Coevolutionary dynamics in a minimal substrate

One of the central difficulties of coevolutionary methods arises from 'intransitive superiority' – in a two-player game, for example, the fact that A beats B, and B beats C, does not exclude the possibility that C beats A. Such cyclic superiority in a coevolutionary substrate is hypothesized to cause cycles in the dynamics of the population such that it 'chases its own tail' - traveling through some part of strategy space more than once despite apparent improvement with each step. It is often difficult to know whether an application domain contains such difficulties and to verify this hypothesis in the failure of a given coevolutionary set-up. In this paper we wish to elucidate some of the issues and concepts in an abstract domain where the dynamics of coevolution can be studied simply and directly. We define three simple 'number games' that illustrate intransitive superiority and resultant oscillatory dynamics, as well as some other relevant concepts. These include the distinction between a player's perceived performance and performance with respect to an external metric, and the significance of strategies with a multidimensional nature. These features alone can also cause oscillatory behavior and coevolutionary failure.