A Survey of Replicator Equations

This survey of the "state of the art" of replicator dynamics covers recent developments in the theory of the difference and differential equations which describe the evolution of population frequencies under the influence of selection. Mathematical models of this type play a central role in population genetics, ecology, prebiotic evolution and ethology. They introduce a dynamic element into the theory of normal form games and may also be applied to models of learning and economic evolution. The mathematical aspects considered include fixed-point analysis, the notions of permanence and exclusion, the gradient systems obtained by the introduction of certain Riemann metrics, Hopf bifurcations, and relations with game-theoretical concepts. This research was undertaken as part of the Feasibility Study on the Dynamics of Macrosystems in the System and Decision Sciences Program

The Economics of Fairness

Experimental games are small, but expanding branch of economics. The major part of economics deals with large-scale phenomena like stock market fluctuations, rates of exchange, and gross national products. The trend in economic life towards globalization leads to ever more abstract and virtual forms of interaction, as can be seen by the rapid growth of global e-commerce, trading in licenses and options, and the like. Yet at the same time, paradoxically, economists become increasingly fascinated by interactions at the most down-to-earth level - the sharing and helping that goes on within office pools, households, families, or children groups. How does economic exchange work in the absence of explicit contracts and regulatory institutions

The Maximum Principle for Replicator Equations

By introducing a non-Euclidean metric on the unit simplex, it is possible to identify an interesting class of gradient systems within the ubiquitous "replicator equations" of evolutionary biomathematics. In the case of homogeneous potentials, this leads to maximum principles governing the increase of the average fitness, both in population genetics and in chemical kinetics. This research was carried out as part of the Dynamics of Macrosystems Feasibility Study in the System and Decision Sciences Program

Grade Dynamics, Mixed Strategies and Gradient Systems

Game dynamics, as a branch of frequency dependent population genetics, leads to replicator equations. If phenotypes correspond to mixed strategies, evolution will affect the frequencies of the phenotypes and of the strategies and thus lead to two dynamical models. Some examples of this, including the sex ratio, will be discussed with the help of a non-Euclidean metric leading to a gradient system. Some other examples from population genetics and chemical kinetics confirm the usefulness of such gradients in describing evolutionary optimization