Cooperative games, replicator dynamics, and stability

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Evolutionary models have been part and parcel of economics for a long time. A specific class of such models has been developed within game theory. In usual parlance, "evolutionary game theory" means the combination of evolutionary approaches with non-cooperative games. Particularly close to the theory of evolution in biology are replicator dynamics.So far, only few efforts have been taken to combine evolutionary approaches with cooperative game theory. This is what this projects aims at. We develop and analyze replicator dynamics that are based on cooperative games, particularly with respect to stable populations.In the first part of the project, we continue the study of the Lovász-Shapley replicator dynamics initiated by Casajus, Kramm, and Wiese (2020, JET). These dynamics are derived from cooperative games with transferable utility (TU games) by help of the Lovász-Shapley solution (Casajus and Wiese, 2017, IJGT). Players are regarded as types of agents and their weights as the sizes of the populations of agents of these types. In order to handle them, we have to apply the theory of differential equations with discontinuous right-hand side (Filippov, 1988). Whereas the relation between the stable populations in these dynamics and the underlying TU games already has been clarified, remains the question of the existence and stability of cycles, for example.The Lovász-Shapley solution is based on a Leontief type technology, i.e., the types are complements. Alternatively, one could consider a technology, where the types are perfect substitutes, or, more generally, technologies that can be described by CES production functions. In the second part of the project, we generalize the Lovász-Shapley solution for these production functions using the construction introduced by Casajus and Wiese (2017, IJGT). We analyze these CES solutions and compare their properties with those of the Lovász-Shapley solution.In the third part of the project, we extend the analysis of the first part to the CES solutions.