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Co-evolutionary algorithms have a wide range of applications, such as inhardware design, evolution of strategies for board games, and patching softwarebugs. However, these algorithms are poorly understood and applications areoften limited by pathological behaviour, such as loss of gradient, relativeover-generalisation, and mediocre objective stasis. It is an open challenge todevelop a theory that can predict when co-evolutionary algorithms findsolutions efficiently and reliable.  This paper provides a first step in developing runtime analysis forpopulation-based competitive co-evolutionary algorithms. We provide amathematical framework for describing and reasoning about the performance ofco-evolutionary processes. An example application of the framework shows ascenario where a simple co-evolutionary algorithm obtains a solution inpolynomial expected time. Finally, we describe settings where theco-evolutionary algorithm needs exponential time with overwhelmingly highprobability to obtain a solution.

Runtime Analysis of Competitive co-Evolutionary Algorithms for Maximin Optimisation of a Bilinear Function