Evolutionary Stability and Dynamic Stability in a Class of Evolutionary Normal Form Games

Abstract

A complete game theoretical and dynamical analysis is given for a dass of evolutionary normal form games which are called ‘RSP—games’ since they include the well—known ‘Rock—ScissorsPaper’ game. RSP—games induce a rich selection dynamics, but they are simple enough to allow a global analysis of their evolutionary properties. They provide an ideal illustration for the incongruities in the evolutionary predictions of evolutionary game theory and dynamic selection theory.

Every RSP—game has a unique interior Nash equilibrium strategy which is an ESS if and only if and only if the average binary payoffs of the game are all positive and not too different from one another. Dynamic stability with respect to the continuous replicator dynamics may be characterized by the much weaker requirement that the equilibrium payoff of the game has to be positive. The qualitative difference between evolutionary stability and dynamic stability is illustrated by the fact that every ESS can be transformed into a non-ESS attractor by means of a transformation which leaves the dynamies essentially invariant.

In all evolutionary normal form games, evolutionary stability of a fixed point implies dynamic stability with respect to the continuous replicator dynamics. Due to ‘overshooting eflects’, this is generally not true for the discrete replicator dynamics. In contrast to all game theoretical concepts (including the ESS concept), discrete dynamic stability is not invariant with resped to positive linear transformations of payofls. In fact, every ESS of an RSP—game can both be stabilized and destabilized by a transformation of payoffs. Quite generally, however, evolutionary stability implies discrete dynamic stability if selection is ‘weak enough7#x2019;.

In the continuous—time case, the interior fixed point of an RSP—game ia either a global attractor, or a global repellor, or a global center. In contrast, the discrete replicator dynamics admits a much richer dynamics including stable non-equilibrium behaviour. The occurrence of stable and unstable limit cycles is demonstrated both numerically and analytically. Some selection experiments in chemostats reveal that competition between different asexual strains of the yeast Saccharom1lces cerevisiae leads to the same cyclical best reply atrudure that is characteristic for RSP7#x2014;games. Possibly, Rock—Scissors—Paper—games are also played in non—human biological populations.