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Lie algebra structure of fitness and replicator control
arXiv - CS - Systems and Control Pub Date : 2020-05-19 , DOI: arxiv-2005.09792 Vidya Raju and P. S. Krishnaprasad
arXiv - CS - Systems and Control Pub Date : 2020-05-19 , DOI: arxiv-2005.09792 Vidya Raju and P. S. Krishnaprasad
For over fifty years, the dynamical systems perspective has had a prominent
role in evolutionary biology and economics, through the lens of game theory. In
particular, the study of replicator differential equations on the standard
(probability) simplex, specified by fitness maps or payoff functions, has
yielded insights into the temporal behavior of such systems. However behavior
is influenced by context and environmental factors with a game-changing quality
(i.e., fitness maps are manipulated). This paper develops a principled
geometric approach to model and understand such influences by incorporating
replicator dynamics into a broader control-theoretic framework. Central to our
approach is the construction of a Lie algebra structure on the space of fitness
maps, mapping homomorphically to the Lie algebra of replicator vector fields.
This is akin to classical mechanics, where the Poisson bracket Lie algebra of
functions maps to associated Hamiltonian vector fields. We show, extending the
work of Svirezhev in 1972, that a trajectory of a replicator vector field is
the base integral curve of a solution to a Hamiltonian system defined on the
cotangent bundle of the simplex. Further, we exploit the Lie algebraic
structure of fitness maps to determine controllability properties of a class of
replicator systems.
中文翻译:
适应度和复制子控制的李代数结构
五十多年来,通过博弈论的视角,动力系统观点在进化生物学和经济学中发挥了重要作用。特别是,对由适应度图或收益函数指定的标准(概率)单纯形的复制器微分方程的研究,已经深入了解了此类系统的时间行为。然而,行为受到上下文和环境因素的影响,具有改变游戏规则的质量(即,操纵健身图)。本文通过将复制器动力学纳入更广泛的控制理论框架,开发了一种有原则的几何方法来模拟和理解这种影响。我们方法的核心是在适应度图空间上构建李代数结构,同态映射到复制向量场的李代数。这类似于经典力学,其中函数的泊松括号李代数映射到相关的哈密顿向量场。我们证明,扩展 Svirezhev 在 1972 年的工作,复制向量场的轨迹是在单纯形的余切丛上定义的哈密顿系统的解的基积分曲线。此外,我们利用适应度图的李代数结构来确定一类复制系统的可控性属性。
更新日期:2020-05-21
中文翻译:
适应度和复制子控制的李代数结构
五十多年来,通过博弈论的视角,动力系统观点在进化生物学和经济学中发挥了重要作用。特别是,对由适应度图或收益函数指定的标准(概率)单纯形的复制器微分方程的研究,已经深入了解了此类系统的时间行为。然而,行为受到上下文和环境因素的影响,具有改变游戏规则的质量(即,操纵健身图)。本文通过将复制器动力学纳入更广泛的控制理论框架,开发了一种有原则的几何方法来模拟和理解这种影响。我们方法的核心是在适应度图空间上构建李代数结构,同态映射到复制向量场的李代数。这类似于经典力学,其中函数的泊松括号李代数映射到相关的哈密顿向量场。我们证明,扩展 Svirezhev 在 1972 年的工作,复制向量场的轨迹是在单纯形的余切丛上定义的哈密顿系统的解的基积分曲线。此外,我们利用适应度图的李代数结构来确定一类复制系统的可控性属性。