  We propose a mathematical framework for natural selection in finite
populations. Traditionally, many of the selection-based processes used to
describe cultural and genetic evolution (such as imitation and birth-death
models) have been studied on a case-by-case basis. Over time, these models have
grown in sophistication to include population structure, differing phenotypes,
and various forms of interaction asymmetry, among other features. Furthermore,
many processes inspired by natural selection, such as evolutionary algorithms
in computer science, possess characteristics that should fall within the realm
of a "selection process," but so far there is no overarching theory
encompassing these evolutionary processes. The framework of $\textit{stochastic
selection processes}$ we present here provides such a theory and consists of
three main components: a $\textit{population state space}$, an
$\textit{aggregate payoff function}$, and an $\textit{update rule}$. A
population state space is a generalization of the notion of population
structure, and it can include non-spatial information such as strategy-mutation
rates and phenotypes. An aggregate payoff function allows one to generically
talk about the fitness of traits without explicitly specifying a method of
payoff accounting or even the nature of the interactions that determine
payoff/fitness. An update rule is a fitness-based function that updates a
population based on its current state, and it includes as special cases the
classical update mechanisms (Moran, Wright-Fisher, etc.) as well as more
complicated mechanisms involving chromosomal crossover, mutation, and even
complex cultural syntheses of strategies of neighboring individuals. Our
framework covers models with variable population size as well as with
arbitrary, measurable trait spaces.
