach for evolutionary games on graphs and provide new proofs for key results. We formalize the method of identity by descent to derive conditions for strategy selection on finite, weighted graphs. We generalize our results to nonzero mutation rates, and to the case where the interaction and competition graphs do not coincide. We conclude with a perspective of open problems and future directions. Mathematics Subject Classification (2010). 91A22; 92B05, 60J20.
Article Dynamic Properties of Evolutionary Multiplayer Games in Finite Populations
, 2013
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Global Migration Can Lead to Stronger Spatial Selection than Local Migration
, 2013
"... Abstract The outcome of evolutionary processes depends on population structure. It is well known that mobility plays an important role in affecting evolutionary dynamics in group structured populations. But it is largely unknown whether global or local migration leads to stronger spatial selection ..."
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Abstract The outcome of evolutionary processes depends on population structure. It is well known that mobility plays an important role in affecting evolutionary dynamics in group structured populations. But it is largely unknown whether global or local migration leads to stronger spatial selection and would therefore favor to a larger extent the evolution of cooperation. To address this issue, we quantify the impacts of these two migration patterns on the evolutionary competition of two strategies in a finite island model. Global migration means that individuals can migrate from any one island to any other island. Local migration means that individuals can only migrate between islands that are nearest neighbors; we study a simple geometry where islands are arranged on a onedimensional, regular cycle. We derive general results for weak selection and large population size. Our key parameters are: the number of islands, the migration rate and the mutation rate. Surprisingly, our comparative analysis reveals that global migration can lead to stronger spatial selection than local migration for a wide range of parameter conditions. Our work provides useful insights into understanding how different mobility patterns affect evolutionary processes.
Contents lists available at SciVerse ScienceDirect Journal of Theoretical Biology
"... journal homepage: www.elsevier.com/locate/yjtbi Multiplayer games on the cycle ..."
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journal homepage: www.elsevier.com/locate/yjtbi Multiplayer games on the cycle
Journal of Theoretical Biology] (]]]])]]]–]]] Contents lists available at ScienceDirect Journal of Theoretical Biology
"... journal homepage: www.elsevier.com/locate/yjtbi How mutation affects evolutionary games on graphs ..."
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journal homepage: www.elsevier.com/locate/yjtbi How mutation affects evolutionary games on graphs
Supplementary data References
, 2010
"... Advance online articles have been peer reviewed and accepted for publication but have not yet appeared in the paper journal (edited, typeset versions may be posted when available prior to final publication). Advance online articles are citable and establish publication priority; they are indexed by ..."
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Advance online articles have been peer reviewed and accepted for publication but have not yet appeared in the paper journal (edited, typeset versions may be posted when available prior to final publication). Advance online articles are citable and establish publication priority; they are indexed by PubMed from initial publication. Citations to Advance online articles must include the digital object identifier (DOIs) and date of initial publication. Emotions as infectious diseases in a large social network: the SISa model