Some Analytical Properties of the Model for Stochastic Evolutionary Games in Finite Populations with Non-uniform Interaction Rate

Traditional evolutionary games assume uniform interaction rate, which means that the rate at which individuals meet and interact is independent of their strategies. But in some systems, especially biological systems, the players interact with each other discriminately. Taylor and Nowak (2006) were the first to establish the corresponding non-uniform interaction rate model by allowing the interaction rates to depend on strategies. Their model is based on replicator dynamics which assumes an infinite size population. But in reality, the number of individuals in the population is always finite, and there will be some random interference in the individuals' strategy selection process. Therefore, it is more practical to establish the corresponding stochastic evolutionary model in finite populations. In fact, the analysis of evolutionary games in a finite size population is more difficult. Just as Taylor and Nowak said in the outlook section of their paper, "The analysis of non-uniform interaction rates should be extended to stochastic game dynamics of finite populations." In this paper, we are exactly doing this work. We extend Taylor and Nowak's model from infinite to finite case, especially focusing on the influence of non-uniform connection characteristics on the evolutionary stable state of the system. We model the strategy evolutionary process of the population by a continuous ergodic Markov process. Based on the limit distribution of the process, we can give the evolutionary stable state of the system. We make a complete classification of the symmetric 2×2 games. For each case game, the corresponding limit distribution of the Markov-based process is given when noise intensity is small enough. In contrast with most literatures in evolutionary games using the simulation method, all our results obtained are analytical. Especially, in the dominant-case game, coexistence of the two strategies may become evolutionary stable states in our model. This result can be used to explain the emergence of cooperation in the Prisoner is Dilemma Games to some extent. Some specific examples are given to illustrate our results.     

基于Moran过程的消费者众筹策略演化动态

考虑混合均匀的N个消费者众筹策略演化动态,每个消费者可选择参与众筹或者不参与众筹两个策略。基于Moran过程的扎根概率,计算出有限消费者群体中,参与众筹策略和不参与众筹策略成功入侵的概率。比较个体扎根的概率和中性入侵的概率大小,分别得到两策略在强选择性和弱选择性强度下占优的条件。研究发现,强选择性条件下,消费者数量小于某个数值时,不参与众筹策略演化稳定侵占整个种群,众筹融资失败。弱选择性条件下,消费者依据净收益和支出成本两因素决定是否参与众筹,净收益大于支出成本三倍,演化稳定于众筹成功,净收益小于支出成本二分之三倍,演化稳定于众筹失败。     

阈值风险下的多企业合作治理污染的演化博弈分析

针对环境污染具有跨区域性,环境资源的公共物品属性,由单一产污企业治理污染物难以取得有效成果,辖区内多个产污企业合作治理是环境保护的必由之路。 本文从演化博弈论的研究视角探讨了多个排污企业之间的决策演化过程,建立了多主体演化博弈模型,并考虑了污染排放总量超过总量阈值时的环境恶化风险,研究阈值风险对产污企业合作治理污染策略选择的影响。 研究结果表明,较高的治理成本很大地约束了逐利企业治理污染的行为选择,造成公地悲剧的社会问题。 企业治污成本较大背景下,环境阈值风险发生概率越高,越能有效促进合作治理环境策略的演化稳定,合作治理占优于不治理。 出于对企业自身初始禀赋的保护,产污企业初始禀赋越大、排污收费率越高,越能有效促进企业治理污染物的积极性。 最后,为促进辖区内污染企业合作治理污染提出了政策建议。     

具有模糊支付的Moran过程演化博弈动态

以往对演化博弈的研究都假设个体从博弈中获得的支付是确定的并以精确的数来表示。然而由于受环境中各种不确定因素的影响,个体博弈时所获得的支付并不是一个精确的数值,而需要用一个模糊数来表示。本文研究模糊支付下2×2的对称博弈, 利用模糊数的运算, 分析具有模糊支付的有限种群Moran过程演化动态。在弱选择下以梯形模糊数和三角模糊数表示博弈支付,计算策略的模糊扎根概率,分析自然选择有利于策略扎根及策略成为模糊演化稳定策略的条件。将经典博弈推广到模糊环境中丰富了演化博弈理论,更具有现实意义。     

Stochastic evolutionary public goods game with first and second order costly punishments in finite populations

We study the stochastic evolutionary public goods game with punishment in a finite size population. Two kinds of costly punishments are considered, i.e., first-order punishment in which only the defectors are punished, and second-order punishment in which both the defectors and the cooperators who do not punish the defective behaviors are punished. We focus on the stochastic stable equilibrium of the system. In the population, the evolutionary process of strategies is described as a finite state Markov process. The evolutionary equilibrium of the system and its stochastic stability are analyzed by the limit distribution of the Markov process. By numerical experiments, our findings are as follows.(i) The first-order costly punishment can change the evolutionary dynamics and equilibrium of the public goods game, and it can promote cooperation only when both the intensity of punishment and the return on investment parameters are large enough.(ii)Under the first-order punishment, the further imposition of the second-order punishment cannot change the evolutionary dynamics of the system dramatically, but can only change the probability of the system to select the equilibrium points in the "C+P" states, which refer to the co-existence states of cooperation and punishment. The second-order punishment has limited roles in promoting cooperation, except for some critical combinations of parameters.(iii) When the system chooses"C+P" states with probability one, the increase of the punishment probability under second-order punishment will further increase the proportion of the "P" strategy in the "C+P" states.     

Evolutionary games in a generalized Moran process with arbitrary selection strength and mutation

By using a generalized fitness-dependent Moran process, an evolutionary model for symmetric 2×2 games in a well-mixed population with a finite size is investigated. In the model, the individuals' payoff accumulating from games is mapped into fitness using an exponent function. Both selection strength β and mutation rate ε are considered. The process is an ergodic birth-death process. Based on the limit distribution of the process, we give the analysis results for which strategy will be favoured when ε is small enough. The results depend on not only the payoff matrix of the game, but also on the population size. Especially, we prove that natural selection favours the strategy which is risk-dominant when the population size is large enough. For arbitrary β and ε values, the 'Hawk--Dove' game and the 'Coordinate' game are used to illustrate our model. We give the evolutionary stable strategy (ESS) of the games and compare the results with those of the replicator dynamics in the infinite population. The results are determined by simulation experiments.