Evolutionary optimisation models and matrix games in the unified perspective of 
adaptive dynamics 

G. Meszena, E. Kisdi, U. Dieckmann, S.A.H. Geritz, and J.A.J. Metz

Selection 2(1-2): 193-210 (2001)

Abstract

Matrix game theory and optimisation models offer two radically different perspectives on 
the outcome of evolution. Optimisation models consider frequency-independent selection 
and envisage evolution as a hill-climbing process on a constant fitness landscape, with 
the optimal strategy corresponding to the fitness maximum. By contrast, in evolutionary 
matrix games selection is frequency-dependent and leads to fitness equality among 
alternative strategies once an evolutionarily stable strategy has been established. In this 
review we demonstrate that both optimisation models and matrix games represent 
limiting cases of the general framework of nonlinear frequency-dependent selection. 
Adaptive dynamics theory considers arbitrary nonlinear frequency and density 
dependence and envisages evolution as proceeding on an adaptive landscape that changes 
its shape according to which strategies are present in the population. In adaptive 
dynamics, evolutionarily stable strategies correspond to conditional fitness maxima: the 
ESS is characterised by the fact that it has the highest fitness if it is the established 
strategy. In this framework it can also be shown that dynamical attainability, evolutionary 
stability, and invading potential of strategies are pairwise independent properties. In 
optimisation models, on the other hand, these properties become linked such that the 
optimal strategy is always attracting, evolutionarily stable and can invade any other 
strategy. In matrix games fitness is a linear function of the potentially invading strategy 
and can thus never exhibit an interior maximum: Instead, the fitness landscape is a plane 
that becomes horizontal once the ESS is established. Due to this degeneracy, invading 
potential is part of the ESS definition for matrix games and dynamical attainability is a 
dependent property. We conclude that nonlinear frequency-dependent theory provides a 
unifying framework for overcoming the traditional divide between evolutionary 
optimisation models and matrix games.