随机演化博弈动力学

有限种群中的多局效应导致两局之间的策略联系

Multigame Effect in Finite Populations Induces Strategy Linkage Between Two Games

研究了有限种群中两个2-策略博弈的演化博弈动力学。传统上，依赖于频率的演化动力学是在假设种群规模为无限大的情况下，用确定性复制子动力学来模拟的。然而，实际上，人口规模是有限的。近年来，有限种群中的随机过程被引入到进化博弈中，以研究进化博弈动力学中的有限规模效应。然而，这些研究大多集中在只玩单一游戏的人群上。在这项研究中，我们研究了一个有限的人口与两个游戏，并表明，一个有限的人口玩两个游戏往往会朝着一个特定的方向发展，形成特定的联系之间的战略，这两个游戏。

有限种群中的确定性进化博弈动力学

Deterministic evolutionary game dynamics in finite populations

进化博弈动力学描述了成功策略在繁殖个体群体中的传播。通常，策略传播的微观定义是随机的，这样，只有在无限大的种群中，动态才会变得确定。在这里，我们介绍了一个新的微观出生-死亡过程，有一个完全确定的强选择限制在任何规模的混合人口。此外，在弱选择下，从这个新的过程恢复频率依赖的Moran过程。这使得它成为弱选择下通常的演化动力学的自然延伸。我们发现简单的表达式的固定概率和平均固定时间的新进程中的进化游戏有两个球员和两个战略。对于两个玩家和三个策略的循环博弈，我们证明了所产生的确定性动力学关键地依赖于初始条件的非平凡的方式。

弱选择条件下进化博弈中的固定时间

Fixation times in evolutionary games under weak selection

在进化博弈动力学中，繁殖成功率随着进化博弈中的表现而增加。如果策略$A$的表现优于策略$B$，则策略$A$将在人群中传播。在随机动力学下，单个突变体迟早会接管整个种群或灭绝。我们分析了与此过程相关的平均退出时间（或平均固定时间）。我们分析表明，这些时间依赖于一个令人惊讶的简单的方式下弱选择，即强随机性的游戏的回报矩阵：回报差$Δπ$是一个线性函数的$A$个人$i$，$Δπ= u i + v$的数量。无条件平均退出时间仅取决于常数项$v$。假设单个$A$突变体接管群体，则相应的条件平均退出时间仅取决于密度依赖项$u$。我们证明了这一发现的两个常用的微观进化过程。

动态种群中的随机进化博弈

Stochastic evolutionary games in dynamic populations

频率依赖选择和人口波动在进化和生态过程中起着重要作用。在频率依赖选择下，群体的平均适应度可以基于群体内个体之间的相互作用而增加或减少。即使在恒定的环境中，种群规模的波动也应该反映出这一点。在这里，我们提出了一个随机模型，它自然地结合了这两个进化的成分，假设频率依赖于不同类型之间的竞争，在一个基于个人的模型。与以往的博弈论模型不同，种群的承载能力和种群规模是由进化博弈和人口随机性介导的个体间的成对竞争决定的。在种群规模为无穷大的情况下，平均随机动力学被确定性竞争Lotka-Volterra方程所捕获。在小种群中，人口随机性反而可能导致整个种群的灭绝。由于种群规模是由进化博弈中的适应度决定的，因此合作者的种群比背叛者的种群更不容易灭绝，而在通常的固定规模系统中，种群会繁荣，而不管其平均收益如何。

多策略进化多人对策的策略丰富度

Strategy abundance in evolutionary many-player games with multiple strategies

演化博弈论是一种抽象而简单，但非常强大的演化动力学建模方法。即使是复杂的生物现象有时也可以抽象为简单的双人游戏。但通常情况下，几方之间的相互作用决定了进化的成功。在这种情况下，我们必须考虑许多玩家之间的互动，而不是两人之间的互动，这比通常的两人游戏更复杂，但仍然可以产生简单的结果。在这篇手稿中，我们得到了一个多人多策略系统的突变选择均衡的组成。这导致一个简单的表达式，可以通过使用聚结理论的递归获得。这种方法可以被修改以适应各种情况，例如，在多态性中找到有限数量的等位基因的平衡频率，或者在文化背景下的社会困境中找到不同策略的平衡频率。

Evolutionary multiplayer games on graphs with edge diversity

Evolutionary game dynamics in structured populations has been extensively explored in past decades. However, most previous studies assume that payoffs of individuals are fully determined by the strategic behaviors of interacting parties and social ties between them only serve as the indicator of the existence of interactions. This assumption neglects important information carried by inter-personal social ties such as genetic similarity, geographic proximity, and social closeness, which may crucially affect the outcome of interactions. To model these situations, we present a framework of evolut…

Evolutionary dynamics with game transitions

The environment has a strong influence on a population's evolutionary dynamics. Driven by both intrinsic and external factors, the environment is subject to continual change in nature. To capture an ever-changing environment, we consider a model of evolutionary dynamics with game transitions, where individuals' behaviors together with the games they play in one time step influence the games to be played next time step. Within this model, we study the evolution of cooperation in structured populations and find a simple rule: weak selection favors cooperation over defection if the ratio of the b…

多元宇宙中的进化博弈

Evolutionary games in the multiverse

两个局中人采用两种策略的演化博弈动力学已经得到了详细的研究。这些游戏已被用于模拟许多生物相关的场景，从哺乳动物的社会困境到微生物多样性。事实上，这些游戏中的一些可能发生在许多人之间，而不仅仅是两个人之间。在这里，我们解决多个玩家的一次性游戏。只要我们只有两个策略，两个参与人博弈的许多结果都可以推广到多个参与人。对于多个玩家和两个以上的策略的游戏，我们表明，成对相互作用的语句不再成立。对于具有任意数量策略的两人博弈，最多只能有一个孤立的内部均衡。对于任意数量的参与人$\boldsymbol{d}$和任意数量的策略n，最多可以有（d-1）^（n-1）个孤立的内部均衡。多人游戏表现出很大的动态复杂性，无法根据成对的相互作用来捕获。我们的结果适用于任何游戏，可以很容易地应用于特定的情况下，例如，公共产品游戏或多人猎鹿。

表型空间中的进化博弈动力学

Evolutionary game dynamics in phenotype space

演化动力学可以在混合良好或结构化的种群中进行研究。人口结构通常是由个体在物理空间或社交网络上的异质分布引起的。在这里，我们引入了一种新的空间类型的进化博弈动力学：表型空间。在这个意义上，群体是很好的混合，每个人都有可能与其他人互动，但行为策略取决于表型空间的距离。个人可能会对那些看起来相似或不相似的人表现出不同的行为。个体突变成附近的表型。我们研究人口的“表型太空行走”。我们提出的分析计算，汇集了聚结理论和进化博弈动力学的想法。作为一个特殊的例子，我们调查的合作表型空间的演变。我们得到了自然选择有利于合作者而不是背叛者的精确条件：对于一维表型空间和大种群规模，临界收益成本比由b/c=1+2/sqrt{3}给出。我们推导出任何进化博弈的基本条件，并探索更高维的表型空间。

随机进化博弈中多态性的出现

Emergence of polymorphism in stochastic evolutionary games

研究随机进化博弈中多态性的出现问题，通过对鹰鸽博弈进行时间尺度分离的理论分析方法，刻画随机效应驱动的多态性，发现特定条件下存在持久多态状态，解决了相关理论与模拟结果的差异。

Evolutionary dynamics of zero-determinant strategies in repeated multiplayer games

Since Press and Dyson's ingenious discovery of ZD (zero-determinant) strategy in the repeated Prisoner's Dilemma game, several studies have confirmed the existence of ZD strategy in repeated multiplayer social dilemmas. However, few researches study the evolutionary performance of multiplayer ZD strategies, especially from a theoretical perspective. Here, we use a newly proposed state-clustering method to theoretically analyze the evolutionary dynamics of two representative ZD strategies: generous ZD strategies and extortionate ZD strategies. Apart from the competitions between the two strateg…

局中人随机匹配进化博弈的均衡选择

Equilibrium selection in evolutionary games with random matching of players

我们讨论的随机动力学种群的个人玩游戏。我们的模型具有两个进化稳定的策略：一个是有效的策略，即种群处于最大收益（适应度）的状态，另一个是风险主导策略，即玩家厌恶风险。我们假设个体与随机选择的对手进行博弈（他们不像标准的复制因子动态那样与平均策略博弈）。我们表明，一个群体的长期行为取决于它的大小和突变水平。

生态进化博弈动力学中带有环境反馈的随机波动

Stochastic fluctuations in an eco-evolutionary game dynamics with environmental feedbacks

本文研究了在环境反馈机制下，生态进化博弈中随机波动的特性，通过Ω展开方法分析种群规模波动及稳定性，验证了在大环境承载能力下动态稳定性在随机情形下的扩展。

关于两个基本进化对策的注记

Notes on two elementary evolutionary games

在本说明的第一部分，我们表明（继霍夫鲍尔和西格蒙德），道金斯的“性别之战”定义了一个有趣的地图，从经济参数的空间，以心理社会坐标。第二部分讨论了一个更基本的游戏，但它并不完全是微不足道的。

Mean-potential law in evolutionary games

The Letter presents a novel way to connect random walks, stochastic differential equations, and evolutionary game theory. We introduce a new concept of potential function for discrete-space stochastic systems. It is based on a correspondence between one-dimensional stochastic differential equations and random walks, which may be exact not only in the continuous limit but also in finite-state spaces. Our method is useful for computation of fixation probabilities in discrete stochastic dynamical systems with two absorbing states. We apply it to evolutionary games, formulating two simple and intu…

On the expected number of real roots of random polynomials arising from evolutionary game theory

In this paper, we obtain finite estimates and asymptotic formulas for the expected number of real roots of two classes of random polynomials arising from evolutionary game theory. As a consequence of our analysis, we achieve an asymptotic formula for the expected number of internal equilibria in multi-player two-strategy random evolutionary games. Our results contribute both to evolutionary game theory and random polynomial theory.

网络上的演化博弈：相变、准平衡与数学原理

Evolutionary Games on Networks: Phase Transition, Quasi-equilibrium, and Mathematical Principles

该研究针对空间演化博弈近似方法得到的“稳定”合作率可能为准平衡态的问题，发现经典博弈模型收敛时间存在相变，并提出数学原理解释该相变与准平衡现象，揭示了合作与背叛长期共存的机制。

Deciphering chaos in evolutionary games

Discrete-time replicator map is a prototype of evolutionary selection game dynamical models that have been very successful across disciplines in rendering insights into the attainment of the equilibrium outcomes, like the Nash equilibrium and the evolutionarily stable strategy. By construction, only the fixed point solutions of the dynamics can possibly be interpreted as the aforementioned game-theoretic solution concepts. Although more complex outcomes like chaos are omnipresent in the nature, it is not known to which game-theoretic solutions they correspond. Here we construct a game-theoreti…

From Darwin to Poincaré and von Neumann: Recurrence and Cycles in Evolutionary and Algorithmic Game Theory

Replicator dynamics, the continuous-time analogue of Multiplicative Weights Updates, is the main dynamic in evolutionary game theory. In simple evolutionary zero-sum games, such as Rock-Paper-Scissors, replicator dynamic is periodic \cite{zeeman1980population}, however, its behavior in higher dimensions is not well understood. We provide a complete characterization of its behavior in zero-sum evolutionary games. We prove that, if and only if, the system has an interior Nash equilibrium, the dynamics exhibit Poincaré recurrence, i.e., almost all orbits come arbitrary close to their initial cond…

空间演化博弈的统计力学

Statistical mechanics of spatial evolutionary games

我们讨论了空间演化博弈中多个相互作用的参与者的随机动态的长期行为。特别地，我们研究了参与者数量和噪声水平对纳什均衡随机稳定性的影响。我们讨论了系统的相互作用的球员最大化他们的个人收益和粒子最小化他们的相互作用能之间的异同。我们使用统计力学的概念和技术来研究博弈论模型。为了得到结果的情况下，所谓的潜在的游戏，我们分析了热力学极限的相互作用粒子的适当的模型。

高聚类系数网络上的进化博弈

Evolutionary game on networks with high clustering coefficient

本研究探讨了演化博弈中格子结构的影响。在具有高聚类系数的网络中考虑了雪堆游戏，使用了四种不同的策略更新。采用对近似的解析解与数值结果进行了比较。结果表明，一般的陈述断言，晶格结构增强合作是误导。

Nonlinear learning and learning advantages in evolutionary games

The idea of incompetence as a learning or adaptation function was introduced in the context of evolutionary games as a fixed parameter. However, live organisms usually perform different nonlinear adaptation functions such as a power law or exponential fitness growth. Here, we examine how the functional form of the learning process may affect the social competition between different behavioral types. Further, we extend our results for the evolutionary games where fluctuations in the environment affect the behavioral adaptation of competing species and demonstrate importance of the starting leve…

The path integral formula for the stochastic evolutionary game dynamics in the Moran process

The Moran process is one of an basic mathematical structure in the evolutionary game theory. In this work, we introduce the formulation of the path integral approach for evolutionary game theory based on the Moran process. We derive the transition probability by the path integral from the initial state to the final state with updating rule of the Moran process. In this framework, the transition probability is the sum of all the evolutionary paths. The path integral formula of the transition probability maybe expected to be a new mathematical tool to explore the stochastic game evolutionary dyn…

The Stabilisation of Equilibria in Evolutionary Game Dynamics through Mutation: Mutation Limits in Evolutionary Games

The multi-population replicator dynamics (RD) can be considered a dynamic approach to the study of multi-player games, where it was shown to be related to Cross' learning, as well as of systems of coevolving populations. However, not all of its equilibria are Nash equilibria (NE) of the underlying game, and neither convergence to an NE nor convergence in general are guaranteed. Although interior equilibria are guaranteed to be NE, no interior equilibrium can be asymptotically stable in the multi-population RD, resulting, e.g., in cyclic orbits around a single interior NE. We introduce a new no…

Stochastic Evolutionary Dynamics of Trust Games with Asymmetric Parameters

Trusting in others and reciprocating that trust with trustworthy actions are crucial to successful and prosperous societies. The Trust Game has been widely used to quantitatively study trust and trustworthiness, involving a sequential exchange between an investor and a trustee. The deterministic evolutionary game theory predicts no trust and no trustworthiness whereas the behavioural experiments with the one-shot anonymous Trust Game show that people substantially trust and respond trustworthily. To explain these discrepancies, previous works often turn to additional mechanisms, which are borr…

Evolutionary games: natural selection of strategies

In this paper, I model and study the process of natural selection between all possible mixed strategies in classical two-player two-strategy games. I derive and solve an equation that is a natural generalization of the Taylor-Jonker replicator equation that describes dynamics of pure strategy frequencies. I then investigate the evolution of not only frequencies of pure strategies but also of total distribution of mixed strategies. I show that the process of natural selection of strategies for all games obeys the dynamical Principle of minimum of information gain.

无标度网络上进化博弈的亚稳定性和异常固定

Metastability and anomalous fixation in evolutionary games on scale-free networks

我们研究了复杂图对一组进化过程的亚稳定性和固定性的影响。在进化博弈论的框架下，其中的适应度和选择是频率依赖的，并随着人口组成而变化，我们分析了无标度网络上的雪堆游戏（其特征在于一个亚稳态共存状态）的动力学。利用弱选择极限下的有效扩散理论，我们证明了无标度结构如何影响系统的亚稳态并导致反常固定。特别是，我们分析和数值表明，固定的概率和平均时间的特点是拉伸指数行为的指数依赖于网络的度分布。

随机演化博弈与随机多项式

Random evolutionary games and random polynomials

本文发现随机非对称演化博弈的均衡多项式恰为 Kostlan-Shub-Smale 系统，并借此解析刻画内部均衡数量的统计规律，证明对称性会增加内部均衡期望数量。

The Reasonable Effectiveness of Agent-Based Simulations in Evolutionary Game Theory

This is a Reply to comments published in Physics of Life Reviews, on our article "Evolutionary game theory using agent-based methods" (Physics of Life Reviews, 2016, arXiv:1404.0994).

Extinction dynamics from meta-stable coexistences in an evolutionary game

Deterministic evolutionary game dynamics can lead to stable coexistences of different types. Stochasticity, however, drives the loss of such coexistences. This extinction is usually accompanied by population size fluctuations. We investigate the most probable extinction trajectory under such fluctuations by mapping a stochastic evolutionary model to a problem of classical mechanics using the Wentzel-Kramers-Brillouin (WKB) approximation. Our results show that more abundant types in a coexistence can be more likely to go extinct first well agreed with previous results, and also the distance bet…

进化勒索与恶作剧：迭代2x2博弈中的零决定性策略

Evolutionary Extortion and Mischief: Zero Determinant strategies in iterated 2x2 games

本文研究了最近发现的一类零行列式（ZD）策略在迭代2 × 2博弈中的机制、含义和潜在应用。据报道，这些策略成功地敲诈了纯经济最大化者，并通过对双方参与者的预期平均得分施加线性约束，恶意地确定了迭代囚犯困境中可行的长期收益集。这些结果推广到所有对称的2x2游戏和一般的性别之战，举例说明了四个常见的游戏。此外，与传统的策略进行了比较，并分析了典型的ZD游戏模拟以及收敛速度。几个响应策略进行了讨论，包括时间偏好如何改变以前的结果一瞥。此外，报复的可能性：当maximin分数超过最小对称支付，它是可能的勒索勒索。最后，从进化博弈论的研究结果总结表明，恶作剧是有限的，其本身的恶意。尽管如此，这并没有挑战盲目的经济最大化是受勒索的结果：ZD策略的研究揭示了令人兴奋的新视角和机会，在博弈论中，无论是进化的还是经典的。

空间演化博弈中的迁移行为

Opportunistic Migration in Spatial Evolutionary Games

我们研究进化游戏中的空间稀释的网格环境中，代理战略互动本地，但也可以机会主义地移动到其他位置在一个给定的迁移半径。使用模仿的最佳规则的战略修订，它表明，合作可能会发展，并在囚徒困境的游戏空间中的几个迁移距离是稳定的，但只有小游戏的互动半径，而雄鹿狩猎类的游戏成为完全合作。我们还表明，只有少数的试验是需要合作的发展，即搜索成本不是一个问题。当使用随机费米策略更新协议时，尽管存在机会主义迁移，但如果选择强度很高，合作就不能在囚徒困境中进化。然而，当模仿变得更加随机，完全或部分合作的状态达到在所有的游戏测试的所有迁移距离和短到中间的相互作用半径。

有限种群中进化博弈的策略固定

Fixation of strategies for an evolutionary game in finite populations

研究了有限种群中由2 × 2矩阵对策给出的两种策略的随机演化动力学.我们专注于固定的随机特性：如何由一个单一的个人代表的战略赢得了整个人口。该过程中讨论的框架内的随机游走与任意的跳跃率。固定的时间被发现是相同的两种策略在任何特定的游戏。还讨论了大种群规模限制下注视时间和注视概率的渐进行为。我们发现，固定是快速的，当至少有一个纯进化稳定策略（ESS）在无限的人口规模限制，而固定是缓慢的ESS是两种策略共存。

Evolutionary games and spatial periodicity

We establish a theoretical framework to address evolutionary dynamics of spatial games under strong selection. As the selection intensity tends to infinity, strategy competition unfolds in the deterministic way of winners taking all. We rigorously prove that the evolutionary process soon or later either enters a cycle and from then on repeats the cycle periodically, or stabilizes at some state almost everywhere. This conclusion holds for any population graph and a large class of finite games. This framework suffices to reveal the underlying mathematical rationale for the kaleidoscopic cooperat…

进化博弈中的所有权和交易

Ownership and Trade from Evolutionary Games

所有权和贸易作为进化稳定的策略从无政府状态中出现。在这些进化博弈模型中，所有权状态提供了一个内生的不对称标准，使财产冲突的解决更便宜。

Difffusion Dynamics on the Coexistence Subspace in a Stochastic Evolutionary Game

Frequency-dependent selection reflects the interaction between different species as they battle for limited resources in their environment. In a stochastic evolutionary game the species relative fitnesses guides the evolutionary dynamics which fluctuate due to random drift. Dependence of species selection advantages on the environment introduces additional possibilities for the evolutionary dynamics. We analyse a simple model in which a random environment allows competing species to coexist for a long time before a fixation of a single species happens. In our analysis we use stability in a lin…

随机博弈对结构群体中进化动态的影响

Effects of Stochastic Games on Evolutionary Dynamics in Structured Populations

研究随机博弈变化规则对合作行为进化的影响，揭示不同博弈类型对合作进化的影响差异。

Evolutionary games under incompetence

The adaptation process of a species to a new environment is a significant area of study in biology. As part of natural selection, adaptation is a mutation process which improves survival skills and reproductive functions of species. Here, we investigate this process by combining the idea of incompetence with evolutionary game theory. In the sense of evolution, incompetence and training can be interpreted as a special learning process. With focus on the social side of the problem, we analyze the influence of incompetence on behavior of species. We introduce an incompetence parameter into a lear…

The replicator equation in stochastic spatial evolutionary games

We study the multi-strategy stochastic evolutionary game with death-birth updating in expanding spatial populations of size $N\to \infty$. The model is a voter model perturbation. For typical populations, we require perturbation strengths satisfying $1/N\ll w\ll 1$. Under appropriate conditions on the space, the limiting density processes of strategy are proven to obey the replicator equation, and the normalized fluctuations converge to a Gaussian process with the Wright-Fisher covariance function in the limiting densities. As an application, we resolve in the positive a conjecture from the bi…

Evolutionary dynamics of group cooperation with asymmetrical environmental feedback

In recent years, there has been growing interest in studying evolutionary games with environmental feedback. Previous studies exclusively focus on two-player games. However, extension to multi-player game is needed to study problems such as microbial cooperation and crowdsourcing collaborations. Here, we study coevolutionary public goods games where strategies coevolve with the multiplication factors of group cooperation. Asymmetry can arise in such environmental feedback, where games organized by focal cooperators may have a different efficiency than the ones by defectors. Our analysis shows …

演化博弈论与种群动力学

Evolutionary game theory and population dynamics

许多社会经济和生物过程可以被建模为相互作用的个体系统。这种系统的行为通常可以用博弈论模型来描述。在这些课堂讲稿中，我们将介绍演化博弈论的基本概念，并回顾有限种群的确定性复制动力学和随机动力学的基本性质。我们讨论了具有迁移、时滞的确定性动力学中平衡点的稳定性，以及具有局部相互作用的混合种群和空间博弈的随机动力学中平衡点的稳定性。我们分析了人口的长期行为对各种参数的依赖性，如时间延迟，噪声水平和人口的大小。

Persistence probability of a random polynomial arising from evolutionary game theory

In this paper, we obtain an asymptotic formula for the persistence probability in the positive real line of a random polynomial arising from evolutionary game theory. It corresponds to the probability that a multi-player two-strategy random evolutionary game has no internal equilibria. The key ingredient is to approximate the sequence of random polynomials indexed by their degrees by an appropriate centered stationary Gaussian process.

随机空间演化对策的确定性方程

Deterministic Equations for Stochastic Spatial Evolutionary Games

空间进化博弈模拟分布在空间域中的个体，并在与邻居进行正常形式博弈时更新策略。我们推导出积分微分方程作为微观更新随机过程的确定性近似。这推广了已知的平均场常微分方程，为研究种群演化的空间效应提供了有力的工具。确定性方程允许识别许多有趣的功能，在人口中的战略配置文件的演变，如驻波和行波，和模式的形成，特别是在复制型进化。

有限局中人网络上进化博弈的一个新数学模型

A New Mathematical Model for Evolutionary Games on Finite Networks of Players

提出了一种新的图上进化博弈的数学模型，将经典的复制子方程推广到具有一般拓扑结构的网络上的有限总体博弈者.经典的结果从对策论，进化对策论和图论。更具体地说，每个参与者被放置在图的一个顶点上，他被看作是一个无穷总体的复制者，在顶点内复制。在每个时刻，属于不同连接顶点的两个复制子进行博弈，博弈的结果影响它们产生后代的能力。然后，顶点玩家的行为由内部复制者使用的策略分布决定。在适当的假设下，该模型等价于经典的复制因子方程。扩展的模拟进行显示的解决方案的动力学行为和开发的模型的潜力。

Evolution of cooperation and costly incentives in the continuous snowdrift game

Game theory research on the snowdrift game has showed that gradual evolution of the continuously varying level of cooperation in joint enterprises can demonstrate evolutionary merging as well as evolutionary branching. However, little is known about the consequences of changes in diversity at the cooperation level. In the present study I consider effects of costly rewards on the continuous snowdrift game. I show that not evolutionary merging but evolutionary branching can promote the emergence of pool reward, which can then enhance social welfare.

简单矩阵博弈中的 q 形变演化动力学

q-deformed evolutionary dynamics in simple matrix games

本文提出 q 形变演化博弈更新机制，分析 2×2 博弈稳定性与固定时间，并扩展到图拓扑及石头—剪刀—布等多策略博弈，揭示 q 可诱导新型动力学流。

Statistics of the number of equilibria in random social dilemma evolutionary games with mutation

In this paper, we study analytically the statistics of the number of equilibria in pairwise social dilemma evolutionary games with mutation where a game's payoff entries are random variables. Using the replicator-mutator equations, we provide explicit formulas for the probability distributions of the number of equilibria as well as other statistical quantities. This analysis is highly relevant assuming that one might know the nature of a social dilemma game at hand (e.g., cooperation vs coordination vs anti-coordination), but measuring the exact values of its payoff entries is difficult. Our d…

Evolutionary games on cycles with strong selection

Evolutionary games on graphs describe how strategic interactions and population structure determine evolutionary success, quantified by the probability that a single mutant takes over a population. Graph structures, compared to the well-mixed case, can act as amplifiers or suppressors of selection by increasing or decreasing the fixation probability of a beneficial mutant. Properties of the associated mean fixation times can be more intricate, especially when selection is strong. The intuition is that fixation of a beneficial mutant happens fast (in a dominance game), that fixation takes very …

利他行为的付出，或波动在演化博弈论中的重要性

Altruistic behavior pays, or the importance of fluctuations in evolutionary game theory

人类行为是进化的主要问题之一，因为人类的行为往往对自己不利，对他人有利。这个谜题的背后是我们基于博弈论的理性行为信念。在这里，我们表明，通过超越标准的博弈论惯例，显然利他行为可以被理解为利己的。我们详细讨论了一个与所谓的最后通牒博弈有关的例子，并说明了波动引起的利他行为的出现。此外，我们认为在一般情况下，波动起着非常重要的作用，我们通过考虑一个完全不同的例子来支持这一说法，即Stag-Hunt博弈。

进化博弈动力学中的混合时间

Mixing times in evolutionary game dynamics

如果没有突变和迁移，演化动力学最终会导致除一个物种外的所有物种灭绝。这种固定过程是很好理解的，可以用统计物理学的方法进行分析。然而，许多生物学的论点集中在突变选择平衡中的平稳分布。在这里，我们解决了在突变的情况下达到平稳所需的平衡时间，这在马尔可夫过程理论中被称为混合时间。我们发现，在进化游戏中的混合时间有相反的行为时，选择的强度增加固定时间：在协调游戏的双稳态，固定时间减少，但混合时间增加。在亚稳态共存博弈中，固定时间增加，但混合时间减少。我们的结果是基于模拟和WKB近似的主方程。

图上的进化博弈

Evolutionary games on graphs

博弈论是从生物学到行为科学再到经济学的许多科学学科背后的关键范式之一。在其进化形式中，特别是当相互作用的代理人在特定的社会网络中联系时，基本的解决方案概念和方法与非平衡统计物理学中应用的非常相似。这篇评论为物理学家提供了该领域的教程式概述。前三节介绍了经典博弈论和进化博弈论的必要背景，从基本定义到最重要的结果。第四部分概述了非平均场型社会网络结构所隐含的拓扑复杂性。最后三节详细讨论了三个突出的模型类的动态行为：囚徒困境，石头剪刀布游戏，竞争协会。的审查的主要主题是在什么意义上，以及如何互动的图形结构可以修改和丰富的长期行为模式出现在进化游戏的图片。

博弈驱动的演化动力学中的季节性Floquet状态

Seasonal Floquet states in a game-driven evolutionary dynamics

许多生物物种的交配偏好不是恒定的，而是依赖于季节。在演化博弈论的框架内，这可以用两个有限的异性种群按照周期性变化的规则相互博弈来建模。结合Floquet理论和拟平稳分布的概念，我们揭示了有限博弈驱动种群演化中亚稳态时间周期态的存在性。演化Floquet状态对应于策略空间中的时间周期概率流，在平均场框架内无法解决。亚稳态Floquet态的寿命随着粒子数$N$的增加而增加，从而在极限$N \rightarrow \infty$内成为吸引子。

The limits of weak selection and large population size in evolutionary game theory

Evolutionary game theory is a mathematical approach to studying how social behaviors evolve. In many recent works, evolutionary competition between strategies is modeled as a stochastic process in a finite population. In this context, two limits are both mathematically convenient and biologically relevant: weak selection and large population size. These limits can be combined in different ways, leading to potentially different results. We consider two orderings: the wN limit, in which weak selection is applied before the large population limit, and the Nw limit, in which the order is reversed.…

具有博弈转移的高阶演化动力学

Higher-order evolutionary dynamics with game transitions

本研究将博弈转移引入高阶囚徒困境模型，发现其持续促进合作，且高阶交互比例对合作有双重影响，理论与模拟均证实双稳态存在。