Cooperative games, replicator dynamics, and stability

长期以来，演化模型一直是经济学的重要组成部分。在博弈论中已经发展出一类特殊的模型。用通常的说法，“进化博弈论”意味着进化方法与非合作博弈的结合。在生物学中与进化理论特别接近的是复制因子动力学。到目前为止，只有很少的努力将进化方法与合作博弈论结合起来。这就是这个项目的目标。我们开发并分析了基于合作游戏的复制因子动态，特别是在稳定种群方面。在项目的第一部分，我们继续研究由Casajus， Kramm和Wiese （2020， JET）发起的Lovász-Shapley复制因子动力学。这些动态是通过Lovász-Shapley解决方案（Casajus和Wiese， 2017， IJGT）从具有可转移效用（TU）的合作游戏中衍生出来的。玩家被视为agent的类型，他们的权重是这些类型agent的总体大小。为了处理它们，我们必须应用右手边不连续的微分方程理论（Filippov, 1988）。虽然这些动态中的稳定种群与潜在的TU博弈之间的关系已经得到澄清，但仍然存在循环存在和稳定性的问题。Lovász-Shapley解决方案基于Leontief类型技术，也就是说，类型是互补的。或者，可以考虑一种技术，其中类型是完全替代的，或者更一般地说，可以用CES生产函数描述的技术。在项目的第二部分，我们使用Casajus和Wiese （2017， IJGT）介绍的结构来推广这些生产函数的Lovász-Shapley解决方案。我们分析了这些CES解决方案，并将它们的特性与Lovász-Shapley解决方案的特性进行了比较。在项目的第三部分，我们将第一部分的分析扩展到CES解决方案。

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.