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The prevailing methodology for analyzing population games and evolutionarydynamics in the large population limit assumes that a Poisson process (orclock) inherent to each agent determines when the agent can revise itsstrategy. Hence, such an approach presupposes exponentially distributedinter-revision intervals, and is inadequate for cases where each strategyentails a sequence of sub-tasks (sub-strategies) that must be completed beforea new revision time occurs. This article proposes a methodology for such casesunder the premise that a sub-strategy's duration is exponentially-distributed,leading to Erlang distributed inter-revision intervals. We assume that aso-called pairwise-comparison protocol captures the agents' revisionpreferences to render our analysis concrete. The presence of sub-strategiesbrings on additional dynamics that is incompatible with existing models andresults. Our main contributions are twofold, both derived for a deterministicapproximation valid for large populations. We prove convergence of thepopulation's state to the Nash equilibrium set when a potential game generatesa payoff for the strategies. We use system-theoretic passivity to determineconditions under which this convergence is guaranteed for contractive games.

Population Games With Erlang Clocks: Convergence to Nash Equilibria For Pairwise Comparison Dynamics