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Stability Concepts: Beyond Nash Equilibria

The concept of an Evolutionarily Stable Strategy (ESS), which is a stronger stability condition than that of a Nash equilibrium, is introduced. A simple evolutionary dynamic, adaptive dynamics, is also introduced. This leads to the concept of convergence stability under adaptive dynamics. It is shown that these two stability criteria are independent for general games: a strategy can be an ESS but not be reachable under adaptive dynamics and a strategy may be an attractor under adaptive dynamics but a fitness minimum and so not an ESS. The latter situation leads to the possibility of evolutionary branching, a phenomenon in which the population splits into a mixture of two or more distinct morphs. Replicator dynamics provide another evolutionary dynamic, although it is argued that it is of limited relevance to biology. In some games, individuals interact with relatives. The effects of kin assortment, and the direct fitness and gene-centred approaches to games between kin are described and illustrated.