### abstract ###
Evolution is shaping the world around us.
At the core of every evolutionary process is a population of reproducing individuals.
The outcome of an evolutionary process depends on population structure.
Here we provide a general formula for calculating evolutionary dynamics in a wide class of structured populations.
This class includes the recently introduced games in phenotype space and evolutionary set theory.
There can be local interactions for determining the relative fitness of individuals, but we require global updating, which means all individuals compete uniformly for reproduction.
We study the competition of two strategies in the context of an evolutionary game and determine which strategy is favored in the limit of weak selection.
We derive an intuitive formula for the structure coefficient,, and provide a method for efficient numerical calculation.
### introduction ###
Constant selection implies that the fitness of individuals does not depend on the composition of the population.
In general, however, the success of individuals is affected by what others are doing.
Then we are in the realm of game theory CITATION CITATION or evolutionary game theory CITATION CITATION.
The latter is the study of frequency dependent selection; the fitness of individuals is typically assumed to be a linear function of the frequencies of strategies in the population.
The population is trying to adapt on a dynamic fitness landscape; the changes in the fitness landscape are caused by the population that moves over it CITATION.
There is also a close relationship between evolutionary game theory and ecology CITATION : the success of a species in an ecosystem depends on its own abundance and the abundance of other species.
The classical approach to evolutionary game dynamics is based on deterministic differential equations describing infinitely large, well-mixed populations CITATION, CITATION.
In a well-mixed population any two individuals interact equally likely.
Some recent approaches consider stochastic evolutionary dynamics in populations of finite size CITATION, CITATION.
Evolutionary game dynamics are also affected by population structure CITATION CITATION.
For example, a well-mixed population typically opposes evolution of cooperation, while a structured population can promote it.
There is also a long standing tradition of studying spatial models in ecology CITATION CITATION, population genetics CITATION, CITATION and inclusive fitness theory CITATION CITATION .
Evolutionary graph theory is an extension of spatial games, which are normally studied on regular lattices, to general graphs CITATION CITATION.
The graph determines who meets whom and reflects physical structure or social networks.
The payoff of individuals is derived from local interactions with their neighbors on the graph.
Moreover, individuals compete locally with their neighbors for reproduction.
These two processes can also be described by separate graphs CITATION .
Games in phenotype space CITATION represent another type of spatial model for evolutionary dynamics, which is motivated by the idea of tag based cooperation CITATION CITATION.
In addition to behavioral strategies, individuals express other phenotypic features which serve as markers of identification.
In one version of the model, individuals interact only with those who carry the same phenotypic marker.
This approach can lead to a clustering in phenotype space, which can promote evolution of cooperation CITATION .
Evolutionary set theory represents another type of spatial model CITATION.
Each individual can belong to several sets.
At a particular time, some sets have many members, while others are empty.
Individuals interact with others in the same set and thereby derive a payoff.
Individuals update their set memberships and strategies by global comparison with others.
Successful strategies spawn imitators, and successful sets attract more members.
Therefore, the population structure is described by an ever changing, dynamical graph.
Evolutionary dynamics in set structured populations can favor cooperators over defectors.
In all three frameworks evolutionary graph theory, games in phenotype space and evolutionary set theory the fitness of individuals is a consequence of local interactions.
In evolutionary graph theory there is also a local update rule: individuals learn from their neighbors on the graph or compete with nearby individuals for placing offspring.
For evolutionary set theory, however, CITATION assumes global updating: individuals can learn from all others in the population and adopt their strategies and set memberships.
Global updating is also a feature of the model for games in phenotype space CITATION.
The approach that is presented in this paper requires global updating.
Therefore, our result holds for evolutionary set theory and for games in phenotype space, but does not apply to evolutionary graph theory.
