Supriya Krishnamurthy, Eric Smith

Paper #: 11-03-010

We consider the role of stochastic evolutionary games as general models for a subset of processes involving the interaction of evolution with development. We are interested in how the change from strategic to evolutionary interpretation of games creates opportunities to use game models to explain empirical phenomena, and also in the problems raised by empirical application. Difficulties include the identifiability of game structure and limits to model mis-specification from statistical inference, and also problems of moment closure and model consistency familiar from attempts to derive the dynamics of associations within population genetics. Evolutionary game models contain examples of important phenomena such as the emergence of multilevel selection and multi-scale dynamics, and so it is desirable to know whether these phenomena can be implied by statistically feasible inferences from population data against errors of mis-specification, and whether they can be computed within consistent closure assumptions. In this review we provide a formulation of evolutionary games motivated by the concept of an effective theory from statistical mechanics. Effective theories are equivalence classes of fully-defined but approximate stochastic models, with the robust properties of each equivalence class generally encoded in its representation of the symmetry group of the problem. We show how even a qualitative treatment of symmetries and symmetry breaking (a change in the representation of a symmetry group by an ordered population state) distinguishes major classes of emergent multi-scale dynamics, and how it identifies these as robust properties of systems entailed by a small number of assumptions. We then derive the generating functionals for stochastic evolutionary games from the methods of Freidlin and Wentzell, and use these to compute large-deviations formulae for escape rates and trajectories of populations between basins of attraction, diffusive dynamics of populations over limit cycles, and large-fluctuation corrections to inclusive fitness that cause the best fit model at the population level to differ systematically from the interaction model experienced by agents, even in infinite-population limits. We show how Freidlin-Wentzell theory may be used to systematically incorporate collective fluctuation effects into algorithms for model estimation and model aggregation, which provides tools to pursue the problem of universality classification and justification of the concept of effective theories for games. Many diverse results from strategic and evolutionary game theory, and from evolutionary dynamics more generally, are given a unifying framework in terms of symmetries and collective-fluctuation mechanics. We show how the notion of a potential arises in a new form as a representation of symmetry and a governor of dynamics in stochastic processes, with mathematical and intuitive relations to the energy potentials of equilibrium thermodynamics. We map the decomposition of genotypes into genes and associations to the conversion of evolutionary dynamics from normalform to extensive-form games, and we recover the mathematical correspondence of epistasis within development to relatedness in population dynamics. Finally, by using extensive-form games to consider repeated interaction in ontogenetic time, we show how evolutionary dynamics provides constructive solutions to problems of coordination that have been the subject of the Folk Theorems of strategic game theory.