Monotonic solution concepts in coevolution

Assume a coevolutionary algorithm capable of storing and utilizing all phenotypes discovered during its operation, for as long as it operates on a problem; that is, assume an algorithm with a monotonically increasing knowledge of the search space. We ask: If such an algorithm were to periodically report, over the course of its operation, the best solution found so far, would the quality of the solution reported by the algorithm improve monotonically over time? To answer this question, we construct a simple preference relation to reason about the goodness of different individual and composite phenotypic behaviors. We then show that whether the solutions reported by the coevolutionary algorithm improve monotonically with respect to this preference relation depends upon the solution concept implemented by the algorithm. We show that the solution concept implemented by the conventional coevolutionary algorithm does not guarantee monotonic improvement; in contrast, the game-theoretic solution concept of Nash equilibrium does guarantee monotonic improvement. Thus, this paper considers 1) whether global and objective metrics of goodness can be applied to coevolutionary problem domains (possibly with open-ended search spaces), and 2) whether coevolutionary algorithms can, in principle, optimize with respect to such metrics and find solutions to games of strategy.