Pandit Varun, Mukhopadhyay Archan, Chakraborty Sagar
Department of Physics, Indian Institute of Technology Kanpur, Kanpur, Uttar Pradesh 208016, India.
Chaos. 2018 Mar;28(3):033104. doi: 10.1063/1.5011955.
Replicator equation-a paradigm equation in evolutionary game dynamics-mathematizes the frequency dependent selection of competing strategies vying to enhance their fitness (quantified by the average payoffs) with respect to the average fitnesses of the evolving population under consideration. In this paper, we deal with two discrete versions of the replicator equation employed to study evolution in a population where any two players' interaction is modelled by a two-strategy symmetric normal-form game. There are twelve distinct classes of such games, each typified by a particular ordinal relationship among the elements of the corresponding payoff matrix. Here, we find the sufficient conditions for the existence of asymptotic solutions of the replicator equations such that the solutions-fixed points, periodic orbits, and chaotic trajectories-are all strictly physical, meaning that the frequency of any strategy lies inside the closed interval zero to one at all times. Thus, we elaborate on which of the twelve types of games are capable of showing meaningful physical solutions and for which of the two types of replicator equation. Subsequently, we introduce the concept of the weight of fitness deviation that is the scaling factor in a positive affine transformation connecting two payoff matrices such that the corresponding one-shot games have exactly same Nash equilibria and evolutionary stable states. The weight also quantifies how much the excess of fitness of a strategy over the average fitness of the population affects the per capita change in the frequency of the strategy. Intriguingly, the weight's variation is capable of making the Nash equilibria and the evolutionary stable states, useless by introducing strict physical chaos in the replicator dynamics based on the normal-form game.
复制者方程——进化博弈动力学中的一个范式方程——将竞争策略的频率依赖选择数学化，这些策略竞相提高其相对于所考虑的进化种群平均适应度（由平均收益量化）的适应度。在本文中，我们处理复制者方程的两个离散版本，用于研究种群中的进化，其中任意两个参与者的相互作用由一个双策略对称正规形式博弈建模。有十二种不同类别的此类博弈，每一类都由相应收益矩阵元素之间的特定序数关系来表征。在这里，我们找到了复制者方程渐近解存在的充分条件，使得这些解——不动点、周期轨道和混沌轨迹——都是严格物理的，这意味着任何策略的频率在任何时候都位于闭区间零到一内。因此，我们详细阐述了十二种博弈类型中的哪些能够展示有意义的物理解以及针对哪两种复制者方程。随后，我们引入适应度偏差权重的概念，它是连接两个收益矩阵的正仿射变换中的缩放因子，使得相应的一次性博弈具有完全相同的纳什均衡和进化稳定状态。该权重还量化了一种策略的适应度超过种群平均适应度的程度对该策略频率的人均变化的影响。有趣的是，权重的变化能够通过在基于正规形式博弈的复制者动力学中引入严格物理混沌，使纳什均衡和进化稳定状态变得无用。
