隨機穩定性：一個新的演算方法及在隨機演化賽局中的應用

Abstract

本篇論文研究演化的動態過程中的隨機穩定性。演化過程中,突變(mutation)或變異隨時可能會發生。因此,演化中不存在安定(steady)或是穩定(stable)的狀態。但是當突變機率趨近於零時,有些狀態在長期間比其他狀態容易出現在過程中為人所觀察到。這些狀態稱為隨機穩定狀態(stochastically stable state)。我們發展出一具有一般性的演算法來找出所有的隨機穩定狀態。有別於傳統演算法,這套演算法大幅降低計算所需次數。透過這套演算法,我們定義了一個集合: stable set。我們發現,stable set包涵了所有的隨機穩定狀態。同時,我們也提出數個隨機穩定狀態的充份條件。這些發現代表著,分析演化模型的假設及均衡(equilibria)性質之間的關係是可行的。

We study the behaviors of the evolutionary models with persistant noises through a general algorithm which describes the relationships among the stochastic potentials. That is, by constructing a closed loop on the graph of the directed trees, we show that the comparison among the stochastic potential is equivalent to the comparison among one-step transition costs. Hence, we are able to systematically analyze the properties of the stochastically stable states. Our main nding is that the set of the stochastically stable states is contained in a set, which we dene as a stable set. Each state in this set is difcult to escape from and is resistant to the attraction of any other states in the stable set. Based on this nding, related sufficient conditions for the stochastically stable states are presented, and some results\r\nin the literature are also reinterpreted. In addition, we show that this algorithm drastically reduces the necessary steps for characterizing the stochastically stable states.\r\nThis means that the analysis on relationships between the assumptions of the model and the properties of equilibria are possible and promising.